DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 1,
HIGHER ALGEBRAIC K -THEORY OF GROUP ACTIONS WITH FINITE STABILIZERS GABRIELE VEZZOSI and ANGELO VISTOLI
Abstract We prove a decomposition theorem for the equivariant K -theory of actions of affine group schemes G of finite type over a field on regular separated Noetherian algebraic spaces, under the hypothesis that the actions have finite geometric stabilizers and satisfy a rationality condition together with a technical condition that holds, for example, for G abelian or smooth. We reduce the problem to the case of a GLn -action and finally to a split torus action. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. General constructions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Morphisms of actions and induced maps on K -theory . . . . . . . 2.2. The basic definitions and results . . . . . . . . . . . . . . . . . . 3. The main theorem: The split torus case . . . . . . . . . . . . . . . . . . 4. The main theorem: The case of G = GLn,k . . . . . . . . . . . . . . . . 5. The main theorem: The general case . . . . . . . . . . . . . . . . . . . 5.1. Proof of Theorem 5.4 . . . . . . . . . . . . . . . . . . . . . . . 5.2. Hypotheses on G . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix. Higher equivariant K -theory of Noetherian regular separated algebraic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction The purpose of this paper is to prove a decomposition theorem for the equivariant K -theory of actions of affine group schemes of finite type over a field on regular separated Noetherian algebraic spaces. Let X be a regular connected separated Noetherian scheme with an ample line bundle, and let K 0 (X ) be its Grothendieck ring of DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 1, Received 27 April 2000. Revision received 13 June 2001. 2000 Mathematics Subject Classification. Primary 19E08, 14L30; Secondary 14A20. Authors’ work partially supported by the University of Bologna funds for selected research topics. 1
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vector bundles. Then the kernel of the rank morphism K 0 (X ) → Z is nilpotent (see [SGA6, Exp. VI, Th. 6.9]), so the ring K 0 (X ) is indecomposable and remains such after tensoring with any indecomposable Z-algebra. The situation is quite different when we consider the equivariant case. Let G be an algebraic group acting on a Noetherian separated regular scheme, or algebraic space, let X be over a field k, and consider the Grothendieck ring K 0 (X, G) of G-equivariant perfect complexes. This is the same as the Grothendieck group of G-equivariant coherent sheaves on X , and it coincides with the Grothendieck ring of G-equivariant vector bundles if all G-coherent sheaves are quotients of locally free coherent sheaves (which is the case, e.g., when G is finite or smooth and X is a scheme). Assume that the action of G on X is connected, that is, that there are no nontrivial invariant open and closed subschemes of X . Still, K 0 (X, G) usually decomposes, after inverting some primes; for example, if G is a finite group and X = Spec C, then K 0 (X, G) is the ring of complex representations of G, which becomes a product of fields after tensoring with Q. In [Vi2] the second author analyzes the case where the action of G on X has finite reduced geometric stabilizers. Consider the ring of representations R(G), and consider the kernel m of the rank morphism rk : K 0 (X, G) → Z. Then K 0 (X, G) is an R(G)-algebra; he shows that the localization morphism K 0 (X, G) ⊗ Q −→ K 0 (X, G)m is surjective and that the kernel of the rank morphism K 0 (X, G)m ⊗ Q → Q is nilpotent. Furthermore, he conjectures that K 0 (X, G) ⊗ Q splits as a product of the localization K 0 (X, G)m and some other ring, and he formulates a conjecture about what the other factor ring should be when G is abelian and the field is algebraically closed of characteristic zero. The proofs of the results in [Vi2] depend on an equivariant Riemann-Roch theorem, whose proof was never published by the author; however, all of the results have been proved and generalized in [EG]. The case where G is a finite group is studied in [Vi1]. Assume that k contains all nth roots of 1, where n is the order of the group G. Then the author shows that, after inverting the order of G, the K -theory ring K ∗ (X, G) of G-equivariant vector bundles on X (which is assumed to be a scheme in that paper) is canonically the product of a finite number of rings, expressible in terms of ordinary K -theory of L appropriate subschemes of fixed points of X . Here K ∗ (X, G) = i K i (X, G) is the graded higher K -theory ring. The precise formula is as follows. Let σ be a cyclic subgroup of G whose order is prime to the characteristic of k; then the subscheme X σ of fixed points of X under the actions of σ is also regular. The representation ring R(σ ) is isomorphic to the ring Z[t]/(t n −1), where t is a generator of the group of characters hom(σ, k ∗ ). We call e R(σ ) the quotient of the ring R(σ ) by the ideal generated by the element 8n (t), where 8n is the nth cyclotomic polynomial;
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this is independent of t. The ring e R(σ ) is isomorphic to the ring of integers in the nth cyclotomic field. Call NG (σ ) the normalizer of σ in G; the group NG (σ ) acts on the scheme X σ and, by conjugation, on the group σ. Consider the induced actions of NG (σ ) on the K -theory ring K ∗ (X σ ) and on the ring e R(σ ). Choose a set C (G) of representatives for the conjugacy classes of cyclic subgroups of G whose order is prime to the characteristic of the field. The statement of the main result of [Vi1] is as follows. THEOREM
There is a canonical ring isomorphism Y K ∗ (X, G) ⊗ Z[1/|G|] '
K ∗ (X σ ) ⊗ e R(σ )
NG (σ )
⊗ Z[1/|G|].
σ ∈C (G)
In the present paper we generalize this decomposition to the case in which G is an algebraic group scheme of finite type over a field k, acting with finite geometric stabilizers on a Noetherian regular separated algebraic space X over k. Of course, we cannot expect a statement exactly like the one for finite groups, expressing equivariant K -theory simply in terms of ordinary K -theory of the fixed point sets. For example, when X is the Stiefel variety of r -frames in n-space, then the quotient of X by the natural free action of GLr is the Grassmannian of r -planes in n-space, and K 0 (X, GLr ) = K 0 (X/ GLr ) is nontrivial, while K 0 (X ) = Z. Let X be a Noetherian regular algebraic space over k with an action of an affine group scheme G of finite type over k. We consider the Waldhausen K -theory group K ∗ (X, G) of complexes of quasi-coherent G-equivariant sheaves on X with coherent bounded cohomology. This coincides on the one hand with the Waldhausen K -theory group K ∗ (X, G) of the subcategory of complexes of quasi-coherent G-equivariant flat sheaves on X with coherent bounded cohomology (and hence has a natural ring structure given by the total tensor product) and on the other hand with the Quillen group K ∗0 (X, G) of coherent equivariant sheaves on X ; furthermore, if every coherent equivariant sheaf on X is the quotient of a locally free equivariant coherent sheaf, it also coincides with the Quillen group K ∗naive (X, G) of coherent locally free equivariant sheaves on X . These K -theories and their relationships are discussed in the appendix. Our result is as follows. First we have to see what plays the role of the cyclic subgroups of a finite group. This is easy; the group schemes whose rings of representations are of the form Z[t]/(t n − 1) are not the cyclic groups, in general, but their Cartier duals, that is, the group schemes that are isomorphic to the group scheme µn of nth roots of 1 for some n. We call these group schemes dual cyclic. If σ is a dual cyclic group, we can define e Rσ as before. A dual cyclic subgroup σ of G is called
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essential if X σ 6 = ∅. The correct substitute for the ordinary K -theory of the subspaces of invariants is the geometric equivariant K -theory K ∗ (X, G)geom , which is defined as follows. Call N the least common multiple of the orders of all the essential dual cyclic subgroups of G. Call S1 the multiplicative subset of the ring R(G) consisting of elements whose virtual rank is a power of N ; then K ∗ (X, G)geom is the localization S1−1 K ∗ (X, G). Notice that K ∗ (X, G)geom ⊗ Q = K ∗ (X, G)m , with the notation above. Moreover, if every coherent equivariant sheaf on X is the quotient of a locally free equivariant coherent sheaf, by [EG], we have an isomorphism of rings K 0 (X, G)geom ⊗ Q = A∗G (X ) ⊗ Q, where A∗G (X ) denotes the direct sum of G-equivariant Chow groups of X. We prove the following. Assume that the action of G on X is connected. Then the kernel of the rank morphism K 0 (X, G)geom → Z[1/N ] is nilpotent (see Cor. 5.2). This is remarkable; we have made what might look like a small step toward making the equivariant K -theory ring indecomposable, and we immediately get an indecomposable ring. Indeed, K ∗ (X, G)geom “feels like” the K -theory ring of a scheme; we want to think of K ∗ (X, G)geom as what the K -theory of the quotient X/G should be, if X/G were smooth, after inverting N (see Conj. 5.8). Furthermore, consider the centralizer CG (σ ) and the normalizer NG (σ ) of σ inside G. The quotient wG (σ ) = NG (σ )/CG (σ ) is contained inside the group scheme of automorphisms of σ , which is a discrete group, so it is also a discrete group. It acts on e R(σ ), by conjugation, and it also acts on the equivariant K -theory ring K ∗ (X σ , CG (σ )) and on the geometric equivariant K -theory ring K ∗ (X σ , CG (σ ))geom (see Cor. 2.5). We say that the action of G on X is sufficiently rational when the following two conditions are satisfied. Let k be the algebraic closure of k. (1) Each essential dual cyclic subgroup σ ⊆ G k is conjugate by an element of G( k ) to a dual cyclic subgroup of G. (2) If two essential dual cyclic subgroups of G are conjugate by an element of G( k ), they are also conjugate by an element of G(k). Obviously, every action over an algebraically closed field is sufficiently rational. Furthermore, if G is GLm , SLm , Spm , or a totally split torus, then any action of G is sufficiently rational over an arbitrary base field (see Prop. 2.3). If G is a finite group, then the action is sufficiently rational when k contains all nth roots of 1, where n is the least common multiple of the orders of the cyclic subgroups of k of order prime to the characteristic, whose fixed point subscheme is nonempty. Denote by C (G) a set of representatives for essential dual cyclic subgroup schemes, under conjugation by elements of the group G(k). Here is the statement of our result.
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MAIN THEOREM
Let G be an affine group scheme of finite type over a field k, acting on a Noetherian separated regular algebraic space X . Assume the following three conditions. (a) The action has finite geometric stabilizers. (b) The action is sufficiently rational. (c) For any essential cyclic subgroup σ of G, the quotient G/CG (σ ) is smooth. Then C (G) is finite, and there is a canonical isomorphism of R(G)-algebras Y w (σ ) K ∗ (X, G) ⊗ Z[1/N ] ' K ∗ (X σ , CG (σ ))geom ⊗ e R(σ ) G . σ ∈C (G)
Conditions (a) and (b) are clearly necessary for the theorem to hold. We are not sure about (c). It is rather mild, as it is satisfied, for example, when G is smooth (this is automatically true in characteristic zero) or when G is abelian. A weaker version of condition (c) is given in Section 5.2. In the case when G is abelian over an algebraically closed field of characteristic zero, the main theorem implies [Vi2, Conj. 3.6]. When G is a finite group, and the base field contains enough roots of 1, as in the statement of Theorem 1, then the conditions of the main theorem are satisfied; since the natural maps K ∗ (X σ , CG (σ ))geom → K ∗ (X σ )CG (σ ) become isomorphisms after inverting the order of G (see Prop. 5.7), the main theorem implies [Vi1, Th. 1]. However, the proof of the main theorem here is completely different from [Vi1, proof of Th. 1]. As B. Toen pointed out to us, a weaker version of our main theorem (with Qcoefficients and assuming G smooth, acting with finite reduced stabilizers) follows from his [To1, Th. 3.15]; the e´ tale techniques he uses in proving this result make it impossible to avoid tensoring with Q (see also [To2]). Here is an outline of the paper. First we define the homomorphism (see Sec. 2.2). Next, in Section 3, we prove the result when G is a totally split torus. Here the basic tool is the result of R. Thomason, which gives a generic description of the action of a torus on a Noetherian separated algebraic space, and we prove the result by Noetherian induction, using the localization sequence for the K -theory of equivariant coherent sheaves. As in [Vi1], the difficulty here is that the homomorphism is defined via pullbacks, and thus it does not commute with the pushforwards intervening in the localization sequence. This is solved by producing a different isomorphism between the two groups in question, using pushforwards instead of pullbacks, and then relating this to our map, via the self-intersection formula. The next step is to prove the result in the case when G = GLn ; here the key point is a result of A. Merkurjev which links the equivariant K -theory of a scheme with a GLn -action to the equivariant K -theory of the action of a maximal torus. This is carried out in Section 4. Finally (see Sec. 5), we reduce the general result to the case
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of GLn , by considering an embedding G ⊆ GLn , and the induced action of GLn on Y = GLn ×G X . It is at this point that condition (c) enters, allowing a clear description of Y σ where σ is an essential dual cyclic subgroup of G (see Prop. 5.6).
2. General constructions Notation. If S is a separated Noetherian scheme, X is a Noetherian separated Salgebraic space (which is most of the time assumed to be regular), and G is a flat affine group scheme over S acting on X , we denote by K ∗ (X, G) (resp., K 0 (X, G)) the Waldhausen K -theory of the complicial bi-Waldhausen (see [ThTr]) category W1,X,G of complexes of quasi-coherent G-equivariant O X -modules with bounded coherent cohomology (resp., the Quillen K -theory of G-equivariant coherent O X -modules). As shown in the appendix, if X is regular, K ∗ (X, G) is isomorphic to K ∗0 (X, G) and has a canonical graded ring structure. When X is regular, the isomorphism K ∗ (X, G) ' K ∗0 (X, G) then allows us to switch between the two theories when needed. 2.1. Morphisms of actions and induced maps on K -theory Let S be a scheme. By an action over S we mean a triple (X, G, ρ) where X is an S-algebraic space, G is a group scheme over S, and ρ : G × S X → X is an action of G on X over S. A morphism of actions ( f, φ) : (X, G, ρ) −→ (X 0 , G 0 , ρ 0 ) is a pair of S-morphisms f : X → X 0 and φ : G → G 0 , where φ is a morphism of S-group schemes, such that the following diagram commutes: ρ
G × S X −−−−→ φ× f y
X f y
G 0 × S X 0 −−−− → X0 0 ρ
Equivalently, f is required to be G-equivariant with respect to the given G-action on X and the G-action on X 0 induced by composition with φ. A morphism of actions ( f, φ) : (X, G, ρ) → (X 0 , G 0 , ρ 0 ) induces an exact functor ( f, φ)∗ : W3,X 0 ,G 0 → W3,X,G , where W3,Y,H denotes the complicial biWaldhausen category of complexes of H -equivariant flat quasi-coherent modules with bounded coherent cohomology on the H -algebraic space Y (see appendix). Let (E ∗ , ε∗ ) be an object of W3,X 0 ,G 0 ; that is, E ∗ is a complex of G 0 -equivariant flat quasicoherent O X 0 -modules with bounded coherent cohomology, and for any i, i εi : pr0∗ g ρ 0∗ E i 2 E −→
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is an isomorphism satisfying the usual cocycle condition. Here pr02 : G 0 × S X 0 → X 0 denotes the obvious projection, and similarly for pr2 : G × S X → X . Since ρ ∗ f ∗ E ∗ = ( fρ)∗ E ∗ = (φ × f )∗ ρ 0∗ E ∗ and ∗ pr∗2 f ∗ E ∗ =(φ × f )∗ pr0∗ 2E ,
. we define ( f, φ)∗ (E ∗ , ε∗ ) = ( f ∗ E ∗ , (φ × f )∗ (ε∗ )) ∈ W3,X,G (the cocycle condition for each (φ × f )∗ (εi ) following from the same condition for εi ); ( f, φ)∗ is then defined on morphisms in the only natural way. Now, ( f, φ)∗ is an exact functor and, if X and X 0 are regular so that the Waldhausen K -theory of W3,X,G (resp., of W3,X 0 ,G 0 ) coincides with K ∗ (X, G) (resp., K ∗ (X 0 , G 0 )) (see appendix), it defines a ring morphism ( f, φ)∗ : K ∗ (X 0 , G 0 ) −→ K ∗ (X, G). A similar argument shows that if f is flat, ( f, φ) induces a morphism ( f, φ)∗ : K ∗0 (X 0 , G 0 ) −→ K ∗0 (X, G). Example 2.1 Let G and H be group schemes over S, and let X be an S-algebraic space. Moreover, suppose that (1) G and H act on X ; (2) G acts on H by S-group scheme automorphisms (i.e., it is given a morphism G → Aut(GrSch)/S (H ) of group functors over S); (3) the two preceding actions are compatible; that is, for any S-scheme T , any g ∈ G(T ), h ∈ H (T ), and x ∈ X (T ), we have g · (h · x) = h g · (g · x), where (g, h) 7→ h g denotes the action of G(T ) on H (T ). If g ∈ G(S) and if gT denotes its image via G(S) → G(T ), let us define a morphism of actions ( f g , φg ) : (X, H ) → (X, H ) as f g (T ) : X (T ) −→ X (T ) : x 7 −→ gT · x, φg (T ) : H (T ) −→ H (T ) : h 7 −→ h gT . This is an isomorphism of actions and induces an action of the group G(S) on K ∗0 (X, H ) and on K ∗ (X, H ). This applies, in particular, to the case where X is an algebraic space with a G action and G B H , G acting on H by conjugation.
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2.2. The basic definitions and results Let G be a linear algebraic k-group scheme G acting with finite geometric stabilizers on a regular Noetherian separated algebraic space X over k. We denote by R(G) the representation ring of G. A (Cartier) dual cyclic subgroup of G over k is a k-subgroup scheme σ ⊆ G such that there exist an n > 0 and an isomorphism of k-groups σ ' µn,k . If σ, ρ are dual cyclic subgroups of G and if L is an extension of k, we say that σ and ρ are conjugate over L if there exists g ∈ G(L) such that gσ(L) g −1 = ρ(L) (where . H(L) = H ×Spec k Spec L, for any k-group scheme H ) as L-subgroup schemes of G (L) . A dual cyclic subgroup σ ⊆ G is said to be essential if X σ 6= ∅. We say that the action of G on X is sufficiently rational if (1) any two essential dual cyclic subgroups of G are conjugated over k if and only if they are conjugated over an algebraic closure k of k; (2) any essential dual cyclic subgroup ρ of G (k) is conjugated over k to a dual cyclic subgroup of the form σ(k) where σ ⊆ G is (essential) dual cyclic. We denote by C (G) a set of representatives for essential dual cyclic subgroups of G with respect to the relation of conjugacy over k. Remark 2.2 Note that if the action is sufficiently rational and if ρ, σ are essential dual cyclic subgroups of G which are conjugate over an algebraically closed extension of k, then they are also conjugate over k. 2.3 Any action of GLn , SLn , Sp2n , or of a split torus is sufficiently rational. PROPOSITION
Proof If G is a split torus, condition (1) is clear because G is abelian, while it follows from the rigidity of diagonalizable groups that any subgroup scheme of G k is in fact defined over k. Let σ ⊆ GLm be a dual cyclic subgroup. Since σ is diagonalizable, we have an eigenspace decomposition M V = km = Vχσ χ ∈b σ
such that the χ with Vχ 6 = 0 generate b σ . Conversely, given a cyclic group C and a decomposition M V = Vχ b χ ∈C
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b there is a corresponding embedding of such that the χ with Vχ 6= 0 generate C, the Cartier dual σ of C into GLn with Vχ = Vχσ for each χ ∈ C = b σ . Now, if σ ⊆ GLm,k is a dual cyclic subgroup defined over k, we can apply an element of GLm (k) to make the Vχ defined over k, and then gσ g −1 is defined over k. If σ ⊆ GLm and τ ⊆ GLm are dual cyclic subgroups that are conjugate over k, pick an element of GLm (k) sending σ to τ . This induces an isomorphism φ : σk ' τk , which by rigidity is defined over k. Then if χ and χ 0 are characters that correspond under the isomorphism of b σ and b τ induced by φ, then the dimension of Vχσ is equal to the τ dimension of Vχ 0 , so we can find an element g of GLm which carries each Vχσ onto the τ ; conjugation by this element carries σ onto τ . For SL , the proof corresponding Vχ0 m is very similar if we remark that to give a dual cyclic subgroup σ ⊆ SLm ⊆ GLm corresponds to giving a decomposition M V = km = Vχσ χ ∈b σ
Q dim Vχσ such that the χ with Vχσ 6 = 0 generate b σ , with the condition χ∈b =1∈b σ. σ χ For Spm ⊆ GL2m , a dual cyclic subgroup σ ⊆ Spm gives a decomposition M V = k 2m = Vχσ χ∈b σ
such that the χ with Vχσ 6 = 0 generate b σ , with the condition that for v ∈ Vχσ and 0 σ v ∈ Vχ0 the symplectic product of v and v 0 is always zero, unless χχ 0 = 1 ∈ b σ . Both conditions then follow rather easily from the fact that any two symplectic forms over a vector space are isomorphic. Let N(G,X ) denote the least common multiple of the orders of essential dual cyclic subgroups of G. Notice that N(G,X ) is finite: since the action has finite stabilizers, the group scheme of stabilizers is quasi-finite over X ; therefore the orders of the stabilizers of the geometric points of X are globally bounded. . We define 3(G,X ) = Z[1/N(G,X ) ]. If H ⊆ G is finite, we also write 3 H for Z[1/|H |]. Note that, if σ ⊆ G is dual cyclic, then 3σ = 3(σ,Spec k) , and if, moreover, σ is essential, 3σ ⊆ 3(G,X ) . If H ⊆ G is a subgroup scheme and if A is a ring, we write R(H ) A for R(H ) ⊗Z A. We denote by rk H : R(H ) −→ Z and by rk H,3(G,X ) : R(H )3(G,X ) −→ 3(G,X ) the rank ring homomorphisms. We let K ∗0 (X, G)3(G,X ) = K ∗0 (X, G) ⊗ 3G,X and K ∗ (X, G)3(G,X ) = K ∗ (X, G) ⊗ 3G,X
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(for the notation, see the beginning of this section). Recall that K ∗ (X, G)3(G,X ) is an R(G)-algebra via the pullback R(G) ' K 0 (Spec k, G) → K 0 (X, G) and that K ∗ (X, G) ' K ∗0 (X, G) since X is regular (see appendix). If σ is a dual cyclic subgroup of G of order n, the choice of a generator t for the . dual group b σ = HomGrSch/k (σ, Gm,k ) determines an isomorphism R(σ ) '
Z[t] . (t n − 1)
Let pσ be the canonical ring surjection Y Z[t], Z[t] −→ , (t n − 1) (8d ) d|n
and let f pσ be the induced surjection Z[t] Z[t] −→ , (t n − 1) (8n ) where 8d is the dth cyclotomic polynomial. If mσ is the kernel of the composition R(σ ) '
Z[t] Z[t] −→ , − 1) (8n )
(t n
the quotient ring R(σ )/mσ does not depend on the choice of the generator t for b σ. Notation. We denote by e R(σ ) the quotient R(σ )/mσ . We remark that if σ is dual cyclic of order n and if t is a generator of b σ , there are isomorphisms Y 3σ [t] 3σ [t] R(σ )3σ ' n ' . (1) (t − 1) (8d ) d|n
Let π f R(σ )3(G,X ) be the canonical ring homomorphism. The σ : R(G)3(G,X ) → e σ -localization K ∗0 (X, G)σ of K ∗0 (X, G)3(G,X ) is the localization of the R(G)3(G,X ) . −1 (1). The σ -localizaf module K ∗0 (X, G)3(G,X ) at the multiplicative subset Sσ = π σ tion K ∗ (X, G)σ is defined in the same way. If H ⊆ G is a subgroup scheme, we also write R(H )σ for Sσ−1 (R(H )3(G,X ) ). If σ is the trivial group, we denote by K ∗0 (X, G)geom the σ -localization of K ∗0 (X, G)3(G,X ) and call it the geometric part or geometric localization of K ∗0 (X, G)3(G,X ) . Note that πe1 coincides with the rank morphism rkG,3(G,X ) : R(G)3(G,X ) −→ 3(G,X ) . We have the same definition for K ∗ (X, G)geom . Let NG (σ ) (resp., CG (σ ) ⊆ NG (σ )) be the normalizer (resp., the centralizer) of σ in G; since Aut(σ ) is a finite constant group scheme, . NG (σ ) WG (σ ) = CG (σ )
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is also a constant group scheme over k associated to a finite group wG (σ ). 2.4 Let H be a k-linear algebraic group, let σ ' µn,k be a normal subgroup, and let Y be an algebraic space with an action of H/σ . Then there is a canonical action of w H (σ ) on K ∗0 (Y, C H (σ )). LEMMA
Proof Let us first assume that the natural map H (k) −→ w H (σ )
(2)
is surjective (which is true, e.g., if k is algebraically closed). Since C H (σ )(k) acts trivially on K ∗0 (Y, C H (σ )) and, by Example 2.1, H (k) acts naturally on K ∗0 (Y, C H (σ )), we may use (2) to define the desired action. In general, (2) is not surjective and we proceed as follows. Suppose that we can find a closed immersion of k-linear algebraic groups H ,→ H 0 such that (i) σ is normal in H 0 ; (ii) H 0 /C H 0 (σ ) ' W H (σ ); (iii) H 0 (k) → w H (σ ) is surjective. Consider the open and closed immersion Y × C H (σ ) ,→ Y × H ; this induces an open and closed immersion Y ×C H (σ ) C H 0 (σ ) ,→ Y ×C H (σ ) H 0 whose composition with the e´ tale cover Y ×C H (σ ) H 0 → Y × H H 0 is easily checked (e.g., on geometric points) to be an isomorphism. Therefore, K ∗0 Y × H H 0 , C H 0 (σ ) ' K ∗0 Y ×C H (σ ) C H 0 (σ ), C H 0 (σ ) ' K ∗0 Y, C H (σ ) , where the last isomorphism is given by the Morita equivalence theorem (see [Th3, Prop. 6.2]). By (i) and (iii) we can apply the argument at the beginning of the proof and get an action of w H (σ ) on K ∗0 (Y × H H 0 , C H 0 (σ )) and therefore on K ∗0 (Y, C H (σ )), as desired. It is not difficult to check that this action does not depend on the chosen immersion H ,→ H 0 . Finally, let us prove that there exists a closed immersion H ,→ H 0 satisfying conditions (i) – (iii) above. First choose a closed immersion j : H ,→ GLn,k for some n. Clearly, H/C H (σ ) ,→ GLn,k /CGLn,k (σ ), and, embedding σ in a maximal torus of GLn,k , it is easy to check that GLn,k (k) → GLn,k /CGLn,k (σ ) (k) is surjective. Now define H 0 as the inverse image of H/C H (σ ) in the normalizer NGLn,k (σ ).
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COROLLARY 2.5 There is a canonical action of wG (σ ) on K ∗0 (X σ , CG (σ )) which induces an action on K ∗0 (X σ , CG (σ ))geom .
Proof Since CG (σ ) = C NG (σ ) (σ ), Lemma 2.4, applied to Y = X σ (resp., Y = Spec k) and H = N G (σ ), yields an action of wG (σ ) on K ∗0 X σ , CG (σ ) resp., on K 0 (Spec k, CG (σ )) = R(CG (σ )) . The multiplicative system S1 = rk−1 (1) is preserved by this action so that there is an induced action on the ring S1−1 R(CG (σ )). The pullback K 0 Spec k, CG (σ ) → K 0 X σ , CG (σ ) is wG (σ )-equivariant, and then wG (σ ) acts on K ∗0 (X σ , CG (σ ))geom . Remark 2.6 If Y is regular, Lemma 2.4 also gives an action of w H (σ ) on K ∗ (Y, C H (σ )) since K ∗ (Y, C H (σ )) ' K ∗0 (Y, C H (σ )) (see appendix). In particular, since by [Th5, Prop. 3.1], X σ is regular, Corollary 2.5 still holds for K ∗ (X σ , CG (σ ))geom . Note also that the embedding of k-group schemes WG (σ ) ,→ Autk (σ ) induces, by Example 2.1, an action of wG (σ ) on K 0 (Spec k, σ ) = R(σ ). The product in σ induces a morphism of k-groups, σ × CG (σ ) −→ CG (σ ), which in its turn induces a morphism m σ : K ∗ X σ , CG (σ ) −→ K ∗ X σ , σ × CG (σ ) . LEMMA 2.7 If H ⊆ G is a subgroup scheme and if σ is contained in the center of H , there is a canonical ring isomorphism
K ∗ (X σ , σ × H ) ' K ∗ (X σ , H ) ⊗ R(σ ). Proof Since σ acts trivially on X σ , we have an equivalence (see [SGA3, Exp. I, par. 4.7.3]) M (σ × H ) − Coh X σ ' (H − Coh X σ ) (3) b σ
(where b σ is the character group of σ ) which induces an isomorphism K ∗0 (X σ , σ × H ) ' K ∗0 (X σ , H ) ⊗ R(σ ).
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
13
We conclude since K ∗ (Y, H ) ' K ∗0 (Y, H ) and K ∗ (X σ , σ × H ) ' K ∗0 (X σ , σ × H ) (see appendix). . For any essential dual cyclic subgroup σ ⊆ G, let 3 = 3(G,X ) , and consider the composition K ∗ (X, G)3 → K ∗ X, CG (σ ) 3 → K ∗ X σ , CG (σ ) 3 mσ −→ K ∗ X σ , CG (σ ) 3 ⊗3 R(σ )3 −→ K ∗ X σ , CG (σ ) geom ⊗3 e R(σ )3 , (4) where the first map is induced by group restriction, the last one is the geometric localization map tensored with the projection R(σ )3 → e R(σ )3 , and we have used Lemma 2.7 with H = CG (σ ); the second map is induced by restriction along X σ ,→ X which is a regular closed immersion (see [Th5, Prop. 3.1]) and therefore has finite Tor-dimension, so that the pullback on K -groups is well defined (see appendix). It is not difficult to show that the image of (4) is actually contained in the invariant submodule w (σ ) K ∗ (X σ , CG (σ ))geom ⊗3 e R(σ )3 G , so that we get a map ψσ,X : K ∗ (X, G)3 −→ K ∗ (X σ , CG (σ ))geom ⊗3 e R(σ )3
wG (σ )
.
Our basic map is . 9 X,G =
Y
ψσ,X : K ∗ (X, G)3
σ ∈C (G)
−→
Y
K ∗ (X σ , CG (σ ))geom ⊗3 e R(σ )3
wG (σ )
. (5)
σ ∈C (G)
Note that 9 X,G is a morphism of R(G)-algebras as a composition of morphisms of R(G)-algebras. The following technical lemma is used in Propositions 3.5 and 4.6. LEMMA 2.8 Let G be a linear algebraic k-group acting with finite stabilizers on a Noetherian . separated k-algebraic space X , and let 3 = 3(G,X ) . Let H ⊆ G be a subgroup, and let σ be an essential dual cyclic subgroup contained in the center of H . Consider the composition
K ∗0 (Y σ , H )3 −→ K ∗0 (Y σ , H )3 ⊗3 R(σ )3 −→ K ∗0 (Y σ , H )geom ⊗3 e R(σ )3 , (6)
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VEZZOSI and VISTOLI
where the first morphism is induced by the product morphism σ × H → H (recall Lem. 2.7) and the second is the tensor product of the geometric localization morphism with the projection R(σ )3 → e R(σ )3 . Then (6) factors through K ∗0 (Y σ , H )3 → K ∗0 (Y σ , H )σ , yielding a morphism θ H,σ : K ∗0 (Y σ , H )σ −→ K ∗0 (Y σ , H )geom ⊗3 e R(σ )3 .
(7)
Proof Let S1 (resp., Sσ ) be the multiplicative subset in R(H )3 consisting of elements going to 1 via the homomorphism rk H,3 : R(H )3 → 3 (resp., R(H )3 → e R(σ )3 ). Ob0 σ 0 σ e serve that K ∗ (X , H )3 ⊗3 R(σ )3 (resp., K ∗ (X , H )geom ⊗3 R(σ )3 ) is canonically an R(H )3 ⊗ R(σ )3 -module (resp., an S1−1 R(H )3 ⊗ e R(σ )3 -module) and therefore an R(H )-module via the coproduct ring morphism 1σ : R(H )3 −→ R(H )3 ⊗ R(σ )3 resp., via the ring morphism 1σ f σ : R(H )3 −→ R(H )3 ⊗ R(σ )3 −→ S1−1 R(H )3 ⊗ e R(σ )3 . If we denote by A0 the R(H )3 -algebra f σ : R(H )3 −→ S1−1 R(H )3 ⊗ e R(σ )3 , it is enough to show that the localization homomorphism lσ0 : A0 −→ Sσ−1 (A0 ) is an isomorphism, because in this case the morphism (7) is induced by the Sσ localization of (6). Let A denote the R(H )3 -algebra λ1 ⊗ 1 : R(H )3 −→ S1−1 R(H )3 ⊗ e R(σ )3 , where λ1 : R(H )3 → S1−1 R(H )3 denotes the localization homomorphism. It is a well-known fact that there is an isomorphism of R(H )3 -algebras ϕ : A0 → A; this is exactly the dual assertion to “the action H × σ → σ is isomorphic to the projection on the second factor H × σ → σ .” Therefore, we have a commutative diagram A0 lσ0 y
ϕ
−−−−→
A l yσ
Sσ−1 A0 −−−−→ Sσ−1 A Sσ−1 ϕ
where lσ denotes the localization homomorphism, and it is enough to prove that lσ is an isomorphism. To see this, note that the ring e R(σ )3 is a free 3-module of finite rank (equal to φ(|σ |), φ being the Euler function), and there is a norm homomorphism N:e R(σ )3 −→ 3
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
15
sending an element to the determinant of the 3-endomorphism of e R(σ )3 induced by multiplication by this element; obviously, we have ∗ N−1 (3∗ ) = e R(σ )3 . Analogously, there is a norm homomorphism N0 : A0 = S1−1 R(H )3 ⊗ e R(σ )3 −→ S1−1 R(H )3 , and ∗ N−1 (S1−1 R(H )3 )∗ = S1−1 R(H )3 ⊗ e R(σ )3 . There is a commutative diagram N0
S1−1 R(H )3 ⊗ e R(σ )3 −−−−→ S1−1 R(H )3 rk rk H,3 ⊗ idy y H,3 e R(σ )3
−−−−→ N
3
−1 ∗ ∗ and, by definition of S1 , we get rk−1 H,3 (3 ) = (S1 R(H )3 ) . Therefore, by definition of Sσ , Sσ /1 consist of units in A, and we conclude the proof of the lemma.
The following lemma, which is an easy consequence of a result of Merkurjev, is the main tool in reducing the proof of the main theorem from G = GLn,k to its maximal torus T . 2.9 Let X be a Noetherian separated algebraic space over k with an action of a split reductive group G over k such that π1 (G) (see [Me, Par. 1.1]) is torsion free. Then if T denotes a maximal torus in G, the canonical morphism LEMMA
K ∗0 (X, G) ⊗R(G) R(T ) −→ K ∗0 (X, T ) is an isomorphism. Proof Let B ⊇ T be a Borel subgroup of G. Since R(B) ' R(T ) and K ∗0 (X, B) ' K ∗0 (X, T ) (see [Th4, proof of Th. 1.13, p. 594]), by [Me, Prop. 4.1], the canonical ring morphism K ∗0 (X, G) ⊗R(G) R(T ) −→ K ∗0 (X, T ) is an isomorphism.
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VEZZOSI and VISTOLI
Since Merkurjev states his theorem for a scheme, we briefly indicate how it extends to a Noetherian separated algebraic space X over k. By [Th1, Lem. 4.3], there exists an open dense G-invariant separated subscheme U ⊂ X . Since Merkurjev’s map commutes with localization, by the localization sequence and Noetherian induction it is enough to know the result for U . And this is given in [Me, Prop. 4.1]. Note that by [Me, Prop. 1.22], R(T ) is flat (actually free) over R(G), and therefore the localization sequence remains exact after tensoring with R(T ). The following is [Vi1, Lem. 3.2]. It is used frequently in the rest of the paper, and it is stated here for the convenience of the reader. LEMMA 2.10 Let W be a finite group acting on the left on a set A , and let B ⊆ A be a set of representatives for the orbits. Assume that W acts on the left on a product of abelian Q groups of the type α∈A Mα in such a way that
s Mα = Msα for any s ∈ W . For each α ∈ B , let us denote by Wα the stabilizer of α in W . Then the canonical projection Y Y Mα −→ Mα α∈A
α∈B
induces an isomorphism Y
Mα
W
−→
α∈A
Y
(Mα )Wα .
α∈B
3. The main theorem: The split torus case In this section, T is a split torus over k. PROPOSITION 3.1 Let T 0 ⊂ T be a closed
subgroup scheme (diagonalizable, by [SGA3, Exp. IX, par. 8.1]), finite over k. Then the canonical morphism Y e δ : R(T 0 )3T 0 −→ R(σ )3T 0 σ dual cyclic σ ⊆T 0
is a ring isomorphism. Proof Q Since both R(T 0 )3T 0 and e R(σ )3T 0 are free 3T 0 -modules of finite rank, it is enough
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
17
to prove that, for any nonzero prime p - |T 0 |, the induced morphism of F p -vector spaces Y e R(T 0 )3T 0 ⊗Z F p −→ R(σ )3T 0 ⊗Z F p (8) σ dual cyclic σ ⊆T 0
is an isomorphism. Now, for any finite abelian group A, we have an equality |A| = P AC ϕ(|C|), where ϕ denotes the Euler function, |H | denotes the order of the group H , and the sum is extended to all cyclic quotients of A; applying this to the group of characters Tb0 (so that the corresponding cyclic quotients C are exactly the group of characters b σ for σ dual cyclic subgroups of T 0 ), we see that the ranks of both sides in (8) coincide with |T 0 |, and it is then enough to prove that (8) is injective. Define a morphism Y Y e f : R(σ )3T 0 R(τ )3T 0 −→ τ dual cyclic τ ⊆T 0
σ dual cyclic σ ⊆T 0
of R(T 0 )3T 0 -modules by requiring, for any dual cyclic subgroup σ ⊆ T 0 , the commutativity of the following diagram: R(τ )3T 0 τ dual cyclic e τ ⊆T 0
Q
f
−−−−→
σ dual cyclic R(σ )3T 0 σ ⊆T 0
Q
Prσ y Q
τ ⊆σ
pr y σ
e R(τ )3T 0
R(σ )3T 0
]
←−−−− ϕ
where Prσ and prσ are the obvious projections and ϕ is the isomorphism Q
τ ⊆σ
resστ
R(σ )3T 0 −−−−−−→
Y τ ⊆σ
(e prτ )τ
R(τ )3T 0 −−−→
Y τ ⊆σ
e R(τ )3T 0
induced by (1). Obviously, f ◦ δ coincides with the map Y Y 0 resσT : R(T 0 )3T 0 −→ R(σ )3T 0 , σ dual cyclic σ ⊆T 0
σ dual cyclic σ ⊆T 0
so we are reduced to proving that R(T 0 )3T 0 ⊗Z F p −→
Y
R(σ )3T 0 ⊗Z F p
σ dual cyclic σ ⊆T 0
is injective, that is, that if A is a finite abelian group and p - |A|, then Y ϕ : F p [A] −→ F p [C] C∈{cyclic quotients ofA}
(9)
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VEZZOSI and VISTOLI
b = HomAbGrps (A, C∗ ) denotes the complex characters group of A, is injective. If A b then R( A) ' Z[A] and Y b Y A b −→ b ϕ= resCb : R( A) R(C). b C
b b C∈{cyclic subgroups of A}
b ⊗ ˙ Z Z[1/|A|] has image via Since p - |A|, it is enough to prove that if ξ ∈ R( A) A b ⊗Z Z[1/|A|] −→ R(C) b ⊗Z Z[1/|A|] resCb ⊗ Z[1/|A|] : R( A) b ⊗Z Z[1/|A|] for each cyclic C b ⊆ A, b then ξ ∈ contained in p R(C) b ⊗ ˙ Z Z[1/|A|] . p R( A) Q 0 ) ∈ b ⊗Z Z[1/|A|] By [Se, p. 73], there exists (θCb b b b R(C) C C∈{cyclic subgroups of A} such that X 0 b 1= indCAb ⊗ Z[1/|A|] (θCb ); b
b C
therefore ξ=
X
=
X
0 b ξ indCAb ⊗ Z[1/|A|] (θCb )
b C
0 b b A indCAb ⊗ Z[1/|A|] θCb (resCb ⊗ Z[1/|A|])(ξ )
b C
(by the projection formula), and we conclude the proof of the proposition. Remark 3.2 The proof of Proposition 3.1 is similar to [Vi1, proof of Prop. 1.5], which is, however, incomplete; that is why we have decided to give all the details here. COROLLARY 3.3 We have the following. (i) If σ 6 = σ 0 are dual cyclic subgroups of T , we have e R(σ )σ 0 = 0 and e R(σ )σ = e R(σ ). (ii) If T 0 ⊂ T is a closed subgroup scheme, finite over k, and if σ is a dual cyclic subgroup of T , we have R(T 0 )σ = 0 if σ * T 0 . (iii) If T 0 ⊂ T is a closed subgroup scheme, finite over k, the canonical morphism of R(T )-algebras Y R(T 0 )3T 0 −→ R(T 0 )σ σ dual cyclic σ ⊆T 0
is an isomorphism.
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
19
Proof (i) Suppose σ 6= σ 0 , and let T 0 ⊂ T be the closed subgroup scheme of T generated by σ and σ 0 . The obvious morphism π : R(T )3T 0 → e R(σ )3T 0 × e R(σ 0 )3T 0 factors through R(T 0 )3T 0 → e R(σ )3T 0 × e R(σ 0 )3T 0 , which is an epimorphism by Proposition 3.1. If ξ ∈ R(T )3T 0 with π(ξ ) = (0, 1) ⊗ 1, we have ξ ∈ Sσ 0 ∩ ker R(T )3T 0 → e R(σ )3T 0 . Then e R(σ )σ 0 = 0. The second assertion is obvious. (ii) and (iii) These follow immediately from (i) and Proposition 3.1. Now let X be a regular Noetherian separated k-algebraic space on which T acts with . finite stabilizers, and let 3 = 3(T,X ) . Obviously, C (T ) is just the set of essential dual cyclic subgroups of T since T is abelian. PROPOSITION 3.4 We have the following. (i) If jσ : X σ ,→ X denotes the inclusion, the pushforward ( jσ )∗ induces an isomorphism K ∗0 (X σ , T )σ −→ K ∗0 (X, T )σ .
(ii)
The canonical localization morphism K ∗0 (X, T )3 −→
Y
K ∗0 (X, T )σ
σ ∈C (T )
is an isomorphism, and the product on the left is finite. Proof (i) The proof is the same as that of [Th5, Th. 2.1], but we substitute Corollary 3.3(ii) for [Th5, Th. 2.1] since we use a localization different from Thomason’s. (ii) By the generic slice theorem for torus actions (see [Th1, Prop. 4.10]), there exist a T -invariant nonempty open subspace U ⊂ X , a closed (necessarily diagonalizable) subgroup T 0 of T , and a T -equivariant isomorphism 0
U ' T /T 0 × (U/T ) ' (U/T ) ×T T. Since U is nonempty and T acts on X with finite stabilizers, T 0 is finite over k and K ∗0 (U, T ) ' K ∗0 (U/T ) ⊗Z R(T 0 ), by Morita equivalence theorem (see [Th3, Prop. 6.2]) and [Th1, Lem. 5.6]. By Corollary 3.3(ii), the proposition for X = U follows from Corollary 3.3(iii). By Noetherian induction and the localization sequence for K 0 -groups (see [Th3, Th. 2.7]), the statement for U implies the same for X . Again using Noetherian induction, Thomason’s generic slice theorem for torus Q actions, and (i), one similarly shows that the product σ ∈C (T ) K ∗0 (X, T )σ is finite.
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VEZZOSI and VISTOLI
By Proposition 3.4, there is an induced isomorphism (of R(T )-modules, not a ring isomorphism due to the composition with pushforwards) Y K ∗0 (X σ , T )σ −→ K ∗0 (X, T )3 . (10) σ ∈C (T )
As shown in Lemma 2.8, the product morphism σ × T → T induces a morphism θT,σ : K ∗0 (X σ , T )σ −→ K ∗0 (X σ , T )geom ⊗ e R(σ )3 .
(11)
PROPOSITION 3.5 For any σ ∈ C (T ), θT,σ is an isomorphism.
Proof We write θ X,σ for θT,σ in order to emphasize the dependence of the map on the space. We proceed by Noetherian induction on X σ . Let X 0 ⊆ X σ be a T -invariant closed subspace, and let us suppose that (11) is an isomorphism with X replaced by any T invariant proper closed subspace Z of X 0 . By Thomason’s generic slice theorem for torus actions (see [Th1, Prop. 4.10]), there exist a T -invariant nonempty open subscheme U ⊂ X 0 , a (necessarily diagonalizable) subgroup T 0 of T , and a T -equivariant isomorphism 0 U σ ≡ U ' T /T 0 × (U/T ) ' (U/T ) ×T T. Since U is nonempty and T acts on X with finite stabilizers, T 0 is finite over k and, . obviously, 3T 0 ⊆ 3. Let Z σ ≡ Z = X 0 U . Since /
θ Z ,σ
/
/
K ∗0 (Z σ , T )σ
θY 0 ,σ
e )3 K ∗0 (Z σ , T )geom ⊗ R(σ
/
/
K ∗0 (X 0σ , T )σ
e )3 K ∗0 (X 0σ , T )geom ⊗ R(σ
θU,σ
/
/
K ∗0 (U σ , T )σ
e )3 K ∗0 (U σ , T )geom ⊗ R(σ
/
is commutative, by the induction hypothesis and the five-lemma it is enough to show that θU,σ is an isomorphism. By Morita equivalence theorem (see [Th3, Prop. 6.2]) and [Th1, Lem. 5.6], K ∗0 (U, T ) ' K ∗0 (U/T ) ⊗Z R(T 0 ), so it is enough to prove that θSpec k,σ : K 00 (Spec k, T 0 )σ = R(T 0 )σ → K 00 (Spec k, T 0 )geom ⊗ e R(σ )3 0 = R(T )geom ⊗ e R(σ )3 is an isomorphism. But this follows immediately from Corollary 3.3(i) and (iii). Combining Proposition 3.5 with (10), we get an isomorphism Y 8 X,T : K ∗0 (X σ , T )geom ⊗ e R(σ )3 −→ K ∗0 (X, T )3 . σ ∈C (T )
(12)
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21
The following lemma is a variant of [Th5, Lem. 3.2], which already proves it after tensoring with Q. LEMMA 3.6 Let X be a Noetherian regular separated algebraic space over k on which a split k-torus acts with finite stabilizers, and let σ ∈ C (T ). Let X σ denote the regular σ -fixed subscheme, let jσ : X σ ,→ X be the regular closed immersion (see [Th5, Prop. 3.1]), and let N ( jσ ) be the corresponding locally free conormal sheaf. Then, for any T -invariant algebraic subspace Y of X σ , the cap-product λ−1 N ( jσ ) ∩ (−) : K ∗0 (Y, T )σ −→ K ∗0 (Y, T )σ
is an isomorphism. Proof We proceed by Noetherian induction on closed T -invariant subspaces Y of X σ . The statement is trivial for Y = ∅, so let us suppose Y nonempty and λ−1 N ( jσ ) ∩ (−) : K ∗0 (Z , T )σ −→ K ∗0 (Z , T )σ an isomorphism for any proper T -invariant closed subspace Z of Y . By Thomason’s generic slice theorem for torus actions (see [Th1, Prop. 4.10]), there exist a T -invariant nonempty open subscheme U ⊂ Y , a closed (necessarily diagonalizable) subgroup T 0 of T , and a T -equivariant isomorphism U σ ≡ U ' T /T 0 × (U/T ). Since U is nonempty and T acts on X with finite stabilizers, T 0 is finite over k. Using the localization sequence and the five-lemma, we reduce ourselves to showing that λ−1 N ( jσ ) ∩ (−) : K ∗0 (U, T )σ −→ K ∗0 (U, T )σ is an isomorphism. For this, it is enough to show that (the restriction of) λ−1 (N ( jσ )) is a unit in K 0 (U, T )σ ' K 0 (U/T )3 ⊗ R(T 0 )σ (see [Th3, Prop. 6.2]). Decomposing N ( jσ ) according to the characters of T 0 , we may write, shrinking U if necessary, M rρ N ( jσ ) = OU/T ⊗ Lρ , ρ∈Tb0
where Lρ is the line bundle attached to the T 0 -character ρ and rρ ≥ 0, and thereQ fore λ−1 (N ( jσ )) = ρ∈Tb0 (1 − ρ)rρ in K 0 (U/T ) ⊗ R(T 0 ). The localization map R(T 0 )3 → R(T 0 )σ ' e R(σ )3 coincides with the composition πσ
pσ
R(T 0 )3 −→ R(σ )3 −→ e R(σ )3
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VEZZOSI and VISTOLI
of the restriction to σ followed by the projection (see Cor. 3.3), and then M rχ (id K 0 (U/T )3 ⊗πσ ) N ( jσ ) = OU/T ⊗ Lχ , χ∈b σ \{0}
in K 0 (U/T )3 ⊗ R(σ )3 , where the summand omits the trivial character since the decomposition of N ( jσ ) according to the characters of σ has vanishing fixed subsheaf N ( jσ )0 (see, e.g., [Th5, Prop. 3.1]). Therefore, Y λ−1 (id K 0 (U/T )3 ⊗πσ )(N ( jσ )) = (1 − χ )rχ , χ ∈b σ \{0}
and it is enough to show that the image of 1 − χ in e R(σ )3 via pσ is a unit for any nontrivial character χ of σ . Now, the image of such a χ in 3[t] e R(σ )3 ' (8|σ | ) (8|σ | being the |σ |th cyclotomic polynomial) is of the form [t l ] for some 1 ≤ l < |σ |, where [−] denotes the class mod8|σ | ; therefore the cokernel of the multiplication by 1 − [t l ] in 3[t] (8|σ | ) is 3[t] =0 (8|σ | , 1 − t l ) since 8|σ | and (1 − t l ) are relatively prime in 3[t] for 1 ≤ l < |σ |. Thus 1 − [t l ] is a unit in 3[t] (8|σ | ), and we conclude the proof of the lemma. We are now able to prove our main theorem for G = T . THEOREM 3.7 If X is a regular Noetherian separated k-algebraic space, then Y 9 X,T : K ∗ (X, T )3 −→ K ∗ (X σ , T )geom ⊗ e R(σ )3 σ ∈C (T )
is an isomorphism of R(T )-algebras. Proof Recall (see appendix) that K ∗ (X, T ) ' K ∗0 (X, T ) and K ∗ (X σ , T ) ' K ∗0 (X σ , T ), since both X and X σ are regular (see [Th5, Prop. 3.1]). Since 8 X,T is an isomorphism of R(T )-modules, it is enough to show that the composition 9 X,T ◦ 8 X,T is an isomorphism. A careful inspection of the definitions of 9 X,T and 8 X,T easily reduces the problem to proving that, for any σ ∈ C (T ), the composition jσ ∗
jσ∗
K ∗0 (X σ , T )σ −→ K ∗0 (X, T )σ −→ K ∗0 (X σ , T )σ
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
23
is an isomorphism, jσ : X σ ,→ X being the natural inclusion. Since jσ is regular, there is a self-intersection formula jσ∗ ◦ jσ ∗ (−) = λ−1 N ( jσ ) ∩ (−), (13) N ( jσ ) being the conormal sheaf associated to jσ , and we conclude by Lemma 3.6. To prove the self-intersection formula (13), we adapt [Th5, proof of Lem. 3.3]. First, by Proposition 3.4(i), jσ ∗ is an isomorphism, so it is enough to prove that jσ ∗ jσ∗ jσ ∗ (−) = jσ ∗ (λ−1 (N ( jσ )) ∩ (−)). By the projection formula (see Prop. A.5), we have
jσ ∗ jσ∗ jσ ∗ (−) = jσ ∗ jσ∗ (1) ∩ jσ ∗ (−) = jσ ∗ jσ∗ (O X ) ∩ jσ ∗ (−) = jσ ∗ (O X σ ) ∩ jσ ∗ (−) = jσ ∗ jσ∗ (O X σ ) ∩ (−) . Now, as explained in the appendix, to compute jσ∗ (O X σ ) we choose a complex F ∗ of flat quasi-coherent G-equivariant modules on X which is quasi-isomorphic to O X σ , and then X jσ∗ (O X σ ) = [ jσ∗ (F ∗ )] = [F ∗ ⊗ O X σ ] = (−1)i [H i (F ∗ ⊗ O X σ )]. i
But F ∗ is a flat resolution of O X σ , so H i (F ∗ ⊗ O X σ ) = ToriO X (O X σ , O X σ ) ' Vi N ( jσ ), where the last isomorphism (see [SGA6, Exp. VII, par. 2.5]) is natural and hence T -equivariant. Therefore, jσ∗ (O X σ ) = λ−1 (N ( jσ )), and we conclude the proof of the theorem. 4. The main theorem: The case of G = GLn,k In this section we use the result for 9 X,T to deduce the same result for 9 X,GLnk . THEOREM 4.1 Let X be a Noetherian regular separated algebraic space over a field k on which G = GLn,k acts with finite stabilizers. Then the map defined in (5),
9 X,G : K ∗ (X, G)3(G,X ) Y −→
K ∗ (X σ , C(σ ))geom ⊗3(G,X ) e R(σ )3(G,X )
wG (σ )
, (14)
σ ∈C (G)
is an isomorphism of R(G)-algebras and the product on the right is finite. Throughout this section, entirely devoted to the proof of Theorem 4.1, we simply write G for GLn,k , 3 for 3(G,X ) , and T for the maximal torus of diagonal matrices in GLn,k . First, let us observe that we can choose each σ ∈ C (G) contained in T . Moreover, 3(T,X ) = 3(G,X ) .
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VEZZOSI and VISTOLI
We need the following three preliminary lemmas (Lems. 4.2, 4.3, 4.4). If σ, τ ⊂ T are dual cyclic subgroups, they are conjugate under the G(k)-action if and only if they are conjugated via an element in the Weyl group Sn . For any group scheme H with a dual cyclic subgroup σ ⊆ H , we denote by mσH the kernel of \ R(H )3 → e R(σ )3 and by R(H )3,σ the completion of R(H )3 with respect to the H ideal mσ . The following lemma is essentially a variant of Lemma 2.9 for σ -localizations. LEMMA 4.2 Let G = GLn,k , let T be the maximal torus of G consisting of diagonal matrices, and let X be an algebraic space on which G acts with finite stabilizers. (i) For any essential dual cyclic subgroup σ ⊆ T , the morphisms ωσ,geom : K ∗0 X σ , CG (σ ) geom ⊗R(CG (σ ))3 R(T )3 −→ K ∗0 (X σ , T )geom , ωσ : K ∗0 X σ , CG (σ ) σ ⊗R(CG (σ ))3 R(T )3 −→ K ∗0 (X σ , T )σ
(ii)
induced by T ,→ CG (σ ) are isomorphisms. For any essential dual cyclic subgroup σ ⊆ T , G (σ ) ) N · K 0 X σ , C (σ ) (mC = 0, G σ ∗ σ
N 0,
and the morphism induced by T ,→ CG (σ ), \ ω cσ : K ∗0 X σ , CG (σ ) σ ⊗R(C\ R(T )3,σ −→ K ∗0 (X σ , T )σ , (σ )) G
3,σ
is an isomorphism. Proof (i) Since CG (σ ) is isomorphic to a product of general linear groups over k and since T is a maximal torus in CG (σ ), by Lemma 2.9 the canonical ring morphism K ∗0 X, CG (σ ) ⊗R(CG (σ )) R(T ) −→ K ∗0 (X, T ) (15) is an isomorphism. If H ⊆ G is a subgroup scheme, we denote by SσH the multiplicative subset of R(H )3 consisting of the elements sent to 1 by the canonical ring homomorphism R(H )3 → e R(σ )3 . By (15), ωσ coincides with the composition K ∗0 X σ , CG (σ ) σ ⊗R(CG (σ ))3 R(T )3 ' K ∗0 (X σ , T ) ⊗R(CG (σ ))3 (SσCG (σ ) )−1 R(CG (σ ))3 ⊗R(CG (σ ))3 R(T )3 id ⊗νσ −−−→ K ∗0 X σ , CG (σ ) ⊗R(CG (σ ))3 (SσT )−1 R(T )3 ' K ∗0 (X σ , T )σ ,
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
25
where νσ : (SσCG (σ ) )−1 R CG (σ )
3
⊗R(CG (σ ))3 R(T )3 → (SσT )−1 R(T )3
(16)
is induced by T ,→ CG (σ ) and the last isomorphism follows from (15); the same is true for ωσ,geom . Therefore, it is enough to prove that νσ and C (σ ) −1
νσ,geom : (S1 G
)
R CG (σ ) 3 ⊗R(CG (σ ))3 R(T )3 → (S1T )−1 R(T )3 C (σ )
are isomorphisms; that is, if Sτ denotes the image of S1 G R CG (σ ) 3 −→ R(T )3 ,
via the restriction map
then SτT /1 consists of units in (Sτ )−1 R(T )3 for τ = 1 and τ = σ . If 1σ denotes the Weyl group of CG (σ ), which is a product of symmetric groups, we have R(CG (σ )) ' R(T )1σ and therefore 1 (SτCG (σ ) )−1 R CG (σ ) 3 ' (Sτ )−1 R(T )3 σ since R(T ) is torsion free. Moreover, there is a commutative diagram
/ (Sτ )−1 R(T )3 j j j jjjj ψ jjj. j j j j =ϕ tjjj T −1 T )−1 R(T ) o e R(τ )3 ' (Sτ ) R(τ )3 (S 3 τ T −1 T C (σ ) −1 ) R
(Sτ G
CG (σ ) 3
(Sτ )
resτ
where ψ is induced by e πτ and the isomorphism e R(τ )3 ' (SτT )−1 R(τ )3 is obtained from Proposition 3.1 and Corollary 3.3. If we define the map 1 M : (Sτ )−1 R(T )3 −→ (Sτ )−1 R(T )3 σ , Y ξ 7 −→ g·ξ , g∈1σ
it is easily checked that for ξ ∈ (Sτ )−1 R(T )3 , ξ is a unit if M(ξ ) is a unit, and that ψ(M(ξ )) = 1 implies that ξ is a unit in ((Sτ )−1 R(T )3 )1σ . But ϕ is 1σ -equivariant, and therefore SτT /1 consists of units in (Sτ )−1 R(T )3 for τ = 1 or σ . (ii) Since R(CG (σ )) → R(T ) is faithfully flat, by (i) it is enough to prove that (mσT ) N K ∗0 (X σ , T )σ = 0
for N 0.
(17)
But (17) can be proved using the same technique as in the proof of, for example, Proposition 3.5, that is, Noetherian induction together with Thomason’s generic slice
26
VEZZOSI and VISTOLI
theorem for torus actions. The second part of (ii) follows, arguing as in (i), from the fact that (16) is an isomorphism since K ∗0 X σ , CG (σ )
σ
⊗R
CG
0 σ R C\ (σ ) ' K X , C (σ ) , G G ∗ 3,σ σ (σ ) 3
K ∗0 (X σ , T )σ
\ ⊗R(T )3 R(T )3,σ ' K ∗0 (X σ , T )σ .
If σ, τ ⊂ T are dual cyclic subgroups conjugated under G(k), they are conjugate through an element of the Weyl group Sn and we write τ ≈ Sn σ ; moreover, we have mσG = mτG because conjugation by an element in Sn (actually, by any element in G(k)) induces the identity morphism on K -theory and, in particular, on the representation ring. Then there are canonical maps Y \ \ R(T )3,τ , (18) R(G) 3,σ ⊗R(G)3 R(T )3 −→ τ dual cyclicτ ≈ Sn σ
\ R C\ )3,σ . G (σ ) 3,σ ⊗ R(CG (σ ))3 R(T )3 −→ R(T
(19)
4.3 Maps (18) and (19) are isomorphisms. LEMMA
Proof \ Since R(G) = R(T ) Sn → R(T ) is finite, the canonical map R(G) 3,σ ⊗R(G)3 mG
mG
σ σ \ \ R(T )3 → R(T )3 (where R(T )3 denotes the mσG -adic completion of the R(G)3 -module R(T )3 ) is an isomorphism. Moreover, R(G)3 = (R(T )3 ) Sn implies that q \ q \ G mσ R(T )3 = mτT = mτT , τ dual cyclic τ ≈ Sn σ
τ dual cyclic τ ≈ Sn σ
and, by Corollary 3.3(i), mτT + mτT0 = R(T )3 if τ 6= τ 0 . By the Chinese remainder lemma, we conclude that (18) is an isomorphism. Arguing in the same way, we get that the canonical map R C\ G (σ ) 3,σ ⊗R(CG (σ ))3 R(T )3 −→
Y
\ R(T )3,τ
τ dual cyclic τ ≈1σ σ
is an isomorphism, where 1σ = Sn ∩ CG (σ ) is the Weyl group of CG (σ ) with respect to T and we write τ ≈1σ σ to denote that τ and σ are conjugate through an element of 1σ . But 1σ ⊂ CG (σ ), so that τ ≈1σ σ if and only if τ = σ , and we conclude that (19) is an isomorphism.
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
27
LEMMA 4.4 For any essential dual cyclic subgroup σ ⊆ G, the canonical morphism \ \ R(G) 3,σ −→ R CG (σ ) 3,σ
is a finite e´ tale Galois cover (see [SGA1, Exp. V]) with Galois group wG (σ ). Proof Since R(T ) is flat over R(G) = R(T ) Sn , we have \ \ R(G) 3,σ ' R(G) 3,σ ⊗R(G)3 R(T )3
Sn
Sn \ ⊗ ' R(G) 3,σ R(G)3 R(T )3 Sn Y \ R(T )3·τ , ' τ dual cyclic τ ≈ Sn σ
the last isomorphism being given in Lemma 4.3. By Lemma 2.10, we get S \σ ' R(T \ R(G) )σ n,σ , where Sn acts on the set of dual cyclic subgroups of T which are Sn -conjugated to σ and where Sn,σ denotes the stabilizer of σ . Analogously, denoting by 1σ the Weyl group of CG (σ ), by Lemma 4.2(ii) we have \ R(T )3 1σ R C\ G (σ ) 3,σ ' R CG (σ ) 3,σ ⊗R C (σ ) G
3
' R(C\ G (σ ))3,σ ⊗R(CG (σ ))3 R(T )3 1 \ ' R(T )3,σ σ ,
1σ
where the last isomorphism is given by Lemma 4.3. From the exact sequence 1 → 1σ −→ Sn,σ −→ wG (σ ) → 1, we conclude that \ R(G) 3,σ ' R(C\ G (σ ))3,σ
wG (σ )
By [SGA1, Prop. 2.6, Exp. V], it is now enough to prove ometric points (i.e., the inertia groups of points) of Spec wG (σ )-action are trivial. First, let us observe that Spec(e R(σ )3 ) is a Spec(R(C\ G (σ ))3,σ ). This can be seen as follows. It show that if s denotes the order of σ , the map πσ : R CG (σ )
3
−→ R(σ )3 =
.
(20)
that the stabilizers of ge R(C\ G (σ ))3,σ under the closed subscheme of is obviously enough to
3[t] (t s − 1)
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VEZZOSI and VISTOLI
is surjective. First, consider the case where σ is contained in the center of G. Since R(σ )3 is of finite type over 3, we show that for any prime∗ p - s the induced map πσ, p : R CG (σ ) 3 ⊗ F p −→ R(σ )3 ⊗ F p is surjective. Note that if E denotes the standard n-dimensional representation of G, V πσ sends r E to nr t r . If p - n, then πσ, p is surjective (in fact, πσ (E) = nt and n is invertible in F p ). If p | n, let us write n = qm, with q = pi and p - m. Since (s, q) = 1, t q is a ring generator of R(σ )3 , and to prove πσ, p is injective, it is enough to show that p - qn . This is elementary since the binomial expansion of (1 + X )n = (1 + X q )m Q in F p [X ] yields qn = m in F p . For a general σ ⊆ T , let CG (σ ) = li=1 GLdi ,k , P where d = n, and let σi denote the image of σ in GLdi ,k , i = 1, . . . , l. Since Ql i σ ⊆ i=1 σi is an inclusion of diagonalizable groups, the induced map l l O Y σi = R(σi ) −→ R(σ ) R i=1
i=1
is surjective (e.g., see [SGA3, Vol. II]). But R(CG (σ ))3 → R(σ )3 factors as R CG (σ )
3
=
l O
R(GLdi ,k )3 −→
i=1
l O
R(σi )3 −→ R(σ )3 ,
i=1
and also the first map is surjective (by the previous case, since σi is contained in the center of GLdi ,k and |σi | divides |σ |). This proves that Spec(e R(σ )3 ) is a closed sub\ scheme of Spec(R(C\ (σ )) ). Since R(C (σ )) is the completion of R(CG (σ ))3 G 3,σ G 3,σ along the ideal ker R(CG (σ ))3 → e R(σ )3 , any nonempty closed subscheme of Spec R(C\ G (σ ))3,σ meets the closed subscheme Spec(e R(σ )3 ). To prove that wG (σ ) acts freely on the geometric points of \ Spec R(CG (σ ))3,σ , it is then enough to show that it acts freely on the geometric points of Spec(e R(σ )3 ). Actually, more is true: the map q : Spec(e R(σ )3 ) → Spec(3) is a (Z/sZ)∗ torsor† . In fact, if Spec() → Spec(3) is a geometric point, the corresponding geometric fiber of q is isomorphic to the spectrum of Y Y [t] ' (t − αi ) αi ∈e µs ()
∗ Recall † Recall
αi ∈e µs ()
that σ is essential; hence s is invertible in 3. that the constant group scheme associated to (Z/sZ)∗ is isomorphic to Autk (σ ).
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
29
and (Z/sZ)∗ acts by permutation on the primitive roots e µs (), by α 7→ α k , (k, s) = 1. In particular, the action of the subgroup wG (σ ) ⊂ (Z/sZ)∗ on Spec(e R(σ )3 ) is free. PROPOSITION 4.5 The canonical morphism
K ∗0 (X σ , CG (σ ))σ
Y
K ∗0 (X, G)3 −→
wG (σ )
σ ∈C (G)
is an isomorphism. Proof By Lemma 2.9, the canonical ring morphism K ∗0 (X, G) ⊗R(G) R(T ) −→ K ∗0 (X, T ) is an isomorphism. Since R(G) → R(T ) is faithfully flat, it is enough to show that K ∗0 (X, T )3 ' K ∗0 (X, G)3 ⊗R(G)3 R(T )3 Y w (σ ) −→ K ∗0 (X σ , CG (σ ))σ G ⊗R(G)3 R(T )3 σ ∈C (G)
is an isomorphism. By Proposition 3.4(ii), we are left to prove that K ∗0 (X σ , CG (σ ))σ
Y
wG (σ )
⊗R(G)3 R(T )3 '
σ ∈C (G)
Y
K ∗0 (X, T )σ .
(21)
σ dual cyclic σ ⊂T
For any τ ∈ C (G) (τ ⊆ T , as usual), we have \ K ∗0 X τ , CG (τ ) τ ⊗R(G) \ R(T )3,τ 3,τ
'
K ∗0 (X τ , CG (τ ))τ
⊗R(C\ (τ ))
'
K ∗0 (X τ , CG (τ ))τ
⊗ R(G) \
G
3,τ
\ R(T )3,τ R(C\ G (τ ))3,τ ⊗R(G) \ 3,τ 3,τ \ R(C\ R(T )3,τ . G (τ ))3,τ ⊗R(C\ (τ )) 3,τ
G
\ By Lemma 4.4, for any R(G) 3,τ -module M, we have \ M ⊗R(G) \ R CG (τ ) 3,τ ' wG (τ ) × M 3,τ
since a torsor is trivial when base changed along itself. Therefore, \ K ∗0 X τ , CG (τ ) τ ⊗R(G) \ R(T )3,τ 3,τ
' wG (τ ) × K ∗0 (X τ , CG (τ ))τ ⊗R(C\ (τ )) G
3,τ
\ R(T )3,τ
(22)
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VEZZOSI and VISTOLI
with wG (τ ) acting on left-hand side by left multiplication on wG (τ ). Applying Lemma 4.2(ii) to the left-hand side, we get 0 τ \ K ∗0 X τ , CG (τ ) τ ⊗R(G) \ R(T )3,τ ' wG (τ ) × K ∗ (X , T )τ , 3,τ
and taking invariants with respect to wG (τ ), \ K ∗0 (X τ , CG (τ ))τ ⊗R(G) \ R(T )3,τ
wG (τ )
3,τ
' K ∗0 (X τ , T )τ .
(23)
Comparing (21) to (23), we are reduced to proving that for any σ ∈ C (G) there is an isomorphism w (σ ) K ∗0 (X σ , CG (σ ))σ G ⊗R(G)3 R(T )3 Y \ wG (τ ) . ' K ∗0 (X τ , CG (τ ))τ ⊗R(G) \ R(T )3,τ 3,τ
τ dual cyclic τ ≈ Sn σ
\ \ Since R(T )3,τ is flat over R(G) 3,τ and wG (τ ) acts trivially on it, we have (see [SGA1]) w (τ ) \ K ∗0 (X τ , CG (τ ))τ ⊗R(G) R(T )3,τ G \ 3,τ w (τ ) \ ' K ∗0 (X τ , CG (τ ))3,τ G ⊗R(G) \ R(T )3,τ . 3,τ
By Lemma 4.3, we have isomorphisms w (σ ) K ∗0 (X σ , CG (σ ))σ G ⊗R(G)3 R(T )3 w (σ ) \ ' K ∗0 (X σ , CG (σ ))σ G ⊗R(G) R(G) 3,σ ⊗R(G)3 R(T )3 \ 3,σ Y w (σ ) \ ' K ∗0 (X σ , CG (σ ))σ G ⊗R(G) \ R(T )3,τ . 3,σ
τ dual cyclic τ ≈ Sn σ
G G \ \ (Recall that R(G) 3,σ = R(G) 3,τ for any τ ≈ Sn σ , since mσ = mτ .) For each −1 τ , choosing an element g ∈ Sn such that gσ g = τ determines an isomorphism K ∗0 (X σ , CG (σ ))σ ' K ∗0 (X τ , CG (τ ))τ whose restriction to invariants w (σ ) w (τ ) K ∗0 (X σ , CG (σ ))σ G ' K ∗0 (X τ , CG (τ ))τ G
is independent of the choice of g. Therefore, we have a canonical isomorphism w (σ ) K ∗0 (X σ , CG (σ ))σ G ⊗R(G)3 R(T )3 Y w (σ ) \ ' K ∗0 (X σ , CG (σ ))σ G ⊗R(G) \ R(T )3,τ 3,σ
τ dual cyclic τ ≈ Sn σ
'
Y τ dual cyclic τ ≈ Sn σ
K ∗0 (X τ , CG (τ ))τ
wG (τ )
\ ⊗R(G) \ R(T )3,τ , 3,τ
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
31
as desired. Since K ∗ (X, G) ' K ∗0 (X, G) and K ∗ (X σ , CG (σ )) ' K ∗0 (X σ , CG (σ )), comparing Proposition 4.5 with (14), we see that the proof of Theorem 4.1 can be completed by the following. PROPOSITION 4.6 For any σ ∈ C (G), the morphism given by Lemma 2.8 and induced by the product CG (σ ) × σ → CG (σ ), θCG (σ ),σ : K ∗0 X σ , CG (σ ) σ −→ K ∗0 X σ , CG (σ ) geom ⊗ e R(σ )3 ,
is an isomorphism. Proof To simplify the notation, we write θσ for θCG (σ ),σ . As usual, we may suppose σ contained in T . Since CG (σ ) is isomorphic to a product of general linear groups over k, we can take T as its maximal torus, and by Lemma 2.9, the canonical ring morphism K ∗0 X, CG (σ ) ⊗R(CG (σ )) R(T ) −→ K ∗0 (X, T ) is an isomorphism. Moreover, R(CG (σ )) → R(T ) being faithfully flat, it is enough to prove that θσ ⊗ idR(T ) is an isomorphism. To prove this, let us consider the commutative diagram K ∗0 X σ , CG (σ ) σ ⊗R(CG (σ ))3 R(T )3
θσ ⊗id
/
K ∗0 (X, CG (σ ))geom ⊗ e R(σ )3 ⊗R(CG (σ ))3 R(T )3
ωσ
K ∗0 (X σ , T )σ
γf σ
θT,σ
/
K ∗0 (X σ , T )geom ⊗ e R(σ )3
where •
•
• •
K ∗ (X σ , CG (σ ))geom ⊗ e R(σ )3 is an R(CG (σ ))3 -module via the coproduct ring morphism 1CG (σ ) : R(CG (σ ))3 → R(CG (σ ))3 ⊗ e R(σ )3 (induced by the product CG (σ ) × σ → CG (σ )); ωσ is the canonical map induced by the inclusion T ,→ CG (σ ) and is an isomorphism by Lemma 4.2; θT,σ is an isomorphism as shown in the proof of Theorem 3.7; γeσ sends (x ⊗ u) ⊗ t to (1T (t) · x|T ) ⊗ u, for x ∈ K ∗ (X σ , CG (σ ))geom , u ∈e R(σ )3 , t ∈ R(T )3 , 1T : R(T )3 → R(T )3 ⊗ e R(σ )3 being the coproduct induced by the product T × σ → T . So we are left to prove that γeσ is an isomorphism.
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VEZZOSI and VISTOLI
First, let us observe that if R is a ring, A → A0 is a ring morphism, and M is an A-module, there is a natural isomorphism (M ⊗Z R) ⊗ A⊗Z B (A0 ⊗Z R) −→ (M ⊗ A A0 ) ⊗Z R, (m ⊗ r1 ) ⊗ (a 0 ⊗ r2 ) 7 −→ (m ⊗ a 0 ) ⊗ r1r2 . Applying this to M = K ∗ (X σ , CG (σ ))geom , R = e R(σ )3 , A = R(CG (σ ))3 , A0 = R(T )3 and using Lemma 4.2, we get a canonical isomorphism f : K ∗0 (X σ , T )geom ⊗ e R(σ )3 0 e −→ K ∗0 (X σ , CG (σ ))geom ⊗ e R(σ )3 ⊗R(CG (σ ))3 ⊗e R(σ )3 R(T )3 ⊗ R(σ )3 , (24) where we have denoted by (R(T )3 ⊗ e R(σ )3 )0 the R(CG (σ ))3 ⊗ e R(σ )3 -algebra res ⊗ id : R CG (σ ) 3 ⊗ e R(σ )3 −→ R(T )3 ⊗ e R(σ )3 . It is an elementary fact that there are mutually inverse isomorphisms αCG (σ ) , βCG (σ ) , and αT , βT fitting into the commutative diagrams R CG (σ ) 6 1C G (σ ) mmmmm m m mmm mmm
3
⊗e R(σ )3
(25)
αC G (σ ) R CG (σ ) 3 QQQ QQQ QQQ id⊗1 QQQQ( R CG (σ ) 3 ⊗ e R(σ )3 R(T 7)3 ⊗ e R(σ )3 o 1T oooo ooo ooo αT R(T )3 O OOO OOO O id⊗1 OOO' R(T )3 ⊗ e R(σ )3
(26)
and compatible with restriction maps induced by T ,→ CG (σ ). This is exactly the dual assertion to the general fact that “an action H × Y → Y is isomorphic over X to the projection on the second factor pr2 : H × Y → Y ,” for any group scheme H and any algebraic space Y . From (25) we get an isomorphism 0 e α : R(CG (σ ))3 ⊗ e R(σ )3 ⊗R(CG (σ ))3 R(T )3 −→ R(T )3 ⊗ e R(σ )3 ,
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
33
0 where R(CG (σ ))3 ⊗ e R(σ )3 denotes the R(CG (σ ))3 -algebra 1CG (σ ) : R CG (σ ) 3 → R CG (σ ) 3 ⊗ e R(σ )3 . Therefore, if we denote by (R(T )3 ⊗ e R(σ )3 )00 the R(CG (σ ))3 ⊗ e R(σ )3 -algebra (res ⊗ id) ◦ αT : R CG (σ ) 3 ⊗ e R(σ )3 −→ R(T )3 ⊗ e R(σ )3 , the composition 0 K ∗0 (X σ , CG (σ ))geom ⊗ e R(σ )3 ⊗R(CG (σ ))3 R(T )3 = K ∗0 (X σ , CG (σ ))geom ⊗ e R(σ )3 O 0 R(CG (σ ))3 ⊗ e R(σ )3 ⊗R(CG (σ ))3 R(T )3 R(CG (σ ))3 ⊗e R(σ )3 id ⊗e α
R(σ )3 −→ K ∗0 (X σ , CG (σ ))geom ⊗ e O 00 R(T )3 ⊗ e R(σ )3 R(CG (σ ))3 ⊗e R(σ )3
id ⊗βT
−→
K ∗0 (X σ , CG (σ ))geom ⊗ e R(σ )3 O 0 R(T )3 ⊗ e R(σ )3 R(CG (σ ))3 ⊗e R(σ )3
'
K ∗0 (X σ , T )geom
⊗e R(σ )3
is an isomorphism and it can be easily checked to coincide with γeσ . 5. The main theorem: The general case In this section, we use Theorem 4.1 to deduce the same result for the action of a linear algebraic k-group G, having finite stabilizers, on a regular separated Noetherian k-algebraic space X . We write 3 for 3(G,X ) . We start with a general fact. PROPOSITION 5.1 Let X be a regular Noetherian separated k-algebraic space on which a linear algebraic k-group G acts with finite stabilizers. Then there exists an integer N > 0 such that if a1 , . . . , a N ∈ K 0 (X, G)geom have rank zero on each connected component of QN X , then the multiplication by i=1 ai on K ∗0 (X, G)geom is zero.
In particular, we have the following.
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VEZZOSI and VISTOLI
COROLLARY 5.2 Let X be a regular Noetherian separated k-algebraic space with a connected action of a linear algebraic k-group G having finite stabilizers. Then the geometric localization
rk0,geom : K 0 (X, G)geom −→ 3 of the rank morphism has a nilpotent kernel. Proof of Proposition 5.1 Let us choose a closed immersion G ,→ GLn,k (for some n > 0). By Morita equivalence, K ∗0 (X, G) ' K ∗0 (X ×G GLn,k , GLn,k ) and K 0 (X, G) ' K 0 (X ×G GLn,k , GLn,k ). Moreover, 3(X ×G GLn,k ,GLn,k ) = 3. Let ξ = x/s ∈ K ∗0 (X, G)geom with x ∈ K ∗0 (X, G)3 and s ∈ rk−1 (1) where rk : R(G) → 3 is the rank morphism, and let ai = αi /si with αi ∈ K 0 (X, G)3 and si ∈ rk−1 (1) for i = 1, . . . , N . Let us consider the elements x/1 in K ∗0 (X ×G GLn,k , GLn,k )geom , and αi /1 in K 0 (X ×G GLn,k , GLn,k )geom for i = 1, . . . , N . Since the canonical homomorphism K ∗0 (X ×G GLn,k , GLn,k )geom −→ K ∗0 (X, G)geom is a morphism of modules over the ring morphism K 0 (X ×G GLn,k , GLn,k )geom −→ K 0 (X, G)geom , if the theorem holds for G = GLn,k and if N is the corresponding integer, the product Q Q 0 i αi /1 in K 0 (X, G)geom annihilates x/1 ∈ K ∗ (X, G)geom and a fortiori i ai an0 nihilates ξ in K ∗ (X, G)geom . So, we may assume G = GLn,k . Let T be the maximal torus of diagonal matrices in G. By Lemma 4.2(i) with σ = 1, there are isomorphisms ω1,geom : K 0 (X, GLn,k )geom ⊗R(GLn,k )3 R(T )3 ' K 0 (X, T )geom , K ∗0 (X, GLn,k )geom ⊗R(GLn,k )3 R(T )3 ' K ∗0 (X, T )geom . Since R(GLn,k ) → R(T ) is faithfully flat and the diagram rk0,geom ⊗ id
K 0 (X, GLn,k )geom ⊗R(GLn,k )3 R(T )3 −−−−−−−→ 3 ⊗R(GLn,k ) R(T )3 id ⊗ rk ω1,geom y y 3 T K 0 (X, T )geom
−−−−→ rk0,geom
3
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
35
commutes, we reduce ourselves to proving the proposition for G = T , a split torus. To handle this case, we proceed by Noetherian induction on X . By [Th1, Prop. 4.10], there exist a T -invariant nonempty open subscheme j : U ,→ X , a closed diagonalizable subgroup T 0 of T , and a T -equivariant isomorphism U ' T /T 0 × (U/T ). Since U is nonempty and T acts on X with finite stabilizers, T 0 is finite over k and K ∗0 (U, T ) ' K ∗0 (U/T ) ⊗ R(T ) R(T 0 ), by Morita equivalence theorem (see [Th3, Prop. 6.2]). Let i : Z ,→ X be the closed complement of U in X , and let N 0 be an integer satisfying the proposition for both Z and U . Consider the geometric localization sequence i∗
j∗
K ∗0 (Z , T )geom −→ K ∗0 (X, T )geom −→ K ∗0 (U, T )geom , and let ξ ∈ K ∗0 (X, T )geom . Let a1 , . . . , a2N 0 ∈ K 0 (X, T )geom . By our choice of N 0 , j ∗ (a N 0 +1 · . . . · a2N 0 ∪ ξ ) = 0; thus a N 0 +1 · . . . · a2N 0 ∩ ξ = i ∗ (η) for some η in K ∗0 (Z , T )geom . By the projection formula, we get a1 · . . . · a2N 0 ∪ ξ = i ∗ i ∗ (a1 ) · . . . · i ∗ (a N 0 ) ∪ η , which is zero by our choice of N 0 and by the fact that rank morphisms commute with . pullbacks. Thus, N = 2N 0 satisfies our proposition. Remark 5.3 By Corollary 5.2, K ∗ (X, G)geom is isomorphic to the localization of K ∗ (X, G)3 at the multiplicative subset (rk0 )−1 (1), where rk0 : K 0 (X, G)3 → 3 is the rank morphism. Therefore, if X is regular, K ∗ (X, G)geom depends only on the quotient stack [X/G] (see [LMB]) and not on its presentation as a quotient. The main theorem of this paper is the following. 5.4 Let X be a Noetherian regular separated algebraic space over a field k, and let G be a linear algebraic k-group with a sufficiently rational action on X having finite stabilizers. Suppose, moreover, that for any essential dual cyclic k-subgroup scheme σ ⊆ G, the quotient algebraic space G/CG (σ ) is smooth over k (which is the case if, e.g., G is smooth or abelian). Then C (G) is finite, and the map defined in (5), Y w (σ ) 9 X,G : K ∗ (X, G)3 −→ K ∗ (X σ , C(σ ))geom ⊗ e R(σ )3 G , THEOREM
σ ∈C (G)
is an isomorphism of R(G)-algebras.
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VEZZOSI and VISTOLI
Remark 5.5 In Section 5.1 we also give less restrictive hypotheses on G under which Theorem 5.4 still holds. Note also that if X has the “G-equivariant resolution property” (i.e., any Gequivariant coherent sheaf is the G-equivariant epimorphic image of a G-equivariant locally free coherent sheaf), then in Theorem 5.4 one can replace our K ∗ with Quillen K -theory of G-equivariant locally free coherent sheaves. This happens, for example, if X is a scheme and G is smooth or finite (see [Th3]). 5.1. Proof of Theorem 5.4 Let us choose, for some n, a closed immersion G ,→ GLn,k and consider the algebraic space quotient . Y = GLn,k ×G X. We claim that if the theorem holds for Y with the induced GLn,k -action, then it holds for X with the given G-action. First, let us note that Y is separated. The action map ψ : G ×(GLn,k ×X ) → (GLn,k ×X )×(GLn,k ×X ) is proper (hence a closed immersion) since its composition with the separated map p123 : GLn,k ×X × GLn,k ×X → GLn,k ×X × GLn,k (here we use that X is separated) is just id X ×a, where a is the action map of G on GLn,k ; hence it is proper (see [EGAI, Rem. 5.1.7], which obviously carries over to . algebraic spaces). Let P = X × GLn,k . In the Cartesian diagram j
P ×Y P −−−−→ P × P π×π y y Y
−−−−→ Y × Y 1Y
P ×Y P ' G × P since π : P → Y is a G-torsor, and π is faithfully flat; therefore, 1Y is a closed immersion; that is, Y is separated. Note that 3(Y,GLn,k ) = 3. Consider the morphism defined in (5), Y w (σ ) 9 X,G : K ∗ (X, G)3 −→ K ∗ (X σ , CG (σ ))geom ⊗ e R(σ )3 G . (27) σ ∈C (G)
By Theorem 4.1, the map 9Y,GLn,k : K ∗ (Y, GLn,k )3 −→
Y ρ∈C (GLn,k )
K ∗ (Y ρ , CGLn,k (ρ))geom ⊗ e R(ρ)3
wGL
n,k
(ρ)
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
37
is an isomorphism, and by the Morita equivalence theorem (see [Th3, Prop. 6.2]), K ∗ (Y, GLn,k )3 ' K ∗ (X, G)3 . We prove the theorem by constructing an isomorphism Y
w (ρ) K ∗ (Y ρ , C(ρ))geom ⊗ e R(ρ) GLn,k
ρ∈C (GLn,k )
−→
Y
K ∗ (X σ , CG (σ ))geom ⊗ e R(σ )3
wG (σ )
(28)
σ ∈C (G)
commuting with the 9’s and Morita isomorphisms. Let α : C (G) → C (GLn,k ) be the natural map. If Y ρ 6 = ∅, there exists a dual cyclic subgroup σ ⊆ G, GLn,k -conjugate to ρ (and X σ 6 = ∅); therefore, Y ρ = ∅ unless ρ ∈ im(α), and we may restrict the first product in (28) to those ρ in the image of α and suppose im(α) ⊆ C (G) as well. The following proposition describes the Y ρ ’s that appear. PROPOSITION 5.6 Let X be a Noetherian regular separated algebraic space over a field k, and let G be a linear algebraic k-group with a sufficiently rational action on X having finite stabilizers. Suppose, moreover, that for any essential dual cyclic k-subgroup scheme σ ⊆ G, the quotient algebraic space G/CG (σ ) is smooth over k. Let G ,→ GLn,k be a closed embedding, let ρ ∈ im(α) be an essential dual cyclic subgroup, and let . Y = GLn,k ×G X be the algebraic space quotient for the left diagonal action of G. If CGLn,k ,G (ρ) ⊆ C (G) denotes the fiber α −1 (ρ), then (i) choosing for each σ ∈ CGLn,k ,G (ρ) an element u ρ,σ ∈ GLn,k (k) such that u ρ,σ σ u −1 ρ,σ = ρ (in the obvious functor-theoretic sense) determines a unique isomorphism of algebraic spaces over k, a jρ : NGLn,k (σ ) ×NG (σ ) X σ −→ Y ρ ; σ ∈CGLn,k ,G (ρ)
(ii)
CGLn,k ,G (ρ) is finite.
Proof Part (ii) follows from (i) since Y ρ is quasi-compact. The proof of (i) requires several steps. (a) Definition of jρ . If σ ∈ CGLn,k ,G (ρ), let Nσ be the presheaf on the category Sch/k of k-schemes which associates to T → Spec k the set σ . NGLn,k (σ )(T ) × X (T ) ; Nσ (T ) = NG (σ )(T )
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VEZZOSI and VISTOLI
since NG (σ ) acts freely on NGLn,k (σ ) × X σ (on the left), the flat sheaf associated to bρ be the presheaf on Sch/k which associates to Nσ is NGLn,k (σ ) ×NG (σ ) X σ . Let Y T → Spec k the set n GLn,k (T ) × X (T ) 0 . bρ (T ) = Y [A, x] ∈ ∀T G(T ) o → T, ∀r ∈ ρ(T 0 ), [r A T 0 , x T 0 ] = [A T 0 , x T 0 ] ; bρ is Y ρ (e.g., see [DG, Chap. II, §1, n. 3]). If u ρ,σ ∈ the flat sheaf associated to Y GLn,k (k) is such that u ρ,σ σ u −1 ρ,σ = ρ (in the obvious functor-theoretic sense), the presheaf map b bρ , jρ,σ : Nσ −→ Y b bρ (T ) jρ,σ (T ) : Nσ (T ) 3 [B, x] −→ [u ρ,σ B, x] ∈ Y is easily checked to be well defined. Let jρ,σ : NGLn,k (σ ) ×NG (σ ) X σ → Y ρ denote . ` the associated sheaf map, and define jρ = σ ∈CGL ,G (ρ) jρ,σ . n,k (b) The map jρ induces a bijection on geometric points. This is an elementary check. Let ξ ∈ Y ρ () be a geometric point. Then there exist an fppf cover T0 → bρ (T0 ) representing ξ . Therefore, for each T → T0 Spec and an element [A, x] ∈ Y and each r ∈ ρ(T ) there exists g ∈ G(T ) such that r A T g −1 = A T , gx T = x T . Then A−1 ρ A defines (functorially over T0 ) a dual cyclic subgroup scheme σ00 of G (T0 ) over T0 . Since σ00 is isomorphic to some µn,T0 , it descends to a dual cyclic subgroup σ 0 of G over k which is GLn,k -conjugate to ρ since T0 → Spec has a section and GLn,k satisfies our rationality condition (RC) (see Rem. 2.2(i)). By definition of CGLn,k ,G (ρ), there exists a unique σ ∈ CGLn,k ,G (ρ) which is G-conjugated to σ 0 over k; that is, gσ 0 g −1 = σ (functorially) for some g ∈ G(k). Since σ ∈ CGLn,k ,G (ρ), there is an element u ∈ GLn,k (k) such that uσ u −1 = ρ. Therefore, u −1 Ag −1 restricted to T0 is in NGLn,k (σ )(T0 ), gx ∈ X σ (T0 ), and if [u −1 Ag −1 , gx]∼ denotes the element in (NGLn,k (σ ) ×NG (σ ) X σ )() represented by the element [u −1 Ag −1 , gx] in Nσ (T0 ), we have jρ,σ ()([u −1 Ag −1 , gx]∼ ) = ξ by definition of jρ,σ . Thus, jρ () is surjective. 0 Now, let η ∈ (NGLn,k (σ ) ×NG (σ ) X σ )() resp., η0 ∈ (NGLn,k (σ 0 ) ×NG (σ ) 0 X σ )() for σ and σ 0 in CGLn,k ,G (ρ). Choosing a common refinement, we can assume that there exists an fppf cover T0 → Spec such that η (resp., η0 ) is represented by an element [B, y] ∈ Nσ (T0 ) (resp., [B 0 , y 0 ] ∈ Nσ 0 (T0 )). If jρ ()(η) =
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
39
jρ ()(η0 ), there exists an fppf cover T1 → T0 such that [u ρ,σ B, y] = [u ρ,σ 0 B 0 , y 0 ] in GLn,k (T1 ) × X (T1 )/G(T1 ); that is, there is an element g ∈ G(T1 ) such that u ρ,σ Bg −1 = u ρ,σ 0 B 0 gy = y
0
in GLn,k (T1 ),
in X (T1 ).
Then it is easy to check that σ = g −1 σ 0 g over T1 and, as in the proof of surjectivity of jρ (), since T1 → Spec has a section and G satisfies our rationality condition (RC), σ and σ 0 are G-conjugated over k as well, and therefore σ = σ 0 as elements in CGLn,k ,G (ρ). In particular, g ∈ NG (σ )(T1 ) and [B, y] = [B 0 , y 0 ] in Nσ (T1 ). Since T1 → Spec is still an fppf cover, we have η = η0 and jρ () is injective. (c) Each jρ,σ is a closed and open immersion. It is enough to show that each jρ,σ is an open immersion because in this case it is also a closed immersion, Y ρ being ` quasi-compact. Since NGLn,k (ρ) acts on both σ ∈CGL ,G (ρ) NGLn,k (σ ) ×NG (σ ) X σ n,k and Y ρ and since jρ is equivariant, it will be enough to prove that jρ,ρ is an open immersion. We prove first that jρ,ρ is injective and unramified and then conclude the proof by showing that it is also flat (in fact, an e´ tale injective map is an open immersion). (c1 ) The map jρ,ρ is injective and unramified. It is enough to show that the inverse image under jρ,ρ of a geometric point is a (geometric) point. Consider the commutative diagram NGLn,k (ρ) × X ρ py
l
−−−−→ GLn,k ×X π y
NGLn,k (ρ) ×NG (ρ) X ρ −−−−→ i ρ ◦ jρ,ρ
Y
where l and i ρ : Y ρ ,→ Y are the natural inclusions and where p, π are the natural projections. Let y0 be a geometric point of Y in the image of i ρ ◦ jρ,ρ ; using the action of NGLn,k (ρ) on NGLn,k (ρ) ×NG (ρ) X ρ and Y ρ , we may suppose that y0 is of the form [1, x0 ] ∈ Y ρ (), with an algebraically closed field and x0 ∈ X ρ (). Obviously, −1 (y ), and, by faithful flatness (1, x0 ) ∈ NGLn,k (ρ) ×NG (ρ) X ρ () is contained in jρ,ρ 0 −1 of p, jρ,ρ (y0 ) = (1, x0 ) if p −1 (1, x0 ) = π −1 (y0 ) ∩ NGLn,k (ρ) ×NG (ρ) X ρ .
(29)
But G() ' π −1 (y0 ) via g 7→ (g −1 , gx0 ) and NG (ρ)() ' p −1 ((1, x0 )) via h 7→ (h −1 , hx0 ); therefore, (29) follows from NG (ρ) = NGLn,k (ρ) ∩ G. (c2 ) The map jρ,ρ is flat. This fact is proved in Section 5.2, where we also single out a more general technical hypothesis for the action of G on X under which Proposition 5.6 still holds.
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VEZZOSI and VISTOLI
The remaining part of this subsection is devoted to the conclusion of the proof of Theorem 5.4 using Proposition 5.6. First we show that Proposition 5.6(ii) allows one to define a canonical isomorphism Y w (ρ) K ∗ (Y ρ , CGLn,k (ρ))geom ⊗ e R(ρ)3 GLn,k ρ∈C (GLn,k )
'
Y
R(σ )3 K ∗ (NGLn,k (σ ) ×NG (σ ) X σ , CGLn,k (σ ))geom ⊗ e
wGL
n,k
(σ )
;
σ ∈C (G)
next we show, using Lemma 2.10, that each factor in the right-hand side is isomorphic to w (σ ) K ∗ (CGLn,k (σ ) ×CG (σ ) X σ , CGLn,k (σ ))geom ⊗ e R(σ )3 G . The conclusion (i.e., the isomorphism (28)) is then accomplished by establishing, for any regular Noetherian separated algebraic space Z on which G acts with finite stabilizers, a “geometric” Morita equivalence K ∗ (GLn,k ×G Z , GLn,k )geom ' K ∗ (Z , G)geom . First, note that the choice of a family {u ρ,σ | σ ∈ CGLn,k ,G (ρ)} of elements u ρ,σ ∈ GLn,k (k) such that u ρ,σ σ u −1 ρ,σ = ρ, which uniquely defines jρ in Proposition 5.6, also determines a unique family of isomorphisms int(u ρ,σ ) : CGLn,k (ρ) → CGLn,k (σ ) σ ∈ CGLn,k ,G (ρ) (where int(u ρ,σ ) denotes conjugation by u ρ,σ ), and this family gives us an action of CGLn,k (ρ) on a NGLn,k (σ ) ×NG (σ ) X σ σ ∈CGLn,k ,G (ρ)
(since NGLn,k (σ ), and then CGLn,k (σ ), acts naturally on NGLn,k (σ ) ×NG (σ ) X σ by left multiplication on NGLn,k (σ )). With this action, jρ becomes a CGLn,k (ρ)-equivariant isomorphism, and since int(u ρ,σ ) induces an isomorphism R(CGLn,k (ρ)) ' R(CGLn,k (σ )) commuting with rank morphisms, jσ induces an isomorphism K ∗ Y ρ , CGLn,k (ρ) geom ⊗ e R(ρ)3 Y K ∗ NGLn,k (σ ) ×NG (σ ) X σ , CGLn,k (σ ) geom ⊗ e R(σ )3 ' σ ∈CGLn,k ,G (ρ)
which, by definition of the action of NGLn,k (ρ) on each NGLn,k (σ )×NG (σ ) X σ , induces
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
41
an isomorphism w (ρ) K ∗ (Y ρ , CGLn,k (ρ))geom ⊗ e R(ρ)3 GLn,k Y w (σ ) ' K ∗ (NGLn,k (σ ) ×NG (σ ) X σ , CGLn,k (σ ))geom ⊗ e R(σ )3 GLn,k . σ ∈CGLn,k ,G (ρ)
(30) Now, if jρ0 is induced, as in Proposition 5.6, by another choice of a family {vρ,σ | −1 = ρ, then σ ∈ CGLn,k ,G (ρ)} of elements vρ,σ ∈ GLn,k (k) such that vρ,σ σ vρ,σ −1 vρ,σ u ρ,σ ∈ NGLn,k (σ )(k) and there is a commutative diagram −1 u (vρ,σ ρ,σ )·
NGLn,k (σ ) ×NG (σ ) X σ PPP PPP PPP jρ PPPP (
Yρ
/
NGLn,k (σ ) ×NG (σ ) X σ nnn nnn n n nn jρ0 v nn n
Therefore, isomorphism (30) on the invariants is actually independent of the choice of the family {u ρ,σ | σ ∈ CGLn,k ,G (ρ)}. Since CGLn,k ,G (ρ) = α −1 (ρ) and, as already observed, Y ρ = ∅ unless ρ ∈ im(α), this gives us a canonical isomorphism Y w (ρ) K ∗ (Y ρ , CGLn,k (ρ))geom ⊗ e R(ρ)3 GLn,k ρ∈C (GLn,k )
'
R(σ )3 K ∗ (NGLn,k (σ ) ×NG (σ ) X σ , CGLn,k (σ ))geom ⊗ e
Y
wGL
n,k
(σ )
.
σ ∈C (G)
Now, let us fix σ ∈ C (G), and let us choose a set A ⊂ NGLn,k (σ )(k) such that the classes in wGLn,k (σ ) of the elements in A constitute a set of representatives for the wG (σ )-orbits in wGLn,k (σ ); A is a finite set. Since CGLn,k (σ ) ×CG (σ ) X σ ,→ NGLn,k (σ ) ×NG (σ ) X σ is an open and closed immersion, the morphism a CGLn,k (σ ) ×CG (σ ) X σ −→ NGLn,k (σ ) ×NG (σ ) X σ , A
[C, x] Ai ∈A 7 −→ [Ai · C, x] (in the obvious functor-theoretic sense), which is easily checked to induce an isomorphism on geometric points, is an isomorphism. Therefore, there is an induced isomorphism Y K ∗ CGLn,k (σ ) ×CG (σ ) X σ , CGLn,k (σ ) geom ⊗ e R(σ )3 A
' K ∗ NGLn,k (σ ) ×NG (σ ) X σ , CGLn,k (σ ) geom ⊗ e R(σ )3 .
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VEZZOSI and VISTOLI
Since wGLn,k (σ ) acts transitively on A with stabilizer wG (σ ), by Lemma 2.10 we get a canonical isomorphism w (σ ) K ∗ (NGLn,k (σ ) ×NG (σ ) X σ , CGLn,k (σ ))geom ⊗ e R(σ )3 GLn,k w (σ ) ' K ∗ (CGLn,k (σ ) ×CG (σ ) X σ , CGLn,k (σ ))geom ⊗ e R(σ )3 G . Since, by Morita equivalence (see [Th3, Prop. 6.2]), K ∗ CGLn,k (σ ) ×CG (σ ) X σ , CGLn,k (σ ) ' K ∗ X σ , CG (σ ) ,
(31)
to conclude the proof of Theorem 5.4 we need only show that the natural morphism K ∗ CGLn,k (σ ) ×CG (σ ) X σ , CGLn,k (σ ) geom ' K ∗ X σ , CG (σ ) geom (32) induced by (31) is still an isomorphism. Since the diagram K ∗ GLn,k ×CG (σ ) X σ , GLn,k geom
α
/ K ∗ CGLn,k (σ ) ×CG (σ ) X σ , CGLn,k (σ )
WWWWW WWWWW WWWWW WWWWW γ W+
geom
β
K ∗ X σ , CG (σ ) geom
is commutative and, by Morita equivalence, K ∗ CGLn,k (σ ) ×CG (σ ) X σ , CGLn,k (σ ) CGLn,k (σ )
CGLn,k (σ ) ×CG (σ ) X σ , GLn,k C (σ ) GLn,k × GLn,k X σ , GLn,k ,
' K ∗ GLn,k × ' K∗
to show that β is an isomorphism it is enough to prove that for any regular separated algebraic space Z on which G acts with finite stabilizers, Morita equivalence induces an isomorphism K ∗ (GLn,k ×G Z , GLn,k )geom ' K ∗ (Z , G)geom
(33)
since in this case both α and γ are isomorphisms. Let π : R(GLn,k ) → R(G) be the restriction morphism, let ρ : R(G) → K 0 (Z , G) be the pullback along Z → Spec k, let rk0 : R(GLn,k ) → 3 and . . rk : R(G) → 3 be the rank morphisms, and let S 0 = (rk0 )−1 (1), S = (rk)−1 (1), . and T = π(S 0 ) ⊆ S; the following diagram commutes: rk0,T
/ r9 3 rr r r rr rr rkgeom r r
T −1 K 0 (Z , G)3 K 0 (Z , G)geom
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
43
where rkgeom and rk0,T denote the localizations of the rank morphism rk0 : K 0 (Z , G)3 → 3. By Morita equivalence, the natural map (which commutes with the induced rank morphisms) K 0 (GLn,k ×G Z , GLn,k )geom −→ T −1 K 0 (Z , G)3 is an isomorphism, and then, by Proposition 5.1, ker(rk0,T : T −1 K 0 (Z , G)3 → 3) is nilpotent. Now, if s ∈ S, then rk0,T (ρ(s)/1) = rk(s) = 1, and therefore T −1 K 0 (Z , G)3 → K 0 (Z , G)geom and K 0 (GLn,k ×G Z , GLn,k )geom → K 0 (Z , G)geom are both isomorphisms. Since K ∗ (GLn,k ×G Z , GLn,k )3 is naturally a K 0 (GLn,k ×G Z , GLn,k )3 -module and an R(GLn,k )3 -module via the pullback ring morphism ρ 0 : R(GLn,k )3 → K 0 (GLn,k ×G Z , GLn,k )3 , we have K ∗0 (GLn,k ×G Z , GLn,k )geom ' K ∗0 (GLn,k ×G Z , GLn,k )3 ⊗ K 0 (GLn,k ×G Z ,GLn,k )3 K 0 (GLn,k ×G Z , GLn,k )geom ' K ∗0 (Z , G)3 ⊗ K 0 (Z ,G)3 K 0 (GLn,k ×G Z , GLn,k )geom ' K ∗0 (Z , G)3 ⊗ K 0 (Z ,G)3 K 0 (Z , G)geom ' K ∗0 (Z , G)geom , which proves (32), and we conclude the proof of Theorem 5.4. 5.2. Hypotheses on G In this subsection we conclude the proof of Proposition 5.6, showing that (this is part (c2 ) of the proof) jρ,ρ : NGLn,k (ρ) ×NG (ρ) X ρ −→ Y ρ is flat. This is the only step in the proof of Proposition 5.6 where we make use of the hypothesis that the quotient algebraic space G/CG (ρ) is smooth over k. Actually, our proof works under the following weaker hypothesis. Let S denote the spectrum of the dual numbers over k, S = Spec(k[ε]), 1
and for any k-group scheme H , let H (S, H ) denote the k-vector space of isomorphism classes of pairs (P → S, y), where P → S is an H -torsor and y is a k-rational point on the closed fiber of P. Then Proposition 5.6, and hence Theorem 5.4, still
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VEZZOSI and VISTOLI
holds with hypothesis (S) for any essential dual cyclic subgroup scheme σ ⊆ G, the quotient G/CG (σ ) is smooth over k replaced by the following: (S0 ) for any essential dual cyclic k-subgroup scheme σ ⊆ G, we have σ 1 1 dim H S, CG (σ ) = dim H (S, G) . First we prove that jρ,ρ is flat assuming (S0 ) holds. Then we show that (S) implies (S0 ). Since p : NGLn,k (ρ) × X ρ → NGLn,k (ρ) ×NG (ρ) X ρ is faithfully flat, it is enough . to prove that j ρ = jρ,ρ ◦ p is flat. Let π : GLn,k ×X → Y be the projection, and let f : GLn,k ×X × G −→ GLn,k ×X, (A, x, g) 7−→ (Ag −1 , gx). Consider the following Cartesian squares: U
/ π −1 (Y ρ )
uρ
NGLn,k (ρ) × X ρ
jρ
/ GLn,k ×X
(34)
π
/ Yρ
/Y
Since π is faithfully flat, it is enough to prove that u ρ is flat. But the squares U
NGLn,k (ρ) × X ρ
/ GLn,k ×X × G pr12
f
/ GLn,k ×X
/ GLn,k ×X
π
/Y
(35)
π
are Cartesian and (in the obvious functor-theoretic sense) U = (A, x, g) ∈ GLn,k ×X × G A−1 ρ A = ρ, x ∈ X ρ ' NGLn,k (ρ) × X ρ × G. . Moreover, if P = {A ∈ GLn,k | A−1 ρ A ⊆ G}, the map −1 π −1 (Y ρ ) = (A, x) ∈ NGLn,k (ρ) × X A−1 ρ A ⊆ G, x ∈ X A ρ A −→ P × X ρ , (A, x) 7−→ (A, Ax) is an isomorphism. Therefore, we are reduced to proving that the map vρ : NGLn,k (ρ) × X ρ × G −→ P × X ρ , (A, x, g) 7−→ (Ag −1 , Ax)
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
45
is flat. But since the diagram vρ
NGLn,k (ρ) × X ρ × G −−−−→ P × X ρ pr pr13 y y 1 NGLn,k (ρ) × G
P
−−−−→ 2ρ
. (where 2ρ (A, g) = (Ag −1 )) is easily checked to be Cartesian, it is enough to show that zρ is flat. To do this, let us observe that ρ acts by conjugation on GLn,k /G (quotient by the G-action on GLn,k by right multiplication), and we have a Cartesian diagram / GLn,k P τ
(GLn,k /G)ρ
/ GLn,k /G
Then τ is a G-torsor and 2ρ is G-equivariant. Thus, the following commutative diagram, in which the vertical arrows are G-torsors, 2ρ
NGLn,k (ρ) × G −−−−→ pr1 y NGLn,k (ρ)
P τ y
−−−−→ (GLn,k /G)ρ χρ
. (where χρ (A) = [A] ∈ GLn,k /G) is Cartesian, and we may reduce ourselves to prove that χρ is flat. Now, observe that NGLn,k (ρ) acts on the left of both NGLn,k (ρ) and (GLn,k /G)ρ in such a way that χρ is NGLn,k (ρ)-equivariant. Therefore, it is enough to prove that χρ is flat when restricted to the connected component of the identity in NGLn,k (ρ), that is, that the map χρ0 : CGLn,k (ρ) −→ (GLn,k /G)ρ is flat. Now, CGLn,k (ρ) = (GLn,k )ρ , where ρ acts by conjugation; both (GLn,k )ρ and (GLn,k /G)ρ are smooth by [Th5, Prop. 3.1] (since GLn,k and GLn,k /G are smooth); each fiber of χρ0 has dimension equal to dim(CG (ρ)) because χρ0 is CGLn,k (ρ)equivariant for the natural actions; and all the fibers are obtained from (χρ0 )−1 ([1]) = CG (ρ) by the CGLn,k (ρ)-action. Therefore, χρ0 is flat if dim CGLn,k (ρ) = dim CG (ρ) + dim (GLn,k /G)ρ . (36) Note that, in any case, dim CGLn,k (ρ) ≤ dim CG (ρ) + dim (GLn,k /G)ρ .
(37)
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Since GLn,k is smooth, dim((GLn,k /G)ρ ) = dimk (T1 (GLn,k /G)ρ ), where T1 de1 notes the tangent space at the class of 1 ∈ GLn,k . Moreover, since H (S, GLn,k ) = 0, there is an exact sequence of k-vector spaces 1
0 → Lie(G) −→ Lie(GLn,k ) −→ T1 (GLn,k /G) −→ H (S, G) → 0
(38)
which, ρ being linearly reductive over k, yields an exact sequence of ρ-invariants 0 → Lie(G)ρ −→ Lie(GLn,k )ρ −→ T1 (GLn,k /G)ρ −→ H (S, G)ρ → 0. (39) 1
But GLn,k is smooth, so dimk Lie(GLn,k )ρ = dim(GLn,k )ρ = dim CGLn,k (ρ) , and, since Lie(CG (ρ)) = Lie(G) ∩ Lie(CGLn,k (ρ)), we get dimk Lie(G)ρ = dimk Lie(CG (ρ)) . By (39), we get 1 dimk H (S, G)ρ = dimk T1 (GLn,k /G)ρ − dim CGLn,k (ρ) + dimk Lie(CG (ρ)) = dim (GLn,k /G)ρ − dim CGLn,k (ρ) + dim CG (ρ) + dimk Lie(CG (ρ)) − dim CG (ρ) ; (40) hence (36) is satisfied if 1 dimk H (S, G)ρ = dimk Lie(CG (ρ)) − dim CG (ρ) . But
(41)
1 dimk H (S, CG (ρ)) = dimk Lie(CG (ρ)) − dim CG (ρ)
by the exact sequence (analogous to (38) with G replaced by CG (ρ)) 0 → Lie CG (ρ) −→ Lie(GLn,k ) −→ T1 GLn,k /CG (ρ) 1 −→ H S, CG (ρ) → 0; hence (41) holds by hypothesis (S0 ). We complete the proof of Proposition 5.6 by showing that (S) implies (S0 ). Since CG (ρ) ⊆ G, we have a natural map 1 1 : H S, CG (ρ) −→ H (S, G)ρ , and by (40) and (37), we get 1 1 dimk H (S, G)ρ ≥ dimk H (S, CG (ρ)) .
(42)
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Now, if (S) holds, that is, if G/CG (ρ) is smooth, and if [P → S, y] is a class in 1 H (S, G)ρ , P/CG (ρ) → S is smooth and y induces a point in the closed fiber of P/CG (ρ) → S; we may reduce the structure group to CG (ρ), thus showing that is surjective. By (42), we conclude that is an isomorphism, and this implies (S0 ).
5.3. Final remarks PROPOSITION 5.7 Let X be a Noetherian regular separated algebraic space over k, and let G be a finite group acting on X . There is a canonical isomorphism of R(G)-algebras K ∗ (X, G)geom ⊗ Z 1/|G| ' K ∗ (X )G ⊗ Z 1/|G| .
Proof Since ker rk : K 0 (X ) → H0 (X, Z[1/|G|]) is nilpotent by Corollary 5.2, the canonical homomorphism π ∗ : K ∗ (X, G) −→ K ∗ (X )G induces a ring homomorphism (still denoted by π ∗ ) π ∗ : K ∗ (X, G)geom ⊗ Z 1/|G| −→ K ∗ (X )G ⊗ Z 1/|G| . Moreover, the functor π∗ : F 7 −→
M
g∗F ,
g∈G
defined on coherent O X -modules, induces a homomorphism π∗ : K ∗0 (X )G ⊗ Z[1/|G|] −→ K ∗0 (X, G)geom ⊗ Z[1/|G|], and (recalling that K ∗ (X, G) ' K ∗0 (X, G)) we obviously get π ∗ π∗ (F ) = |G| · F . On the other hand, we have π∗ π ∗ (F ) ' F ⊗ π∗ O X . But rk(π∗ O X ) = |G|, and therefore π∗ π ∗ is an isomorphism, too, because of Corollary 5.2. As a corollary of this result and of Theorem 5.4, we recover [Vi1, Th. 1], which was proved there in a completely different way.
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We conclude the paper with a conjecture expressing the fact that K ∗ (X, G)geom should be the K -theory of the quotient X/G, if X/G is regular, after inverting the orders of all the essential dual cyclic subgroups of G. CONJECTURE 5.8 Let X be a Noetherian regular separated algebraic space over a field k, and let G be a linear algebraic k-group acting on X with finite stabilizers in such a way that the quotient X/G exists as a regular algebraic space. Let N denote the least common multiple of the orders of all the essential dual cyclic subgroups of G, and let 3 = Z[1/N ]. If p : X → X/G is the quotient map, the composition p∗
K ∗ (X/G)3 −→ K ∗ (X, G)3 −→ K ∗ (X, G)geom is an isomorphism. Remark 5.9 Bertrand Toen pointed out to us that if X/G is smooth, it follows from the results of [EG] that the composition K 0 (X/G) ⊗ Q −→ K 0 (X, G) ⊗ Q −→ K 0 (X, G)geom ⊗ Q is an isomorphism.
Appendix. Higher equivariant K -theory of Noetherian regular separated algebraic spaces In this appendix we describe the K -theories we use in the paper and their relationships. We essentially follow the example of [ThTr, Sec. 3]. We also adopt the language of [ThTr]. Let us remark that it is strongly probable that there exist equivariant versions of most of the results in [ThTr, Sec. 3]. In particular, there should exist a higher K theory of G-equivariant cohomologically bounded pseudocoherent complexes on Z (resp., of G-equivariant perfect complexes on Z ) for any quasi-compact algebraic space Z having most of the alternative models described in [ThTr, Pars. 3.5 – 3.12]. The arguments below can also be considered as a first step toward an extension of [ThTr, Lems. 3.11, 3.12] to the equivariant case on algebraic spaces. However, to keep the paper to a reasonable size, we have decided to give only the results we need, and, moreover, we have made almost no attempt to optimize the hypotheses. We would also like to mention the paper [J] (in particular, Section 1) in which, among many other results, the general techniques of [ThTr] are used as guidelines for the K -theory of arbitrary Artin stacks.
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49
We work in a slightly more general situation than required in the rest of the paper. Let S be a separated Noetherian scheme, and let G be a group scheme affine over S which is finitely presented, separated, and flat over S. We denote by G-AlgSpreg the category of regular Noetherian algebraic spaces separated over S with an action of G over S and equivariant maps. Definition A.1 If X ∈ G-AlgSpreg , we denote by K ∗ (X, G) (resp., K ∗0 (X, G), resp., K ∗naive (X, G)) the Waldhausen K -theory of the complicial bi-Waldhausen category (see [ThTr]) W1,X of complexes of quasi-coherent G-equivariant O X -modules with bounded coherent cohomology (resp., the Quillen K -theory of the abelian category of Gequivariant coherent O X -modules, resp., the Quillen K -theory of the exact category of G-equivariant locally free coherent O X -modules). PROPOSITION A.2 Let Z → S be a morphism of Noetherian algebraic spaces such that the diagonal Z → Z × S Z is affine, and let H → S be an affine group space acting on Z . Let F be an equivariant quasi-coherent sheaf on Z of finite flat dimension; then there exists a flat equivariant quasi-coherent sheaf F 0 on Z together with a surjective H equivariant homomorphism F 0 → F . In particular, if Z is regular, this holds for all equivariant quasi-coherent sheaves F on Z .
The hypotheses of Proposition A.2 ensure that the usual morphism Z × S H → Z × S Z is affine. In fact, the projection Z × S H → Z is obviously affine, the projection Z × S Z → Z has affine diagonal, so this follows from the elementary fact that if Z → U → V are morphisms of algebraic spaces, Z → V is affine, and U → V has affine diagonal, then Z → U is affine. Consider the quotient stack Z = [Z /H ] (see [LMB]); the argument above implies that the diagonal Z → Z × S Z is affine. Since an H -equivariant quasi-coherent O Z -module is the same as a quasi-coherent module over Z , now Proposition 5.3 follows from the more general result below. A.3 Let S be a Noetherian algebraic space, and let X be a Noetherian algebraic stack over S with affine diagonal. Let F be a quasi-coherent sheaf of finite flat dimension on X ; then there exists a flat quasi-coherent sheaf F 0 on X together with a surjective homomorphism F 0 → F . PROPOSITION
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Proof Take an affine scheme U with a flat morphism f : U → X ; then f is affine, and, in particular, the pushforward f ∗ on quasi-coherent sheaves is exact. Consider a quasicoherent sheaf F on X of finite flat dimension, with the adjunction map F → f ∗ f ∗ F . This map is injective; call Q its cokernel. Clearly, the flat dimension of f ∗ f ∗ F is the same as the flat dimension of F ; we claim that the flat dimension of Q is at most equal to the flat dimension of F . Now, if there were a section X → U of f , then the sequence 0 → F → f∗ f ∗F → Q → 0 would split and this would be clear. However, to compute the flat dimension of Q , we can pull back to any flat surjective map to X ; in particular, after pulling back to U , we see that f acquires a section, and the statement is checked. Now U is an affine scheme, so we can take a flat quasi-coherent sheaf P on U with a surjective map u : P → f ∗ F . Call F 0 the kernel of the composition f ∗ P → f ∗ f ∗ F → Q ; then F 0 surjects onto F , and it fits into an exact sequence 0 → F 0 → f ∗ P → Q → 0. But f ∗ P is flat over X , so the flat dimension of F 0 is less than the flat dimension of Q , unless Q is flat. But since the flat dimension of Q is at most equal to the flat dimension of F , we see that the flat dimension of F 0 is less than the flat dimension of F , unless F is flat. The proof is completed with a straightforward induction on the flat dimension of F . THEOREM A.4 Let X be an object in G-AlgSpreg . The obvious inclusions of the following complicial biWaldhausen categories induce homotopy equivalences on the Waldhausen K . theory spectra K (i) (X ) = K (Wi,X ), i = 1, 2, 3. In particular, the corresponding (i) Waldhausen K -theories K ∗ (X, G) coincide. (i) W1,X = (complexes of quasi-coherent G-equivariant O X -modules with bounded coherent cohomology). (ii) W2,X = (bounded complexes in G − Coh X ). (iii) W3,X = (complexes of flat quasi-coherent G-equivariant O X -modules with bounded coherent cohomology). Moreover, the Waldhausen K -theory of any of the categories above coincides with Quillen K -theory K ∗0 (X, G) of G-equivariant coherent O X -modules.
Proof By [Th2, Par. 1.13], the inclusion of W2,X in W1,X induces an equivalence of K -theory spectra. Proposition 5.3, together with [ThTr, Lem. 1.9.5] (applied to D = (flat Gequivariant O X -modules) and A = (G-equivariant O X -modules)), implies that for any object E ∗ in W1,X there exist an object F ∗ in W3,X and a quasi-isomorphism
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
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F ∗ −→ g E ∗ . Therefore, by [ThTr, Par. 1.9.7 and Th. 1.9.8], the inclusion of W3,X in W1,X induces an equivalence of K -theory spectra. The last statement of the theorem follows immediately from [Th2, Par. 1.13, p. 518]. Since any complex in W3,X is degreewise flat and X is regular (hence boundedness of cohomology is preserved under tensor product∗ ), the tensor product of complexes makes the Waldhausen K -theory spectrum of W3,X into a functor K (3) from GAlgSpreg to ring spectra, with product K (3) ∧ K (3) −→ K (3) , exactly as described in [ThTr, Par. 3.15]. In particular, by Theorem 5.3, K ∗ is a functor from G-AlgSpreg to graded rings. In the same way, the tensor product with complexes in W3,X gives a pairing K (3) ∧ K (1) −→ K (1) (1)
between the corresponding functors from G-AlgSpreg to spectra, so that K ∗ (X, G) (3)
becomes a module over the ring K ∗ (X, G) functorially in (X, G) ∈ G-AlgSpreg . We denote the corresponding cap-product by ∩ : K ∗(3) (X, G) ⊗ K ∗(1) (X, G) −→ K ∗(1) (X, G), which becomes the ring product in K ∗ (X, G) with the identifications allowed by Theorem 5.3. Note that there is an obvious ring morphism η : K ∗naive (X, G) → (3) K ∗ (X, G), and if 0
∩naive : K ∗naive (X, G) ⊗ K ∗ (X, G) −→ K ∗0 (X, G) denotes the usual “naive” cap-product on Quillen K -theories, there is a commutative diagram ∩naive
K ∗naive (X, G) ⊗ K ∗0 (X, G) −−−−→ K ∗0 (X, G) u η⊗u y y (3)
(1)
(1)
K ∗ (X, G) ⊗ K ∗ (X, G) −−−−→ K ∗ (X, G) ∩
where u is the isomorphism of Theorem 5.3. Because of that, we simply write ∩ for both the naive and nonnaive cap-products. Note that, as shown in [Th2, Par. 1.13, p. 519], K ∗0 (−, G) (and therefore K ∗ (X, G) under our hypotheses) is a covariant functor ∗ In
fact, this is a nonequivariant statement and a local property in the flat topology, so it reduces to the same statement for regular affine schemes, which is elementary (see also [SGA6]).
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for proper maps in G-AlgSpreg ; on the other hand, since any map in G-AlgSpreg has finite Tor-dimension, K ∗ (−, G) is a controvariant functor from G-AlgSpreg to (graded) rings. In fact, if f : X −→ Y is a morphism in G-AlgSpreg , the same argument in [ThTr, Par. 3.14.1] shows that there is an induced pullback exact functor (3) f ∗ : W3,Y → W3,X , and then we use Theorem A.4 to identify K ∗ (−, G) with K ∗ (−, G). PROPOSITION A.5 (Projection formula) Let j : Z −→ X be a closed immersion in G-AlgSpreg . Then, if α is in K ∗ (X, G) and β in K ∗0 (Z , G), we have j∗ j ∗ (α) ∩ β = α ∩ j∗ (β)
in K ∗0 (X, G). Proof Since j is affine, j∗ is exact on quasi-coherent modules and therefore induces an exact functor of complicial bi-Waldhausen categories j∗ : W1,Z → W1,X (the condition of bounded coherent cohomology being preserved by regularity of Z and X ). Therefore, the maps (α, β) 7 −→ j∗ j ∗ (α) ∩ β , (α, β) 7 −→ α ∩ j∗ (β) from K ∗ (X, G) × K ∗ (Z , G) to K ∗0 (X, G) ' K ∗ (X, G) are induced by the exact functors W3,X × W1,Z −→ W1,X , (F ∗ , E ∗ ) 7 −→ j∗ j ∗ (F ∗ ) ⊗ E ∗ , (F ∗ , E ∗ ) 7 −→ F ∗ ⊗ j∗ (E ∗ ).
(43)
But for any equivariant quasi-coherent sheaf F on X and G on Z , there is a natural (hence, equivariant) isomorphism j∗ ( j ∗ F ⊗ G ) ' F ⊗ j∗ G which, again by naturality, induces an isomorphism between the two functors in (43); therefore, we conclude by [ThTr, Par. 1.5.4]. Remark A.6 Since we need the projection formula only for (regular) closed immersion in this paper, we have decided to state the result only in this case. However, since, by [Th2,
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Par. 1.13 p. 519], K ∗ (X, G) coincides also with Waldhausen K -theory of the category W4,X of complexes of G-equivariant quasi-coherent injective modules on X with bounded coherent cohomology, therefore, by Theorem 5.3, it also coincides with Waldhausen K -theory of the category W5,X complexes of G-equivariant quasicoherent flasque modules on X with bounded coherent cohomology. For any proper map f : X → Y in G-AlgSpreg , we have an exact functor f ∗ : W5,X → W5,Y , which therefore gives a “model” for the pushforward f ∗ : K ∗ (X, G) → K ∗ (Y, G) (cf. [ThTr, Par. 3.16]). Now, the proof of [ThTr, Prop. 3.17] should also give a proof of Proposition 3 with j replaced by any proper map in G-AlgSpreg because it only uses [ThTr, Th. 2.5.5], which obviously holds for X and Y Noetherian algebraic spaces, and [SGA4, Exp. XVII, par. 4.2], which should give a canonical G-equivariant Godement flasque resolution of any complex of G-equivariant modules on any algebraic space in G-AlgSpreg since it is developed in a general topos. It is very probable that Theorem 5.3 and therefore the functoriality with respect to morphisms of finite Tor-dimension still hold without the regularity assumption on the algebraic spaces. On the other hand, it should also be true that with G and X as above (therefore, X regular), the Waldhausen K -theory of the category of G-equivariant perfect complexes on X coincides with K ∗0 (X, G). This last statement should follow (with a bit of work to identify K ∗0 (X, G) with the Waldhausen K -theory of Gequivariant pseudocoherent complexes with bounded cohomology on X ) from [J, Th. 1.6.2]. Acknowledgments. We wish to thank the referee for useful and precise remarks. We also thank Bertrand Toen, who pointed out the content of Remark 5.9 to us. References [SGA4] M. ARTIN, A. GROTHENDIECK, and J. L. VERDIER, Th´eorie des topos et cohomologie e´ tale des sch´emas, Vol. 3, S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie (SGA 4), Lecture Notes in Math. 305, Springer, Berlin, 1973. MR 50:7132 53 [SGA6] P. BERTHELOT, A. GROTHENDIECK, and L. ILLUSIE, Th´eorie des intersections et th´eor`eme de Riemann-Roch, S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie (SGA 6), Lecture Notes in Math. 225, Springer, Berlin, 1971. MR 50:7133 2, 23, 51 [DG] M. DEMAZURE and P. GABRIEL, Groupes alg´ebriques, Vol. 1: G´eom´etrie alg´ebrique, g´en´eralit´es, groupes commutatifs, Masson, Paris, 1970. MR 46:1800 38 [SGA3] M. DEMAZURE and A. GROTHENDIECK, Sch´emas en groupes, S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie (SGA 3), Lecture Notes in Math. 151, 152, 153, Springer, Berlin, 1970. MR 43:223a, MR 43:223b, MR 43:223c 12, 16, 28 [EG] D. EDIDIN and W. GRAHAM, Riemann-Roch for equivariant Chow groups, Duke
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Math. J. 102 (2000), 567 – 594. MR 2001f:14018 2, 4, 48 [SGA1] A. GROTHENDIECK, Revˆetements e´ tales et groupe fondamental, S´eminaire de G´eom´etrie alg´ebrique du Bois-Marie (SGA 1), Lecture Notes in Math. 224, Springer, Berlin, 1971. 27, 30 [EGAI] A. GROTHENDIECK and J. A. DIEUDONNE´ , El´ements de g´eom´etrie alg´ebrique, I: Le langage des sch´emas, Springer, Berlin, 1971. 36 [J] R. JOSHUA, Higher intersection theory on algebraic stacks, II, preprint, 2000, http://math.ohio-state.edu/˜joshua/pub.html 48, 53 [LMB] G. LAUMON and L. MORET-BAILLY, Champs alg´ebriques, Ergeb. Math. Grenzgeb. (3) 39, Springer, Berlin, 2000. MR 2001f:14006 35, 49 [Me] A. S. MERKURJEV, Comparison of the equivariant and the standard K -theory of algebraic varieties, St. Petersburg Math. J. 9 (1998), 815 – 850. MR 99d:19003 15, 16 [Q] D. QUILLEN, “Higher algebraic K -theory, I” in Algebraic K -Theory, I: Higher K -Theories (Seattle, 1972), Lecture Notes in Math. 341, Springer, Berlin, 1973. MR 49:2895 [Se] J.-P. SERRE, Linear representations of finite groups, Grad. Texts in Math. 42, Springer, New York, 1977. MR 56:8675 18 [Sr] V. SRINIVAS, Algebraic K -Theory, 2d ed., Progr. Math. 90, Birkh¨auser, Boston, 1996. MR 97c:19001 [Th1] R. W. THOMASON, Comparison of equivariant algebraic and topological K -theory, Duke Math. J. 53 (1986), 795 – 825. MR 88h:18011 16, 19, 20, 21, 35 [Th2] , Lefschetz-Riemann-Roch theorem and coherent trace formula, Invent. Math. 85 (1986), 515 – 543. MR 87j:14028 50, 51, 53 [Th3] , “Algebraic K -theory of group scheme actions” in Algebraic Topology and Algebraic K -Theory (Princeton, 1983), Ann. of Math. Stud. 113, Princeton Univ. Press, Princeton, 1987, 539 – 563. MR 89c:18016 11, 19, 20, 21, 35, 36, 37, 42 [Th4] , Equivariant algebraic vs. topological K -homology Atiyah-Segal-style, Duke Math. J. 56 (1988), 589 – 636. MR 89f:14015 15 [Th5] , Une formule de Lefschetz en K -th´eorie e´ quivariante alg´ebrique, Duke Math. J. 68 (1992), 447 – 462. MR 93m:19007 12, 13, 19, 21, 22, 23, 45 [Th6] , Les K -groupes d’un sch´ema e´ clat´e et une formule d’intersection exc´edentaire, Invent. Math. 112 (1993), 195 – 215. MR 93k:19005 [ThTr] R. W. THOMASON and T. TROBAUGH, “Higher algebraic K -theory of schemes and of derived categories” in The Grothendieck Festschrift, Vol. III, Progr. Math. 88, Birkh¨auser, Boston, 1990, 247 – 435. MR 92f:19001 6, 48, 49, 50, 51, 52, 53 [To1] B. TOEN, Th´eor`emes de Riemann-Roch pour les champs de Deligne-Mumford, K -Theory 18 (1999), 33 – 76. MR 2000h:14010 5 [To2] , Notes on G-theory of Deligne-Mumford stacks, preprint, arXiv:math.AG/9912172 5 [Vi1] A. VISTOLI, Higher equivariant K -theory for finite group actions, Duke Math. J. 63 (1991), 399 – 419. MR 92d:19005 2, 3, 5, 16, 18, 47 [Vi2] , “Equivariant Grothendieck groups and equivariant Chow groups” in Classification of Irregular Varieties (Trento, Italy, 1990), ed. E. Ballico,
K -THEORY OF ACTIONS WITH FINITE STABILIZERS
F. Catanese, and C. Ciliberto, Lecture Notes in Math. 1515, Springer, Berlin, 1992, 112 – 133. MR 93j:14008 2, 5
Vezzosi Dipartimento di Matematica, Universit`a di Bologna, Piazza di Porta San Donato 5, Bologna 40127, Italy;
[email protected] Vistoli Dipartimento di Matematica, Universit`a di Bologna, Piazza di Porta San Donato 5, Bologna 40127, Italy;
[email protected]
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DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 1,
FINITENESS OF DE RHAM COHOMOLOGY IN RIGID ANALYSIS ¨ ELMAR GROSSE-KLONNE
Abstract For a large class of smooth dagger spaces—rigid spaces with overconvergent structure sheaf—we prove finite dimensionality of de Rham cohomology. This is enough to obtain finiteness of P. Berthelot’s rigid cohomology also in the nonsmooth case. We need a careful study of de Rham cohomology in situations of semistable reduction. Introduction Let R be a complete discrete valuation ring of mixed characteristic, let π ∈ R be a uniformizer, and let k = Frac(R), k¯ = R/(π). It is a simple observation that the de Rham cohomology Hd∗R (X ) of a positive-dimensional smooth affinoid k-rigid space X computed with respect to its (usual) structure sheaf is not finite-dimensional. The idea of instead using an overconvergent structure sheaf arises naturally from the paper of P. Monsky and G. Washnitzer [25]. The Monsky-Washnitzer cohomology of a ¯ smooth affine k-scheme Spec(A) is the de Rham cohomology of A˜ † ⊗ R k, where A˜ † is a weakly complete formal lift of A. Monsky-Washnitzer cohomology has recently been shown to be finite-dimensional (independently by Berthelot [2] and, based on common work with G. Christol, by Z. Mebkhout [24]). The algebra A˜ † ⊗ R k can be geometrically interpreted as a k-algebra of overconvergent functions on the rigid space Sp( A˜ ⊗ R k), where A˜ is a lifting of A to a formally smooth π-adically complete R-algebra. In [11] we introduce a category of k-rigid spaces with overconvergent structure sheaf, which we call k-dagger spaces, and we study a functor X 7 → X 0 from this category to the category of k-rigid spaces which is not far from being an equivalence. For example, X and X 0 have the same underlying G-topological space and the same stalks of structure sheaves. Finiteness of Monsky-Washnitzer cohomology implies finiteness of de Rham cohomology for affinoid k-dagger spaces with good reduction; in the above notation, the algebra A˜ † gives rise to the affinoid k-dagger space X with 0(X, O X ) = A˜ † ⊗ R k. Our main result generalizes this statement as follows.
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 1, Received 4 July 2000. Revision received 15 May 2001. 2000 Mathematics Subject Classification. Primary 14F30, 14G22. 57
58
¨ ELMAR GROSSE-KLONNE
THEOREM A (Corollary 3.5 plus Theorem 3.6) Let X be a quasi-compact smooth k-dagger space, let U ⊂ X be a quasi-compact open subset, and let Z → X be a closed immersion. Then T = X − (U ∪ Z ) has finite-dimensional de Rham cohomology Hd∗R (T ).
By [11, Theorem 3.2], this implies finiteness of de Rham cohomology also for certain smooth k-rigid spaces Y , for example, if Y admits a closed immersion i into a polydisk without boundary (at least if i extends to a closed immersion with bigger radius) or if Y is the complement of a quasi-compact open subspace in a smooth proper k-rigid space. But our main corollary is of course the following. COROLLARY B (Corollary 3.8) ∗ (X/k) (see [2]) are finite¯ For a k-scheme X of finite type, the k-vector spaces Hrig dimensional.
We do not re-prove finiteness of Monsky-Washnitzer cohomology; rather, we reduce our Theorem A to it. A big part of this paper is devoted to the study of de Rham cohomology in situations of semistable reduction. We need and prove the following theorem. THEOREM C (Theorem 2.3) S Let X be a strictly semistable formal R-scheme, and let Xk¯ = i∈I Yi be the decomposition of the closed fibre into irreducible components. For K ⊂ I , set Y K = T † i∈K Yi . Let X be a k-dagger space such that its associated rigid space is identified with Xk . For a subscheme Y ⊂ Xk¯ , let ]Y [†X be the open dagger subspace of X † corresponding to the open rigid subspace ]Y [X of Xk . Then for any ∅ 6= J ⊂ I , the S canonical map Hd∗R (]Y J [†X ) → Hd∗R ]Y J − (Y J ∩ ( i∈I −J Yi ))[†X is bijective.
Another important tool is A. de Jong’s theorem on alterations by strictly semistable pairs, in its strongest sense. We proceed as follows. After recalling some facts on dagger spaces in Section 0, we formulate in Section 1 some basic concepts about D -modules on rigid and dagger spaces. This follows the complex analytic case (see, e.g., [23]). Instead of reproducing well-known arguments, we focus only on what is specific to the nonarchimedean case. Then we construct a long exact sequence for de Rham cohomology with supports in blowing-up situations. As in [18] (for algebraic k-schemes), this results from the existence of certain trace maps for proper morphisms; we define such trace maps based on constructions from [8], [3], and [26]. Finally, we prove the important technical fact that the de Rham cohomology Hd∗R (X ) of a smooth dagger
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space X depends only on its associated rigid space X 0 ; hence knowledge of X 0 (e.g., a decomposition into a fibre product) gives information about Hd∗R (X ). In Section 2, we begin to look at R-models of the associated rigid spaces; more specifically, we consider the case of semistable reduction. Its main result is Theorem C. It enables us to reduce Theorem A, in the case where U = ∅ and X has semistable reduction, to the finiteness of Monsky-Washnitzer cohomology. In Section 3, we first prove Theorem A in the case when U = ∅ = Z . After reduction to the case where X is affinoid and defined by polynomials, we apply de Jong’s theorem to an R-model of a projective compactification of X to reduce to the finiteness result of Section 2. The case of general Z is handled by a resolution of singularities (see [4]). Then we treat the case of general U by another application of de Jong’s theorem. The formal appearence of these last arguments bears some resemblance to the finiteness proofs in [2] and [18]. But there are also distinctive features, in the simultaneous control of special and generic fibre, and in particular in our second application of de Jong’s theorem. We apply it to a certain closed immersion of R-schemes X¯ k¯ ∪ Y¯ → X¯ , where the space X − U we are interested in is realized in (the tube ]Y¯ [ of) the compactifying divisor—not in its open complement. 0. Dagger spaces Let k be a field of characteristic zero, complete with respect to a nonarchimedean valuation |.|, with algebraic closure ka , and let 0 ∗ = |ka∗ | = |k ∗ | ⊗ Q. We gather some facts from [11]. For ρ ∈ 0 ∗ , the k-affinoid algebra Tn (ρ) consists P of all series aν X ν ∈ k[[X 1 , . . . , X n ]] such that |aν |ρ |ν| tends to zero if |ν| → ∞. S ∗ The algebra Wn is defined to be Wn = ρ>1 Tn (ρ). A k-dagger algebra A is a ρ∈0 ∗
quotient of some Wn ; a surjection Wn → A endows it with a norm that is the quotient seminorm of the Gauss norm on Wn . All k-algebra morphisms between k-dagger algebras are continuous with respect to these norms, and the completion of a k-dagger algebra A is a k-affinoid algebra A0 in the sense of [6]. There is a tensor product ⊗†k in the category of k-dagger algebras. As for k-affinoid algebras, one has for the set Sp(A) of maximal ideals of A the notions of rational and affinoid subdomains, and for these the analogue of Tate’s acyclicity theorem (see [6, Theorem 8.2.1]) holds. The natural map Sp(A0 ) → Sp(A) of sets is bijective, and via this map the affinoid subdomains of Sp(A) form a basis for the strong G-topology on Sp(A0 ) from [6]. Imposing this G-topology on Sp(A), one gets a locally G-ringed space, an affinoid k-dagger space. (Global) k-dagger spaces are built from affinoid ones precisely as in [6]. The fundamental concepts and properties from [6] translate to k-dagger spaces. ∗ The
notation Wn is taken from [16]. There the author assigns only the name of Washnitzer to this algebra. However, the referee pointed out that the name Monsky-Washnitzer algebra is the usual one.
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There is a faithful functor from the category of k-dagger spaces to the category of k-rigid spaces, assigning to a k-dagger space X a k-rigid space X 0 (to which we refer as the associated rigid space; however, we use the notation (?)0 not only for this functor). X and X 0 have the same underlying G-topological space and the same stalks of structure sheaf. A smooth k-rigid space Y admits an admissible open affinoid S covering Y = Vi such that Vi = Ui0 for uniquely determined (up to noncanonical isomorphisms) affinoid k-dagger spaces Ui . Furthermore, this functor induces an equivalence between the respective subcategories formed by partially proper spaces as defined below. In particular, there is an analytification functor from k-schemes of finite type to k-dagger spaces. For a smooth partially proper k-dagger space X with associated k-rigid space X 0 , the canonical map Hd∗R (X ) → Hd∗R (X 0 ) between the de Rham cohomology groups is an isomorphism. This follows from applying [11, Theorem 3.2] to the morphism between the respective Hodge – de Rham spectral sequences. By a dagger space not specified otherwise, we mean a k-dagger space; we use similarly the terms dagger algebras, rigid spaces, and so on. In the sequel, all dagger spaces and rigid spaces are assumed to be quasiseparated. We denote by D = {x ∈ k; |x| ≤ 1} (resp., D0 = {x ∈ k; |x| < 1}) the unit disk with (resp., without) boundary, with its canonical structure of k-dagger or k-rigid space, depending on the context. For ∈ 0 ∗ , the ring of global functions on the polydisk {x ∈ k n ; all |xi | ≤ }, endowed with its canonical structure of kdagger space, is denoted by kh −1 · X 1 , . . . , −1 · X n i† . The dimension dim(X ) of a dagger space X is the maximum of all dim(O X,x ) for x ∈ X . We say that X is pure-dimensional if dim(X ) = dim(O X,x ) for all x ∈ X . A morphism f : X → Y of rigid or dagger spaces is called partially proper (cf. [19, p. 59]) if f is separated, if there is an admissible open affinoid covering S Y = i Yi , and if for all i there are admissible open affinoid coverings f −1 (Yi ) = S S 0 0 j∈Ji X i j = j∈Ji X i j with X i j ⊂⊂Yi X i j for every j ∈ Ji (where ⊂⊂Yi is defined as in [6]). 1 Let Z → X be a closed immersion into an affinoid smooth dagger space. There is a proper surjective morphism g : X˜ → X with X˜ smooth, g −1 (Z ) a divisor with normal crossings on X˜ , and g −1 (X − Z ) → (X − Z ) an isomorphism. LEMMA
Proof Write X = Sp(Wn /I ), Z = Sp(Wn /J ) with ideals I ⊂ J ⊂ Wn . Since these ideals are finitely generated, there are a ρ > 1 and ideals Iρ ⊂ Jρ ⊂ Tn (ρ) such that I = Iρ · Wn and J = Jρ · Wn , and such that the rigid space X ρ = Sp(Tn (ρ)/Iρ ) is smooth.
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Apply [4, Theorem 1.10] to the closed immersion Z ρ = Sp(Tn (ρ)/Jρ ) → X ρ to get a morphism of rigid spaces X˜ ρ → X ρ with the desired properties. Its restriction S to the partially proper open subspace ρ 0 <ρ Sp(Tn (ρ 0 )/(Iρ )) ⊂ X ρ is a morphism of partially proper spaces (compositions of partially proper morphisms are partially proper; see [19]) and hence by [11, Theorem 2.27] is equivalent to a morphism of dagger spaces. The restriction of the latter to X does the job.
1. D -modules 1.1 For a smooth dagger (or rigid) space X , let I = Ker(O X ×X → O X , f ⊗ g 7→ f g). For n ∈ N, we view P Xn = O X ×X /I n+1 via d1,n : O X → P Xn , f 7→ f ⊗ 1 as an D Xn comes O X -algebra, and we define D Xn = HomO X (P Xn , O X ). Then D X = lim→ n in a natural way with the structure of a sheaf of rings on X (as in [15, Chapitre IV, Corollaire 16.8.10]). By modules over D X which are not explicitly declared as D X right modules we always mean D X -left modules. O X becomes a D X -left module by setting P · f = P(d2,n ( f )) for P ∈ HomO X (P Xn , O X ), f ∈ O X , and d2,n : O X → P Xn , f 7 → 1⊗ f . For a morphism X → Y of smooth spaces with J = Ker(O X ×Y X → O X , f ⊗ g 7 → f g), we define 1X/Y = J/J 2 . By means of d : O X → 1X/Y , a 7 → V (1 ⊗ a − a ⊗ 1), we form in the usual way the complex •X/Y = ( • 1X/Y , d). If Y = Sp(k), we write •X . If X is of pure dimension r , we write ω X = rX , and if f : X → Y is a morphism of smooth and pure-dimensional spaces, we write ⊗(−1) ω X/Y = ω X ⊗O X f ∗ ωY . 1.2 Suppose X = Sp(A) with a regular k-dagger algebra (resp., k-affinoid algebra) A. Then d : A → 1X (X ) = 1A is the universal k-derivation of A into finite A-modules. A dagger space X is smooth if and only if its associated rigid space X 0 is smooth, and 1X 0 is canonically identified with the coherent O X 0 -module obtained by completing on affinoid open pieces the coherent O X -module 1X . D X is a coherent D X -module (see [12, Lemma 5.5]). If X = Sp(A), then D X (X ) = D A is a both-sided Noetherian ring and dim(A) = coh dim(D A ) (see [23, pp. 42–55]). 1.3 Let X be smooth of pure dimension. Then ω X can be equipped with a canonical structure of a D X -right module and there is an equivalence between the category of
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D X -right modules and the category of D X -(left)modules, where a D X -right module E is sent to HomO X (ω X , E ), and a D X -(left)module M is sent to ω X ⊗O X M . This can be seen as over the complex numbers (cf. [23]).
1.4 For a morphism f : Z → Y of smooth pure-dimensional dagger (resp., rigid) spaces, we define D Z →Y = O Z ⊗ f −1 OY f −1 DY , which is a (D Z , f −1 DY )-bimodule, and DY ←Z = ω Z ⊗ f −1 OY f −1 HomOY (ωY , DY ) = ω Z /Y ⊗O Z D Z →Y , which is an ( f −1 DY , D Z )-bimodule (cf. [23]). In particular, if M is a D Z -module, DY ←Z ⊗D Z M becomes an f −1 DY -module, and we get a left-derived functor − − −1 DY ←Z ⊗L DY ). D Z (.) : D (D Z ) → D ( f
1.5 As in the complex case, we have a canonical D X -linear projective resolution 0 → D X ⊗O X
n ^
T X → · · · → D X ⊗O X T X → D X
(∗)
of O X , with T X = HomO X (1X , O X ). Therefore the definition DR(M ) = RHomD X (O X , M ) makes sense for any M ∈ D(D X ). For r ∈ N, we define the k–vector spaces Hdr R (X ) = H r (X, DR(O X )). Application of ω X ⊗O X (.) to (∗) yields DR(M ) ∼ = ω X ⊗L D X M [− dim(X )]. If Y is another smooth space, with Z = X × Y → Y the canonical projection, there is a canonical isomorphism •Z /Y ∼ = DY ←Z ⊗L D Z O Z [− dim(X )]. 1.6 We recall a definition from [19, Section 5.6]. Let f : Z → Y be partially proper, and let F be an abelian sheaf on Z . If Y is quasi-compact, let 0c (Z /Y, F ) = s ∈ 0(Z , F ); there is a quasi-compact admissible open U ⊂ Z such that s ∈ Ker(0(Z , F ) → 0(Z − U, F )) . If Y is arbitrary, let 0c (Z /Y, F ) = s ∈ 0(Z , F ); for all quasi-compact admissible open Y 0 ⊂ Y we have s| f −1 (Y 0 ) ∈ 0c f −1 (Y 0 )/Y 0 , F
.
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Then V 7→ 0c ( f −1 (V )/V, F ) defines a sheaf f ! F on Y . We denote by R f ! : D+ (Z ) → D+ (Y ) the functor induced by the left exact functor f ! (−). In the following we always assume tacitly that R f ! : D+ (Z ) → D+ (Y ) extends to R f ! : D(Z ) → D(Y ) and R f ! : D− (Z ) → D− (Y ). (This is true in the cases relevant for us, namely, if Z can be written as Z = X ×Y for an X which admits a finite and admissible cover by open subspaces that are Zariski closed in some (D0 )m , and if f is the projection—as in Proposition 3, then there is an n ∈ N such that R i f ! F = 0 for all i > n, all F ; see [12, p. 47].) If Z and Y are smooth and pure-dimensional and if M ∈ D(D Z ), we define f + (M ) = R f ! (DY ←Z ⊗L D Z M ) ∈ D(DY ). If M ∈ D− (D X ), we get f + (M ) ∈ D− (DY ). PROPOSITION 1 f g Let X → Y → Z be partially proper morphisms between smooth pure-dimensional dagger (resp., rigid) spaces. We assume that g is a projection or a closed immersion. For M ∈ D− (D X ), there is a canonical isomorphism g+ ( f + M ) ∼ = (g ◦ f )+ M in D− (D Z ).
Proof We have to show the projection formula L −1 ∼ D Z ←Y ⊗L D Z ←Y ⊗Lf −1 D DY ←X ⊗L DY R f ! (DY ←X ⊗D X M ) = R f ! ( f D X M ). Y
If g is a projection Y = W × Z → Z , we see that D Z ←Y = ωY/Z ⊗O Z D Z is a coherent (DY = DW ⊗k D Z )-right module (because ωY/Z = ωW ⊗k O Z is a coherent (DW ⊗k O Z )-right module). Therefore we obtain the above projection formula using the way-out lemma (see [17, Chapter I, Section 7]), since for finite free DY -right modules (instead of the DY -right module D Z ←Y ) the projection formula is evident. If g is a closed immersion, D Z ←Y is a locally free DY -right module. This can be shown, as in the complex analytic case, by using the fact that, locally for an admissible covering of Z , there are isomorphisms Y × Dcodim Z (Y ) ∼ = Z such that g corresponds to the zero section; this is [21, Theorem 1.18] in the rigid case but holds true also in the dagger case, as one observes by examining the proof in [21]. By the commutation of R f ! with pseudo-filtered limits (cf. [19, Section 5.3.7] or [12, Lemma 4.8]), again one reduces the proof of the projection formula to the case of finite free DY -right modules.
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1.7 Let Y → X be a closed immersion into a smooth dagger (resp., rigid) space, defined by the coherent ideal J ⊂ O X . Then 0 ∗Y (E ) = lim HomO X (O X /J n , E ) →
n
and E (∗Y ) = lim HomO X (J n , E ) →
n
for a D X -module E are again D X -modules. We get right-derived functors R0 ∗Y (−) and R(−)(∗Y ) as functors D+ (D X ) → D+ (D X ), but also as functors D(D X ) → D(D X ) and D− (D X ) → D− (D X ). We have distinguished triangles +1
R0 ∗Y (K ) → K → R K (∗Y ) → for all K ∈ D(D X ). 1.8 We list some properties of the above functors. The proofs are similar to those in [23]; the projection formulas needed can be justified as in Proposition 1. (a) If f : Z → Y is a partially proper morphism between smooth puredimensional dagger (resp., rigid) spaces and if M ∈ D− (D Z ), there is a canonical isomorphism DR( f + M )[dim(Y )] ∼ = R f ! DR(M )[dim(Z )]. (b) If in addition T → Y is a closed immersion and if TZ = T ×Y Z , there is a canonical isomorphism R0 ∗T ( p+ M ) ∼ = p+ (R0 ∗TZ M ). (c) Let X be smooth, let Yi → X be closed immersions (i = 1, 2), and let M ∈ D(D X ). There are a canonical isomorphism R0 ∗Y1 R0 ∗Y2 (M ) ∼ = R0 ∗(Y1 ∩Y2 ) (M ) and a distinguished triangle +1
R0 ∗(Y1 ∩Y2 ) (M ) → R0 ∗Y1 (M ) ⊕ R0 ∗Y2 (M ) → R0 ∗(Y1 ∪Y2 ) (M ) → . (d) (e)
Let X be smooth, let Y → X be a closed immersion, and let M ∈ D(D X ). ∼ There is a canonical isomorphism R0 ∗Y (O X ) ⊗L O X M = R0 ∗Y (M ). Let X be smooth and affinoid, let Sp(B) = Y → X be a closed immersion of pure codimension d such that all local rings Bx (for x ∈ Sp(B) = Y ) are locally complete intersections, and let F be a D X -module that is locally free as an O X -module. Then R i 0 ∗Y (F ) = 0 for all i 6 = d (a well-known algebraic fact).
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PROPOSITION 2 s Let Z → Y → X be a chain of closed immersions, where Y and X are smooth and pure-dimensional. If d = codim(s), there is a canonical isomorphism s+ R0 ∗Z OY ∼ = R0 ∗Z O X [d].
Proof (cf. [24, Lemma 3.3-1]) First, Section 1.8(b), (c) allows us to assume that Z = Y . Denote by I ⊂ O X the ideal of Y in X . Observe that DY →X = D X /I.D X and that this, as well as D X ←Y , is locally free over DY (cf. the proof of Proposition 1). We claim that there is a canonical map of D X -right modules ExtdO X (O X /I, ω X ) ⊗DY DY →X → lim ExtdO X (O X /I k , ω X ). →
k
Indeed, choose an injective resolution J • of the D X -right module ω X , and define the morphism of complexes HomO X (O X /I, J • ) ⊗DY DY →X → lim HomO X (O X /I k , J • ) →
k
as follows. Let g be a local section of HomO X (O X /I, J m ), and let P be a local section of DY →X , represented by the local section P˜ of D X . The O X -linear map O X → J m which sends 1O X to g(1O X ) · P˜ actually induces an element of lim→ HomO X (O X /I k , J m ). Indeed, if P˜ is of order n, then g(1O X ) · P˜ is ank
nihilated by I n+1 . The promised map is the one that sends g ⊗ P to the local section of lim→ HomO X (O X /I k , J m ) just described. Now since we have as usual k ωY ∼ = Extd (O X /I, ω X ), we get a map OX
ωY ⊗DY DY →X → lim ExtdO X (O X /I k , ω X ) →
k
of D X -right modules. Applying HomO X (ω X , .) (cf. Section 1.3), it becomes the map D X ←Y ⊗DY OY → lim ExtdO X (O X /I k , O X ) = R d 0 ∗Y O X →
(∗)
k
of D X -left modules. We claim that (∗) is an isomorphism. Indeed, if x1 , . . . , xn are local coordinates on X such that Y is defined by x1 , . . . , xd , and if δ1 , . . . , δn is the basis of HomO X (1X , O X ) dual to d x1 , . . . , d xn , one verifies that both sides in (∗) are identified with D X /(x1 , . . . , xd , δd+1 , . . . , δn ). Since the right-hand side in (∗) is already all of R0 ∗Y O X [d] (due to Section 1.8(e)), and since on the left-hand side we may write ⊗L instead of ⊗, we are done.
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PROPOSITION 3 Let X be smooth, proper, and of pure dimension n, let Y be smooth and purep dimensional, and let Z = X × Y → Y be the canonical projection. Then there is a canonical trace map p+ O Z [n] → OY . If, furthermore, T → Y and S → Z ×Y T = X × T are closed immersions, there are canonical trace maps p+ R0 ∗S O Z [n] → R0 ∗T OY and R0 Z , DR(R0 ∗S O Z ) [2n] → R0 Y, DR(R0 ∗T OY ) ,
which are isomorphisms if the composition S → X × T → T is an isomorphism. Proof We begin with the rigid case. In [3] P. Beyer described a finite admissible open coverS T ing X = i Ui such that all U J = i∈J Ui (for J ⊂ I ) have the following properties: there is a closed immersion U J → (D0 )n J for some n J ∈ N, and for all affinoid Yˆ , all coherent OU J ×Yˆ -modules F , and all j > n, we have R j ( p J,Yˆ )! F = 0, where p ˆ : U J × Yˆ → Yˆ denotes the projection. By means of Mayer-Vietoris sequences, J,Y
we get R j p! F = 0 for all coherent O Z -modules F , all j > n; hence R m p! •Z /Y = 0 for m > 2n and R
2n
p! •Z /Y
=
R n p! nZ /Y n n Im(R n p! n−1 Z /Y → R p! Z /Y )
.
In view of the fact that p+ O Z [n] ∼ = Rp! •Z /Y (cf. Section 1.5), to define p+ O Z [n] → OY it is therefore enough to define a map t p : R n p! ω Z /Y → OY vanishing on n n Im(R n p! n−1 Z /Y → R p! Z /Y ). We take the following map (cf. [26], [3]). For U J → D J = (D0 )n J ⊂ Dn J = Sp khT1 , . . . , Tn J i as above and projection p D J ,Yˆ : D J × Yˆ → Yˆ , there is a canonical identification R n J ( p D J ,Yˆ )! ω D J ×Yˆ /Yˆ (Yˆ ) n X = ω= aµ T µ dT1 ∧ · · · ∧ dTn J ; aµ ∈ OYˆ (Yˆ ) µ∈Zn J
µ<0
o and ω converges on {t ∈ D J ; all |ti | < } × Yˆ for some 0 < < 1 . For elements ω of this module, set t (ω) = a(−1,...,−1) . On the other hand, we have the Gysin map g: R n ( p J,Yˆ )! ωU J ×Yˆ /Yˆ (Yˆ ) ∼ = R n ( p J,Yˆ )! Extn J −n (OU J ×Yˆ , ω D J ×Yˆ /Yˆ )(Yˆ ) → R n J ( p D J ,Yˆ )! ω D J ×Yˆ /Yˆ (Yˆ ).
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Locally, it can be described as η 7 → η˜ ∧ d X n+1 / X n+1 ∧ · · · ∧ d X n J / X n J , where η˜ is a lift of η, and where X n+1 , . . . , X n J are local equations for U J in D J . We get t ◦g : R n ( p J,Yˆ )! ωU J ×Yˆ /Yˆ (Yˆ ) → OYˆ (Yˆ ). This is seen to be independent of n J and of the chosen embedding U J → (D0 )n J . Hence these maps glue for varying J and give the desired map t p . By construction, it vanishes on Im(R n p! n−1 Z /Y → n n R p! Z /Y ). In the dagger case, we argue by comparison with the associated morphism 0 p : Z 0 → Y 0 of rigid spaces. Due to [11, Theorem 3.5], we have again R j p! F = R j p∗ F = 0 for all coherent O Z -modules F and all j > n, and the composition of the canonical map R n p! ω Z /Y → R n p!0 ω Z 0 /Y 0 with t p0 : R n p!0 ω Z 0 /Y 0 → OY 0 has its image in OY ⊂ OY 0 ; hence we obtain a map t p : R n p! ω Z /Y → OY . (This can be checked locally on Y ; if Y is affinoid, then this t p is the direct limit of the maps t p for the morphisms of rigid spaces p : X 0 × Y0 → Y0 , for appropriate extensions Y 0 ⊂ Y0 .) If now in addition T and S are given, we can derive from t p the other promised maps using the isomorphisms from Section 1.8; note that R0 Z , DR(R0 ∗S O Z ) ∼ = R0 Y, Rp! DR(R0 ∗S O Z ) because S → T is quasi-compact. Finally, suppose that S → T is an isomorphism. Our additional statement in this situation is seen to be local on X . By the definition of t p : R n p! ω Z /Y → OY , we may replace our X with X = (D0 )n (dropping the assumption on properness). Passing to an admissible covering of Y , we may assume that there is a section s : Y → Z of p : Z → Y inducing the inverse of S → T . It comes with an isomorphism R0 ∗T OY ∼ = R0 ∗T p+ R0 ∗Y O Z [n] ∼ = p+ R0 ∗S O Z [n] by Section 1.8(b). It is enough to show that its composition with the map in question, p+ R0 ∗S O Z [n] → R0 ∗T OY , is an isomorphism. Of course, this will follow once we know that the underlying map OY ∼ = p+ s+ OY ∼ = p+ R0 ∗Y O Z [n] → p+ O Z [n] → OY
is the identity. Since OY (Y ) is Jacobson, we may assume that Y = Sp(k) for this. The definition of t p as above does not depend on the choice of the closed embedding into some (D0 )n . This tells us that the map tidY we get for X = Y = Sp(k), computed by means of the embedding s, is the identity; but on the other hand, it is precisely the map we are interested in, by the compatibility of the Gysin map in the definition of t p with the Gysin map in Proposition 2.
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COROLLARY 1.9 Let gi : Z → Yi (i = 1, 2) be closed immersions into smooth affinoid dagger spaces Yi of pure dimension n i . Then we have R0 Y1 , DR(R0 ∗Z OY1 ) [2n 1 ] ∼ = R0 Y2 , DR(R0 ∗Z OY2 ) [2n 2 ].
Proof Because of Section 1.8(a) and Proposition 2, applied to closed immersions Yi → Dn i for appropriate m i ∈ N, we may assume that Yi = Dn i . If l : Z → X = Y1 × Y2 is the diagonal embedding, it is enough to give isomorphisms R0 Yi , DR(R0 ∗Z OYi ) ∼ = R0 X, DR(R0 ∗Z O X ) [2n 3−i ]. Let i = 1. The open embedding j : X → W = Y1 × Pnk 2 induced by the open embedding into projective space Y2 = Dn 2 → Pnk 2 induces a closed immersion ( j ◦ l) : Z → W , and we have R0(X, DR(R0 ∗Z O X )) ∼ = R0(W, DR(R0 ∗Z OW )). Now apply Proposition 3. For i = 2 the argument is the same. 1.10 Let Z be an affinoid k-dagger space, q ∈ N. The definition h qd R (Z , k) = h qd R (Z ) = dimk H 2n−q (Y, DR(R0 ∗Z OY )) , where Z → Y is a closed embedding into a smooth affinoid k-dagger space Y , is justified by Corollary 1.9. For a finite field extension k ⊂ k1 , let (?)1 = (?) ×Sp(k) Sp(k1 ). Then h qd R (Z , k) = h qd R (Z 1 , k1 ). Indeed, clearly dimk (Hd∗R (X/k)) = dimk1 (Hd∗R (X 1 /k1 )) for any smooth k-dagger space X ; hence dimk H ∗ (X, DR(R0 Z O X )) = dimk1 H ∗ (X 1 , DR(R0 Z 1 O X 1 )) for closed subspaces Z of smooth k-dagger spaces X . Now use Proposition 4. 1.11 Let f : X → Y be a finite e´ tale morphism of smooth dagger (or rigid) spaces, Y S irreducible, X = X i the decomposition into connected components. Assume all maps f | X i : X i → Y to be surjective. Then there are an l = deg( f ) ∈ N and a trace map t : f ∗ •X → •Y t
such that the composition •Y → f ∗ •X → •Y is multiplication by l. In particular, Hdi R (Y ) → Hdi R (X ) is injective for all i ∈ N.
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Indeed, for admissible open connected U = Sp(A) ⊂ Y with decomposition S f −1 (U ) = j Sp(B j ) such that each B j is free over A, let t j : B j → A be the trace q q map, and for q ∈ Z, let f ∗ X (X ) → Y (Y ) be the A-linear map q
q
q
q
q
f ∗ X (X ) = B j = A ⊗ A B j → A = Y (Y ) which sends ω ⊗ b to t j (b) · ω. By the same computation as in the proof of [18, Proposition 2.2], we see that for varying q it commutes with the differentials. Clearly, P it glues for varying U , and the number l = j l j , where l j denotes the rank of B j over A, is independent of U and fulfills our requirement. In fact, the e´ taleness of f is not really needed; f is flat in any case, by regularity of X and Y . If U is as above, let L j = Frac(B j ), K = Frac(A). Then f induces P finite separable field extensions K ⊂ L j . Let l = j [L j : K ]. The trace maps q q q σ j : L j → K give rise to L j /k = K /k ⊗ K L j → K /k , ω ⊗ f 7→ σ j ( f ) · ω, q
q
restricting to σ j : B j → A . Compare with the discussion in [25, Theorem 8.3]. We do not need this. 1.12 Let X be a smooth dagger (or rigid) space, and let j : U → X be an open immersion with complement Y = X − j (U ). We do not put a structure of dagger (or rigid) space on Y . By R0 Y (.) : D+ (D X ) → D+ (D X ) we denote the right-derived functor of the left exact functor F 7 → Ker(F → j∗ j −1 F ) on abelian sheaves on X , and by R j∗ : D+ (DU ) → D+ (D X ) we denote the rightderived functor of j∗ . Note that R j∗ DR(L ) ∼ = DR(R j∗ L ) for L ∈ D+ (DU ). If 0 0 j : U → X is another open immersion with complement Y 0 = X − j 0 (U 0 ) such that U 0 ∪ U is an admissible covering of an admissible open subset of X , there is a distinguished triangle +1
R0 Y ∩Y 0 (K ) → R0 Y (K ) ⊕ R0 Y 0 (K ) → R0 Y ∪Y 0 (K ) → . 4 We have the following. (a) Let Y → X be a closed immersion into a smooth dagger (or rigid) space X . The canonical map R0 X, DR(R0 ∗Y O X ) → R0 X, DR(R0 Y O X ) (∗) PROPOSITION
is an isomorphism.
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(b)
Let Z → Y be another closed immersion. There is a long exact sequence · · · → H i X, DR(R0 ∗Z O X ) → H i X, DR(R0 ∗Y O X ) → H i X − Z , DR(R0 ∗Y −Z O X −Z ) → H i+1 X, DR(R0 ∗Z O X ) → · · · .
Proof Step 1: The rigid case. First assume that Y is locally defined by a single equation. Then the inclusion j : U = (X − Y ) → X of the complement is a quasi-Stein morphism, hence (see [22]) acyclic for coherent OU -modules. (Since we do not know if the analogue in the dagger case holds, we are forced to distinguish.) It follows • . On the other hand, R O (∗Y ) = O (∗Y ), and by [21, that DR(R j∗ OU ) = j∗ U X X • is an isomorphism. We get Theorem 2.3], the canonical map D R(O X (∗Y )) → j∗ U (a) for this type of Y . For general Y , assertion (a) is now deduced by an induction on the number of defining local equations, using the Mayer-Vietoris sequences from Sections 1.8 and 1.12. Assertion (b) follows from (a) and the fact that for every sheaf F on X we have a natural distinguished triangle +1
R0 Z F → R0 Y F → R j∗ R0 Y −Z j −1 F →, where j : (X − Z ) → X is the open immersion. Take an injective resolution I • of F ; then j −1 I • is an injective resolution of j −1 F , and 0 → 0 Z I • → 0 Y I • → j∗ 0 Y −Z j −1 I • → 0 is exact. Step 2: The dagger case. Again (b) follows from (a). For (a), first assume that Y is also smooth. As in the proof of Proposition 1, we find an affinoid admissible open S covering X = i∈I Ui such that for each i ∈ I either Ui ∩ Y is empty or there exists an isomorphism φi : Ui ∼ = Dm × (Ui ∩ Y ) such that Ui ∩ Y → Ui is the zero section. By a Cech argument, one sees that it is enough to prove that for all finite and T nonempty subsets J of I , if we set U J = i∈J Ui , the canonical map R0 U J , DR(R0 ∗Y O X ) → R0 U J , DR(R0 Y O X ) is an isomorphism. If U J ∩ Y is empty, this is trivial, so we assume that U J ∩ Y is nonempty. Choose one j ∈ J . For ∈ |k ∗ | ∩ ]0, 1], let D() be the closed disk of radius (with its dagger structure), and let m U j,J, = φ −1 j D () × (U J ∩ Y ) .
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The set of open subspaces U j,J, is cofinal in the set of all open neighbourhoods of U J ∩ Y in the affinoid space U j,J,1 . Since U J ∩ U j,J,1 is such a neighbourhood, we find an 0 such that U j,J,0 ⊂ U J . Now observe that the canonical restriction maps R0 U J , DR(R0 ∗Y O X ) → R0 U j,J,0 , DR(R0 ∗Y O X ) , R0 U J , DR(R0 Y O X ) → R0 U j,J,0 , DR(R0 Y O X ) are isomorphisms. Therefore we need to show that R0 U j,J,0 , DR(R0 ∗Y O X ) → R0 U j,J,0 , DR(R0 Y O X ) is an isomorphism. In other words, we may assume from the beginning that X = Dm × Y and that Y → X is the zero section. Let D = Dm ⊂ Pm k = (projective space) = P, let 0 be its origin, let W = P × Y , V = P − {0}, and think of Y = {0} × Y as embedded into W . Since the natural restriction maps R0 W, DR(R0 ∗Y OW ) → R0 X, DR(R0 ∗Y O X ) , R0 W, DR(R0 Y OW ) → R0 X, DR(R0 Y O X ) are isomorphisms, it suffices to show that R0 W, DR(R0 ∗Y OW ) → R0 W, DR(R0 Y OW ) is an isomorphism. The dagger spaces {0}, P, and V are partially proper; therefore Step 1(b) applies to give us the long exact Gysin sequence i−2m+1 i i · · · → Hdi−2m ({0}) → · · · . R ({0}) → Hd R (P) → Hd R (V ) → Hd R
By the K¨unneth formulas (in this case easily derived from Lemma 3), we thus obtain the long exact sequence i−2m+1 i i · · · → Hdi−2m (Y ) → · · · . R (Y ) → Hd R (W ) → Hd R (W − Y ) → Hd R
Because of Hd∗−2m (Y ) ∼ = H ∗ (W, DR(R0 ∗Y OW )), this implies what we want. R Now for arbitrary Y , we may as in Step 1 suppose that Y is defined by a single equation and that X is affinoid. Then we can reduce to the case where Y is a divisor with normal crossings as in [14], considering a proper surjective morphism g : X 0 → X with X 0 smooth, U 0 = g −1 (U ) → U an isomorphism, and g −1 (Y ) a divisor with normal crossings on X 0 (such a g exists by Lemma 1). But in view of the MayerVietoris sequences from Sections 1.8 and 1.12, the normal crossings divisor case is equivalent to the case where Y is smooth, which has been treated above.
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COROLLARY 1.13 Let f : X 0 → X be a proper morphism of dagger (or rigid) spaces, and let Y → X be a closed immersion such that f | X 0 −Y 0 : (X 0 − Y 0 ) → (X − Y ) is an isomorphism, where Y 0 = X 0 × X Y . Let b : X → Z be a closed immersion into a smooth space, and let a : X 0 → W 0 be a locally closed immersion into a smooth proper space of pure dimension n. Then (a, b ◦ f ) : X 0 → W 0 × Z = Z 0 is a closed embedding, and there is a long exact sequence · · · → H i Z 0 , DR(R0 ∗Y 0 O Z 0 ) → H i−2n Z , DR(R0 ∗Y O Z ) ⊕ H i Z 0 , DR(R0 ∗X 0 O Z 0 ) → H i−2n Z , DR(R0 ∗X O Z ) → H i+1 Z 0 , DR(R0 ∗Y 0 O Z 0 ) → · · · .
Proof By Proposition 3, there is a morphism between the acyclic complexes that we get when we apply Proposition 4(b) to Y 0 → X 0 → Z 0 and to Y → X → Z ; observe that H i Z 0 − Y 0 , DR(R0 ∗X 0 O Z 0 ) ∼ = H i W 0 × (Z − Y ), DR(R0 ∗X 0 O Z 0 ) for this, and that by the construction of the trace map in Proposition 3, it is indeed a morphism of complexes; that is, the resulting diagrams commute. Every third rung of this morphism of complexes is bijective (also by Proposition 3); therefore we can perform a diagram chase according to the pattern in the proof of [18, Proposition 4.3]. LEMMA 2 Let X 1 and X 2 be smooth dagger spaces, and let φ : X 10 → X 20 be an isomorphism of the associated rigid spaces. Then φ gives rise to an isomorphism φ † : Hd∗R (X 2 ) ∼ = Hd∗R (X 1 ).
Proof ˜ is Set X = X 1 × X 2 , X 0 = X 10 × X 20 , and φ˜ = (id, φ) : X 10 → X 0 . Then 1 = Im(φ) 0 0 a Zariski closed subspace of X isomorphic to X 1 . The canonical projections X 1 ← X → X 2 induce maps a1
a2
Hd∗R (X 1 ) −→ H ∗ (X, lim jV ∗ •V ) ←− Hd∗R (X 2 ), →
V
where in the middle the term V runs through the open immersions jV : V → X of dagger spaces with 1 ⊂ V 0 and where V 0 is the rigid space associated with V , regarded as an open subspace of X 0 . We claim that the ai are isomorphisms. The
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claim is local, so we may assume that X 1 and X 2 are affinoid and connected, that there are elements t1 , . . . , tm ∈ O X 1 (X 1 ) = A1 such that dt1 , . . . , dtm is a basis of 1X 1 (X 1 ) over A1 , and that there exist an open affinoid subspace U ⊂ X such that 1 ⊂ U 0 , where U 0 ⊂ X 0 is the associated rigid space, an element δ ∈ 0 ∗ , and an isomorphism of rigid spaces ρ : U 0 → Sp khδ −1 · Z 1 , . . . , δ −1 · Z m i × 1, where δ −1 · Z i is sent to δ −1 · (ti ⊗ 1 − 1 ⊗ (φ ∗ )−1 (ti )) ∈ OU 0 (U 0 ) (cf. [21, Theorem 1.18]). For 0 < ≤ δ, let U0 = ρ −1 Sp(kh −1 · Z 1 , . . . , −1 · Z m i) × 1 . This is an open subspace of X 0 , and we let U be the corresponding open subspace of X . Since U0 is a Weierstrass domain in X 0 , the same is true for U in X . (If necessary, modify the defining functions slightly to get overconvergent ones.) In particular, U is • = Rj • affinoid, so jU ∗ U U ∗ U . Since X is quasi-compact, lim → commutes with V formation of cohomology, and since the U are cofinal in {V }, it is now enough to show that for arbitrary 0 < ≤ δ the maps bi, : Hd∗R (X i ) → Hd∗R (U ) are isomorphisms (i = 1, 2). By [5], we can find an isomorphism σ : X 1 → X 2 such that the induced map σ 0 : X 10 → X 20 is close to φ, in particular, so close that for σ˜ = (id, σ ) : X 1 → X we ρ have Im(σ˜ ) ⊂ U/2 . Similarly, we can approximate the map O X 0 (X 10 ) ∼ = O1 (1) → 1 OU 0 (U 0 ) → OU0 (U0 ) by a map O X 1 (X 1 ) → OU (U ). Its extension to the map ∗
O X 1 (X 1 ) ⊗†k kh −1 · Z 1 , . . . , −1 · Z m i† → OU (U )
which sends −1 · Z i to −1 · (ti ⊗ 1 − 1 ⊗ (σ ∗ )−1 (ti )) is an isomorphism since its completion is close to the isomorphism obtained from ρ. So we have an isomorphism U ∼ = Sp(kh −1 · Z 1 , . . . , −1 · Z m i† ) × X 1 , where the closed immersion σ˜ : X 1 → U corresponds to the zero section. Hence the maps Hd∗R (U ) → Hd∗R (X 1 ) induced by σ˜ are isomorphisms, by Lemma 3. Since σ˜ is a section for the canonical map U → X 1 that gives rise to b1, , we derive the bijectivity of b1, . That b2, is bijective is seen symmetrically. Now we define φ † = a1−1 ◦ a2 . One can show that this construction is compatible with compositions: If X 3 is a third dagger space with associated rigid space X 30 , and if γ : X 20 → X 30 is an isomorphism, then φ † ◦ γ † = (γ ◦ φ)† (see [13]). We do not need this fact here.
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2. De Rham cohomology of tubes of a semistable reduction From now on, let R be a complete discrete valuation ring of mixed characteristic (0, p), and let π ∈ R be a uniformizer, k its fraction field, and k¯ = R/(π ) its residue field. 3 We have the following. (a) Let r, n ∈ Z, 0 ≤ r ≤ n, and let µ and γi for 1 ≤ i ≤ r and δi for 1 ≤ i ≤ n Q be elements of 0 ∗ such that γi ≤ δi for all 1 ≤ i ≤ r , and ri=1 δi ≥ µ. Define the open dagger subspace V of the dagger affine space Ank by LEMMA
r n Y V = (x1 , . . . , xn ) ∈ Ank ; |xi | ≥ µ, |xi | ≤ δi i=1
o for all 1 ≤ i ≤ n, and |xi | ≥ γi for all 1 ≤ i ≤ r . q
Let X 1 , . . . , X r be the first r standard coordinates on Ank . Then Hd R (V ) is the kvector space generated by the classes of the q-forms d X i1 / X i1 ∧ · · · ∧ d X iq / X iq q
with 1 ≤ i 1 < · · · < i q ≤ r . In particular, if r = 0, we have Hd R (V ) = 0 for all q > 0. If X is another smooth dagger space, the canonical maps M
q
q
β
q
Hd R1 (X ) ⊗k Hd R2 (V ) → Hd R (X × V )
q1 +q2 =q
are bijective. Q (b) Suppose even γi < δi for all 1 ≤ i ≤ r , and ri=1 δi < µ. Define the open dagger (resp., rigid) subspace V of the dagger (resp., rigid) affine space Ank by r n Y n V = (x1 , . . . , xn ) ∈ Ak ; |xi | > µ, |xi | < δi i=1
o for all 1 ≤ i ≤ n, and |xi | > γi for all 1 ≤ i ≤ r . Then the same assertions as in (a) hold. (Of course, if V is the dagger (resp., rigid) space, then X should be a dagger (resp., rigid) space, too.) Proof (a) Note that V is affinoid. We may assume that X is also affinoid and connected, X = Sp(B). After a finite extension of k, we may assume that there are δ i , γ i , and
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µ in k such that |δ i | = δ i , |γ i | = γ i , and |µ| = µ. We regard O X ×V (X × V ) as a subring of
−1 −1 −1 −1 −1 . B δ −1 1 · X 1 , (δ 1 · X 1 ) , . . . , δ n · X n , (δ n · X n ) (Computing in this ring was suggested by the referee.) (i) We begin with the following observation. Let a=
X
θv
v∈Zn
n Y
v
X j j ∈ O X ×V (X × V ),
j=1
θv ∈ B, θv = 0 whenever there is an r < j ≤ n with v j < 0. Fix 1 ≤ l ≤ n. We claim that X
b=
vl−1 θv
v∈Zn
n Y
v
X j j ∈ O X ×V (X × V ),
j=1
vl 6 =0
that is, that this sum also converges in O X ×V (X × V ). Indeed, consider the surjection of dagger algebras τ
D = B ⊗†k khX 1 , . . . , X n , Y1 , . . . , Yr , Z i† −→ O X ×V (X × V ), r −1 Y −1 −1 −1 −1 X i 7→ δi · X i , Yi 7→ (γ i · X i ) , Z 7→ µ · Xi . i=1
By the definition of ⊗†k , we have ˆ k khδ −1 · X 1 , . . . , δ −1 · X n , δ −1 · Y1 , . . . , δ −1 · Yr , δ −1 · Z i, D = lim O X (X )⊗ →
X ,δ
where the X run through the strict neighbourhoods of X 0 , the rigid space associated with X , in an appropriate affinoid rigid space that contains X 0 as a relatively compact open subset, and where δ runs through all δ > 1. Hence, a given c=
X w∈Zn+m+1 ≥0
βw
n Y j=1
wj
Xj
r Y
w j+n
Yj
Z wn+r +1 ∈ D
j=1
(βw ∈ B) is an element of ˆ k khδ −1 · X 1 , . . . , δ −1 · X n , δ −1 · Y1 , . . . , δ −1 · Yr , δ −1 · Z i O X (X )⊗
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for some X , δ, and one easily sees that d=
X
(wl − wl+n − wn+r +1 )−1 βw
n Y
wj
Xj
r Y
j=1
w∈Zn+m+1 ≥0
w j+n
Yj
Z wn+r +1
j=1
wl −wl+n −wn+r +1 6 =0
is then an element of ˆ k khδ1−1 · X 1 , . . . , δ1−1 · X n , δ1−1 · Y1 , . . . , δ1−1 · Yr , δ1−1 · Z i O X (X )⊗ for any 1 < δ1 < δ; in particular, d ∈ D, too. Clearly, if τ (c) = a, then τ (d) = b, and the claim follows. (ii) For 1 ≤ i 1 < · · · < i t ≤ n, we write d X i / X i = d X i1 / X i1 ∧ · · · ∧ d X it / X it , ˆ
(d X i / X i )it = d X i1 / X i1 ∧ · · · ∧ d X it−1 / X it−1 . q
Every ω ∈ X ×V (X × V ) can be uniquely written as a convergent series ω=
X
X
X
σt,i,v
0≤t≤q 1≤i 1 <···
n Y
v X j j d Xi / Xi
(∗)
j=1
q−t
with σt,i,v ∈ X (X ) and σt,i,v = 0 whenever there is a j with r < j ≤ n and q q v j ≤ 0. For 0 ≤ l ≤ n, let l ⊂ X ×V (X × V ) be defined by the additional condition that σt,i,v = 0 whenever l < i t or v j 6 = 0 for some j with l < j ≤ n. This condition means that no X j and no d X j / X j for l < j occur in ω. We claim q that if l > 0, every ω ∈ l with dω = 0 can be written modulo exact forms as q q−1 ω = ω0 + ω1 d X l / X l with ω0 ∈ l−1 , ω1 ∈ l−1 , dω0 = 0, dω1 = 0, and even ω1 = 0 if l > r . Indeed, if ω is represented as in (∗), then by (i) the series X
η=
X
0≤t≤q 1≤i 1 <···
X
vl−1 σt,i,v
v∈Zn
vl 6 =0
q−1
n Y
v ˆ X j j (d X i / X i )it
j=1
converges in l , and subtracting dη from ω, we see that we may suppose σt,i,v = 0 whenever i t = l and vl 6= 0. But dω = 0 implies σt,i,v = 0 whenever vl 6= 0 and i t < l, so in any case we have σt,i,v = 0 if only vl 6 = 0. From this the claim follows. q q (iii) Let ω ∈ X ×V (X × V ) = n be such that dω = 0. Repeated application of (ii) shows that modulo exact forms, ω can be written as X X ω= σt,i d X i / X i 0≤t≤q 1≤i 1 <···
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q−t
with uniquely determined σt,i ∈ X (X ) such that dσt,i = 0. This provides us with q an inverse map for β and proves the assertion on Hd R (V ). (b) Again we may assume that X is affinoid. In the dagger case, we then exhaust V by affinoid dagger spaces of the type considered in (a), and we conclude by using our result in (a) (passing to the limit). In the rigid case, X × V is quasi-Stein, hence acyclic for coherent modules (see [22]). Therefore, in this case, we can argue literally as in (a); since here V is defined by strict inequalities, we do not need overconvergence. 2.1 We call a closed immersion Z → X of Noetherian π -adic formal R-schemes a strictly semistable formal pair (X , Z ) over R if there are an n ∈ N, a Zariski open S covering X = i Ui , and for all i a pair s(i), r (i) ∈ N with n ≥ s(i) ≥ r (i) ≥ 1 and an e´ tale morphism qi : Ui → Spf RhX 1 , . . . , X n i/(X 1 . . . X r (i) − π ) S such that Z |Ui = j=r (i)+1,...,s(i) V (qi∗ X j ). We call X a strictly semistable formal R-scheme if (X , ∅) is a strictly semistable formal pair over R. 2.2 For a π-adic topologically finite type (tf) formal R-scheme X with generic fibre (see [7]) the rigid space Xk , there is a specialization map s : Xk → X , and if Y → Xk¯ is an immersion into the closed fibre, then ]Y [X = s −1 (X ) is an admissible open subspace of Xk , the tube of Y . 2.3 S Let X be a strictly semistable formal R-scheme, and let Xk¯ = i∈I Yi be the decomposition of the closed fibre into irreducible components. For K ⊂ I , set T Y K = i∈K Yi . Let X † be a k-dagger space such that its associated rigid space is identified with Xk . For a subscheme Y ⊂ Xk¯ , let ]Y [†X be the open dagger subspace of X † corresponding to the open rigid subspace ]Y [X of Xk . Then for any ∅ 6 = J ⊂ I , the canonical map i [ h† Hd∗R (]Y J [†X ) → Hd∗R Y J − Y J ∩ Yi THEOREM
i∈I −J
is bijective.
X
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Proof (i) Suppose that I 6= J . For L ⊂ (I − J ), set i [ h† G L = YJ − YJ ∩ Yi . i∈L
X
For m ∈ N, let Pm (I − J ) be the set of subsets of I − J with m elements, and let S G m = L∈Pm (I −J ) G L . Let G 0 =]Y J [†X . Then G m+1 ⊂ G m for all m ≥ 0, and i [ h† G |I −J | = Y J − Y J ∩ Yi . i∈I −J
X
It is enough to show that Hd∗R (G m ) → Hd∗R (G m+1 )
(∗)m
S is bijective for all m ≤ |I − J | − 1. Since G m = L∈Pm (I −J ) G L is an admissible covering, it is enough to show that \ \ (∗∗)m Hd∗R G L → Hd∗R G L ∩ G m+1 L∈Q
L∈Q
is bijective for all Q ⊂ Pm (I − J ). But \ L∈Q
G L ∩ G m+1 =
T
L∈Q
[ i∈(I −(M∪J ))
G L = G M for M =
S
L∈Q
L, and
G M − (G M ∩ ]Yi [†X ) .
If X is now replaced by its open formal subscheme that is the complement of S i∈M Yi in Xk¯ on the underlying topological space, then this means the replacement of ]Y J [†X by G M and of I by I − M (but J stays the same). In this way, (∗∗)m takes the form (∗)0 ; therefore it suffices to prove (∗)0 . Note that G 1 =]Y J [†X −]Y I [†X ; that is, we must prove that Hd∗R (]Y J [†X ) → Hd∗R (]Y J − Y I [†X ) is bijective. S (ii) Let X = s∈S Us be an open covering of X as in the definition of strict T semistability. For a finite nonempty subset T of S, let UT = s∈T Us . It is enough to show that for each such T the map Hd∗R ]Y J ∩ UT [†UT → Hd∗R ](Y J − Y I ) ∩ UT [†UT is bijective. This is trivial if Y I ∩ UT is empty. If Y I ∩ UT is nonempty, the irreducible components of the reduction (UT )k¯ of UT correspond bijectively to those of Xk¯ ,
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so we can replace X by UT . In other words, it is enough to prove the statement in (i) in the following case: X = Spf(A) is affine, and there is an e´ tale morphism φ
X = Spf(A) → Spf(RhX 1 , . . . , X n i/(X 1 . . . X r − π)). Let f i = φ ∗ (X i ) ∈ A, inducing f¯i ∈ A/(π). Passing to an open covering of X , we may suppose that each V ( f¯i ) is irreducible (and nonempty), so we derive an identification {1, . . . , r } = I . For λ ∈ 0 ∗ with λ < 1, set Fλ = x ∈ G 0 ; | f j (x)| ≤ λ for all j ∈ J S and E λ = Fλ ∩ G 1 . Then G 0 = λ<1 Fλ is an admissible covering, and to prove (∗)0 it suffices to show the bijectivity of Hd∗R (Fλ ) → Hd∗R (E λ ) for all such λ. (iii) For β ∈ 0 ∗ with β < 1 and i ∈ I − J , set [ i i Fλ,β = x ∈ Fλ ; | f i (x)| ≥ β and Fλ,β = Fλ,β . i∈I −J
We have E λ ⊂ Fλ,β ⊂ Fλ , and it suffices now to prove that the following maps are bijective: lim Hd∗R (Fλ,β ) → Hd∗R (E λ ),
(1)
→
β→1
Hd∗R (Fλ ) → Hd∗R (Fλ,β )
for β < 1.
(2)
Note that G {i} = ]Y J − (Y J ∩ Yi )[†X = {x ∈]Y J [†X ; | f i (x)| = 1} for i ∈ I − J . S We compare the admissible covering E λ = i∈I −J (G {i} ∩ Fλ ) with the admissible S i . Since the direct limit is exact, to prove the bijectivity covering Fλ,β = i∈I −J Fλ,β of (1) it is enough to prove the bijectivity of \ \ i lim Hd∗R Fλ,β → Hd∗R G {i} ∩ Fλ (3) →
β→1
i∈K
i∈K
T for all ∅ 6= K ⊂ (I − J ). But the affinoid dagger space i∈K G {i} ∩ Fλ is the inverse T i ; in particular, limit of the affinoid dagger spaces i∈K Fλ,β \ \ i 0 G {i} ∩ Fλ , • = lim 0 Fλ,β , • , →
i∈K
β→1
i∈K
so the bijectivity of (3) follows from the exactness of direct limits. (iv) It remains to prove the bijectivity of the maps (2). Set Sλ,β = x ∈ Fλ ; | f i (x)| ≤ β for all i ∈ I − J . Then Fλ = Sλ,β ∪ Fλ,β is an admissible covering, and the bijectivity of (2) is equivalent to that of Hd∗R (Sλ,β ) → Hd∗R (Sλ,β ∩ Fλ,β ). (4)
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(v) We claim that there is a π-adic affine formally smooth tf formal Spf(R)scheme Spf(D) and an isomorphism of rigid spaces ]Y I [X = V ( f¯i )i=1,...,r X ∼ = Sp(D ⊗ R k) × x ∈ Sp(khT1 , . . . , Tr i/(T1 . . . Tr − π)); |Ti (x)| < 1 for all i such that f i ∈ A corresponds to Ti . This is constructed as follows. Let Aˆ be the ˆ ( f 1 , . . . , fr )-adic completion of A. Since A/(π, f 1 , . . . , fr ) = A/(π, f 1 , . . . , fr ) = s¯ ¯ ˆ for the canonical D¯ is a smooth k-algebra (φ is e´ tale), we have a section D¯ → A/(π) ˆ ¯ a lift of D¯ to a smooth R-algebra D˜ (see [10]), and a lift surjection A/(π ) → D, s ˆ where D is the π-adic completion of D. ˜ The extension of s¯ to a morphism D → A, ˆ Ti 7 → f i of s induces an isomorphism D[[T1 , . . . , Tr ]] → A, ˆ D[[T1 , . . . , Tr ]]/(T1 , . . . , Tr − π ) ∼ = A. This gives the desired isomorphism of rigid spaces (compare [1, Proposition 0.2.7], for the computation of tubes). (vi) To prove the bijectivity of (4), we may now, in view of Lemma 2, assume that there is a smooth k-dagger algebra B and an isomorphism of dagger spaces † ]Y I [†X = V ( f¯i )i=1,...,r X ∼ = Sp(B) × x ∈ Sp(khT1 , . . . , Tr i† /(T1 . . . Tr − π )); |Ti (x)| < 1 for all i such that f i ∈ A corresponds to Ti . Let N = {x ∈ Sλ,β ; | f i (x)| = β for all i ∈ I − J }. It suffices to show the bijectivity of the maps Hd∗R (Sλ,β ) → Hd∗R (N ), Hd∗R (Sλ,β
∩ Fλ,β ) →
Hd∗R (N ).
(5) (6)
The bijectivity of (5) follows immediately from Lemma 3. Finally, consider the adS i ). To prove the bijectivity of missible covering Sλ,β ∩ Fλ,β = i∈I −J (Sλ,β ∩ Fλ,β (6), it is enough to prove that of \ i Hd∗R Sλ,β ∩ Fλ,β → Hd∗R (N ) (7) i∈K
for all ∅ 6 = K ⊂ (I − J ), which again can be done using Lemma 3. THEOREM 2.4 In Theorem 2.3, suppose in addition that X is quasi-compact. Then for every J ⊂ I q q and q ∈ N, one has dimk (Hd R (]Y J [†X )) < ∞ and also dimk (Hd R (X † )) < ∞.
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Proof S Since X † = i∈I ]Yi [†X is a finite admissible covering, the second claim follows from the first. For the first we may, due to Theorem 2.3, assume that J = I , shrinking X if necessary. Passing to a finite covering, we may assume that X is affine and that φ
there is an e´ tale morphism X → Spf(RhX 1 , . . . , X n i/(X 1 . . . X r − π )) such that T T Y I = i∈I Yi = ri=1 V ( f i ) with f i = φ ∗ (X i ). From the proof of Theorem 2.3, we ˜ see that we may assume, again due to Lemma 2, that there is a smooth R-algebra D, † † with weak completion D , such that if B denotes the k-dagger algebra D ⊗ R k, we have an isomorphism of dagger spaces † ]Y I [†X = V ( f¯i )i=1,...,r X ∼ Sp(B) × x ∈ Sp(khT1 , . . . , Tr i† /(T1 . . . Tr − π )); |Ti (x)| < 1 for all i = such that f i corresponds to Ti . By Lemma 3 this yields isomorphisms M q q q q Hd R (]Y I [†X ) ∼ Hd R1 Sp(B) ⊗k Hd R2 (V ), = Hd R Sp(B) × V ∼ = q=q1 +q2
where we write n
V = (x1 , . . . , xr −1 ) ∈
Ark−1 ;
rY −1
o |xi | > |π|, |xi | < 1 for all 1 ≤ i ≤ r − 1 .
i=1 ∗ (Y ), and the latter ˜ Now since Y I = Spec( D/(π )), we have Hd∗R (Sp(B)) = HM W I is known to be finite-dimensional by [2] and [24]. But by Lemma 3 also Hd∗R (V ) is finite-dimensional. We are done.
LEMMA 4 Let n ≥ s ≥ r ≥ 1, let
q : Spf(A) → Spf RhX 1 , . . . , X n i/(X 1 . . . X r − π ) be an e´ tale morphism, and let B = A/(q ∗ X j ) j=r +1,...,s . There is an isomorphism Spec B/(π ) Spf(A) ∼ = Sp(B ⊗ R k) × (D0 )s−r such that the standard coordinates on (D0 )s−r correspond to q ∗ X r +1 , . . . , q ∗ X s . Proof A strictly semistable formal R-scheme X carries a canonical log structure: the log structure MX associated with the divisor X , the reduction modulo (π ) of X . In particular, Spf(R) gives rise to a formal log scheme S, and (X , MX ) → S is a log
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smooth morphism of formal log schemes. In our situation, Spf(A) and Spf(B) are strictly semistable formal R-schemes, and the embedding (U , MU ) → (V , MV ) is an exact closed immersion of formal S-schemes. Now we prove Lemma 4, using the above log structures. Because of [15, Chapitre IV, Lemme 18.3.2.1], we may work over the truncations mod (π n ). Due to the extension property of morphisms from exact nilimmersions to log smooth objects provided by [20, Corollary 3.11], which is analogous to the classical extension property of morphisms from nilimmersions to smooth objects, one easily verifies the following transposition of [15, Chapitre 0, Corollaire 19.5.4] to the log context. Let g
f
(Spec(B), M 0 ) → (Spec(A), M) → (Spec(C), N ) be morphisms of affine log schemes such that g is an exact closed immersion defined by the ideal I ⊂ A and such that f and f ◦ g are log smooth. Then I /I 2 is a projective B-module, and if Aˆ (resp., Sˆ) is the respective I -adic completion of A (resp., Sym B (I /I 2 )), then ˆ there is a section B → Aˆ together with an isomorphism of B-algebras Sˆ ∼ = A. The lemma follows because by the method of [1, Proposition 0.2.7], for computing tubes all we have to do is to construct an R-isomorphism B[[T1 , . . . , Ts−r ]] ∼ = Aˆ with ∗ Ti 7 → q X i+r , where Aˆ is the Ker(A → B)-adic completion of A.
3. The finiteness theorem 3.1 We recall the terminology from [9]. An R-variety is an integral separated flat Rscheme of finite type. Let g : X → Spec(R) be an R-variety, and let X i , i ∈ I , be the T irreducible components of the closed fibre X k¯ . For J ⊂ I , set X J = i∈J X i . Then X is called strictly semistable over R if the following four conditions are fulfilled. (1) The generic fibre X k is smooth over k. S (2) X k¯ is reduced; that is, X k¯ = i∈I X i scheme theoretically. (3) X i is a divisor on X for all i ∈ I . (4) X J is smooth over k¯ of codimension |J | for all J ⊂ I , J 6= ∅. Let Z ⊂ X be Zariski closed with its reduced structure, such that X k¯ ⊂ Z . Then (X, Z ) is called a strictly semistable pair over R if the following three conditions are fulfilled. (1) X is strictly semistable over R. (2) Z is a divisor with normal crossings on X . S (3) Decompose Z = Z f ∪ X k¯ with Z f → Spec(R) flat, and let Z f = i∈K Z i T be the decomposition into irreducible components. Then Z L = i∈L Z i is a union of strictly semistable R-varieties for all L ⊂ K , L 6 = ∅.
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Note that for a strictly semistable pair (X, Z ) over R, the π-adic completion of Z f → X is a strictly semistable formal pair over R in the sense of Section 2.1. 3.2 We call a dagger space H quasi-algebraic if there is an admissible covering of H by dagger spaces U that admit an open embedding into the dagger analytification of a k-scheme of finite type. If H is quasi-algebraic, there is even an admissible covering of H by open affinoids U such that for each U there are n, r ∈ N, polynomials f j ∈ k[X 1 , . . . , X n ], and isomorphisms U ∼ = Sp(khX 1 , . . . , X n i† /( f 1 , . . . , fr )) (see [11, Section 2.18]). THEOREM 3.3 ([9, Theorem 6.5]) If Y is a proper R-variety and Z ⊂ Y is a proper Zariski closed subset with Yk¯ ⊂ Z , then there is a finite extension R → R 0 of complete discrete valuation rings, an R 0 variety X , a proper surjective morphism of R-schemes f : X → Y , and an open dense subscheme U ⊂ Y such that f −1 (U ) → U is finite and (X, f −1 (Z )red ) is a strictly semistable pair over R 0 . THEOREM 3.4 For an affinoid quasi-algebraic dagger space H , the numbers h qd R (H ) are finite for all q ≥ 0.
Proof We proceed by induction on m = dim(H ). We may suppose that H = Sp khX 1 , . . . , X n i† /( f 1 , . . . , fr ) with polynomials f j ∈ R[X 1 , . . . , X n ], and we set V = Spec(R[X 1 , . . . , X n ]/ ( f 1 , . . . , fr )). We regard H as an open subspace of the dagger analytification of the generic fibre Vk of V . The decomposition of V into irreducible components induces a decomposition of H into Zariski closed quasi-algebraic subspaces. Therefore we can reduce our claim by means of Section 1.8(c) and the induction hypothesis to the case where V is irreducible, m = dim(Vk ). Since h qd R (H ) depends only on the reduced structure of H , we may assume that V is reduced and hence integral. Choose an open immersion V → Y into a projective R-variety Y . For the pair (Y, Yk¯ ), choose R 0 and f : X → Y and U ⊂ Y as in Theorem 3.3. Since h qd R (H ) gk
is not affected by finite extensions of k, we may suppose that R = R 0 . Let X k → πk X k0 → Yk be the Stein factorization of the map f k : X k → Yk of generic fibres; note that X k0 is integral. From [15, Chapitre III, Proposition 4.4.1] it follows that there is a closed subscheme T 0 ⊂ X k0 , dim(T 0 ) < m, such that, for T = gk−1 (T 0 ),
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gk | X k −T : (X k − T ) → (X k0 − T 0 ) is an isomorphism. Since the locus of smoothness over k is dense in X k0 and in Yk and since πk is finite, we find a closed subscheme S ⊂ Yk , dim(S) < m, such that, for S 0 = X k0 ×Yk S, the open subschemes Yk − S and X k0 − S 0 are smooth over k and (X k0 − S 0 ) → (Yk − S) is e´ tale. For a k-scheme L of finite type, we denote by L † its analytification as a dagger space. We regard H as an open subspace of Yk† and set H X = H ×Y † X k† , k
H X 0 ,S 0 = H X 0 × X 0† S , 0†
k
H X 0 = H ×Y † X k0† , k
H X,T = H X × X † T , †
k
H S = H ×Y † S † , k
H X 0 ,T 0 = H X 0 × X 0† T 0† . k
As auxiliary data we choose closed embeddings H → N (resp., H X 0 → M) into smooth dagger spaces of pure dimension n (resp., h). R (H ). By Proposition 4 there is We have dimk H i (N , DR(R0 ∗H O N )) = h d2n−i a long exact sequence · · · → H i N , DR(R0 ∗HS O N ) → H i N , DR(R0 ∗H O N ) → H i N − HS , DR(R0 ∗(H −HS ) O N −HS ) → H i+1 . . . . R (H ) are finite by The numbers dimk H i (N , DR(R0 ∗HS O N )) = h d2n−i S the induction hypothesis, so it is enough to show that dimk H i (N − HS , DR(R0 ∗(H −HS ) O N −HS )) < ∞, which by smoothness of H − HS is equivalent to dimk (Hdi−2l R (H − HS )) < ∞ (where l = codim N (H )). The canonical maps j j Hd R (H − HS ) → Hd R (H X 0 − H X 0 ,S 0 ) are injective (see Section 1.11), so it is enough j to show that dimk (Hd R (H X 0 − H X 0 ,S 0 )) < ∞. Since H X 0 − H X 0 ,S 0 is smooth, one has j Hd R (H X 0 − H X 0 ,S 0 ) ∼ = H j+2t M − H X 0 ,S 0 , DR(R0 ∗(H X 0 −H X 0 ,S0 ) O M−H X 0 ,S0 ) (where t = codim M (H X 0 )). By Proposition 4 there is a long exact sequence · · · → H i M, DR(R0 ∗H X 0 ,S0 O M ) → H i M, DR(R0 ∗H X 0 O M ) → H i M − H X 0 ,S 0 , DR(R0 ∗(H X 0 −H X 0 ,S0 ) O M−H X 0 ,S0 ) → H i+1 . . . . R (H 0 0 ), a finite number by the Again, dimk H i (M, DR(R0 ∗H X 0 ,S0 O M )) = h d2h−i X ,S i induction hypothesis. It remains to show that dimk H (M, DR(R0 ∗H X 0 O M )) < ∞. Set P = X k† × M, and consider the closed immersion H X → P. From Corollary 1.13 we get (with b = dim(X k )) an exact sequence · · · → H i P, DR(R0 ∗H X,T O P ) → H i−2b M, DR(R0 ∗H X 0 ,T 0 O M ) ⊕ H i P, DR(R0 ∗H X O P ) → H i−2b M, DR(R0 ∗H X 0 O M ) → H i+1 P, DR(R0 ∗H X,T O P ) → · · · .
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Here dimk H 2h− j (M, DR(R0 ∗H X 0 ,T 0 O M )) = h dj R (H X 0 ,T 0 ) is finite by the induction hypothesis. Furthermore, H 2h+i P, DR(R0 ∗H X,T O P ) = H i H X , DR(R0 ∗H X,T O H X ) since H X is smooth, and for all open affinoid U ⊂ H X R dimk H i (U, DR(R0 ∗H X,T O H X )) = h d2b−i (U ∩ H X,T ), a finite number by the induction hypothesis. Since H X is quasi-compact, it follows that dimk H i (H X , DR(R0 ∗H X,T O H X )) < ∞. So we are left with showing that dimk H 2h+i (P, DR(R0 ∗H X O P )) = dimk (Hdi R (H X )) is finite. By construction, the rigid space associated with H is the generic fibre of an open formal subscheme of the π -adic completion of Y . It follows that the rigid space associated with H X is the generic fibre of an open formal subscheme of the π -adic completion of X . Thus dimk (Hdi R (H X )) < ∞ follows from Theorem 2.4. COROLLARY 3.5 If X is a smooth quasi-compact dagger space and if i : Z → X is a closed immersion, then the k-vector spaces Hdi R (X − Z ) are finite-dimensional.
Proof We may assume that X is affinoid. Choose a proper surjective morphism f : X˜ → X such that X˜ is smooth, Z˜ = f −1 (Z ) is a divisor with normal crossings on X˜ , and ( X˜ − Z˜ ) → (X − Z ) is an isomorphism (see Lemma 1). Passing to an appropriate finite affinoid admissible open covering of X˜ , we see that we may assume from the beginning that X is affinoid and that Z is a normal crossings divisor on X such that all its irreducible components are smooth. Now note that if W = X or if W is the intersection of some irreducible components of Z , then W is quasi-algebraic. Indeed, since it is smooth and affinoid, it follows from [10, Theorem 7] that the associated rigid space W 0 can be defined by polynomials. In particular, W 0 is the rigid space associated to a quasi-algebraic affinoid dagger space W1 , and by [11, Lemma 1.15] there exists a (noncanonical) isomorphism W ∼ = W1 . Now Theorem 3.4 says q that H (X, DR(R0 ∗W O X )) is finite-dimensional for all q and all such W . Repeated application of Proposition 4 gives the corollary. THEOREM 3.6 Let X be a quasi-compact smooth dagger space, and let U ⊂ X be a quasi-compact q admissible open subset. Then Hd R (X − U ) is finite-dimensional for all q ∈ N.
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Proof We prove the following by induction on n ∈ N. (an ) For every quasi-compact smooth dagger space X with dim(X ) ≤ n, every q quasi-compact open U ⊂ X , and every q ∈ N, we have dimk (Hd R (X − U )) < ∞. (bn ) For every quasi-compact smooth dagger space Y , every closed immersion T ,→ Y with dim(T ) ≤ n, every quasi-compact open V ⊂ Y , and every q ∈ N, we have dimk H q (Y − V, DR(R0 ∗T OY )) < ∞. Here (a0 ) is evident, and so is (b0 ), because of Proposition 2. (bn−1 ) ⇒ (an ) We may suppose that dim(X ) = n and that X is affinoid and connected. Since U is the union of finitely many rational subdomains of X , MayerVietoris sequences allow us to reduce to the case where U is a rational subdomain of X . Choose an affine formal R-scheme X such that its generic fibre Xk is the rigid space associated with X . For subschemes Z ⊂ Xk¯ , we denote by ]Z [†X the open dagger subspace of X corresponding to ]Z [X ⊂ Xk . By [7], after an admissible blowing up and further localization, we may suppose that there is a closed subscheme Z ⊂ Xk¯ such that ]Z [†X = X −U . From [10, Theorem 7], it follows that X is locally defined by polynomials; that is, we may suppose that X = Spf RhX 1 , . . . , X m i/( f 1 , . . . , fr ) with f i ∈ R[X 1 , . . . , X m ]. We view X as an open subspace of the dagger analytification of the generic fibre Tk of T = Spec(R[X 1 , . . . , X m ]/( f 1 , . . . , fr )). Since X is smooth, it is contained in the dagger analytification of a single irreducible component of Tk . This component is the generic fibre of a closed subscheme of T ; dividing out the π-torsion and the nilpotent elements of its coordinate ring, we see that we may suppose that T is integral. We may also suppose that Z is defined by a single equation in Tk¯ = Xk¯ . We choose a closed subscheme Y ⊂ T defined by a single equation such that Yk¯ = Z . Furthermore, we choose an open embedding T → X¯ into a projective R-variety X¯ , and we define Y¯ to be the schematic closure of Y in X¯ . Since (an ) is proven in the case U = ∅ by Theorem 3.4, we may suppose that Y¯ ∪ X¯ k¯ is a proper subset of X¯ . Therefore we can apply Theorem 3.3 to ( X¯ , Y¯ ∪ X¯ k¯ ). Performing a base change with a finite extension of R, we may assume that there is a surjective proper morphism φ¯ : V¯ → X¯ of R-varieties such that there is an open dense subscheme of X¯ over which φ¯ is finite, and such that (V¯ , φ¯ −1 (Y¯ ∪ X¯ k¯ )red ) is a strictly semistable pair over R. Let V¯ (resp., W¯ , resp., X¯ ) be the π-adic formal completion of V¯ (resp., φ¯ −1 (Y¯ )red , resp., X¯ ), and set V = V¯ ×X¯ X and W = W¯ ×V¯ V . Let V¯k† be the dagger analytification of the generic fibre of V¯ , and let V be its open dagger subspace whose associated rigid space is Vk . As before, for subschemes S ⊂ Vk¯ , we denote by ]S[†V the open dagger subspace of V corresponding to ]S[V ⊂ Vk . Now φ¯ induces a morphism φk : V → X of dagger spaces, and if F = X − U , we have ]Wk¯ [†V = φk−1 (F).
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CLAIM
It is enough to show that dimk (Hd∗R (]Wk¯ [†V )) < ∞ for all q ∈ N. Using the induction hypothesis (bn−1 ), this can be shown in a way similar to the f
g
proof of Theorem 3.4. Let V → D → X be the Stein factorization of φk (for example, ¯ Let Q = g −1 (F), obtained from the Stein factorization of the algebraic morphism φ). let D → P 0 be a closed immersion into a smooth affinoid dagger space P 0 , and let P ⊂ P 0 be an open subspace such that P 0 − P is quasi-compact and open in P 0 and such that Q → P 0 factorizes through a closed immersion Q → P. Let T → F be a closed immersion with dim(T ) < n, such that for L = g −1 (T ) we have that Q − L is smooth and (Q − L) → (F − T ) is e´ tale. By Proposition 4 there are long exact sequences · · · → H i F, DR(R0 ∗T O F ) → Hdi R (F) → Hdi R (F − T ) → H i+1 F, DR(R0 ∗T O F ) → · · · and · · · → H i P, DR(R0 ∗L O P ) → H i P, DR(R0 ∗Q O P ) → H i P − L , DR(R0 ∗Q−L O P−L ) → H i+1 P, DR(R0 ∗L O P ) → · · · . Now H i (F, DR(R0 ∗T O F )) and H i (P, DR(R0 ∗L O P )) are finite-dimensional by the induction hypothesis (bn−1 ). Since Q − L is smooth, we have H i P − L , DR(R0 ∗Q−L O P−L ) ∼ = Hdi−2l R (Q − L) q
(where l = codim P (Q)), and by Section 1.11 the canonical maps Hd R (F − T ) → q Hd R (Q − L) are injective. q Together we obtain that to prove that dimk (Hd R (F)) < ∞ it is enough to prove that dimk H q (P, DR(R0 ∗Q O P )) < ∞ for all q ∈ N. Now choose a closed immersion G → Q with dim(G) < n such that for H = G× Q ]Wk¯ [†V we have that (]Wk¯ [†V −H ) → (Q − G) is an isomorphism. Let E = V¯k† × P with ]Wk¯ [†V diagonally embedded. By Corollary 1.13 there is a long exact sequence · · · → H i E, DR(R0 ∗H O E ) → H i−2n P, DR(R0 ∗G O P ) ⊕ H i E, DR(R0 ∗]W [† O E ) k¯ V → H i−2n P, DR(R0 ∗Q O P ) → H i+1 E, DR(R0 ∗H O E ) → · · · . We have dimk H i (E, DR(R0 ∗H O E )) < ∞ and dimk H j (P, DR(R0 ∗G O P )) < ∞ by the induction hypothesis (bn−1 ). On the other hand, † H i E, DR(R0 ∗]W [† O E ) = Hdi−2n R (]Wk¯ [V ), k¯ V
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and altogether the claim follows. q Now we prove that dimk (Hd R (]Wk¯ [†V )) < ∞. Note that W is the π-adic formal −1 completion of φ¯ (Y )red . Let W = W f ∪ W0 be the decomposition into the R-flat part W f and the π-torsion part W0 . Since Y is defined by a single equation in T , the same is true for φ¯ −1 (Y )red in φ¯ −1 (T )red . Since (V¯ , φ¯ −1 (Y¯ ∪ X¯ k¯ )red ) is a strictly semistable pair over R, this means that φ¯ −1 (Y )red is the union of some irreducible components of φ¯ −1 (Y ∪Tk¯ )red . After passing to a finite Zariski open covering of V , we may therefore suppose that there is an e´ tale morphism ψ V → Spf RhX 1 , . . . , X m i/(X 1 . . . X r − π ) for some m ≥ r ≥ 1, and if we set W i = V (ψ ∗ X i ) for i ≤ m, there are subsets J ⊂ S S {r + 1, . . . , m} and N ⊂ {1, . . . , r } such that W f = i∈J W i and W0 = i∈N W i . S The covering ]Wk¯ [†V = i∈J ∪N ]Wk¯i [†V is admissible, so it is enough to show that for all I ⊂ J , all M ⊂ N , all q ∈ N, we have \ q i † dimk Hd R ]Wk¯ [V < ∞. i∈I ∪M i Set R = i∈I W , a strictly semistable formal R-scheme. By construction, equations defining the closed immersion R → V are contained in OV ; they fine a Zariski closed dagger subspace of V whose associated rigid space is generic fibre of R . As before, we define its open subspaces ]S[†R for S ⊂ Rk¯ . T i C = Rk¯ ∩ i∈M W . We claim that \ q q q † ]Wk¯i [†V = Hd R (]C[†V ) ∼ Hd R = Hd R (]C[R ).
T
the dethe Let
i∈I ∪M
Indeed, from Lemma 4 we derive an isomorphism ]C[R ×(D0 )|I | ∼ =]C[V of rigid spaces. Because of Lemma 2, to prove our claim we may therefore assume that there is an isomorphism of dagger spaces † ]C[†R ×(D0 )|I | ∼ =]C[V , q
and then the claim is obvious. But Hd R (]C[†R ) is finite-dimensional by Theorem 2.4, because C is the intersection of some irreducible components of Rk¯ . (an ) + (bn−1 ) ⇒ (bn ) Passing to a finite affinoid covering of Y , we may suppose f
by Lemma 1 that there is a smooth P and a proper surjective P → Y such that S = f −1 (T ) is a divisor with normal crossings on P and (P − S) → (Y − T ) is an isomorphism. As in (bn−1 ) ⇒ (an ), one shows that it suffices to prove that dimk H q ( f −1 (Y − V ), DR(R0 ∗S O P )) < ∞, so we suppose that T is a divisor with normal crossings
FINITENESS OF DE RHAM COHOMOLOGY IN RIGID ANALYSIS
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on Y . But then, in view of Section 1.8(c), the problem is equivalent to the one where T is smooth (of arbitrary codimension). Thus it is reduced by means of Section 1.8 and Proposition 2 to the induction hypothesis (an ). 3.7 As an application of Theorem 3.6, we will show in [13] that the de Rham cohomology groups of smooth rigid Stein spaces are topologically separated for their canonical topology and hence are Fr´echet spaces. We also define reasonable de Rham cohomology groups for arbitrary rigid spaces (the underlying idea is that of Lemma 2) and derive from Theorem 3.6 K¨unneth and duality formulas for them. COROLLARY 3.8 q ¯ For a k-scheme Y of finite type, the k-vector spaces Hrig (Y/k) are finite-dimensional for all q ∈ N.
Proof Since Y is of finite type, we may assume, after passing to a finite open covering, that Y is affine. Let X be a proper smooth π-adic tf formal Spf(R)-scheme, and let Y → Xk¯ be an immersion with schematic closure j : Y → Y¯ in Xk¯ . Then ]Y¯ [X is a partially proper rigid space and is therefore equivalent to a dagger space Q. Let X be the open subspace of Q whose underlying set is identified with ]Y [X . q q q From [11, Theorem 5.1] we get an isomorphism Hrig (Y/k) ∼ = Hd R (X ), but Hd R (X ) is finite-dimensional by Theorem 3.6. Acknowledgments. I wish to thank Peter Schneider, who patiently read early drafts of my attempts on this subject and gave helpful comments. I am also very grateful to Pierre Berthelot for an invitation to the University of Rennes I, for discussions of decisive importance during this visit, and for further remarks on the text later on. I thank Annette Huber-Klawitter and Klaus K¨unnemann for answering various questions for me. Finally, I thank the referee for careful reading and helpful advice. References [1]
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P. BERTHELOT, Cohomologie rigide et cohomologie rigide a` supports propres,
Premi`ere partie, pr´epublication IRMAR 96 – 03, Universit´e de Rennes, 1996. 80, 82 , Finitude et puret´e cohomologique en cohomologie rigide, Invent. Math. 128 (1997), 329 – 377. MR 98j:14023 57, 58, 59, 81 P. BEYER, On Serre-duality for coherent sheaves on rigid-analytic spaces, Manuscripta Math. 93 (1997), 219 – 245. MR 99b:32009 58, 66
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E. BIERSTONE and P. D. MILMAN, Canonical desingularization in characteristic zero
by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), 207 – 302. MR 98e:14010 59, 61 S. BOSCH, A rigid analytic version of M. Artin’s theorem on analytic equations, Math. Ann. 255 (1981), 395 – 404. MR 82k:32067 73 ¨ S. BOSCH, U. GUNTZER, and R. REMMERT, Non-Archimedean Analysis, Grundlehren Math. Wiss. 261 (1984), Springer, Berlin, 1984. MR 86b:32031 59, 60 ¨ S. BOSCH and W. LUTKEBOHMERT , Formal and rigid geometry, I: Rigid spaces, Math. Ann. 295 (1993), 291 – 317. MR 94a:11090 77, 86 B. CHIARELOTTO, “Duality in rigid analysis” in p-adic Analysis (Trento, Italy, 1989), Lecture Notes in Math. 1454, Springer, Berlin, 1990, 142 – 172. MR 92d:32015 58 ´ A. J. DE JONG, Smoothness, semi-stability and alterations, Inst. Hautes Etudes Sci. Publ. Math. 83 (1996), 51 – 93. MR 98e:14011 82, 83 R. ELKIK, Solutions d’´equations a` coefficients dans un anneau hens´elien, Ann. Sci. ´ Ecole Norm. Sup. (4) 6 (1973), 553 – 603. MR 49:10692 80, 85, 86 ¨ E. GROSSE-KLONNE , Rigid analytic spaces with overconvergent structure sheaf, J. Reine Angew. Math. 519 (2000), 73 – 95. MR 2001b:14033 57, 58, 59, 60, 61, 67, 83, 85, 89 , DeRham-Kohomologie in der rigiden Analysis, Preprintreihe SFB 478, Universit¨at M¨unster, Heft 39 (1999). 61, 63 , De Rham cohomology of rigid spaces, in preparation. 73, 89 A. GROTHENDIECK, On the de Rham cohomology of algebraic varieties, Inst. Hautes ´ Etudes Sci. Publ. Math. 29 (1966), 95 – 103. MR 33:7343 71 A. GROTHENDIECK AND J. DIEUDONNE´ , El´ements de g´eom´etrie alg´ebrique, I – IV, ´ Inst. Hautes Etudes Sci. Publ. Math. 4, 8, 11, 17, 20, 24, 28, 32 (1960) – (1967). MR 29:1207, MR 29:1208, MR 29:1209, MR 29:1210, MR 30:3885, MR 33:7330, MR 36:178, MR 39:220 61, 82, 83 ¨ U. GUNTZER , Modellringe in der nichtarchimedischen Funktionentheorie, Indag. Math. 29 (1967), 334 – 342. MR 37:470 59 R. HARTSHORNE, Residues and Duality, Lecture Notes in Math. 20, Springer, Berlin, 1966. MR 36:5145 63 ´ , On the de Rham cohomology of algebraic varieties, Inst. Hautes Etudes Sci. Publ. Math. 45 (1975), 5 – 99. MR 55:5633 58, 59, 69, 72 ´ R. HUBER, Etale Cohomology of Rigid Analytic Varieties and Adic Spaces, Aspects Math. E30, Vieweg, Braunschweig, Germany, 1996. MR 2001c:14046 60, 61, 62, 63 K. KATO, “Logarithmic structures of Fontaine-Illusie” in Algebraic Analysis, Geometry, and Number Theory, Johns Hopkins Univ. Press, Baltimore, 1989, 191 – 224. MR 99b:14020 82 R. KIEHL, Die de Rham Kohomologie algebraischer Mannigfaltigkeiten u¨ ber einem ´ bewerteten K¨orper, Inst. Hautes Etudes Sci. Publ. Math. 33 (1967), 5 – 20. MR 37:5218 63, 70, 73 , Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie,
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Invent. Math. 2 (1967), 256 – 273. MR 35:1834 70, 77 [23]
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D X -modules coh´erents, Travaux en Cours 35 (1989), Hermann, Paris, 1989. MR 90m:3206 58, 61, 62, 64 , Sur le th´eor`eme de finitude de la cohomologie p-adique d’une vari´et´e affine non singuli`ere, Amer. J. Math. 119 (1997), 1027 – 1081. MR 99a:14027 57, 65, 81 P. MONSKY and G. WASHNITZER, Formal cohomology, I, Ann. of Math. (2) 88 (1968), 181 – 217. MR 40:1395 57, 69 M. VAN DER PUT, Serre duality for rigid analytic spaces, Indag. Math. (N.S.) 3 (1992), 219 – 235. MR 93d:32054 58, 66
Mathematisches Institut der Universit¨at M¨unster, Einsteinstrasse 62, 48149 M¨unster, Germany;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 1,
BRASCAMP-LIEB-LUTTINGER INEQUALITIES FOR CONVEX DOMAINS OF FINITE INRADIUS ´ ´ PEDRO J. MENDEZ-HERN ANDEZ
Abstract We prove a multiple integral inequality for convex domains in Rn of finite inradius. This inequality is a version of the classical inequality of H. Brascamp, E. Lieb, and J. Luttinger, but here, instead of fixing the volume of the domain, one fixes its inradius r D and the ball is replaced by (−r D , r D ) × Rn−1 . We also obtain a sharper version of our multiple integral inequality, which generalizes the results in [6], for two-dimensional bounded convex domains where we replace infinite strips by rectangles. It is well known by now that the Brascamp-Lieb-Luttinger inequality provides a powerful and elegant method for obtaining and extending many of the classical geometric and physical isoperimetric inequalities of G. P´olya and G. Szeg¨o. In a similar fashion, the new multiple integral inequalities in this paper yield various new isoperimetric-type inequalities for Brownian motion and symmetric stable processes in convex domains of fixed inradius which refine in various ways the results in [2], [3], [4], [5], and [20]. These include extensions to heat kernels, heat content, and torsional rigidity. Finally, our results also apply to the processes studied in [10] whose generators are relativistic Schr¨odinger operators. 1. Introduction Let D ⊂ Rn be a domain of finite volume, and denote by D ∗ the ball in Rn centered at the origin with the same volume as D. As described in C. Bandle [1] and P´olya and Szeg¨o [18], there is a large class of analytical quantities related to the Dirichlet Laplacian in D which are maximized or minimized by the corresponding quantities for D ∗ . These inequalities are often called generalized isoperimetric inequalities. Perhaps the most famous example of these inequalities is the celebrated Rayleigh-Faber-Krahn inequality, which asserts that among all bounded domains of fixed volume the first Dirichlet eigenvalue is minimized by the ball. That is, if λ D and λ D ∗ are the first eigenvalues for the Laplacian with Dirichlet boundary conditions in D and D ∗ , reDUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 1, Received 27 November 2000. Revision received 6 June 2001. 2000 Mathematics Subject Classification. Primary 31C60. Author’s work supported in part by Purdue Research Foundation grant number 690-1395-3149.
93
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spectively, then λ D∗ ≤ λ D .
(1)
In [15], [16], and [17], Luttinger provided a new method, based on the FeynmanKac representation of the heat kernel in terms of multiple integrals, to prove (1) as well as some of the other generalized isoperimetric inequalities of P´olya and Szeg¨o [18]. The following inequality, proved by Brascamp, Lieb, and Luttinger in [9], is a refinement of the original inequality of Luttinger. THEOREM 1.1 ([9]) ∗ be their symLet p1 , . . . , p2m be nonnegative functions in Rn , and let p1∗ , . . . , p2m metric decreasing rearrangement. Then, for any z 0 ∈ Rn , we have
Z
m Y D m j=1
pm+ j (z j ) p j (z j − z j−1 ) dz 1 · · · dz m Z ≤ {D ∗ }m
∗ p1∗ (z 1 ) pm+1 (z 1 )
m Y
∗ ∗ pm+ j (z j ) p j (z j − z j−1 ) dz 1 · · · dz m .
j=2
In [6] we showed not only that Theorem 1.1 proves generalized isoperimetric inequalities for the classical Dirichlet Laplacian operator but also that this approach yields, without any change, the same results for symmetric stable processes (the Dirichlet fractional Laplacian) or for any other L´evy processes that have right continuous paths and whose transition probabilities are radially symmetric and nonincreasing. In addition, this method works equally well for Schr¨odinger operators, provided that we make the right assumptions on the potentials. There has been substantial interest for many years now in obtaining isoperimetric-type inequalities for convex domains in Rn where, instead of fixing the volume of D, one fixes its inradius r D . Recall that the inradius is the supremum of the radius of all the balls contained in D. J. Hersh [14] (n = 2) and M. Protter [20] (n ≥ 3) proved that if D ⊂ Rn is a convex domain of inradius r D , then π2 = λ I (D) = λ S(D) ≤ λ D , 2 4r D
(2)
where I (D) = (−r D , r D ) and S(D) is the infinite slab (infinite strip for n = 2) of inradius r D in Rn given by S(D) = (x1 , . . . , xn ) ∈ Rn : xn ∈ I (D) . In [6], R. Ba˜nuelos, R. Latała, and P. M´endez proved a version of Theorem 1.1 for convex domains in R2 of fixed inradius. This inequality was then used to ex-
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tend to two-dimensional symmetric stable processes the isoperimetric-type inequalities proved in [5], which include (2) and inequalities for integrals of heat kernels (distribution of exit times of Brownian motion). Our first result in this paper is an extension of the inequality in [6] to convex domains in Rn . 1.2 Let D be a convex domain in Rn of finite inradius r D , and let S(D) be the infinite slab as defined above. Let p1 , . . . , pm be nonnegative nonincreasing radially symmetric functions on Rn . Then, for any t1 , . . . , tm > 0 and any z 0 ∈ Rn , we have THEOREM
Z
Z ··· t1 D
m Y
p j (z j − z j−1 ) dz 1 · · · dz m
tm D j=1
Z
...
≤ t1 S(D)
Z
p1 (z 1 ) tm S(D)
m Y
p j (z j − z j−1 ) dz 1 · · · dz m . (3)
j=2
In §5 we use this theorem and a probabilistic representation of the heat kernel in terms of multiple integrals to extend (2) and the results of Ba˜nuelos and P. Kr¨oger [5] to a symmetric stable process in all dimensions. In particular, if we denote the Dirichlet heat kernel of the Laplacian on D by p D (t, z, w) and its Green function by G D (z, w), then Theorem 1.2 implies that Z Z rD p D (t, z, w) dw ≤ p I (D) (t, 0, w) dw (4) D
−r D
for all z ∈ D and all t > 0. Here and for the rest of the paper, we use 0 to denote the origin in Rn . Upon integrating (4) in time, we obtain R. Sperb’s inequality [23] on integrals of Green’s functions. It is important to note here that inequalities (2) and (4) are false if the convexity assumption of the domain is removed. This can be seen by taking a slit disk. We refer the reader to [2] for details. One of the deficiencies of the above sharp inequalities for convex domains is that they ignore (by comparing to infinite slabs) the diameter of the domain. It would be better if we had sharp inequalities that took into account not only the inradius of the domain but also its diameter. As we shall see in §4, to prove Theorem 1.2 we can assume that p1 , . . . , pm are indicator functions of balls centered at the origin. This suggests that if D is a bounded convex domain, one should be able to replace S(D) by a convex domain of finite diameter. Yet another fact that points to this sharper result is the theorem of Protter [20] which asserts that if D is a convex domain in Rn , then π2 1 n − 1 + ≤ λD , (5) 2 2 4 rD dD
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where d D is the diameter of D. This means that one can replace S(D) by the ndimensional rectangle R(D) = (−r D , r D ) × (−d D , d D )n−1 in (2). That is, λ R(D) ≤ λ D , which takes into account not only the inradius but also the diameter of the domains. Our second result, Theorem 1.3, is a sharper version of Theorem 1.2 for bounded convex domains in R2 where we replace S(D) by the bounded region C(D) = S(D) ∩ B(0, d D − r D ), where B(0, d D − r D ) is the disk centered at the origin of radius d D − r D . Note that C(D) is strictly contained in the rectangle used by Protter in R2 . This result enables us to obtain a new set of isoperimetric-type inequalities for bounded convex domains in R2 which sharpen the results in [3], [5], [6], [14], [20], and [23]. In order to state our result precisely, we need to set up some notation. A simple geometric argument shows that any convex domain D in R2 of finite inradius is contained in a strip S of the same inradius or in a triangle H of the same inradius (see [3]). After translating and rotating, if necessary, we may suppose that S = S(D) and that the largest disk in H is centered at the origin. Let us first assume that D is contained in H . In this case, D is bounded and we can check that D is contained in B(0, d D − r D ). In addition, since the largest disk contained in H is centered at the origin, there exist a1 , a2 , a3 such that |a1 | = |a2 | = |a3 | = r D and 2 H = {x ∈ R2 : x · ai < r D , 1 ≤ i ≤ 3}.
For 1 ≤ i ≤ 3 we define 2 5i = {z ∈ R2 : z · ai = r D }, 2 5i0 = {z ∈ R2 : z · ai = −r D }, 2 2 Si (D) = {z ∈ R2 : −r D < z · ai < r D },
Ci (D) = Si (D) ∩ B(0, d D − r D ). Let 5i,0 j be the plane that passes through 5i ∩ 50j and 5i0 ∩ 5 j , and take Hi, j to be the intersection of H with the closed half-space determined by 5i,0 j containing 5i ∩ 5 j (see Figure 1).
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B S2 (D) B BB
B
501,2 B B '$ B B B B q S1 (D) B H1,2B B B&% B B B B B 5 0 B 52 B 2 B B B
B
B
B
501
51
Figure 1
Let us suppose now that D is contained in S(D). If D is unbounded, we adopt the convention that B(0, d D − r D ) = R2 and we take Hi, j = D and Ci (D) = S(D) for all i, j ∈ {1, 2, 3}. On the other hand, if D is bounded, it must be contained in a ball of radius d D − r D . Thus, after a translation if necessary, D is contained in C(D) and we take Ci (D) = C(D) and Hi, j = D for all i, j ∈ {1, 2, 3}. We are now ready to state our result. 1.3 Let D be a convex domain in R2 of finite inradius r D and diameter d D (which may be infinite). Let p1 , . . . , pm , q1 , . . . , qm be nonnegative nonincreasing radially symmetric functions on R2 . Then, for any z 0 ∈ Hi, j , we have THEOREM
Z ··· D
Z Y m
q j (z j ) p j (z j − z j−1 ) dz 1 · · · dz m
D j=1
Z ≤ Cs (D)
...
Z
m Y
Cs (D) j=1
q j (z j ) p j (z j − z j−1 ) dz 1 · · · dz m ,
where s ∈ {i, j}. As mentioned before, Theorem 1.3 is sharp enough to yield new isoperimetric-type inequalities for the Dirichlet Laplacian in planar convex domains. These inequalities and their extensions to symmetric stable processes are proved in §5. Here we give
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some examples of the results to be proved below. Let D be a convex domain in R2 with finite inradius r D and diameter d D . Recall that if d D = ∞, then D is contained in a strip. Since C(D) = S(D) if d D = ∞, inequalities (6) – (11) immediately follow, after a rotation and a translation, from the domain monotonicity of the Dirichlet heat kernel. Thus for the remainder of this section we may assume that d D < ∞. As we see later, Theorem 1.3 implies that for all z ∈ D and all t > 0, Z Z p D (t, z, w) dw ≤ pC(D) (t, 0, w) dw. (6) D
C(D)
This inequality is a refinement of the Ba˜nuelos-Kr¨oger inequality (4). Furthermore, (6) implies that the inequality in (4) is strict unless D is a strip, which is [5, Theorem 1]. As before, we can integrate (6) in time from zero to ∞ to get an inequality for integrals of Green’s functions which improves Sperb’s result in [23]. Using the well-known eigenfunction expansion of the Dirichlet heat kernel in D, it immediately follows from (6) that 1 π2 1 + < λC(D) ≤ λ D , (7) 2 4 rD (d D − r D )2 sharpening Protter’s inequality for n = 2. In addition, we are able to obtain other inequalities for Green’s functions which refine the inequality of Ba˜nuelos, T. Carroll, and E. Housworth in [3] by replacing S(D) by C(D). More precisely, let zˆ be the center of the largest disk contained in D, and let 8 : R+ → R+ be an increasing function. Then it follows from Theorem 1.3 that, for all t > 0, Z Z 8 p D (t, zˆ , w) dw ≤ 8 pC(D) (t, 0, w) dw (8) D
and
Z
C(D)
8 G D (ˆz , w) dw ≤ D
Z
8 G C(D) (0, w) dw.
(9)
C(D)
We should mention here that (8) is new even if we replace C(D) by S(D). These two inequalities are the counterparts of the classical isoperimetric inequalities of Bandle (see [1, Theorems 4.2 and 4.17]). Finally, we obtain new isoperimetric-type inequalities for the heat content and the torsional rigidity. The heat content of D at time t is given by the integral Z Z Q t (D) = p D (t, z, w) dz dw, D
D
and the torsional rigidity of D is the integral of the heat content in time from zero to ∞. (For more on these quantities, see [18] and [24].) We show that Theorem 1.3
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implies that for all t > 0 the heat content of D is less than the heat content of C(D). That is, for all t > 0, Q t (D) ≤ Q t C(D) . (10) Integrating (10) in time, we obtain the following inequality for the torsional rigidity: Z Z Z Z G D (z, w) dz dw ≤ G C(D) (z, w) dz dw. (11) D
D
C(D) C(D)
This inequality is similar to a classical isoperimetric inequality of P´olya and Szeg¨o which asserts that the torsional rigidity of D is less than the torsional rigidity of D ∗ . As we have seen in this section, Luttinger’s idea of combining multiple integral inequalities with the probabilistic representation of the heat kernel, originally designed for extremal problems for domains of fixed volume, provides a powerful and versatile tool for studying extremal inequalities for Dirichlet heat kernels in various other settings. This method was also used by Ba˜nuelos and M´endez in [7] to study extremal inequalities for ratios of heat kernels and spectral gaps in convex domains of fixed diameter. Not only do Theorems 1.2 and 1.3 yield isoperimetric-type inequalities for the Dirichlet Laplacian, but this method applies, without change, to any L´evy processes with right continuous paths and whose transition probability densities are radially symmetric and nonincreasing. This class of processes includes the symmetric stable processes, the geometric stable processes, the stochastic processes studied in [10] whose generators are the so-called “relativistic” Schr¨odinger operators, and any processes of the form B At , where Bt is a Brownian motion and At is a subordinator (see [13] and [22]). The paper is organized as follows. We prove Theorem 1.3 in §2. In §3 we give some preliminary results on polyhedral domains in Rn which are fundamental in the proof of Theorem 1.2. The proof of Theorem 1.2 is given in §4, and the applications to Dirichlet heat kernels of the Laplacian and symmetric stable processes are discussed in §5. 2. Proof of Theorem 1.3 Throughout this section D is a convex domain in R2 of finite inradius r D . Recall that any planar convex domain of finite inradius is contained in a strip of the same inradius or in a triangle of the same inradius. This follows from a simple geometric argument that can be found in [3]. Hence, if D is unbounded, it is contained in a strip of the same inradius and there is nothing to prove because in this case C(D) = S(D). On the other hand, if D is bounded, it contains a largest disk that, after a translation if necessary, we may take to be centered at the origin. Notice that in this case the distance from any point of D to the origin is less than d D − r D . Then either D is contained in a rotation
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of C(D) = S(D) ∩ B(0, d D − r D ) or D is contained in a triangle H with inradius r D . Therefore, without loss of generality, we may assume that D is contained in a triangle H whose largest incircle is centered at (0, 0) and a1 = (0, −1). That is, we assume that C1 (D) = C(D). The next result corresponds to the case m = 1 in Theorem 1.3. The proof of this result is very similar to the proof of [6, Lemma 1]. We include it here for completeness and because we believe it is helpful in understanding the idea behind the proof of Theorem 1.3. LEMMA 2.1 Let H be a triangle with inradius r D such that the largest circle contained in H is centered at the origin. Then, for all z ∈ Hi, j , 0 < r1 , and 0 < r2 ≤ d D − r D , we have m H ∩ B(z, r1 ) ∩ B(0, r2 ) ≤ m Cs (D) ∩ B(z, r1 ) ∩ B(0, r2 ) ,
where s ∈ {i, j} and m is the Lebesgue measure in R2 . Proof Suppose z ∈ H1,2 . We must prove that, for all r1 > 0 and d D − r D ≥ r2 > 0, m H ∩ B(z, r1 ) ∩ B(0, r2 ) ≤ m C(D) ∩ B(z, r1 ) ∩ B(0, r2 ) . We refer the reader to Figure 2 for more clarity in the geometric arguments of this proof. Let P2 be the intersection of S2 (D) and the half-space determined by 501 which does not contain the origin, and let P1 be the intersection of S1 (D) and the half-space determined by 52 which does not contain the origin.
52 B B 0
51
51
B 0 P2 B 52 B w B qB 501,2 ZB B Z BZ B Z Z B B B Zq w ∗ B O P1 B qz B B BB B
Figure 2
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Then m B(z, r1 ) ∩ B(0, r2 ) ∩ H \ S1 (D) ≤ m B(z, r1 ) ∩ B(0, r2 ) ∩ P2 . On the other hand, since C1 (D) = [R × (−r D , r D )] ∩ B(0, d D − r D ), we have m B(z, r1 ) ∩ B(0, r2 ) ∩ P1 ≤ m B(z, r1 ) ∩ B(0, r2 ) ∩ C1 (D) \ H . Hence, it is enough to prove that m B(z, r1 ) ∩ B(0, r2 ) ∩ P2 ≤ m B(z, r1 ) ∩ B(0, r2 ) ∩ P1 .
(12)
Notice that the reflection of P2 ∩ B(0, r2 ) with respect to the line 501,2 is P1 ∩ B(0, r2 ). Now if w ∈ P2 ∩ B(0, r2 ), let w∗ be its reflection with respect to 501,2 , and let lw be the line that passes through w and w∗ . Then 501,2 and lw are orthogonal, and they intersect at the midpoint of the segment ww∗ . Since z and w∗ are in the same halfspace determined by 501,2 , we have |w − z| ≥ |w∗ − z| with equality if and only if z ∈ 501,2 . Hence, if w ∈ P2 ∩ B(0, r ) and |z − w| < r1 , then w∗ ∈ P1 ∩ B(0, r ) and |z − w∗ | < r1 . Therefore, ∗ P2 ∩ B(0, r2 ) ∩ B(z, r1 ) ⊂ P1 ∩ B(0, r2 ) ∩ B(z, r1 ), and (12) follows from the rotation and translation invariance of the Lebesgue measure. The inequality m H ∩ B(z, r1 ) ∩ B(0, r2 ) ≤ m C2 (D) ∩ B(z, r1 ) ∩ B(0, r2 ) follows by the symmetry of the argument. Proof of Theorem 1.3 The proof is by induction on m. First, recall that a nonnegative radially symmetric nonincreasing function p can be expressed in the form Z ∞ p(z) = I B(0,r ) (z) dµ(r ) 0
for some nonnegative measure µ on (0, ∞]. Thus we may assume that, for all i ∈ {1, . . . , m}, there exist li > 0 and ri > 0 such that pi = I B(0,ri ) and qi = I B(0,li ) . As explained before, we may assume that D is contained in H˜ = H ∩ B(0, d D − r D ), where H is a triangle of inradius r D . Let Z Z Y m f m (z 0 ) = ··· p j (z j − z j−1 )q j (z j ) dz 1 · · · dz m , H˜
i gm (z 0 )
Z = Ci (D)
H˜ j=1
...
Z
m Y
Ci (D) j=1
p j (z j − z j−1 )q j (z j ) dz 1 · · · dz m .
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Then it is enough to prove that, for all z ∈ Hi, j , i j f m (z) ≤ min gm (z), gm (z) .
(13)
If z ∈ Hi, j , Lemma 2.1 implies that, for s ∈ {i, j}, Z p1 (z 1 − z)q1 (z 1 ) dz 1 = m H˜ ∩ B(z, r1 ) ∩ B(0, l1 ) H˜
≤ m Cs (D) ∩ B(z, r1 ) ∩ B(0, l1 ) Z = p1 (z − z 1 )q1 (z 1 ) dz 1 , Cs (D)
which proves (13) for m = 1. Let us suppose now that i j f m−1 (w) ≤ min gm−1 (w), gm−1 (w) ,
(14)
provided that w ∈ Hi, j and 1 ≤ i, j ≤ 3. Let z ∈ H1,2 , pm (w) = p(w) = I B(0,r ) (w), and qm (w) = q(w) = I B(0,l) (w). We denote by w∗ the reflection of w with respect to 501,2 . A simple computation shows that 2 1 2 1 gm−1 (w) = gm−1 (w∗ ), gm−1 (w∗ ) = gm−1 (w). (15) Take A = H˜ ∩ S1 (D) \ (H1,2 ∪ H1,3 ); then H˜ = A ∪ H˜ \ S1 (D) ∪ H˜ ∩ (H1,2 ∪ H1,3 ) . We now use this decomposition of H˜ to prove (13). Let z 1 ∈ H˜ \ S1 (D) ⊂ H˜ \ (H1,2 ∪ H1,3 ). Lemma 3.4 asserts that z 1 ∈ H2,3 . Then it follows from (14) and (15) that 2 1 f m−1 (z 1 ) ≤ gm−1 (z 1 ) = gm−1 (z 1∗ ).
(16)
Moreover, |z 1 | = |z 1∗ | and z ∈ H1,2 imply that q(z 1 ) = q(z 1∗ ) and p(z 1 − z) ≤ p(z 1∗ − z). Thus, q(z 1 ) p(z 1 − z) ≤ q(z 1∗ ) p(z 1∗ − z). (17) Combining (16) and (17), we obtain Z Z q(z 1 ) p(z − z 1 ) f m−1 (z 1 ) dz 1 ≤ H˜ \S1 (D)
H˜ \S1 (D)
1 q(z 1∗ ) p(z − z 1∗ )gm−1 (z 1∗ ) dz 1 .
Notice that ( H˜ \ S1 (D))∗ ⊂ C1 (D) \ H . Thus, making the substitution w = z 1∗ in the right-hand side of the last inequality, we conclude that Z Z 1 q(z 1 ) p(z − z 1 ) f m−1 (z 1 ) dz 1 ≤ q(w) p(z −w)gm−1 (w) dw. (18) H˜ \S1 (D)
C1 (D)\H
BRASCAMP-LIEB-LUTTINGER INEQUALITIES
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On the other hand, A ⊂ S1 (D) ∩ S2 (D) \ H1,2 ; then A∗ ⊂ H1,2 and 1 f m−1 (z 1∗ ) ≤ gm−1 (z 1∗ ),
(19)
2 1 f m−1 (z 1∗ ) ≤ gm−1 (z 1∗ ) = gm−1 (z 1 ),
(20)
for all z 1 ∈ A. We claim that, for all z ∈ H1,2 , p(z − z 1 ) f m−1 (z 1 ) + p(z − z 1∗ ) f m−1 (z 1∗ ) 1 1 ≤ p(z − z 1 )gm−1 (z 1 ) + p(z − z 1∗ )gm−1 (z 1∗ ). (21)
Recall that p is an indicator function; hence p = 1 or p = 0, and by (17) it suffices to consider two cases. Case 1. If p(z − z 1 ) = p(z − z 1∗ ) = 1, then (21) follows from (16) and (20). Case 2. If p(z − z 1 ) = 0 and p(z − z 1∗ ) = 1, then (21) follows from (19). Multiplying (21) by q(z 1 ) = q(z 1∗ ) and integrating over A, we have Z q(z 1 ) p(z − z 1 ) f m−1 (z 1 ) + q(z 1∗ ) p(z − z 1∗ ) f m−1 (z 1∗ ) dz 1 A Z 1 1 ≤ q(z 1 ) p(z − z 1 )gm−1 (z 1 ) + q(z 1∗ ) p(z − z 1∗ )gm−1 (z 1∗ ) dz 1 . A
Since Z A
q(z 1∗ ) p(z
−
z 1∗ ) f m−1 (z 1∗ ) dz 1
Z
q(z 1 ) p(z − z 1 ) f m−1 (z 1 ) dz 1
= A∗
and Z A
q(z 1∗ ) p(z
−
1 z 1∗ )gm−1 (z 1∗ ) dz 1
Z =
we obtain Z Z q(z 1 ) p(z − z 1 ) f m−1 (z 1 ) dz 1 ≤ A∪A∗
A∗
1 q(z 1 ) p(z − z 1 )gm−1 (z 1 ) dz 1 ,
A∪A∗
1 q(z 1 ) p(z − z 1 )gm−1 (z 1 ) dz 1 .
(22)
Finally, let us suppose that z 1 ∈ ( H˜ \ A∗ ) ∩ (H1,2 ∪ H1,3 ). Then by (14) we have 1 f m−1 (z 1 ) ≤ gm−1 (z 1 ) and Z q(z 1 ) p(z − z 1 ) f m−1 (z 1 ) dz 1 ( H˜ \A∗ )∩(H1,2 ∪H1,3 )
Z ≤ ( H˜ \A∗ )∩(H1,2 ∪H1,3 )
1 q(z 1 ) p(z − z 1 )gm−1 (z 1 ) dz 1 . (23)
1 (z) for all z ∈ H . Putting (18), (22), and (23) together, we conclude that f m (z) ≤ gm 1,2 The other cases follow in a similar fashion.
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3. Some results on polyhedral domains In the previous section we showed that it suffices to prove Theorem 1.3 for D being a triangle. Similarly, in higher dimensions we see in §4 that it is enough to prove Theorem 1.2 for polyhedral domains of finite inradius. In this section we prove several geometric lemmas on these domains which are instrumental in §4. Throughout this section, H is a polyhedral domain in Rn such that the largest ball contained in H is centered at the origin and has radius 1. In addition, we assume that there exist k ≤ n + 1 and a1 , a2 , . . . , ak ∈ Sn−1 such that H is given by the set of solutions of the system x · a1 < 1, x · a2 < 1, (24) .. . x · ak < 1. Let L be the linear subspace generated by {a1 , . . . , ak }, and denote its dimension by dim L. Note that if n = 1 and k = 2, then H must be the interval (−1, 1). In this section we frequently use the following lemma, which is a corollary of the well-known Kuhn-Tucker theorem (see [21]). LEMMA 3.1 Let c1 , . . . , cr ∈ Rn \ {0}, and let f be the convex function given by
f (x) = max{x · ci − b : 1 ≤ i ≤ q}, where b ∈ R and q ≤ r . Consider the problem (P): minimize f (x) in Rn subject to x · c1 ≤ 1, . . . , x · cl ≤ 1,
x · cl+1 = 1, . . . , x · cr = 1,
q ≤ l.
(25)
Assume that there exists xˆ ∈ Rn satisfying (25). Then x0 is a solution of (P) if and only if x0 satisfies (25) and there exist λ1 ≥ 0, . . . , λr ≥ 0 such that λi (x0 · ci − 1) = 0
for 1 ≤ i ≤ l,
r X
0 ∈ ∂ f (x0 ) +
λi ci ,
i=1
where ∂ f (x0 ) =
n
X
o
αi ci : αi ≥ 0
(26)
{i: f (x0 )=x0 ·ci −b,1≤i≤q}
is the subgradient of f at x0 . Throughout this section we often consider the set of constraints given by H , the closure of H , that is, x · a1 ≤ 1, . . . , x · ak ≤ 1. (27) The following lemma describes some of the properties of {a1 , . . . , an }.
BRASCAMP-LIEB-LUTTINGER INEQUALITIES
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LEMMA 3.2 Let H be a polyhedral domain in Rn such that the largest ball contained in H is centered at the origin and has radius 1. If H is given by (24), then either (1) H is bounded, dim L = n, k = n + 1, and for every i ∈ {1, . . . , n + 1}, n+1 X
−ai =
λij a j ,
(28)
j=1, j6 =i
(2)
where λij > 0 for all 1 ≤ j, i ≤ n + 1; or H is contained in a rotation of Hˆ × Rq , where 0 ≤ q ≤ n − 1 and Hˆ is a polyhedral domain in Rn−q satisfying part (1) and such that the largest ball contained in Hˆ is centered at the origin and has radius 1.
Proof Consider the convex function f (y) = max{y ·ai −1 : 1 ≤ i ≤ k} for y ∈ H . A simple Sk computation shows that the distance from any point y ∈ H to ∂ H ⊂ i=1 {x ∈ Rn : x · ai = 1} is given by d(y, ∂ H ) = min{1 − y · ai : 1 ≤ i ≤ k} = − f (y).
(29)
Thus, B(0, 1) is the largest ball contained in H if and only if f (y) ≥ −1 = f (0) for all y ∈ H . That is, the origin is a minimum of the convex function f subject to (27). Therefore, by Lemma 3.1 there exist nonnegative real numbers λ˜ 1 , . . . , λ˜ k such that 0 ∈ ∂ f (0) +
k X
λ˜ i ai .
i=1
Using the definition of the subgradient at the origin, we conclude that there exist λ1 ≥ 0, . . . , λk ≥ 0 such that k X λi ai = 0. i=1
We prove Lemma 3.2 by induction on the dimension n. Let us consider the twodimensional case. It is clear that k = 2 if and only if H is an infinite strip. If k = 3 and λi = 0 for some 1 ≤ i ≤ 3, say λ3 = 0, then (30) implies that a1 = −a2 and H is contained in a strip. We conclude that H is contained in a rotation of (−1, 1) × R unless k = 3 and λi > 0 for all 1 ≤ i ≤ 3. On the other hand, if λ1 , λ2 , and λ3 are positive, H must be a triangle and dim L = 2. This finishes the proof for n = 2. Let us now suppose that the result is true for any polyhedral domain in Rq for q ≤ n − 1. Let λ1 , . . . , λk be as in (30), and let λk1 , . . . , λkr be the nonzero elements of {λ1 , . . . , λk }. Take Lˆ to be the linear space generated by {ak1 , . . . , akr }, and take
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ˆ Note that dim Lˆ is at most r − 1. If q = dim Lˆ < n, dim Lˆ to be the dimension of L. ˜ ˆ then H = L ∩ H is a polyhedral domain in Rq such that the largest ball contained in H is centered at the origin and has radius 1. Rotating H if necessary, we have H = H˜ × Rn−q , and we can apply the induction hypothesis to H˜ to obtain the desired result. We must now consider the case q = n. Then r = k = n + 1, dim L = n, and λi > 0 for all i ∈ {1, . . . , n + 1}. Thus n+1 X
−ai =
λij a j ,
(30)
j=1, j6 =i
where λij = (λ j /λi ) > 0. It remains to prove that H is bounded. Given that dim L = n, it is enough to prove that the functions x · ai , 1 ≤ i ≤ n + 1, are bounded in H (the closure of H ). Let x ∗ be the solution of the system of equations x · a2 = 1, . . . , x · an+1 = 1. By (30), there exist λ12 , . . . , λ1n+1 ∈ (0, ∞) such that −a1 =
n+1 X
λ1j a j .
j=2
Then x ∗ · a1 < 0. Lemma 3.1 implies that the minimum of x · a1 subject to (27) is x ∗ · a1 > −∞. Since x · a1 ≤ 1 in H , we conclude that x · a1 is bounded in H . A similar argument proves that x · ai , 2 ≤ i ≤ n + 1, are all bounded. If A ⊂ Rn , we denote Rn \ A by Ac . For 1 ≤ i ≤ k, define 5i = {z ∈ Rn : z · ai = 1} and Si = {z ∈ Rn : −1 < z · ai < 1}. In the proof of Theorem 1.3, we used the fact that if H is a triangle, then H is contained in the union of any two of the strips S1 , S2 , S3 . The next lemma provides an n-dimensional version of this property. LEMMA 3.3 Let H be a polyhedral domain in Rn such that the largest ball contained in H is centered at the origin and has radius 1, and H satisfies Lemma 3.2(1). If there exist x0 ∈ H , 1 ≤ i ≤ n + 1, and 1 < r ≤ n such that c
c
x0 ∈ 5i ∩ Sk1 ∩ · · · ∩ Skr −1 ∩ Skr ∩ · · · ∩ Skn ,
BRASCAMP-LIEB-LUTTINGER INEQUALITIES
107
where {i, k1 , . . . , kn } = {1, . . . , n + 1}, then H ⊂ Si ∩ Sk1 ∩ · · · ∩ Skr −1 . Proof Without loss of generality, we can suppose that i = 1 and k j = j + 1 for j ∈ {1, . . . , n}. Let fr (x) = max{x · a1 , . . . , x · ar }. Note that H ⊂ S1 ∪ S2 ∪ · · · ∪ Sr if and only if fr (x) > −1 for all x ∈ H . Thus we must prove that the minimum of fr on H is larger than or equal to −1. Consider the following problem: minimize fr subject to (27). Since 0 ∈ H and H is bounded, Lemma 3.1 implies that there exist z 0 ∈ H and nonnegative numbers λ1 , . . . , λn+1 such that n+1 X 0 ∈ ∂ fr (z 0 ) + λi ai i=1
and λi (z 0 · ai − 1) = 0,
1 ≤ i ≤ n + 1.
Given that fr (z 0 ) ≤ fr (0) ≤ 0, we have z 0 · ai 6 = 1 for 1 ≤ i ≤ r . Thus λ1 = · · · = λr = 0. Therefore, (28) and (26) imply that λr +1 6 = 0, . . . , λn+1 6 = 0. We conclude that if fr (z 0 ) is the minimum of fr subject to (27), then z 0 · ar +1 = · · · = z 0 · an+1 = 1.
(31)
We now proceed by induction on r . Let r = 2; then there exists c
c
x0 ∈ H ∩ 51 ∩ S2 ∩ S3 ∩ · · · ∩ Sn+1 . By (31) it is enough to prove that M = {x ∈ Rn : x · a3 = · · · = x · an+1 = 1, x ∈ H } is contained in S2 ∪ S1 . Let y0 be the solution of x · a2 = · · · = x · an+1 = 1, and let y1 be the solution of x · a1 = x · a3 = · · · = x · an+1 = 1.
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We can easily check that y0 ∈ S2 , y1 ∈ S1 , and M = yλ ∈ Rn : yλ = λy1 + (1 − λ)y0 , λ ∈ [0, 1] . If y0 ∈ S1 , then M ⊂ S1 and we are done. On the other hand, if y0 ∈ / S1 , then y0 · a1 < −1 and there exists λ0 ∈ (0, 1) such that yλ0 = λ0 y1 + (1 − λ0 )y0 ∈ 501 . If yλ0 ∈ S2 , it is easy to check that yλ ∈ S1 for all λ ∈ [λ0 , 1] and that yλ ∈ S2 for all λ ∈ [0, λ0 ]. Thus it is enough to prove that yλ0 ∈ S2 . That is, we must prove that the solution of x · a1 = −1, x · a3 = · · · = x · an+1 = 1 (32) is in S2 . Consider the following problem: minimize f (x) = x · an+1 subject to x · a1 = 1,
x · a2 ≤ 1,
x · a3 ≤ −1, . . . , x · an+1 ≤ −1.
(33)
Since H is compact and x0 is a solution of (33), there exists a point x ∗ that solves this problem. By Lemma 3.1 there exist λ1 ≥ 0, . . . , λn+1 ≥ 0 such that −an+1 =
n+1 X
λi ai ,
λ2 (x ∗ · a2 − 1) = 0,
i=1
λi (x ∗ · ai + 1) = 0,
3 ≤ i ≤ n + 1.
If λn+1 6 = 0, then x ∗ ·an+1 = −1 and f (x ∗ ) = −1 > f (x0 ), which is a contradiction. Then λn+1 = 0, and by Lemma 3.2 we conclude that λi > 0 for 1 ≤ i ≤ n. Therefore, x ∗ · a1 = x ∗ · a2 = 1,
x ∗ · a3 = · · · = x ∗ · an = −1,
x ∗ · an+1 < −1.
Applying Lemma 3.1 and Lemma 3.2, we can see that if xˆ is the solution of x · a1 = 1,
x · a2 = −1, . . . , x · an = −1,
then xˆ solves the following problem: minimize f (x) = −x · an+1 subject to 1 − x · a1 = 0,
−x · a2 ≤ 1, . . . , −x · an ≤ 1,
x · a2 ≤ 1, . . . , x · an ≤ 1. (34)
Given that a1 satisfies (34), we obtain −xˆ · an+1 ≤ −a1 · an+1 < 1. Take x λ = λx ∗ + (1 − λ)xˆ for 0 ≤ λ ≤ 1. A simple computation shows that x λ ∈ S2 and x λ · a1 = 1, x λ · a3 = · · · = x λ · an = −1,
BRASCAMP-LIEB-LUTTINGER INEQUALITIES
109
for all λ ∈ (0, 1). On the other hand, x 0 · an+1 = xˆ · an+1 > −1,
x 1 · an+1 = x ∗ · an+1 < −1.
Therefore, there exists λ0 ∈ (0, 1) such that x λ0 · an+1 = −1. We conclude that the solution of x · a1 = 1, x · a3 = · · · = x · an+1 = −1 is in S2 , and (32) follows from the symmetry with respect to the origin of S2 . Let us now suppose that the result is true for r − 1 and that there exists x0 ∈ H such that c c x0 ∈ 51 ∩ S2 ∩ · · · ∩ Sr ∩ Sr +1 ∩ · · · ∩ Sn+1 . Assume that there exists x ∈ H such that x ∈ / S1 ∪ · · · ∪ Sr . From (31) we have that z 0 ∈ H is a solution of the system z · a1 ≤ −1, . . . , z · ar ≤ −1.
z · ar +1 = · · · = z · an+1 = 1,
(35)
Consider now the following problem: minimize f (x) = x ·a2 subject to (35). Since z 0 satisfies (35), we can apply Lemma 3.1 to conclude that the solution of this problem z ∗ is such that z ∗ · ar +1 = · · · = z ∗ · an+1 = 1,
z ∗ · a1 = z ∗ · a3 = · · · = z ∗ · ar = −1, (36)
and z ∗ · a2 < −1. We prove that this yields a contradiction. Let x ∗ be a solution to the following problem: minimize f (x) = x · an+1 subject to x · ar +1 ≤ −1, . . . , x · an+1 ≤ −1. (37) Then, just as in the case r = 2, we obtain that x ∗ · an+1 < −1 and x ∗ is a solution of the system x · a1 = 1,
x · a2 ≤ 1, . . . , x · ar ≤ 1,
x · a1 = · · · = x · ar = 1,
x · ar +1 = · · · = x · an = −1.
Let xˆ be the solution of x · a1 = x · a3 = · · · = x · ar = 1,
x · a2 = x · ar +1 = · · · = x · an = −1.
If xˆ · an+1 < −1, let w be the solution of the system x · a1 = · · · = x · an = 1. Then v = µ(xˆ − w) + x, ˆ for µ > 0, is such that v · a1 = v · a3 = · · · = v · ar = 1,
v · a2 = v · ar +1 = · · · = v · an = −1 − 2µ.
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Taking µ > 0 small enough, we obtain v · an+1 < −1. Therefore, there is a point c c c v ∈ H such that v ∈ 51 ∪ S3 · · · ∪ Sr and v ∈ S2 ∪ Sr +1 · · · ∪ Sn+1 . But this implies that H is contained in S1 ∪ S3 ∪ · · · ∪ Sr , which is a contradiction. We now suppose that xˆ · an+1 ≥ −1. Then an argument similar to the one used in the case r = 2 proves that λxˆ + (1 − λ)x ∗ ∈ S2 for all λ ∈ [0, 1] and that there exists λ0 ≥ 0 such that λ0 xˆ + (1 − λ0 )x ∗ is the solution of the system x · a1 = x · a3 = · · · = x · ar = 1,
x · ar +1 = · · · = x · an+1 = −1.
But this contradicts (36) and finishes the proof. The next result is the corollary of Lemma 3.3 which we used in the proof of Theorem 1.3. LEMMA 3.4 Let n = 2, and suppose that H is a triangle. Then
H = H1,2 ∪ H1,3 ∪ H2,3 . Proof Let x ∈ H , and take λ0 > 0 such that λx ∈ H for all λ ∈ [0, λ0 ) and λ0 x ∈ ∂ H . Let us suppose that λ0 x ∈ 51 . Given that H is not contained in a strip, Lemma 3.3 implies that λ0 x ∈ S2 or λ0 x ∈ S3 . Moreover, if 2 ≤ i ≤ 3, then λ0 x ∈ 51 ∩ Si ⇔ λ0 x ∈ H1,i ⇔ x ∈ H1,i , and the result follows. We end this section with some elementary inequalities that we use in the proof of Theorem 1.2. 3.5 Let θ3 , θ2 , θ1 ∈ [0, π). If θ1 > θ2 > θ3 , LEMMA
θ2 + θ3 > θ1 = θ1,3 + θ1,2 , and
θ2 + θ3 + θ1 < 2π,
sin(θ1,2 ) sin(θ2 ) = . sin(θ3 ) sin(θ1,3 )
Then θ1,2 < θ2 and either θ1,3 < θ3 or θ2 ≥ π/2 and θ3 ≤ π/2.
(38)
BRASCAMP-LIEB-LUTTINGER INEQUALITIES
111
Proof Consider the function
sin(x) sin(θ1 − x) for x ∈ [0, θ1 ). A straightforward computation shows that g(x) is strictly increasing in [0, θ1 ). Then θ1,2 < θ2 if and only if g(x) =
sin(θ2 ) sin(θ1,2 ) sin(θ2 ) = = g(θ1,2 ) < g(θ2 ) = . sin(θ3 ) sin(θ1,3 ) sin(θ1 − θ2 ) Hence, we must prove that sin(θ3 ) > sin(θ1 − θ2 ).
(39)
If θ3 ≤ π/2, (39) holds because θ3 > θ1 − θ2 . Let us suppose now that θ3 > π/2. Recall that θ2 ≥ θ3 > π/2 and θ1 ≤ π. Then θ1 − θ2 ≤ π/2, and (39) follows from the fact that π − θ3 ≥ θ1 − θ2 . Thus θ1,2 < θ2 . In a similar way, we see that θ1,3 < θ3 if and only if sin(θ2 ) > sin(θ1 − θ3 ).
(40)
As before, (40) immediately follows if θ2 ≤ π/2. On the other hand, if θ2 > π/2 and θ3 ≥ π/2, then θ2 + θ1 < 2π − θ3 ≤ π + θ3 . Thus θ1 − θ3 ≤ π − θ2 < π/2, and (40) follows. We conclude that θ1,3 < θ3 holds if either θ2 ≤ π/2 or θ2 > π/2 and θ3 ≥ π/2, which completes the proof of the lemma. 4. Proof of Theorem 1.2 Throughout this section, D is a convex domain in Rn of finite inradius r D for n > 2. As explained in §2, any planar convex domain D of inradius r D is contained either in a triangle or in a strip of inradius r D . For convex domains in Rn , n > 2, we have the following lemma of D. Gale [20]. LEMMA 4.1 Let D ⊂ Rn be a convex domain in Rn with inradius r D . Then for some integer k, 2 ≤ k ≤ n + 1, there is a polyhedral domain H consisting of the intersection of k half-spaces which contains D and whose inradius is r D .
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Thus, as before, it is enough to prove Theorem 1.2 for polyhedral domains of finite inradius. Without loss of generality, we may assume that the largest ball contained in H is centered at the origin and that there exist a1 , a2 , . . . , ak ∈ Rn such that |a1 | = · · · = |ak | = r D , and H is given by the system of inequalities 2, x · a1 < r D 2 x · a2 < r , D H= .. . 2. x · ak < r D Consider the polyhedral domain D = H × Rq in Rn+q , and let p1 , . . . , pm be radially symmetric and radially decreasing functions in Rn+q . If u 1 , . . . , u m ∈ Rn and v1 , . . . , vm ∈ Rq , then for all (u 0 , v0 ) ∈ Rn+q , Z Z Y m ··· p j |(u j − u j−1 , v j − v j−1 )| du 1 dv1 · · · du m dvm D
D j=1
Z
Z
≤
··· Rmq
H
Z Y m H j=1
q × pi |u j − u j−1 |2 + |v j − v j−1 |2 du 1 · · · du m dv1 · · · dvm . q Notice that for each v j , v j−1 ∈ Rn , the function pˆ i (u) = pi |u|2 + |v j − v j−1 |2 is radially symmetric and decreasing. Thus, to prove (3) for D, it is enough to prove Theorem 1.2 for H . Therefore, by Lemma 3.2 we can assume that H is bounded, k = n + 1, the dimension of the subspace generated by {a1 , . . . , an+1 } is n, and for every i ∈ {1, . . . , n + 1}, n+1 X −ai = λij a j , j=1, j6 =i
λij
where > 0 for all 1 ≤ j, i ≤ n + 1. As in the two-dimensional case, we define for 1 ≤ i ≤ n + 1, 2 5i = {z ∈ Rn : z · ai = r D }, 2 5i0 = {z ∈ Rn : z · ai = −r D },
and 2 2 Si (D) = {z ∈ Rn : −r D < z · ai < r D }.
We denote by 5i,0 j the plane that passes through 5i ∩ 50j and 5i0 ∩ 5 j , and we denote by 5i,+j the closed half-space determined by 5i,0 j which contains 5i ∩ 5 j .
BRASCAMP-LIEB-LUTTINGER INEQUALITIES
113
We now give a decomposition of H that is essential in the proof of Theorem 1.2. Let 5i, j be the plane that passes through 5i ∩ 5 j and 5i0 ∩ 50j . It is easy to check that z ∈ 5i, j if and only if dist(z, 5i ) = dist(z, 5 j ), where dist(z, A) is the distance between z and A. Take 5ii, j to be the closed halfspace determined by 5i, j which contains 5i ∩ 50j . For r1 , . . . , rm ∈ {1, . . . , n + 1}, with m ≤ n, we define m \
H (r1 , . . . , rm ) = ri
5rrii ,r j ∩ H ∩ Sri .
j=1, j6 =i
Then z ∈ H ri (r1 , . . . , rm ) if and only if z ∈ H and dist(z, 5ri ) = min dist(z, 5r j ) : 1 ≤ j ≤ m . Hence, we have H=
m [
H ri (r1 , . . . , rm ).
(41)
i=1
In Lemma 4.3 we reduce Theorem 1.2 to prove various inequalities for areas involving S1 (D), S2 (D). To deal with the difficulty of working in several dimensions, we need a convenient representation of the planes that arise in this case. For A, B ∈ Rn , let ∠(A, B) be the angle between A and B, and denote by C(A, B) = {x ∈ Rn : λ1 A + λ2 B, λi ≥ 0} the wedge generated by A and B. It is easy to prove that if A, B, C ∈ Rn are linear independent, then ∠(A, B) + ∠(A, C) + ∠(C, B) < 2π.
(42)
Let a ∈ 501 ∩ 52 ∩ 53 , and consider V = {x ∈ Rn : x · a1 = x · a2 = x · a3 = 0}. Then the dimension of the linear subspace V is n − 3 and a + V = 501 ∩ 52 ∩ 53 = {x ∈ Rn : x · a1 = −1, x · a2 = x · a3 = 1}. Let us consider the linear subspace V2,3 = {x ∈ Rn : x · a2 = x · a3 = 0}. Clearly, the dimension of V2,3 is n − 2 and V ⊂ V2,3 . Thus there exists A1 ∈ V2,3 such that A1 is orthogonal to V , and 52 ∩ 53 = {x ∈ Rn : x · a2 = x · a3 = 1} = a + V2,3 = a + V + RA1 ,
(43)
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where RA1 = {x ∈ Rn : x = λA1 , λ ∈ R}. Moreover, we can normalize A1 to satisfy a + A1 ∈ 51 . In a similar way, we see that there exist A2 , A3 ∈ Rn with the following properties: A2 , A3 are orthogonal to V , a + A2 ∈ 503 , a + A3 ∈ 502 , 501 ∩ 52 = a + {x ∈ Rn : x · a1 = x · a2 = 0} = a + V + RA2 ,
(44)
501
(45)
∩ 53 = a + {x ∈ R : x · a1 = x · a3 = 0} = a + V + RA3 . n
Since a + A1 ∈ 52 ∩ 53 , there exists d2,3 ∈ Rn orthogonal to V such that a + d2,3 ∈ 501 ∩ 502 ∩ 503 and 52,3 ∩ S2 (D) ∪ S3 (D) = a + V + x ∈ Rn : x = λ1 A1 + λ2 d2,3 , λ1 ∈ R, λ2 ∈ [0, 1] . (46) Notice that d2,3 ∈ C(A2 , A3 ). The next geometric lemma is fundamental in the proof of Theorem 1.2. 4.2 Let θ3 = ∠(A3 , A1 ), θ2 = ∠(A2 , A1 ), θ1,3 = ∠(A3 , d2,3 ), and θ1,2 = ∠(A2 , d2,3 ). Then |A3 | sin(θ2 ) sin(θ1,2 ) = = . (47) |A2 | sin(θ3 ) sin(θ1,3 ) LEMMA
Proof Let P be the polyhedral domain S1 (D) ∩ · · · ∩ Sn (D), and let m n be the Lebesgue measure in Rn . Fubini’s theorem implies that m n (P) = 2m n−1 (P ∩ 52 ) = 2m n−1 (P ∩ 502 ) = 2m n−1 (P ∩ 53 ) = 2m n−1 (P ∩ 503 ) and m n−1 (P ∩ 52 ) = dist(52 ∩ 53 , 52 ∩ 503 )m n−2 (P ∩ 52 ∩ 53 ), m n−1 (P ∩ 53 ) = dist(52 ∩ 53 , 502 ∩ 53 )m n−2 (P ∩ 52 ∩ 53 ). Thus we have dist(52 ∩ 53 , 52 ∩ 503 ) = dist(52 ∩ 53 , 502 ∩ 53 ). Since 53 and 503 are parallel planes and a + A2 ∈ 52 ∩ 503 , we have dist(52 ∩ 53 , 52 ∩ 503 ) = dist(52 ∩ 53 , a + A2 ).
BRASCAMP-LIEB-LUTTINGER INEQUALITIES
115
Using (43), we easily check that dist(52 ∩ 53 , a + A2 ) = |A2 | sin(θ2 ). In the same way, we obtain dist(52 ∩ 53 , 502 ∩ 53 ) = |A3 | sin(θ3 ). We conclude that
|A3 | sin(θ2 ) = . |A2 | sin(θ3 )
(48)
On the other hand, 52,3 divides P into two polyhedral domains of equal volume. Then a similar argument gives |A2 | sin(θ1,2 )m n−2 (52,3 ∩ P ∩ 501 ) = |A3 | sin(θ1,3 )m n−2 (52,3 ∩ P ∩ 501 ), and the lemma follows from the last equation and (48). The next lemma is the case m = 1 in Theorem 1.2. 4.3 Sm
LEMMA
If H ⊂
j=1 Sr j (D)
and z ∈
Tm
j=1, j6 =i
5r+i ,r j , then
m B(z, r ) ∩ H ≤ m B(z, r ) ∩ Sri (D) , where m is the Lebesgue measure in Rn . Proof Without loss of generality, we can assume that ri = 1 and r j = j, 1 ≤ j ≤ m. We must prove that m B(z, r ) ∩ H ∩ S1 (D)c ≤ m B(z, r ) ∩ S1 (D) ∩ H c , T provided that z ∈ mj=2 5+ 1, j . As in Lemma 2.1, the basic idea is to use reflections to fold H into an infinite slab. Let 2 ≤ i ≤ m, let A∗i be the reflection of a Borel set A with respect to the plane 501,i , and let Pi (D) be the intersection of S1 (D) and {x ∈ Rn : x · ai ≥ 1}. Since c ∗i is contained in P (D), we have z ∈ 5+ i 1, j and H ∩ Si (D) ∩ S1 (D) ∗ m B(z, r ) ∩ H ∩ Si (D) ∩ S1 (D)c ≤ m B(z, r ) ∩ H ∩ Si (D) ∩ S1 (D)c i ≤ m B(z, r ) ∩ Pi (D) .
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Unfortunately, this argument implies the result only when m = 2 because the intersection of ∗ ∗ H ∩ Si (D) ∩ S1 (D)c i and H ∩ S j (D) ∩ S1 (D)c j may have positive measure. Therefore, we need a better decomposition of H ∩S1 (D)c . Recall that m [ H ∩ S1 (D)c = H i (1, . . . , m) ∩ S1 (D)c . i=2
Then we have ∗ m B(z, r ) ∩ H i (1, . . . , m) ∩ S1 (D)c ≤ m B(z, r ) ∩ H i (1, . . . , m) ∩ S1 (D)c i , provided that z ∈ 5i,+j . The result follows if we prove that whenever i 6= j,
H i (1, . . . , m) ∩ S1 (D)c
∗i
∗ ∩ H j (1, . . . , m) ∩ S1 (D)c j
is contained in a set of measure zero. Since H i (1, . . . , m) =
m \
H i (i, k),
k=1,k6 =i
it is enough to prove that i ∗ ∗ H (i, j) ∩ S1 (D)c i ∩ H j (i, j) ∩ S1 (D)c j
(49)
is contained in a plane for all i, j ∈ {2, . . . , m}, i 6 = j. Without loss of generality, we can assume that m ≥ 3, i = 2, and j = 3. Let 5− 1 be the closed half-space determined by 501 which does not contain the origin. Then by (44) and (45) we have that if {i, j} = {2, 3}, i 5i ∩ 5− 1 ∩ 5i, j = a + V + C(Ai , −A1 )
and i ∗i 5i ∩ 5− = a + V + C(Ai , −A∗1i ). 1 ∩ 5i, j Note that a − A∗1i is in 501 ∩ {x ∈ Rn : x · ai > 1} for 2 ≤ i ≤ 3. Since reflections ∗ ∗ preserve angles, A∗12 and A13 are orthogonal to V . Thus −A∗12 and −A13 are in the subspace generated by A2 and A3 . On the other hand, if θ1 = ∠(A2 , A3 ), then (42) implies that θ1 + π − θ2 + π − θ3 = ∠(A2 , A3 ) + ∠(A2 , −A1 ) + ∠(A3 , −A1 ) < 2π, and it follows that ∗
C(A2 , −A∗12 ) ∩ C(A3 , −A13 ) = 0.
(50)
BRASCAMP-LIEB-LUTTINGER INEQUALITIES
117
This is illustrated in Figure 3.
A2 ∗3 C(A2 , A3 ) C(A3 , −A1 ) θ1 π − θ 3 A3 π − θ2 ZZ Z −A∗3 C(A2 , −A∗12 ) Z ~ 1 Z −A∗12 ?
Figure 3 We conclude that 3 ∗3 2 ∗2 ∩ 53 ∩ 5− = a + V. 52 ∩ 5− 1 ∩ 52,3 1 ∩ 53,2
(51)
Let {i, j} = {2, 3}, and define 5i1 = 501 ∩ 5ii, j . Then by (43) – (46) we obtain 5i1 ∩ Si (D) = a + V + x ∈ Rn : x = λ1 d2,3 + λ2 Ai , λ1 ∈ [0, 1], λ2 ≥ 0 and i ∗ ∗i 51 ∩ Si (D) i = a + V + x ∈ Rn : x = λ1 d2,3 + λ2 Ai , λ1 ∈ [0, 1], λ2 ≥ 0 . Notice that ∗ i 51 ∩ Si (D) i ⊂ 5i ∩ S1 (D). From (44) and (45) we have 5i ∩ S1 (D) ∩ {x ∈ Rn : x · a j ≤ 1} = a + V + x ∈ Rn : x = λ1 A1 + λ2 Ai , λ1 ∈ [0, 1], λ2 ≥ 0 . Thus i ∗ 51 ∩ Si (D) i ⊂ 5i ∩ S1 (D) ∩ {x ∈ Rn : x · a j ≤ 1} if and only if ∗i ∠(Ai , d2,3 ) = θ1,i ≤ θi = ∠(Ai , A1 )
(see Figure 4).
(52)
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aq θ1,3 q ∗3 a + d2,3
aq + A3
501 -
531 ∩ S3 (D)
∗3
51
-
aq 501 aq + A3 C θ C 3 S1 (D) ∩ 53 ∩ {x · a2 ≤ 1} C Cq 51 a + A1
Figure 4 Notice that H i (2, 3) ∩ S1 (D)c is contained in the set of all convex combinations of i elements of 5i1 ∩ Si (D) union 5i ∩ 5− 1 ∩ 52,3 . If θ1 ≤ θ2 , (51) and (52) imply that ∗ 2 ∗2 52 ∩ 5− ∪ 521 ∩ S2 (D) 2 ⊂ {x ∈ Rn : x · a3 ≤ 1}. 1 ∩ 52,3 The result follows because 3 ∗ 53,2 ∩ S1 (D)c ∩ S3 (D) 3 ⊂ {x ∈ Rn : x · a3 ≥ 1}. On the other hand, if θ1,3 ≤ θ3 and θ1,2 ≤ θ2 , (52) asserts that 2 ∗ ∗ 51 ∩ S2 (D) 2 ∩ 531 ∩ S3 (D) 3 ⊂ 52 ∩ 53 , and we conclude from (51) that H 2 (2, 3) ∩ S1 (D)c
∗2
∩ H 3 (2, 3) ∩ S1 (D)c
∗3
⊂ 52 ∩ 53 .
Hence, we must study the case θ1 > θ3 , θ1 > θ2 , and either θ1,3 > θ3 or θ1,2 > θ2 . Without loss of generality, we can assume that θ2 ≥ θ3 . Then, by Lemmas 4.2 and 3.5, we have θ1,3 > θ3 ,
θ1,2 < θ2 ,
θ3 ≤ π/2,
π/2 ≤ θ2 ≤ θ1 .
Thus (52) implies that 2 ∗ 51 ∩ S2 (D) 2 ⊂ 52 ∩ S1 (D) ∩ {x ∈ Rn : x · a3 ≤ 1}, and ∗ 53 ∩ S1 (D) ∩ {x ∈ Rn : x · a2 ≤ 1} ⊂ 531 ∩ S3 (D) 3 . It is easy to check that (see Figure 5) ∗ 52 ∩ S1 (D) ∩ {x ∈ Rn : x · a3 ≤ 1} \ 521 ∩ S2 (D) 2 ∗2 ⊂ a + V + {x ∈ Rn : x = λ1 A1 + λ2 d2,3 , λ j ≥ 0, λ1 + λ2 ≤ 1}
BRASCAMP-LIEB-LUTTINGER INEQUALITIES
119
and
∗ 531 ∩ S3 (D) 3 \ 53 ∩ S1 (D) ∩ {x ∈ Rn : x · a2 ≤ 1} ∗
3 ⊂ a + V + {x ∈ Rn : x = λ1 A1 + λ2 d2,3 , λ j ≥ 0, λ1 + λ2 ≤ 1}.
aq + A2 aq
S0 θ S1 (D) ∩ 52 ∩ {x · a3 ≤ 1} 2 S S q Sq ∗ 2 a + d2,3
a + A1
aq q a + A3 S 0 θ3 S S1 (D) ∩ 53 ∩ {x · a2 ≤ 1} S q Sq ∗
3 a + d2,3
a + A1
Figure 5. θ30 = θ1,3 − θ3 , θ20 = θ2 − θ1,2 ∗i Notice that Ti = {x ∈ Rn : x = λ1 A1 + λ2 d2,3 , λ j ≥ 0, λ1 + λ2 ≤ 1} is a triangle with sides |A1 |, |d2,3 |, and
di2 = |A1 |2 + |d2,3 |2 − 2 cos(|θ1,i − θi |)|A1 ||d2,3 |. By (50), θ3 + θ2 ≥ θ1 = θ1,3 + θ1,2 , and thus θ2 − θ1,2 ≥ θ1,3 − θ3 > 0 and d2 > d3 . Consider now the plane ∗
3 5 = a + V + {x ∈ Rn : x = λ1 d2,3 + λ2 A∗12 , λi ∈ R}.
From (50) we have that (H 3 (2, 3) ∩ S1 (D)c )∗3 is contained in the closed halfspace determined by 5 which does not contain 501 ∩ 52 ∩ 522,3 . We now prove that (H 2 (2, 3) ∩ S1 (D)c )∗2 is contained in the half-space determined by 5 which contains ∗3 501 ∩ 52 ∩ 522,3 . Let x0 = a + v + λ1 d2,3 + λ2 A∗12 ∈ 5 ∩ 52 ∩ 51 . Then we have ∗
3 1 = x0 · a1 = (a + v) · a1 + λ1 (d2,3 + a) · a1 + λ2 (A∗12 + a) · a1
− λ1 a · a1 − λ2 a · a1 = −1 + λ1 − λ2 + λ1 + λ2 = −1 + 2λ1 . On the other hand, θ1,3 > θ3 and θ1 > θ3 imply that ∗
3 (d2,3 + a) · a2 > 1,
(−A∗12 + a) · a2 > 1.
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Since ∗
3 1 = x0 · a2 = (a + v) · a2 + (d2,3 + a) · a2 − λ2 (−A∗12 + a) · a2 − a · a2 + λ2 a · a2 ,
we obtain ∗3 −(d2,3 + a) · a2 + 1 = λ2 − (−A∗12 + a) · a2 + 1 . ∗
3 We conclude that x0 = a + v + d2,3 + λA∗12 , where v ∈ V and λ ≥ 0. Let B = ∗3 ∗2 d2,3 + λA1 ; then B is orthogonal to V , a + B ∈ 51 , and
5 ∩ 52 = a + V + RB. In a similar fashion, we can check that ∗
3 d2,3 = A1 − µA3
for some µ ≥ 0
and B = A1 + µ1 A2
for some µ1 ≥ 0.
Therefore, we have ∗
3 A1 + µ1 A2 = d2,3 + λA∗12 = A1 − µA3 + λA∗12 ,
and by simplifying, we obtain µ1 A2 = −µA3 + λA∗12 .
(53)
Notice that d3 = µ|A3 |. Consider the triangle T = {x ∈ Rn : x = a + α1 B + α2 A1 , αi ≥ 0, 1 ≥ α1 + α2 }. It is clear that T ⊂ 52 ∩ S1 ∩ {x ∈ Rn : x · a3 ≤ 1}. Since B = A1 + µ1 A2 , the length of the sides of T are d = |µ1 A2 |, B, and |A1 |. Thus we obtain from (53) d sin(θ1 − θ2 ) = d3 sin(θ2 ) (see Figure 6). But this implies that d ≤ d3 , since θ2 ≥ π/2 and π/2 ≥ π − θ2 ≥ θ1 − θ2 ≥ 0. Therefore, T ⊂ T2 , and we conclude that [H 2 (2, 3) ∩ S1 (D)c ]∗2 and [H 3 (2, 3) ∩ S1 (D)c ]∗3 are in different half-spaces of 5. Thus ∗ ∗ 2 3 H2,3 ∩ S1 (D)c 2 ∩ H2,3 ∩ S1 (D)c 3 ⊂ 5, which completes the proof of Lemma 4.3.
BRASCAMP-LIEB-LUTTINGER INEQUALITIES
λA∗12
A K A A
121
Aµ1 A2 A A A θ2 A A α2A-A A α1 −µA3
Figure 6. α1 = π − θ1 , α2 = θ1 − θ2 Proof of Theorem 1.2 As in the two-dimensional case, the proof is by induction on m. The beginning of the proof of Theorem 1.2 is copied from the beginning of the proof of Theorem 1.3. First, recall that a nonnegative radially symmetric nonincreasing function f in Rn can be expressed in the form Z ∞
f (z) =
0
I B(0,r ) (z) dµ(r )
for some nonnegative measure on (0, ∞]. Therefore, we assume that for all i ∈ {1, . . . , m} there exists ri > 0 such that f i = I B(0,ri ) . Let z 0 ∈ H . Since H is bounded, there exists t > 0 such that t z 0 ∈ ∂ H . Thus, t z 0 ∈ 5i ∩ H for some i ∈ {1, . . . , n + 1}. Without loss of generality, we can assume that i = 1. Let r1 , . . . , rk be such that k \ t z 0 ∈ S1 (D) ∩ Sri (D) \
[
Si (D) .
i ∈{1,r / 1 ,...,rk }
i=1
By Lemma 3.3, we have H ⊂ S1 (D) ∩ Sr1 (D) ∩ · · · ∩ Srk (D). On the other hand, (54) implies that t z0 ∈
k \
5+ 1,ri .
i=1
Since each of the 5+ 1,ri is a half-space containing the origin, we have z0 ∈
k \ i=1
5+ 1,ri =
k \ i=1
s5+ 1,ri
(54)
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for all s > 0. Hence, we conclude from Lemma 4.3 that Z f 1 (z 1 − z 0 ) dz 1 = m t1 H ∩ B(z 0 , r1 ) t1 H
≤ m t1 S1 (D) ∩ B(z 0 , r1 ) ≤ m t1 S(D) ∩ B(0, r1 ) Z = f 1 (z 1 ) dz 1 . t1 S(D)
Now assume that the inequality is true for m − 1, and consider Z
Z
A :=
··· t1 H
Z ··· t1 H
f j (z j − z j−1 ) dz 1 · · · dz m
tm H j=1
Z =
m Y
m−1 Y
f j (z j − z j−1 )m tm H ∩ B(z m−1 , rm ) dz 1 · · · dz m−1 .
tm−1 H j=1
For any convex domain D ⊂ Rn containing the origin, consider the following Minkowski pseudonorm: o n 1 kxk D = inf{t > 0 : x ∈ t D} = inf t > 0 : x ∈ D . t The argument used to obtain (54) implies that for each z 0 ∈ H there exists i ∈ {1, . . . , n + 1} such that kz 0 k H = kz 0 k Si (D) and m tm H ∩ B(z 0 , rm ) ≤ m tm Si (D) ∩ B(z 0 , rm ) . Moreover, it is easy to check that if kyk S(D) = kzk Si (D) , 1 ≤ i ≤ n + 1, then m tm S(D) ∩ B(y, rm ) = m tm Si (D) ∩ B(z, rm ) . Therefore, the function g on R+ given by the formula g(kzk H ) = m tm S(D) ∩ B(y, rm ) , for any y such that kyk S(D) = kzk H , is well defined. Notice that g is continuous and nonincreasing with g(∞) = 0; so Z ∞ g(t) = I[0,s) (t) dν(s) 0
BRASCAMP-LIEB-LUTTINGER INEQUALITIES
123
for some nonnegative measure ν. Hence, applying Lemma 4.3 and the induction assumption, Z
m−1 Y
Z
A≤
··· t1 H ∞Z
Z
f j (z j − z j−1 )g(kz m−1 k H ) dz 1 · · · dz m−1
tm−1 H j=1
Z
=
m−1 Y
Z
··· 0
t1 H
tm−2 H
∞Z
Z
(s∧tm−1 )H j=1
Z
≤
f j (z j − z j−1 ) dz 1 · · · dz m−1 dν(s)
Z
··· 0
×
t1 S(D) m−1 Y
tm−2 S(D) (s∧tm−1 )S(D)
f 1 (z 1 )
f j (z j − z j−1 ) dz 1 · · · dz m−1 dν(s)
j=2
Z =
...
t1 S(D)
Z
f 1 (z 1 ) tm S(D)
m Y
f j (z j − z j−1 ) dz 1 · · · dz m .
j=2
This completes the proof of Theorem 1.2. 5. Applications In this section we give several applications of Theorems 1.2 and 1.3. Let X t be an n-dimensional symmetric α-stable process of order α ∈ (0, 2]. The process X t has stationary independent increments, and its transition density ptα (z, w) = f tα (z − w) is determined by its Fourier transform Z exp(−t|z|α ) = ei z·w f tα (w) dw. Rn
These processes have right continuous sample paths and scaling properties similar to those of the Brownian motion. The operator associated to X t is (−1)α/2 , where 1 is the Laplace operator in Rn . When α = 2, the process X t is just the n-dimensional Brownian motion Bt running at twice the speed. That is, if α = 2, then X t = B2t and so h −|z − w|2 i 1 . pt2 (z, w) = exp 4t (4πt)n/2 If 0 < α < 2, then X t = B2σt ,
(55)
where σt is a stable subordinator of index α/2 independent of Bt (see [8]). Therefore, Z ∞ α pt (z, w) = pu2 (z, w)gα/2 (t, u) du, 0
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where gα/2 (t, u) is the transition density of σt . Thus, for every positive t, ptα (z, w) = f tα (|z − w|) and the function f tα (r ) is decreasing. For any Borel set D ⊂ Rn , we denote by τ D,α the first exit time of X t from D. Some of our results below also hold for Schr¨odinger operators, and hence we present our heat kernel formula in such generality. Let V be a nonnegative continuous potential in D. We denote the Dirichlet heat kernel of (−1)α/2 + V on D by p αD,V (t, z, w) and its Green function by G αD,V (z, w). The potential theory of these processes has been extensively studied for several years. For some of the recent developments and basic properties of p αD,V (t, z, w) and G αD,V (z, w), we refer the reader to Z.-Q. Chen and R. Song [11], [12]. If V = 0, we just write p αD (t, z, w) for the heat kernel and G αD (z, w) for the Green function. Let us denote the space of C ∞ -functions with compact support in D by C0∞ (D). By the Feynman-Kac formula, if f ∈ C0∞ (D), we have Z f (w) p αD (t, z, w) dw D n Z t o = E z f (X t ) exp − V (X s ) ds ; τ D > t 0 h n Z t oi = E z f (X t )E z exp − V (X s ) ds ; τ D > t|X t 0 Z n Z t o α = pt (z, w) f (w)E z exp − V (X s ) ds ; τ D > t | X t = w dw. D
0
Therefore, we have n Z t o V (X s ) ds , τ D > t | X t = w . p αD,V (t, z, w) = ptα (z, w)E z exp − 0
Let us suppose now that D ⊂ Rn is a convex set of inradius r D . Take an increasS∞ ing sequence {Dk }∞ k=1 Dk = D, the closure of each Dk k=1 of convex sets such that is strictly contained in D, and the inradius of Dk is r D − r D /k. Since X t is a L´evy process with right continuous paths, we have o n Z t V (X s ) ds ; τ D > t | X t = w E z exp − 0 o n Z t = E z exp − V (X s ) ds ; X s ∈ D, ∀s ∈ [0, t] | X t = w 0
= lim lim E z k→∞ m→∞
n
m−1 t X exp − V (X it/m ) ; m
i=1
o X it/m ∈ Dk , i = 1, . . . , m − 1 | X t = w .
BRASCAMP-LIEB-LUTTINGER INEQUALITIES
125
Let 0 = t0 < t1 < · · · < tn < t. Then the conditional finite-dimensional distribution Pz 0 {X t1 ∈ dz 1 , . . . , X tn ∈ dz n | X t = w} is given by
n α (z , w) Y pt−t n n
ptα (z 0 , w)
ptαi −ti−1 (z i , z i−1 ).
i=1
Hence, for m ≥ 2, ptα (z, w)E z
n
m−1 o t X exp − V (X it/m ) ; X it/m ∈ Dk , i = 1, . . . , m − 1 | X t = w m
i=1
Z
t
= Dkm−1
e− m
Pm−1 i=1
m Y
V (z i )
α pt/m (z i − z i−1 ) dz 1 · · · dz m−1 ,
i=1
where z 0 = z and z m = w. We conclude that if z 0 = z and z m = w, then p αD,V (t, z, w) Z
t
= lim lim
k→∞ m→∞ D m−1 k
e− m
Pm−1 i=1
V (z i )
m Y
α pt/m (z i − z i−1 ) dz 1 · · · dz m−1 . (56)
i=1
We now combine this representation of the heat kernel (for V = 0) with Theorem 1.2 to obtain the following theorem, which extends the result of Ba˜nuelos and Kr¨oger [5] to symmetric stable processes. 5.1 Let D ⊂ Rn be a convex domain with finite inradius r D . Then, for all 0 < α ≤ 2, all z ∈ D, and all t > 0, THEOREM
Pz {τ D,α > t} ≤ P0 {τ S(D),α > t} = Pz {τ I (D),α > t},
(57)
where, as above, I (D) = (−r D , r D ) and S(D) = Rn−1 × I (D). Proof Let S(Dk ) = {(x1 , . . . , xn ) ∈ Rn : −r D + r D /k < xn < r D − r D /k}. From (56) we have, for z 0 ∈ D, Z Pz 0 {τ D,α > t} = p αD (t, z m , z 0 ) dz m D
Z = lim lim
k→∞ m→∞ Dk
...
Z
m Y Dk j=1
α pt/m (z j − z j−1 ) dz 1 · · · dz m .
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On the other hand, Theorem 1.2 and the fact that the transition density functions are radially symmetric and decreasing imply that Z Z Y m α lim lim ... pt/m (z j − z j−1 ) dz 1 · · · dz m k→∞ m→∞ Dk
Dk j=1
Z ≤ lim lim
k→∞ m→∞ S(Dk )
Z = S(D)
...
Z S(Dk )
α pt/m (z 1 )
m Y
α pt/m (z j − z j−1 ) dz 1 · · · dz m
j=2
p αS(D) (t, z m , 0) dz m
= P0 {τ S(D),α > t}. Note that (55) implies that P0 {τ S(D),α > t} = P0 {τ I (D),α > t}, and this concludes the proof of Theorem 5.1. As shown in the proof of Theorem 5.1, inequality (57) is equivalent to Z Z rD p αD (t, z, w) dw ≤ p αD (t, 0, w) dw D
−r D
for all z ∈ D and all t > 0. Upon integrating this inequality in t, we obtain for all z ∈ D, Z Z rD α G D (z, w) dw ≤ G αI (D) (0, w) dw, D
−r D
which extends Sperb’s result (see [23]) to symmetric stable processes. This inequality is equivalent to E z τ D,α ≤ E 0 τ I (D),α for all z ∈ D. In fact, it follows from (57) that, for all nonnegative increasing functions ϕ and all z ∈ D, we have E z ϕ(τ D,α ) ≤ E 0 ϕ(τ I (D),α ) . If λ D,α denotes the first Dirichlet eigenvalue of (−1)α/2 in D, it follows from the eigenfunction expansion of the heat kernel p αD (t, z, w) (see [11]) that 1 log Pz {τ D,α > t} t→∞ t
−λ D,α = lim
for all bounded domains D. Thus (57) implies that λ I (D),α ≤ λ D,α
(58)
BRASCAMP-LIEB-LUTTINGER INEQUALITIES
127
for any bounded convex domain of finite inradius r D . However, as mentioned earlier, if the domain is not bounded, then it is contained in H ×Rn−m , where H is a bounded convex set in Rm of inradius r D and (58) trivially holds. This gives the extension of Hersh’s [14] and Protter’s [20] result to symmetric stable processes. It is well known that π2 λ I (D),2 = 2 . 4r D However, for α ∈ (0, 2), very little seems to be known about the explicit value of λ I (D),α . We refer the reader to [6] and [19] for some estimates on these eigenvalues. We now use Theorem 1.3 to obtain a new set of isoperimetric-type inequalities that include sharper versions of Theorem 5.1 and its consequences for convex planar domains. Suppose that D is an unbounded planar convex domain of finite inradius r D ; then D is contained in a strip S of inradius r D and we have p αD (t, w, z) ≤ p αS (t, w, z) for all z, w ∈ D. Since in this case C(D) = S(D), this inequality trivially implies inequalities (59) – (66) for this type of domain after a rotation and a translation. Thus, for the remainder of this section, we may assume that D is bounded and that the largest circle contained in D is centered at the origin. We remark here that all the inequalities below are new even in the Brownian motion case. Taking q1 = · · · = qm = 1 in Theorem 1.3, we can follow the proof of Theorem 5.1 to obtain the following sharper version of (57). THEOREM 5.2 Let D ⊂ R2 be a convex domain of finite inradius r D and finite diameter d D . Then, for all 0 < α ≤ 2, all z ∈ D, and all t > 0,
Pz {τ D,α > t} ≤ P0 {τC(D),α > t}, where C(D) = R × I (D) ∩ B(0, d D − r D ).
(59)
As before, integrating (59) in time, we obtain for all z ∈ D, Z Z α G αD (z, w) dw ≤ G C(D) (0, w) dw, D
C(D)
which improves Sperb’s result for n = 2. In addition, for all nonnegative increasing functions ϕ, it follows that E z ϕ(τ D,α ) ≤ E 0 ϕ(τC(D),α ) .
´ ´ PEDRO J. MENDEZ-HERN ANDEZ
128
Our next result, another corollary of Theorem 1.3, gives pointwise inequalities for heat kernels and Green’s functions of Schr¨odinger operators of symmetric stable processes. However, these inequalities are new even if we replace C(D) by S(D) and take V = 0. 5.3 Let D ⊂ R2 be a convex domain of finite inradius r D and finite diameter d D . Let V be a nonnegative continuous potential that is radially symmetric, decreasing, and defined in all of R2 . Let Hi, j and Cs (D) be as in the introduction. If z ∈ Hi, j , then for all t > 0 and all 0 < α ≤ 2, p αD,V (t, z, 0) ≤ min pCα s (D),V (t, z, 0) : s ∈ {i, j} (60) THEOREM
and
α G αD,V (z, 0) ≤ min G C (z, 0) : s ∈ {i, j} . s (D),V
(61)
Proof It is clear that (61) follows from (60) upon integrating in time. Since exp(−t V (x)) is radially symmetric and decreasing for all t > 0, we can take α q1 (x) = · · · = qm−1 (x) = exp −(t/m)V (x) , qm (x) = pt/m (x) in Theorem 1.3 and apply (56) to obtain (60). If we denote the first Dirichlet eigenvalue of the operator (−1)α/2 + V in D by λαD,V , it follows from (60) that α λC(D),V ≤ λαD,V . (62) Taking α = 2 and V = 0, we have λ D ≥ λC(D) >
1 π2 1 + 2 4 rD (d D − r D )2
for all convex domains D of finite inradius r D and finite diameter d D , which improves Protter’s result in the two-dimensional case. Moreover, if we combine (60) and (61) with the induction step in the proof of Theorem 1.3, we obtain the following sharper version of the results of Ba˜nuelos, Carroll, and Housworth in [3]. As before, inequality (63) is new even if we replace C(D) by S(D) and take V = 0. THEOREM 5.4 Let D and V be as in the statement of Theorem 5.3. Let 8 : R+ → R+ be an increasing function. Then, for all 0 < α ≤ 2 and all t > 0, Z Z α 8 p αD,V (t, z, 0) dz ≤ 8 pC(D),V (t, z, 0) dz (63) D
C(D)
BRASCAMP-LIEB-LUTTINGER INEQUALITIES
and
Z D
129
Z
8 G αD,V (z, 0) dz ≤
C(D)
α 8 G C(D),V (z, 0) dz.
(64)
As explained in [24], the amount of heat contained in D at time t when D has temperature 1 at t = 0 and the boundary of D is kept at temperature zero at all times is given by Z Z Q t (D) =
p D (t, z, w) dz dw,
D
D
where in the case of α = 2 we simply write p D (t, z, w) for p 2D (t, z, w). Also (see [18]), the torsional rigidity of D is given by Z ∞ Z Z Q t (D)dt = G D (z, w) dz dw. 0
D
D
Hence, the next result, which follows after taking p1 = q1 = · · · = qm = 1 in Theorem 1.3, gives new inequalities for the heat content and torsional rigidity of any bounded convex domain of finite inradius r D and finite diameter d D . THEOREM 5.5 Let D and V be as in the statement of Theorem 5.3. Then, for all 0 < α ≤ 2 and all t > 0, Z Z Z Z α α p D,V (t, z, w) dz dw ≤ pC(D),V (t, z, w) dz dw (65) D
and
D
Z Z D
D
C(D) C(D)
G αD,V (z, w) dz dw ≤
Z
Z
C(D) C(D)
α G C(D),V (z, w) dz dw.
(66)
Acknowledgment. I would like to thank Professor Rodrigo Ba˜nuelos, my academic advisor, for his guidance on this paper. References [1]
C. BANDLE, Isoperimetric Inequalities and Applications, Monog. Stud. Math. 7,
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˜ R. BANUELOS and T. CARROLL, Brownian motion and the fundamental frequency of a
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˜ R. BANUELOS, T. CARROLL, and E. HOUSWORTH, Inradius and integral means for
Pitman, Boston, 1980. MR 81e:35095 93, 98 drum, Duke Math. J. 75 (1994), 575 – 602. MR 96m:31003 93, 95
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Green’s Functions and conformal mappings, Proc. Amer. Math. Soc. 126 (1998), 577 – 585. MR 98g:30016 93, 96, 98, 99, 128 ˜ R. BANUELOS and E. HOUSWORTH, An isoperimetric-type inequality for integrals of Green’s functions, Michigan Math. J. 42 (1995), 603 – 611. MR 96j:30038 93
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´ ´ PEDRO J. MENDEZ-HERN ANDEZ ¨ ˜ R. BANUELOS and P. KROGER , Isoperimetric-type bounds for solutions of the heat
equation, Indiana Univ. Math. J. 46 (1997), 83 – 91. MR 98k:35081 93, 95, 96, 98, 125 ˜ ´ ´ R. BANUELOS, R. LATAŁA, and P. J. MENDEZ-HERN ANDEZ ,A Brascamp-Lieb-Luttinger-type inequality and applications to symmetric stable processes, Proc. Amer. Math. Soc. 129 (2001), 2997 – 3008. MR 2002c:60125 93, 94, 95, 96, 100, 127 ˜ ´ ´ R. BANUELOS and P. J. MENDEZ-HERN ANDEZ , Sharp inequalities for heat kernels of Schr¨odinger operators and applications to spectral gaps, J. Funct. Anal. 176 (2000), 368 – 399. MR 2001f:35096 99 R. M. BLUMENTHAL and R. K. GETOOR, Some theorems on symmetric stable processes, Trans. Amer. Math. Soc. 95 (1960), 263 – 273. MR 22:10013 123 H. J. BRASCAMP, E. H. LIEB, and J. M. LUTTINGER, A general rearrangement inequality for multiple integrals, J. Funct. Anal. 17 (1974), 227 – 237. MR 49:10835 94 R. CARMONA, W. C. MASTERS, and B. SIMON, Relativistic Schr¨odinger operators: Asymptotic behavior of the eigenfunctions, J. Funct. Anal. 91 (1990), 117 – 142. MR 91i:35139 93, 99 Z.-Q. CHEN and R. SONG, Intrinsic ultracontractivity and conditional gauge for symmetric stable processes, J. Funct. Anal. 150 (1997), 204 – 239. MR 98j:60103 124, 126 , Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann. 312 (1998), 465 – 501. MR 2000b:60179 124 H. GEMAN, D. B. MADAN, and M. YOR, Time changes hidden in Brownian subordination, preprint, 2000, http://rhsmith.umd.edu/Finance/dmadan 99 J. HERSH, Sur la fr´equence fondamentale d’une membrane vibrante: e´ valuations par d´efaut et principe de maximum, Z. Angew. Math. Phys. 11 (1960), 387 – 413. MR 23:A2622 94, 96, 127 J. M. LUTTINGER, Generalized isoperimetric inequalities, J. Math. Phys. 14 (1973), 586 – 593. MR 49:1969 94 , Generalized isoperimetric inequalities, II, J. Math. Phys. 14 (1973), 1444 – 1447. MR 49:6012 94 , Generalized isoperimetric inequalities, III, J. Math. Phys. 14 (1973), 1448 – 1450. MR 49:6012 94 ´ G. POLYA and G. SZEGO¨ , Isoperimetric Inequalities in Mathematical Physics, Ann. of Math. Stud. 27, Princeton Univ. Press, Princeton, 1951. MR 13:270d 93, 94, 98, 129 S. M. POZIN and L. A. SAKHNOVICH, A two-sided bound on the lowest eigenvalue of an operator that characterizes stable processes, Theory Probab. Appl. 36 (1992), 385 – 388. MR 92k:60176 127 M. H. PROTTER, A lower bound for the fundamental frequency of a convex region, Proc. Amer. Math. Soc. 81 (1981), 65 – 70. MR 82b:35113 93, 94, 95, 96, 111, 127 R. T. ROCKAFELLAR, Convex Analysis, Princeton Math. Ser. 28, Princeton Univ.
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Press, Princeton, 1970. MR 43:445 104 [22]
K. SATO, L´evy Processes and Infinitely Divisible Distributions, Cambridge Stud. Adv.
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R. SPERB, Maximum Principles and Their Applications, Math. Sci. Engrg. 157,
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Math. 68, Cambridge Univ. Press, Cambridge, 1999. CMP 1 739 520 99 Academic Press, New York, 1981. MR 84a:35033 95, 96, 98, 126 Z. 215 (1994), 437 – 464. MR 94m:58237 98, 129
Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112-0090;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 1,
UNIFORM POINTWISE BOUNDS FOR MATRIX COEFFICIENTS OF UNITARY REPRESENTATIONS AND APPLICATIONS TO KAZHDAN CONSTANTS HEE OH
Abstract Let k be a local field of characteristic not 2, and let G be the group of k-rational points of a connected reductive linear algebraic group defined over k with a simple derived group of k-rank at least 2. We construct new uniform pointwise bounds for the matrix coefficients of all infinite-dimensional irreducible unitary representations of G. These bounds turn out to be optimal for SLn (k), n ≥ 3, and Sp2n (k), n ≥ 2. As an application, we discuss a simple method of calculating Kazhdan constants for various compact subsets of semisimple G. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Temperedness and Howe’s strategy . . . . . . . . . . . . . . . . . . . . 3. The subgroups Hα , strongly orthogonal systems, and the Harish-Chandra function 4PGL2 (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Temperedness of ρ| Hα . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Uniform pointwise bound ξS . . . . . . . . . . . . . . . . . . . . . . 6. Representations with the slowest decay . . . . . . . . . . . . . . . . . . 7. Uniform L p -bound for matrix coefficients: p K (G) . . . . . . . . . . . . 8. Kazhdan constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Maximal strongly orthogonal systems . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
133 139 148 156 163 170 176 178 187 190
1. Introduction Let k be a local field, and let G be the group of k-rational points of a connected reductive linear algebraic group over k with k-semisimple rank(G) ≥ 2. Let K be a good maximal compact subgroup of G. For a unitary representation ρ of G, a vecDUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 1, Received 17 May 2001. 2000 Mathematics Subject Classification. Primary 22E46, 22E50; Secondary 22D10, 22E30, 22E35. Author’s work partially supported by the Lady Davis Foundation and the German Israel Foundation. 133
134
HEE OH
tor v in ρ is called K -finite if the subspace spanned by K v is finite-dimensional. We use the term K matrix coefficients (resp., K -finite matrix coefficients) of ρ to refer to its matrix coefficients with respect to K -invariant (resp., K -finite) unit vectors. Following [BT], we denote by G + the subgroup generated by the unipotent k-split subgroups of G. The main goal of the present paper is to construct a class of uniform pointwise bounds for the K -finite matrix coefficients of all infinite-dimensional irreducible unitary representations of G or, more generally, of all unitary representations of G without a nonzero G + -invariant vector. Let A be a maximal k-split torus, and let A+ be the closed positive Weyl chamber of A such that the Cartan decomposition G = K A+ K holds, where is a finite subset of the centralizer of A (see Section 2.1). Denote by 8 the set of nonmultipliable roots of A, and denote by 8+ the set of positive roots in 8. A subset S of 8+ is called a strongly orthogonal system of 8 if any two distinct elements α and β of S are strongly orthogonal, that is, if neither of α±β belongs to 8. The notation 4PGL2 (k) denotes the Harish-Chandra function of PGL2 (k) (see Section 2.2). For simplicity, we use the notation x 0 . 4PGL2 (k) (x) := 4PGL2 (k) 0 1 We denote by Z (G) the center of G and by ss rank the semisimple rank of G. 1.1 Let k be any local field with char(k) 6 = 2 and k 6 = C. Let G be the group of k-rational points of a connected reductive linear algebraic group over k with k-ss rank(G) ≥ 2 and G/Z (G) almost k-simple. Let S be a strongly orthogonal system of 8. Then for any unitary representation ρ of G without a nonzero G + -invariant vector and with K -finite unit vectors v and w, Y hρ(g)v, wi ≤ [K : K ∩ d K d −1 ] · dimhK vi dimhK wi 1/2 4PGL2 (k) α(a) THEOREM
α∈S
for any g = k1 adk2 ∈ K A+ K = G. When k is archimedean, we have = {e}, and for nonarchimedean k, K ∩ gK g −1 is an open compact subgroup of K for any g ∈ G, and hence [K : K ∩ gK g −1 ] < ∞. For k nonarchimedean, we fix a uniformizer q so that |q| = p −1 , where p is the cardinality of the residue field of k. The Harish-Chandra function 4PGL2 (k) has the formula Z π/2 2 −1/2 cos t 2 2 4PGL2 (R) (x) = √ + sin t dt for x ≥ 1, x2 π x 0
UNIFORM POINTWISE BOUNDS
135
and for k nonarchimedean, 1 n( p − 1) + ( p + 1) 4PGL2 (k) (q n ) = √ n p p+1
for n ∈ N.
Note that 4PGL2 (k) (q n ) = 4PGL2 (k) (q −n ) for any n ∈ Z. THEOREM 1.2 Let G be a connected reductive complex algebraic group with semisimple rank at least 2, and let G/Z (G) be almost simple. Let S be a strongly orthogonal system of 8. Let ρ be any unitary representation of G without a nonzero G + -invariant vector. (1) If G/Z (G) Sp2n (C) (locally), then for any K -finite unit vectors v and w, Y hρ(g)v, wi ≤ dimhK vi dimhK wi 1/2 4PGL2 (C) α(a) α∈S
(2)
for any g = k1 ak2 ∈ K A+ K = G. If G/Z (G) ∼ = Sp2n (C) (locally), let n α = 1/2 if α is a long root and n α = 1 otherwise. Then for any K -invariant unit vectors v and w, Y n α hρ(g)v, wi ≤ 4PGL α(a) 2 (C) α∈S
for any g = k1 ak2 ∈ K A+ K = G. The Harish-Chandra function 4PGL2 (C) is as follows: 1 4PGL2 (C) (x) = πx
π/2 cos2 t
Z 0
x2
+ sin2 t
−1
sin (2t) dt
for x ≥ 1.
Remark. We have the following remarks. (1) Note that in the above theorems G/G + is finite mod center. It follows that any infinite-dimensional irreducible unitary representation of G has no nonzero G + -invariant vector. (2) Note that G = G + in the case when G is almost k-simple and simply connected. If k = R and G is semisimple, then G + coincides with the connected component of the identity in G. To simplify the explanation, we assume that G/Z (G) Sp2n (C) in the rest of the introduction. Definition For a strongly orthogonal system S of 8, we set a bi-K -invariant function ξS of G
136
HEE OH
as follows: ξS (k1 adk2 ) =
Y
4PGL2 (k) α(a)
for any k1 adk2 ∈ K A+ K = G.
α∈S
By Theorems 1.1 and 1.2, ξS presents a uniform pointwise bound for all K matrix coefficients (resp., K -finite matrix coefficients) of G (resp., up to a constant). Properties of ξS The following are additional properties of ξS : (1) 0 < ξS (g) ≤ 1 for any g ∈ G; (2) for any > 0, there are constants d1 > 0 and d2 () > 0 such that d1
Y
−1/2 Y −(1/2)+ |α(a)| ≤ ξS (g) ≤ d2 () |α(a)|
α∈S
α∈S
for any g = k1 adk2 ,
(3)
where | · | denotes the absolute value on k in the sense of [We, Chap. 1]; ξS (g) = 1 if and only if α(g) = 1 for all α ∈ S .
A strongly orthogonal system S is called maximal if the coefficient of each simple P P root in the formal sum α∈S α is not less than the one in α∈O α for any strongly orthogonal system O of 8. A maximal strongly orthogonal system for each irreducible root system has been constructed in [Oh1] (see the appendix for the list). Let Q denote a maximal strongly orthogonal system of 8. In view of the inequality in (2), the uniform pointwise bound function ξQ gives the sharpest bound in this construction. We remark that in general there exist more than one maximal strongly orthogonal system P in 8. Note, however, that the formal sum η(8) := (1/2) α∈Q α, which determines the decay rate of ξQ , does not depend on the choice of a maximal strongly orthogonal system. Moreover, it turns out that for G = SLn (k) or Sp2n (k), ξQ is in fact the best possible uniform pointwise bound for K -finite matrix coefficients; more precisely, there exists an irreducible class-one unitary representation of G whose K matrix coefficient 1+ is bounded below by the function ξQ up to some constant. In the following theorem, ¯ the group Sp2n (k) is defined by the bilinear form 0 In , where I¯n denotes the skew diagonal (n × n) identity matrix.
− I¯n 0
THEOREM 1.3 Let G be either SLn (k), n ≥ 3, or Sp2n (k), n ≥ 2, char k 6 = 2, k 6= C. Let P be the maximal parabolic subgroup of G which stabilizes ke1 , and let v be a unique
UNIFORM POINTWISE BOUNDS
137
K -invariant unit vector in IndGP (1). Then for any > 0, there exists a constant C depending on such that
1+ C · ξQ (g) ≤ IndGP (1)(g)v, v ≤ ξQ (g) for any g ∈ G. For Sp2n (C), see Theorem 6.6. In our proofs of Theorems 1.1 and 1.2, the following definition is a crucial notion. Definition Let M be the group of k-rational points of a connected reductive linear algebraic group over k with a good maximal compact subgroup K . A unitary representation ρ of M is said to be tempered if for any K -finite unit vectors v and w, hρ(g)v, wi ≤ dimhK vi dimhK wi 1/2 4 M (g) for any g ∈ M, where 4 M denotes the Harish-Chandra function of M (see Section 2.2). A unitary representation ρ being tempered is equivalent to saying that ρ is weakly contained in the regular representation of M. (See Theorem 2.4 for more equivalent definitions for temperedness.) ˜ Let G˜ be the underlying algebraic group of G; that is, G = G(k). Also, let A˜ ˜ be the maximal k-split torus of G˜ such that A = A(k). Denote by HS the group of k-rational points of the connected semisimple k-subgroup generated by the 1dimensional root subgroups U˜ ±α corresponding to ±α, α ∈ S , and denote by G S the group of k-rational points of the connected reductive k-subgroup generated by U˜ ±α , α ∈ S , and A˜ (see Section 5.1). The following theorem then plays a key role in the proofs of Theorems 1.1 and 1.2. THEOREM 1.4 Let S be any strongly orthogonal system of 8. Then for any unitary representation ρ of G without a nonzero G + -invariant vector, the restrictions ρ| HS and ρ|G S are tempered. In fact, ξS | HS = 4 HS and ξS |G S = 4G S .
Note that for any α ∈ 8, the singleton {α} is a strongly orthogonal system. We set Hα = H{α} . In particular, Hα is isomorphic to either SL2 (k) or PGL2 (k) (see Section 3.1). Hence the following corollary is a special case of Theorem 1.4. COROLLARY 1.5 Let α ∈ 8 be any root. Then for any unitary representation ρ of G without a nonzero G + -invariant vector, the restriction ρ| Hα is tempered.
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Theorems 1.1 and 1.2 cover all the groups of k-rational points of a connected almost simple algebraic k-group (char k 6 = 2) with Kazhdan property (T), except for the two rank 1 real groups Sp(1, n), F4−20 . In fact, Theorem 1.1 also holds for Sp(1, n) (see Theorem 4.11). We remark that if one can provide an analogue of Theorem 4.8 for F4−20 , the same theorem holds for this group as well. The pointwise bound ξS provides us with a simple and general method of calculating Kazhdan constants (see Section 8.1 for a definition) for various compact subsets of semisimple G, in particular for any compact subset properly containing K . For instance, in SLn (R), n ≥ 3, for any m ∈ N, the number 0.08/m is a Kazhdan constant with respect to the Kazhdan set {SO(2), diag(41/m , 4−1/m )} (embedded in the left upper corner of SLn (R); see Example 8.7.1). We also have the following interesting example. THEOREM
1.6
We have inf
inf
inf
p=prime n≥3 s ∈SL / n (Z p )
κ SLn (Q p ), {SLn (Z p ), s} > 0.10,
where κ(G, Q) is the best Kazhdan constant for Q; that is, κ(G, Q) = inf max kρ(g)v − vk, g∈Q
where the infimum is taken over all unitary representations ρ of G without a nonzero invariant vector and for all unit vectors v of ρ. The problem of calculating Kazhdan constants was first raised by J.-P. Serre (see [Bu], [HV]). Kazhdan constants for the group of k-rational points of a semisimple algebraic group over k, and for its lattices, have been obtained with special choices of Kazhdan sets (see [Bu], [CMS], [Sh2], [Sh1], [Zu]). The paper is organized as follows. In Section 2, after recalling Cartan decomposition and the definition of the Harish-Chandra functions, we study when a unitary representation of a reductive algebraic group is tempered and recall Howe’s strategy. In Section 3, we study strongly orthogonal systems, subgroups Hα , and the HarishChandra function of PGL2 (k). In Section 4, we show the temperedness of ρ| Hα . In Section 5, we prove Theorem 1.4 as well as our main results on uniform pointwise bounds. In Section 6, we show that the bounds given in Section 5 are optimal for SLn (k) and for Sp2n (k). In Section 7, we give some upper bounds for the constant p K (G) as an application of our main theorems. Finally, in Section 8, we discuss another application to computation of Kazhdan constants. Some of the above results for k = R were announced in [Oh1], where the application of these results to the classification of non-Riemannian homogeneous spaces
UNIFORM POINTWISE BOUNDS
139
not admitting compact quotients by discrete subgroups (cf. [Ma2], [Oh1]) is discussed as well. Let G be a connected almost simple simply connected Q-group, and let 0 ⊂ G(Q) be a congruence subgroup of G. Combined with some results of L. Clozel and E. Ullmo in [CU, Th. 1.1], in the case of p-adic fields, Theorem 1.1 yields an application in obtaining the equidistribution of Hecke points on G(R)/ 0 with the rate estimate (see [COU]; see also [GO] and [Oh2] for other applications).
2. Temperedness and Howe’s strategy 2.1. Cartan decomposition Let k be a local field with the standard absolute value | · | in the sense of [We, Chap. 1]. Let G˜ be a connected linear reductive algebraic group defined over a local field k, ˜ and let G = G(k). Let A˜ be a maximal k-split torus, and let B˜ be a minimal parabolic ˜ ˜ Set A = A(k) ˜ ˜ k-subgroup of G containing A. and B = B(k). Denote by 80 the set of roots of A˜ in G˜ and by 8 the set of nonmultipliable roots in 80 with the ordering ˜ Let X ( A) ˜ denote the set of characters of A˜ defined over k whose ordering given by B. is induced from 8. Denote by X + (resp., 8+ ) the set of positive characters (resp., ˜ with respect to that ordering. roots) in X ( A) If k is archimedean, that is, isomorphic to R or C, we set k 0 = {x ∈ R | x ≥ 0}
and
kˆ = {x ∈ R | x ≥ 1}.
When k is nonarchimedean, we fix a uniformizer q of k such that |q|−1 is the cardinality of the residue field of k, and we set k 0 = {q n | n ∈ Z}
and
kˆ = {q −n | n ∈ N}.
We set ˜ A0 = a ∈ A α(a) ∈ k 0 for each α ∈ X ( A) and A+ = a ∈ A α(a) ∈ kˆ for each α ∈ 8+ . Equivalently, A+ = {a ∈ A0 | |α(a)| ≥ 1 for each α ∈ 8+ }. We call A+ a positive Weyl chamber of G. ˜ and let Z = Z˜ (k). Since X ( Z˜ ) can be Let Z˜ denote the centralizer of A˜ in G, ˜ in a natural way, it has an induced ordering from this considered as a subset of X ( A) inclusion. Define Z + = z ∈ Z |α(z)| ≥ 1 for each α ∈ X (Z )+
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and Z 0 = z ∈ Z |α(z)| = 1 for each α ∈ X (Z )+ . For any subgroup H of G, N G (H ) denotes the normalizer of H , C G (H ) denotes the centralizer of H , and Z (H ) denotes the center of H . PROPOSITION
There exists a maximal compact subgroup K of G such that the following are true: (1) N G (A) ⊂ K A; (2) the Cartan decomposition G = K (Z + /Z 0 )K and the Iwasawa decomposition G = K (Z /Z 0 )Ru (B) hold in the sense that, for any g ∈ G, there are elements a ∈ Z + (unique up to mod Z 0 ) and b ∈ Z (unique up to mod Z 0 ) such that g ∈ K a K and g ∈ K b Ru (B); (3) for any subset 1 of the set of simple roots of 8 and for the subgroup M := Z G {a ∈ A | α(a) = 1 for all α ∈ 1} , M ∩ K satisfies properties (1) and (2) with respect to (M, M ∩ A). See [GV, Sec. 2.2] for the archimedean case, and see [BT], [Ti2], and [Si] for the nonarchimedean case. In general, the positive Weyl chamber A+ has finite index in (Z + /Z 0 ). Hence for some finite subset ⊂ Z + , G = K (A+ )K , that is, for any g ∈ G, there exist unique elements a ∈ A+ and d ∈ such that g ∈ K ad K . A maximal compact subgroup K is called a good maximal compact subgroup of G if it satisfies the properties listed in the above proposition. Remark. We have G = K A+ K and G = K A0 Ru (B) (1) if k is archimedean, that is, k ∼ = R or C; (2) if G is split over k; or (3) if G is quasi-split and split over an unramified extension over a nonarchimedean local field k; for example, G = SU( f ), where f is a hermitian form of dimension 2n or 2n + 1 over an unramified quadratic extension over k with Witt index n, so that G ∼ = SU(n, n) or G ∼ = SU(n, n + 1). Example 2.1.1 For G = SLn (k), let A be the subgroup of all diagonal matrices, and let B be the subgroup of all upper triangular matrices. If k is archimedean, set a1 + A = 0
K = {g ∈ G | t gg ¯ = 1}, 0 .. ∈ G | ai ∈ R, a1 ≥ · · · ≥ an > 0 ; . an
UNIFORM POINTWISE BOUNDS
141
otherwise, set K = SLn (O ), where O is the ring of integers of k, and k 1 q + A = 0
0 ..
.
∈ G | ki ∈ Z, k1 ≤ · · · ≤ kn
q kn
.
Note that the condition on the ki ’s is equivalent to saying that the norms of the q ki ’s are decreasing as the index i increases. Then K is a good maximal compact subgroup of G, and G = K A+ K . Example 2.1.2 Let D be a central simple division algebra of degree m over a nonarchimedean local field k, and let G = SLn (D) be the group of all (n × n)-matrices with entries in D which have reduced norm 1. We may choose A to be the group of all diagonal matrices in G with entries in k. Then Z is the full diagonal group of G. If d 6= 1, then Z 6 = A. Denote by q D a uniformizer of D; that is, any element x in D ∗ can be written k u for some integer k and a unit u in D. Then the representatives of Z /Z can as q D + 0 be taken as k1 0 qD . .. ∈ G | ki ∈ Z, k1 ≤ · · · ≤ kn . kn 0 qD Here
k 1 q + A = 0
0 ..
.
∈ G | ki ∈ Z, k1 ≤ · · · ≤ kn
q kn
,
where q is a uniformizer of k, and k 1 qD = 0
0 ..
. kn qD
where m is the degree of D over k.
∈ G | −m + 1 ≤ k1 ≤ · · · ≤ kn
,
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2.2. The Harish-Chandra function 4G We denote by δ B the modular function of B; in particular, for a ∈ A0 , Y δ B (a) = |α(a)|m α , all positive α∈80
where m α denotes the multiplicity of α. The Harish-Chandra function 4G is defined by Z 4G (g) =
δ B (gk)−1/2 dk.
K
As is well known, 4G is the diagonal matrix coefficient g 7 → hIndG B (1)(g) f 0 , f 0 i, G where Ind B (1) is the representation that is unitarily induced from the trivial representation 1 and f 0 is its unique (up to scalar) K -invariant unit vector. In fact, f 0 is given by 1/2 f 0 (kb) = δ B (b) for k ∈ K , b ∈ B. We list some well-known properties of 4G (see [Wa], [GV], [Ha2]) which are frequently used in this paper. PROPOSITION
We have the following: (1) 4G is a continuous bi-K Z (G)-invariant function of G with values in (0, 1]; (2) for any > 0, there exist constants c1 and c2 () such that −1/2
c1 δ B (3)
−1/2+
(b) ≤ 4G (b) ≤ c2 () δ B
(b)
for all b ∈ B;
4G is L 2+ (G/Z (G))-integrable for any > 0.
2.3 We now recall the definition of strongly L p (cf. [Li]). Definition For a locally compact group M, a (continuous) unitary representation ρ of M is said to be strongly L p if there is a dense subset V in the Hilbert space attached to ρ such that for any x and y in V , the matrix coefficient g 7 → hρ(g)x, yi lies in L p (M). We say ρ is strongly L p+ if it is strongly L q for any q > p. For an irreducible unitary representation ρ, the center Z (M) acts by a character (Schur’s lemma). Hence for any vectors v and w of ρ, [g] 7→ |hρ(g)v, wi| is a welldefined function on M/Z (M). Therefore the notion of its matrix coefficient being in L p (M/Z (M)) and that of ρ being strongly L p (M/Z (M)) is well defined.
UNIFORM POINTWISE BOUNDS
143
Since the matrix coefficients of a unitary representation with respect to unit vectors are bounded by 1, a strongly L q representation is also strongly L p for any p ≥ q. Any unitary representation ρ of G is decomposed into a direct integral R X ρx dµ(x) of irreducible unitary representations of G for some measure space (X, µ). (We refer to [Zi, Sec. 2.3] or [Ma1] for a more detailed account of the direct R integral theory.) If ρ has no invariant vector and v = X vx dµ(x) is a K -invariant unit vector of ρ, then for almost all x ∈ X , ρx is nontrivial and vx is a K -invariant unit vector. We say that ρ is weakly contained in a unitary representation σ of G if any diagonal matrix coefficients of ρ can be approximated, uniformly on compact sets, by convex combinations of diagonal matrix coefficients of σ (see [Fe]; also see [CHH], [Sh2]). Note that ρ is weakly contained in a countable direct sum ∞ · ρ, and vice versa.
2.4. Temperedness Definition A unitary representation ρ of G is said to be tempered if for any K -finite unit vectors v and w, hρ(g)v, wi ≤ dimhK vi dimhK wi 1/2 4G (g) for any g ∈ G, where hK vi denotes the subspace spanned by K v and hK wi is denoted similarly. The following theorem establishes equivalent definitions of a tempered unitary representation of a reductive algebraic group over a local field, generalizing the results in [CHH] for the semisimple case. THEOREM
R For any unitary representation ρ = X ρx d x of G, the following are equivalent: (1) for almost all x ∈ X , the (irreducible) representation ρx is strongly L 2+ (G/Z (G)); (2) ρ is weakly contained in the regular representation L 2 (G); (3) for almost all x ∈ X , ρx is weakly contained in the regular representation L 2 (G); (4) ρ is tempered; (5) for almost all x ∈ X , ρx is tempered. In the case when G is semisimple, the above is equivalent to saying that ρ is strongly L 2+ .
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Proof The equivalence of (2) and (3) is well known (e.g., see [Zi, Prop. 7.3.8]). We show that (2) ⇒ (4) ⇒ (1) ⇒ (2) and (3) ⇒ (5) ⇒ (1) ⇒ (2). The directions (2) ⇒ (4) and (3) ⇒ (5) follow from [CHH, Th. 2]. Even though it is assumed that G is semisimple in [CHH, Th. 2], the proof works for any reductive group case as well without any change. To see the direction (4) ⇒ (1), we may assume that ρx is weakly contained in ρ (up to equivalence) for each x ∈ X since G is of type I (see Section 2.6). Then by [Ho, Lem. 6.2], (4) implies that for almost all x ∈ X , the K -finite matrix coefficients of ρx are bounded by 4G up to a constant multiple. Since 4G is L 2+ (G/Z (G))integrable for any > 0 and the K -finite vectors of ρx are dense by the Peter-Weyl theorem, this proves that (4) implies (1). The direction (5) ⇒ (1) clearly follows from the above argument. It is now enough to show the direction (1) ⇒ (2). Since (2) and (3) are equivalent, we may assume that ρ is irreducible. For the semisimple case, it is a direct consequence of [CHH, Th. 1]. We now make some modification of the proof in [CHH] for our claim. For a unitary representation ρ and its attached Hilbert space H , define an operator ρ( f ) : H → H for f ∈ Cc (G) as follows: Z hρ( f )v, wi = f (g)hρ(g)v, wi dg for v, w ∈ H . G
By [Ey], (2) is equivalent to saying that for any f ∈ Cc (G), kρ( f )k ≤ kλ( f )k, where λ denotes the regular representation L 2 (G), kρ( f )k denotes the operator norm of ρ( f ), and kλ( f )k is denoted similarly. Let µ1 and µ2 be Haar measures on Z (G) and G/Z (G), respectively. Since Z (G) is a normal subgroup of G, without loss of generality, we may assume that G = Z (G) × G/Z (G) and dg = dµ1 × dµ2 (cf. [Pa, Sec. 1.11]). For f ∈ Cc (G) such that supp f = Y1 Y2 , where Y1 ⊂ Z (G) and Y2 ⊂ G/Z (G) are compact subsets, Z Z Z f (g) dg = f (zh) dµ1 (z) dµ2 (h). G
Y2
Y1
Then the Haar measure µ2 grows at most exponentially; hence, for some constants C and M, µ2 (Y2n ) ≤ C · M n for any n ≥ 1. On the other hand, since Z (G) is an abelian locally compact group, the Haar measure µ1 grows polynomially (cf. [Pa, Th. 6.17]); hence for some d and r , µ1 (Y1n ) ≤ d · nr
for any n ≥ 1.
As is shown in [CHH, p. 102], 1/4n
kλ( f )k = lim k( f ∗ ∗ f )∗2n k2 n→∞
UNIFORM POINTWISE BOUNDS
145
and kρ( f )k = sup lim
Z
θ ∈ρ n→∞
1/4n
( f ∗ ∗ f )∗2n (x) ρ(x)θ, θ dµ(x) . G
Let v be a (unit) vector in ρ such that the matrix coefficient hρ(x)v, vi is L 2+ (G/Z (G))-integrable. Since Gv is a dense subset in ρ (ρ being irreducible), it suffices to consider vectors θ in G · v in the above formula for kρ( f )k. For θ ∈ Gv, set ψ(x) = hρ(x)θ, θi. Then we have Z Z Z |ψ(x)|2 dµ(x) ≤ 1 dµ1 (x) · |ψ(x)|2 dµ2 (x) (Y1 ×Y2 )n
Y1n
Y2n
≤ µ1 (Y1n ) ·
Z Y2n
2/(2+) |ψ(x)|2+ dµ2 (x) µ2 (Y2n )/(2+)
≤ µ1 (Y1n ) µ2 (Y2n )/(2+)
Z Y2n
2/(2+) |ψ(x)|2+ dµ2 (x)
≤ dnr (C M n )/(2+) kψk2L 2+ (G/Z (G)) . By plugging this into the formula for kρ( f )k and extracting roots, we obtain kρ( f )k ≤ M /(8(2+)) lim inf k( f ∗ ∗ f )∗2n k2
1/(4n)
n→∞
;
since is arbitrary, kρ( f )k ≤ lim inf k( f ∗ ∗ f )∗2n k2 n→∞
1/(4n)
.
Hence kρ( f )k ≤ kλ( f )k. Since the subset of functions in Cc (G) with support of the form Y1 Y2 as above is dense in Cc (G), this proves that ρ is weakly contained in L 2 (G). For semisimple G, the center Z (G) is finite, and hence a matrix coefficient of an irreducible unitary representation ρ is L p (G)-integrable if and only if it is L p (G/Z (G))-integrable. Therefore ρ (not necessarily irreducible) being tempered is equivalent to saying that ρ is strongly L 2+ (G), as was shown in [CHH]. It is well known that the trivial representation (in fact, any unitary representation) of an amenable group is weakly contained in its regular representation (cf. [Zi, Prop. 7.3.6]). It follows that IndG B (1) is weakly contained in the regular representation of G because B is amenable and the induction map is continuous. The above theorem therefore implies that the Harish-Chandra function 4G gives the sharpest pointwise bound for the K -finite matrix coefficients of unitary representations weakly contained in the regular representation of G.
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2.5 The next theorem follows from [CHH, proof of corollary on p. 108]. THEOREM
For semisimple G and its unitary representation ρ without a nonzero invariant vector, (1) if ρ is strongly L 2+ , then every nonzero matrix coefficients of ρ is L 2+ integrable; (2) if ρ is strongly L 2k+ for some positive integer k, then for any K -finite unit vectors v and w, hρ(g)v, wi ≤ dimhK vi dimhK wi 1/2 41/k (g) for any g ∈ G. G 2.6 The group of k-rational points of a connected reductive algebraic group over k is known to be of type I (see [Wa], [Be]). Hence we have the following proposition. (cf. [Ho, Prop. 6.3]) Let k be a local field. Let I be a finite set; for each i ∈ I , let G i be the group of krational points of a connected reductive algebraic group over k, and let K i be a good maximal compact subgroup of G i . Then for any irreducible unitary representation Q Q N ρ of i∈I G i having i∈I K i -invariant unit vector v, we have ρ = i∈I ρi and N v = v , where ρ is an irreducible unitary representation of G and vi is a i i i∈I i K i -invariant unit vector of ρi for each i ∈ I . PROPOSITION
2.7. Howe’s strategy In the spirit of Howe’s strategy (cf. [Ho, Prop. 6.3], [LZ, Th. 3.1]), we have the following proposition. PROPOSITION
For 1 ≤ i ≤ k, let Hi be the group of k-rational points of a connected reductive k-subgroup of G such that Hi ∩ B, Hi ∩ A, and Hi ∩ K are, respectively, a minimal parabolic subgroup, a maximal split torus, and a good maximal compact subgroup of Hi . (1) Suppose that for all 1 ≤ i 6 = j ≤ k, xi x j = x j xi for xi ∈ Hi and x j ∈ H j , and that Hi ∩ H j is a finite subset of K ∩ Hi . (2) Suppose that for each 1 ≤ i ≤ k, there exists a bi-(Hi ∩ K )-invariant positive function φi of Hi such that for any nontrivial irreducible unitary representation σ of G, the (Hi ∩ K ) matrix coefficients of σ | Hi are bounded by φi . Then for any unitary representation ρ of G without a nonzero invariant vector and
UNIFORM POINTWISE BOUNDS
147
with K -invariant unit vectors v and w, k k D Y E Y ρ h c v, w ≤ φi (h i ), i i=1
i=1
where h i ∈ Hi for each 1 ≤ i ≤ k and c ∈
Tk
i=1 C G (Hi ).
Proof Qk We denote by H I the formal direct product of the Hi ’s; that is, H I = i=1 Hi . Let H be the subgroup of G generated by the Hi ’s. Then we have a natural homomorphism Qk f from H I onto H , whose kernel is in i=1 (Hi ∩ K ). Let ρ˜ = ρ ◦ f and ρ| ˜ HI = R Tk Qk ρ dµ(x). Since c ∈ C (H ), the vector ρ(c)v is obviously (H i ∩ K )i=1 GR i i=1 X x R invariant. Write ρ(c)v and w as vx dµ(x) and wx dµ(x), respectively, where vx and wx are vectors in ρx . Without loss of generality, we may assume that for all x ∈ X , Qk ρx is a nontrivial unitary representation of H I and vx and wx are i=1 (Hi ∩ K )invariant unit vectors. Hence by assumption (2), hρx (h)vx , wx i ≤ φi (h) = φi f (h) for all h ∈ Hi . The last equality holds since ker f | Hi ⊂ K ∩ Hi and φi is bi-(K ∩ Hi )-invariant. Nk Nk Fixing x ∈ X , by Proposition 2.6 we have ρx | HI = i=1 ρxi , vi = i=1 vxi , and Nk wx = i=1 wxi , where ρxi is an irreducible class-one representation of Hi , and vxi and wxi are (K ∩ Hi )-invariant unit vectors for each 1 ≤ i ≤ k. If h i ∈ Hi , then k k D k D Y E Y E Y h i vx , wx = φi (h i ). ρx ρxi (h i )vxi , wxi ≤ i=1
i=1
i=1
Hence for any h i0 ∈ Hi such that f (h i0 ) = h i , k k k D Y E E Z D Y E D Y 0 h i0 vx , wx dµ(x) h i (cv), w ≤ ρx h i c v, w = ρ˜ ρ x
i=1
i=1
≤
k Y i=1
φi (h i0 ) =
k Y
i=1
φi (h i ).
i=1
This proves our claim. Remark. In fact, the proof of [Ho, Prop. 6.3] shows that if, for any (K ∩ Hi )-finite vectors v and w, the matrix coefficient hσ | Hi (h)v, wi is bounded by Cvw φi (h), where
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Cvw is some constant depending only on dimhK vi and dimhK wi, the above proposition holds also for K -finite unit vectors v and w, provided we multiply the function Qk i=1 φi by some constant depending on dimhK vi and dimhK wi.
3. The subgroups Hα , strongly orthogonal systems, and the Harish-Chandra function 4PGL2 (k) 3.1 We keep the notation from Section 2.1. For each α ∈ 8, set U˜ (α) to be the unipotent subgroup of G˜ attached to the root α so that Lie(U˜ (α) ) coincides with the root sub˜ | Ad a(x) = α(a)x for each x ∈ A}. ˜ The group G˜ contains a conspace {x ∈ Lie(G) ˜ G˜ 0 ), nected reductive k-split k-subgroup G˜0 such that we have A˜ ⊂ G˜ 0 , 8 = 8( A, ˜ ˜ and for each α ∈ 8, the subgroup U(α) ∩ G 0 coincides with the 1-dimensional ksplit root subgroup U˜ α in G˜ 0 (cf. [BT, Th. 7.2]). As usual, we set G 0 = G˜ 0 (k) and Uα = U˜ α ∩ G 0 . PROPOSITION ([Ti1]) For any α ∈ 8, there is a homomorphism φα : SL2 → G˜ 0 defined over k such that the kernel of φα is contained in the center of SL2 (k) and 1 x 1 0 z 0 φα ∈ Uα , φα ∈ U−α , and φα ∈ A, 0 1 y 1 0 z −1
for any x, y ∈ k and z ∈ k ∗ . For each α ∈ 8, we set H˜ α = φα (SL2 ) and Hα = H˜ α (k). Denote by U + and U − the upper and lower triangular subgroup of SL2 . Since SL2 is generated by U ± and φα (U ± ) = U˜ ±α , H˜ α is the closed subgroup of G˜ 0 generated by U˜ ±α . Note that Hα is isomorphic to either SL2 (k) (when ker φα = 1) or PGL2 (k) (when ker φα = ±1). Denote by A¯ the diagonal subgroup of SL2 and by B¯ the upper triangular subgroup of SL2 . Consider the simple root α¯ of SL2 defined by z 0 α¯ = z2. 0 z −1 Denote by K¯ the good maximal compact subgroup of SL2 (k) as defined in Example 2.1.1, so that the Cartan decomposition SL2 (k) = K¯ A¯ + K¯ and the Iwasawa decomposition SL2 (k) = K¯ A¯ 0 U + hold. Here A¯ + and A¯ 0 are defined as in Section 2.1 with respect to α. ¯
UNIFORM POINTWISE BOUNDS
149
LEMMA
We have the following: (1) α¯ = α ◦ φα ; ¯ = A˜ ∩ H˜ α ; (2) φα ( A) (3) φα ( A¯ +,0 ) ⊂ {a ∈ A ∩ Hα | α(a) ∈ k +,0 }; (4) φα ( K¯ ) ⊂ K ∩ Hα ; (5) the set {α, −α} is a root system of Hα with respect to A ∩ Hα , and the Cartan decomposition Hα = (K ∩ Hα )A+ (Hα )(K ∩ Hα ) holds, where A+ (Hα ) = a ∈ A ∩ Hα α(a) ∈ k +,0 . Proof ¯ and let u ∈ U + . Then It suffices to verify claim (1) by Proposition 2.1. Let a ∈ A, −1 aua = α(a)u. ¯ Hence, applying φα on both sides, we get φα (a)φα (u)φα (a −1 ) = α(a)φ ¯ α (u). (Note that φα |U + is a k-linear map.) On the other hand, since φα (a) ∈ A and φα (u) ∈ Uα , it follows that φα (a)φα (u)φα (a −1 ) = α φα (a) φα (u) . ¯ proving claim (1). Hence we have α(φα (a)) = α(a) ¯ for any a ∈ A, 3.2 Since PGL2 is the adjoint group of type A1 , there exists a unique k-rational isogeny ψα : H˜ α → PGL2 such that ψα ◦ φα = j, where j denotes the natural projection map j : SL2 → PGL2 . The group PGL2 (k) has the Cartan decomposition and the Iwasawa decomposition that are compatible with those of SL2 (k) as described in Section 3.1. LEMMA
For any a ∈ A ∩ Hα , α(a) 0 4 Hα (a) = 4PGL2 (k) . 0 1 Proof Since 4 Hα (a) = 4PGL2 (k) (ψα (a)) for any a ∈ A ∩ Hα , it is enough to show that 0 ψα (a) = α(a) . Since 0 1 a = φα
y 0
0 y −1
for some y ∈ k ∗ ,
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we have ψα (a) = ψα ◦ φα
y 0 0
0
y −1
y = j 0
0 y −1
.
Since α(a) = α ◦ φα
y 0
y −1
we have
y ψα (a) = j 0
0 y −1
=
y = α¯ 0 y2 0
0 y −1
= y2,
0 α(a) 0 = , 1 0 1
proving the lemma. 3.3 We recall that two distinct roots α and β in 8+ are said to be strongly orthogonal if neither of α ± β is a root. Definition (cf. [Oh1]) We have the following definitions. (1) A subset S of 8+ is called a strongly orthogonal system of 8 if any two elements of S are strongly orthogonal to each other. (2) A strongly orthogonal system S is called large if every simple root of 8+ has P a nonzero coefficient in the formal sum α∈S α. (3) A strongly orthogonal system S is called maximal if the coefficient of each P P simple root in the formal sum α∈S α is not less than the one in α∈O α for any strongly orthogonal system O of 8. Example Let G = SL3 (k), and let α1 , α2 be the simple roots. Then there are three strongly orthogonal systems: {α1 }, {α2 }, {α1 + α2 }. For G = SL4 (k), let α1 , α2 , α3 be the simple roots. Then the following is a complete list of strongly orthogonal systems: {a positive root}, {α1 , α3 }, {α1 +α2 , α2 + α3 }, and {α1 + α2 + α3 , α2 }. Clearly, for any α ∈ 8+ , a singleton {α} is a strongly orthogonal system. If γ is the highest root of 8+ , then the singleton {γ } is a large strongly orthogonal system. See the appendix for a list of a maximal strongly orthogonal system for each irreducible root system constructed in [Oh1]. We remark that a maximal strongly orthogonal system is not unique in general (see [Oh1, remark, Sec. 2.3]). We note that a priori it is not clear from the definition whether a maximal strongly orthogonal system always exists. It will be interesting to give an intrinsic explanation for its existence.
UNIFORM POINTWISE BOUNDS
151
LEMMA 3.4 Let S be a subset of 8+ . Then S is a strongly orthogonal system if and only if for any α 6 = β in S , xα xβ = xβ xα for any xα ∈ Hα and xβ ∈ Hβ .
Proof Since H˜ α is generated by U±α for any α ∈ 8, the claim follows from the Chevalley commutator relations (see [St]). Recall that for a group H , the notation Z (H ) denotes the center of H . 3.5 If S is a strongly orthogonal system of 8, COROLLARY
Hα ∩ Hβ ⊂ Z (Hα ) ⊂ K ∩ Z (Hα ) for any α 6= β in S . Proof If x ∈ Hα ∩ Hβ , x centralizes Hα by Lemma 3.4, and hence x ∈ Z (Hα ). Note that Z (Hα ) is nontrivial only when φα is injective and hence when Hα is isomorphic to SL2 (k). Since Z (SL2 (k)) ⊂ K¯ , we have φα (Z (SL2 (k)) = Z (Hα ) ⊂ K ∩ Hα by Lemma 3.2. 3.6 We now show that for any root α ∈ 8+ , the subgroup Hα is embedded in G in one of the following four ways, as the lemma below describes. We denote by k 2 n k (resp., k 4 n k) the Heisenberg group of dimension 3 (resp., 5). LEMMA
Assume that G/Z (G) is almost k-simple with k-ss rank(G) ≥ 2. Then for any α ∈ 8+ , there exist a connected almost simple k-split subgroup G˜ α of rank 2 and a unipotent algebraic k-subgroup N˜ α of G˜ such that C G α ( H˜ α ) N˜ α is a Levi decomposition of a parabolic subgroup of G˜ α . Moreover, if we set G α = G˜ α (k) and Nα = N˜ α (k), one of the following holds for the type of G α , Nα , and the action of Hα on Nα /[Nα , Nα ] up to local isomorphism: (1) A2 , Nα ∼ = k 2 : the standard representation of SL2 (k) on k 2 ; (2) C 2 , Nα ∼ = k 3 : the adjoint representation of SL2 (k) on k 3 ; (3) C 2 , Nα ∼ = k 2 n k: the standard representation of SL2 (k) on k 2 ; (4) G 2 , Nα ∼ = k 4 n k: the symplectic representation of SL2 (k) on k 4 . Moreover, in the root system of G α , α is a short root in cases (2) and (4), and a long root in case (3).
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Proof Since G/Z (G) is almost k-simple, the group G˜ 0 /Z (G˜ 0 ) (see Section 3.1 for notation) is almost simple and split over k. Since 8 is a reduced irreducible root system, there exists a root β ∈ 8 such that one of α±β belongs to 8. Consider 9 = {iα+ jβ ∈ 8 | i, j ∈ Z}. Then 9 is a reduced irreducible root system of rank 2. Denote by G˜ 00 the closed subgroup of G˜ 0 generated by the 1-dimensional root sub-subgroups U˜ γ of G˜ 0 , γ ∈ 9. Then G˜ 00 is of type A2 , C2 , or G 2 . Denote by {α1 , α2 } the set of simple roots of G˜ 00 . Let α1 be the short simple root if the lengths of α1 and α2 are different. Since any root is conjugate to a simple root by a Weyl element, we may assume that α = αi for i = 1 or 2. In the following case-by-case proof according to the type of (G˜ α , α), except in the last case, we set G˜ α = G˜ 00 . For simplicity, we omit the notation ˜ in the rest of proof. (1) For A2 , α1 , or α2 , since α1 and α2 are conjugate in A2 , it suffices to consider α = α1 . Then it is enough to consider the unipotent subgroup Nα generated by Uα2 and Uα1 +α2 . Note that Nα is a 2-dimensional abelian subgroup. (2) For C2 , α1 , it suffices to consider the unipotent subgroup Nα generated by Uα2 , Uα1 +α2 , and U2α1 +α2 . Note that Nα is a 3-dimensional abelian subgroup. (3) For C2 , α2 , consider the unipotent subgroup Nα generated by Uα1 , Uα1 +α2 and U2α1 +α2 . Hence Nα is a Heisenberg subgroup with the center U2α1 +α2 . (4) For G 2 , α1 , consider the unipotent subgroup Nα generated by Uα2 , Uα1 +α2 , U2α1 +α2 , U3α1 +α2 , and U3α1 +2α2 . Hence Nα is a Heisenberg subgroup with the center U3α1 +2α2 . (5) For G 2 , α2 , since the root system generated by long roots of G 2 is A2 , it suffices to set G α to be the corresponding subgroup of type A2 and to apply the case of A2 . Now the proof of lemma is straightforward from the above list. Remark. Unless 8 = Cn (n ≥ 2) and α is a long root (simultaneously), we can always take β in the proof so that case (3) in the above lemma does not occur. This can be seen as follows. First, we may assume that α is a simple root up to conjugation. If 8 = G 2 , case (4) or (5) happens as explained in the above proof. In all other cases, except when α is a short (resp., long) root in 8 = Bn , n ≥ 3 (resp., Cn , n ≥ 2), it is clear from the Dynkin diagram that we can take β so that 9 is A2 (here we regard B2 as C2 ). Finally, if α is a short root in Bn , case (2) arises. 3.7. The Harish-Chandra function of PGL2 (k) We explicitly calculate the Harish-Chandra function of PGL2 (k). Let A (resp., B) be the image of the diagonal (resp., the upper triangular) subgroup of GL2 (k) under the natural projection GL2 (k) → PGL2 (k). Then the set of representatives of
UNIFORM POINTWISE BOUNDS
153
A can be taken to be a0 01 a ∈ k ∗ . We set a˜ = a0 01 ∈ A+ . Denote by α the simple root of PGL2 (k) defined by α(a) ˜ = a. In the Iwasawa decomposition PGL2 (k) = K A0 Ru (B), the A0 -part of an element of PGL2 (k) is uniquely determined. The modular function δ B satisfies ˜ = b1 for k ∈ K , b˜ = diag(b1 , b2 ) ∈ A0 , and n ∈ Ru (B). δ B (k bn) b2 Recall from Section 2.2 that 4PGL2 (k) (a) ˜ =
Z
δ B (ak) ˜ −1/2 dk, K
where dk is a normalized Haar measure on K . 3.7.1. For k = R, since √ a a 0 4PGL2 (R) = 4SL2 (R) 0 1 0
! 0 √ −1 , a
we may calculate 4SL2 (R) . Let G = SL2 (R), and let cos t sin t K = SO2 = 0 ≤ t < 2π , − sin t cos t where dt is the Lebesgue measure on [0, 2π ). Let a˜ =
dk =
1 dt, 2π
√ a 0 √ −1 . 0 a
To compute the
A0 -component of ak, ˜ let 0˜
ak ˜ = k bn,
b 0 ˜ where b = , b > 0, 0 b−1
and apply the standard vector e1 on both sides. Then ak.e ˜ 1 = k 0 b.e1 . Hence cos t sin t b2 = a cos2 t + a −1 sin2 t, where k = − sin t cos t 2 + k 2 = 1 for any k = (k ) ∈ K . since k11 ij 21 Since δ B (ak) ˜ −1/2 = b−1 , ! √ a 0 a 0 = 4SL2 (R) 4PGL2 (R) √ −1 0 1 0 a −1/2 Z π/2 2 2 cos t 2 + sin t dt = √ a2 π a 0
for a ≥ 1.
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3.7.2. For k = C, we can parameterize K = SU2 by cos t e(i(φ+ψ)/2) i sin t e(i(φ−ψ)/2) k(ψ, t, φ) = i sin t e(i(ψ−φ)/2) cos t e(i(φ+ψ)/2) 0 ≤ t < π/2, 0 ≤ ψ < 2π, −2π ≤ ψ < 2π
and
1 sin(2t) dψ dt dφ, 8π 2 where each of dψ, dt, and dφ is the Lebesgue measure on the corresponding domain of each variable (see [VK]). By the same argument as in Section 3.7.1, now using the fact that the equation k11 k11 + k21 k21 = 1 for every k = (ki j ) ∈ K (where ki j denotes the (i, j)th entry of ˜ that b1 b2 = a and k), we deduce from the Iwasawa decomposition ak ˜ = k 0 bn dk =
b12 = a 2 cos2 t + a −2 sin2 t, where k = k(ψ, t, φ) (here recall that a, b1 , and b2 are positive reals). Since b −1/2 b −1 1 1 δ B (ak) ˜ −1/2 = = b2 b2 (recall that |z| = z z¯ ), Z Z 2π Z π/2 Z 2π −1 1 b1 −1/2 sin(2t) dψ dt dφ. δ B (ak) ˜ dk = 2 b2 8π K −2π 0 0 Hence 4PGL2 (C)
Z π/2 2 −1 1 cos t a 0 = + sin2 t sin (2t) dt 2 0 1 πa 0 a
for a ≥ 1.
3.7.3. For k nonarchimedean with a uniformizer q such that |q|−1 = p, in the realization of IndGP on L 2 (k) (see Section 6.2), the K -invariant unit vector f 0 is given by r −1 p f 0 (x) = · max 1, |x| for any x ∈ k. p+1 Hence 4PGL2 (k) (a) ˜ = hIndGP (a) ˜ f0, f0i Z −1 −1 p = |a|−1/2 max |a|−1 , |ax| max 1, |x| d x, p+1 k
UNIFORM POINTWISE BOUNDS
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where d x is a normalized Haar measure on k. For a = q −n , we split this integral into three parts—the integral over |x| < p −n , the integral over p −n < |x| ≤ 1, and the integral over |x| > 1. Then the integral becomes Z Z Z p p −n |x|−2 d x + p −n |x|−1 d x + pn d x . √ n p ( p + 1) |x|>1 p −n <|x|≤1 |x|< p −n Since Z |x|
−2
dx = p
−1
Z
,
p −n <|x|≤1
|x|>1
|x|−1 d x =
n( p − 1) , p
and Z
d x = p −n
|x|< p −n
(cf. [GGP, Chap. 3.10]), we obtain 4PGL2 (k) Since have
qn 0 0 1
1 0 0 q −n
=
q −n 0 0 1
=
n( p − 1) + ( p + 1) . √ n p ( p + 1)
in PGL2 (Q p ) and the latter belongs to K
n q 4PGL2 (k) 0
−n 0 q = 4PGL2 (k) 1 0
q −n 0 0 1
K , we
0 . 1
3.8 In summary, the function 4PGL2 (k) is a bi-K -invariant function of PGL2 (k) such that
4PGL2 (k)
x 0
0 1
=
2 √ π x
1 πx
R π/2 cos2 t 0
x2
R π/2 cos2 t 0
x2
+ sin2 t
−1/2
dt
for x ≥ 1 when k = R, −1 + sin2 t sin (2t) dt for x ≥ 1 when k = C,
n( p−1)+( p+1) √ n p ( p+1)
for x = q ±n and n ∈ N when k is nonarchimedean.
(Recall that |q| = p −1 , where p is the cardinality of the residue field of k.)
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4. Temperedness of ρ| Hα 4.1 We continue the notation from Sections 2.1 and 3.1. We also denote by G + the subgroup generated by the subgroups U (k), where U runs through the set of unipotent k-split subgroups of G˜ (cf. [BT]). The goal of Section 4 is to show the theorems in Sections 4.1 (for k 6= C) and 4.2 (for k = C), which play key roles in the construction of our uniform pointwise bounds for K matrix coefficients. THEOREM (k 6 = C) Let k be a local field not of characteristic 2. Assume that G/Z (G) is almost k-simple with k-ss rank(G) ≥ 2. For any α ∈ 8 and for any unitary representation ρ of G without a nonzero G + -invariant vector, the restriction ρ| Hα is tempered. Moreover, if G/Z (G) ∼ = SLn (k) (locally), k can be any local field.
4.2 When k is the complex field, there appears a different phenomenon, mainly due to the fact that the oscillator representation occurs as a representation of SL2 (C); whereas if k 6 = C, it is a representation of a double covering of SL2 (k) but not a representation of SL2 (k). More precisely, when Hα ⊂ G = Sp2n (C) and α is a long root, the restrictions to Hα of the (K ∩ Hα ) matrix coefficients of a unitary representation of G without a nonzero G + -invariant vector are in general strongly L 4+ but not strongly L 2+ (see Theorem 6.6 for an example of such a representation). (k = C) Let G be a connected reductive complex algebraic group with ss rank(G) ≥ 2, and let G/Z (G) be almost simple. For any α ∈ 8 and for any unitary representation ρ of G without a nonzero G + -invariant vector, the restriction ρ| Hα is tempered, except in the case when G/Z (G) ∼ = Sp2n (C), n ≥ 2 (locally), and α is a long root. In the latter case, ρ| Hα is strongly L 4+ . THEOREM
4.3 Recall the following direct consequence of the well-known Howe-Moore theorem on vanishing of the matrix coefficients at infinity (see [HM]): if G is the group of krational points of a connected reductive algebraic group over any local field k with G/Z (G) almost k-simple, ρ is a unitary representation of G without a nonzero G + invariant vector, and M is a closed subgroup of G with M/(M ∩ Z (G)) noncompact, then ρ has no M-invariant vector. In view of this, we may assume that G = G α (see Lemma 3.6 for notation) in the proofs of Theorems 4.1 and 4.2. We then carry out
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a case-by-case analysis based on the position of Hα in G α along with the type of G α given in Lemma 3.6. By the following lemma we may also assume that G α is of simply connected type and hence that Hα is isomorphic to SL2 (k), without loss of generality. LEMMA
Let Gα be the simply connected covering of G˜ α , and let π : Gα → G˜ α be the kisogeny. Set Hα := π −1 ( H˜ α ). Suppose that for any unitary representation ρ of Gα (k) with no nonzero invariant vector, ρ|Hα (k) is tempered. Then for any unitary representation ρ of G α with no nonzero G + α -invariant vector, ρ| Hα is tempered. Proof Let ρ be a unitary representation of G α with no nonzero G + α -invariant vector. Note that Gα (k) = Gα (k)+ and π(Gα (k)) = G α (k)+ . Hence ρ ◦ π|Gα (k) is a unitary representation of Gα (k) with no nonzero Gα (k)-invariant vector. Note that π(Hα (k)) is a subgroup with finite index in Hα . Let h 1 , . . . , h k be a set of representatives in Hα /π(Hα (k)). Let v and w be nonzero vectors in ρ, and set wi = ρ(h i−1 )w for each 1 ≤ i ≤ k. Denote by dh a Haar measure on Hα . Since π(Hα (k)) is an open subgroup of Hα , the restriction of dh defines a Haar measure on π(Hα (k)), which is also denoted by dh. Let dm denote the Haar measure on Hα (k) which is the pullback of dh under the covering map π : Hα (k) → π(Hα (k)). Fix > 0. Then Z k Z X hρ(h i h)v, wi 2+ dh hρ(h)v, wi 2+ dh = Hα
π(Hα (k))
i=1 k
=
1X c i=1
Z Hα (k)
hρ ◦ π(m)v, wi i 2+ dm,
where c is the cardinality of ker(π ) ∩ Hα (k). Since any nonzero matrix coefficient of ρ ◦ π |Hα (k) is L 2+ -integrable for any > 0 by the assumption (see Theorem 2.5), it follows that ρ| Hα is tempered by Theorem 2.4. Our main tool is the theory of G. Mackey on representations of a semidirect product of groups (cf. [Zi, Exam. 7.3.4], [Ma1, Chap. III, Sec. 4.7], [LZ]). The first two cases of Lemma 3.6 are handled by the following proposition. 4.4 Let k be any local field, and let H = SL2 (k). Let G be the group H nk 2 (resp., H nk 3 ), where H acts on k 2 (resp., k 3 ) as the standard (resp., adjoint) representation. Let K be a good maximal compact subgroup of H . Then for any unitary representation of ρ without any k 2 - (resp., k 3 -) invariant vector, ρ| H is tempered. PROPOSITION
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Proof Set N = k 2 or N = k 3 accordingly. Consider the action of H on the character group Nˆ of N defined by h · χ(n) := χ(h −1 nh) for any h ∈ H , χ ∈ Nˆ , and n ∈ N . Let N 0 denote the space of k-linear forms on N , and fix a nontrivial additive character λ of k. Then the map φ : N 0 → Nˆ defined by φ(n 0 )(x) = λ(n 0 (x)) for any x ∈ n is a bijection (cf. [We, Chap. II, Sec. 5, Th. 3]). Through this identification of Nˆ with N 0 ∼ = N , the action of H on Nˆ is equivalent to the standard SL2 (k)-action 2 on k if N = k 2 , and it is equivalent to the standard SL2 (k)-action on the symmetric power Sym2 (k 2 ) of k 2 if N = k 3 . Therefore the actions of H in Nˆ are algebraic, and hence the H -orbits on Nˆ are locally closed (see [BZ, Secs. 6.15 and 6.8]). We can easily check that the zero element in Nˆ is the only H -fixed point in Nˆ . Hence the stabilizer in H of any nonzero element in Nˆ is amenable since any proper algebraic subgroup of SL2 (k) is amenable. Now assume that ρ is irreducible. Applying Mackey’s theory, we conclude that ρ is induced from an irreducible unitary representation σ of the stabilizer in G of an element, say, χ , of Nˆ , and if χ is trivial, then ρ| N contains the trivial representation (cf. [Zi, Th. 7.3.1]). It then follows from the assumption that χ must be nontrivial and hence that the stabilizer of χ in G, which is the semidirect product of the stabilizer of χ in H with N , is amenable. Recall the well-known fact that any irreducible unitary representation of an amenable group is weakly contained in its regular representation (cf. [Ma1, Chap. 1, Sec. 5.5.3]). Hence ρ is weakly contained in the regular representation of G since the induced representation of a regular representation is the regular representation and the induction map is continuous (cf. [Zi, Prop. 7.3.7]). It now follows that ρ| H is weakly contained in a multiple of the regular representation R L 2 (H ). In general, in the direct integral decomposition ρ = X ρx dµ(x), where ρx is irreducible, for almost all x ∈ X , ρx | N has no nonzero invariant vector. Hence ρx | H is weakly contained in the regular representation L 2 (H ) for almost all x ∈ X , by the above argument. By Theorem 2.4, this finishes the proof. 4.5 Now the last two two cases of Lemma 3.6 are based on Propositions 4.5 and 4.6. In both cases, Hα is a subgroup (in fact, the derived subgroup of a Levi subgroup) of a parabolic subgroup of G α whose unipotent radical Nα is a Heisenberg subgroup. Letting W = Nα /[Nα , Nα ], Hα acts on W as a sympletic representation, and hence Hα ,→ Sp(W ). Fix a subset 1 of 8+ such that W admits a polarization X X W = Uβ ⊕ Uβ ∗ . β∈1
β∈1
UNIFORM POINTWISE BOUNDS
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Denote by Hˆα the double cover of Hα for k 6 = C, and set Hˆα = Hα for k = C. Applying Mackey’s theory and the theory of oscillator representation, we can conclude that for any unitary representation of G α without a nonzero invariant vector, there exist unitary representations µt of Hˆ α , t ∈ k ∗ /(k ∗ )2 , such that X ρ| Hα = ωt | Hˆ α ⊗ µt t∈k ∗ /(k ∗ )2
(up to equivalence), where ωt is the oscillator representation of the symplectic group Sp(W ) corresponding to t ∈ k ∗ /(k ∗ )2 (cf. [LZ, Prop. 2.1]). Here the representation ωt | Hˆ α ⊗ µt factors through Hˆ α → Hα . Therefore there exists a dense subset V of ρ| Hα such that for any x, y ∈ V , the matrix coefficient hρ| Hα (a)x, yi is bounded by a constant multiple of Y −1/2 φ(a) = max(|β(a)|, |β(a)|−1 ) β∈1
for any a ∈ A(Hα ) (cf. [LZ, Prop. 2.2]). Here φ is essentially a matrix coefficient of ωt | Hˆ α . We keep this notation in the proofs of the following two propositions. PROPOSITION
Let k be a local field not of characteristic 2. Let G be the group of k-rational points of a connected simply connected almost simple k-split group of type G 2 , and let α be a short root in 8. For any unitary representation ρ of G without a nonzero invariant vector, the restriction ρ| Hα is tempered. Proof Let α1 and α2 be as in the the proof of Lemma 3.6 (in particular, α1 is short). Then we may assume that α = α1 (up to conjugation by a Weyl element). We may assume that ρ is irreducible, without loss of generality. The maximal split torus A of G 2 can be identified as A = a = diag(a1 , a2 , a3 ) ∈ SL3 (k) ai ∈ k ∗ with α1 (a) = a2
and
α2 (a) =
a1 . a2
Then A+ ∩ Hα1 = b = diag(b1−1 , b12 , b1−1 ) b1 ∈ kˆ . Since 1 can be taken as {α2 , α1 + α2 }, φ(b) = |α2 (b)|−1/2 |α1 α2 (b)|−1/2 = |b1 |−2 . Therefore ρ| Hα is strongly L 2+ ; hence ρ| Hα is tempered by Theorem 2.4.
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PROPOSITION 4.6 Let k be a local field not of characteristic 2. Let G be the group of k-rational points of a connected simply connected almost simple k-split group of type C2 , and let α be a long root in 8. Let ρ be a unitary representation of G without a nonzero invariant vector. (1) For k 6= C, ρ| Hα is tempered. (2) For k = C, ρ| Hα is strongly L 4+ .
Proof We may realize G by Sp4 (k), so that A = a = diag(a1 , a2 , a2−1 , a1−1 ) a1 , a2 ∈ k ∗ with
a1 and α2 (a) = a22 a2 are simple roots. We may assume that α = α2 and that A+ (Hα ) = b = diag(1, b1 , b1−1 , 1) b1 ∈ kˆ . α1 (a) =
Then φ(b) = |α1 (b)|−1/2 = |b1 |−1/2 for any b ∈ A(Hα ). Hence ρ| Hα is strongly L 4+ . Now consider the case when k 6 = C. By the remark in Section 4.1, we may assume that ρ is irreducible. By [Ho, Cor. 2.15], any nontrivial irreducible unitary representation of Sp4 (k) has pure rank 2. By [Ho, Cor. 2.12], the representations ωt are representations of Hˆ α of pure rank 1. Since ρ| Hα is a representation of Hα and ωt acts on the kernel of Hˆ α → Hα nontrivially, it follows that µt is a genuine representation of Hˆ α . It is well known that the genuine ˆ 2 (k) is strongly L 4+ . Hence so is µt . irreducible unitary representation of Hˆ α = SL Since both ωt and µt are strongly L 4+ , ρ| Hα is strongly L 2+ . By Theorem 2.4, ρ| Hα is tempered. The above argument for k 6 = C is borrowed from the proof of [DHL, Prop. 4.4]. 4.7. Proof of Theorems 4.1 and 4.2 Let G α and Nα be as in Lemma 3.6. By the Howe-Moore theorem, ρ|G α has no G+ α -invariant vectors. Hence we may assume that G = G α . Suppose that G α contains Hα nk 2 and Hα nk 3 as the first two cases of Lemma 3.6. Since ρ|k i (i = 2, 3, respectively) has no nonzero invariant vector, again by Howe and Moore, we apply Proposition 4.4 to prove the claim. For the last two cases of Lemma 3.6, Propositions 4.5 and 4.6 imply the claim. Since Lemma 3.6(3) can be avoided unless 8 = Cn , n ≥ 2 (i.e., G/Z (G) ∼ = Sp2n (C)), and α is a long root (see the remark following Lemma 3.5), we
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need to set n α = 1/2 only in the case when k = C, as in Theorem 4.2. The condition that the characteristic of k is not 2 is required only in using oscillator representations of the double cover of a symplectic group. For example, if 8 = An−1 , that is, if G/Z (G) ∼ = SLn (k), this case does not happen. Hence Theorem 4.1 is valid in this case for any local field k. Remark. The proof of [Oh1, Prop. 3.4] is incomplete since only the cases when Nα can be taken to be abelian were treated. The above proof fills up its missing part. 4.8. Sp(1, n) case We also show a theorem analogous to Theorem 4.1 for the real rank 1 group G = Sp(1, n), n ≥ 2. In this case, we have 8+ = {α}. THEOREM
Let ρ be a unitary representation of G = Sp(1, n) without a nonzero invariant vector. Then ρ| Hα is tempered. Proof Consider the natural embedding Sp(1, 2) ,→ G. Thanks to the Howe-Moore theorem again, we may assume that G = Sp(1, 2). If we denote 8 by {α, −α}, we have G 0 = Hα . It is known that any nontrivial irreducible unitary representation of G is strongly L 5+ from the classification in [Ko]. By Theorem 2.5, any K -finite matrix 1/3 1/3 coefficient of ρ is bounded by 4G up to a constant multiple. But 4G ≈ α −5/6 (see 1/3 Section 5.6 for notation). Hence 4G | Hα is L 2 -integrable and ρ| Hα is strongly L 2 . This proves the claim. 4.9 We now obtain the following theorem from Theorems 4.1 and 4.2 and Proposition 2.7. THEOREM
Let k be a local field with char(k) 6 = 2. Let G be the group of k-rational points of a connected reductive linear algebraic group over k with k-ss rank(G) ≥ 2 and G/Z (G) almost k-simple, or with G = Sp(1, n) (over R). Let S be a strongly orthogonal system of 8. Let n α = 1/2 if k = C, 8 = Cn , and α is a long root, and let n α = 1 otherwise. If ρ is a unitary representation of G without a nonzero G + -invariant vector and v and w are (K ∩ HS )-invariant unit vectors, then for any T h α ∈ Hα and c ∈ α∈S C G (Hα ), we have D Y E Y n h α c v, w ≤ 4 Hαα (h α ). ρ α∈S
α∈S
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Definition 4.10 For a strongly orthogonal system S , define a bi-K -invariant function ξS of G = K A+ K as follows: Y n α(a) 0 α ξS (k1 adk2 ) = 4PGL2 (k) for a ∈ A+ , d ∈ , k1 , k2 ∈ K , 0 1 α∈S
where n α = 1/2 if k = C, 8 = Cn , and α is a long root (all at the same time), and n α = 1 in all other cases. LEMMA
For any element a = have
Q
α∈S
aα c, where aα ∈ A+ (Hα ) and c ∈ ξS (a) =
Y α∈S
T
α∈S
C A0 Hα , we
4nHαα (aα ).
Proof It suffices to show that 4PGL2 (k) Since
α(a) 0 = 4 Hα (aα ). 0 1
α(aα ) 0 4 Hα (aα ) = 4PGL2 (k) 0 1
for aα ∈ A ∩ Hα
by Lemma 3.2, it suffices to show that α(a) = α(aα ). Since the element Q β∈S ,β6=α aβ c lies in C G (Hα ), we only need to show that α(b) = 1 for any b ∈ A ∩ C G (Hα ). Note that for any u ∈ Uα , bub−1 = α(b)u. Therefore if b commutes with Hα , and hence with u, we have α(b)u = u, yielding α(b) = 1. 4.11 Recall from Section 2.1 that if k is archimedean, we have the Cartan decomposition G = K A+ K . Therefore the following theorem presents a uniform pointwise bound on G for all K matrix coefficients of unitary representations of G without nonzero G + -invariant vectors in the archimedean field case. THEOREM
Let k = R or C, and let G be the k-points of a connected reductive linear algebraic group over k with k-ss rank(G) ≥ 2 and G/Z (G) almost k-simple or G = Sp(1, n). Let S be a strongly orthogonal system of 8. Then for any unitary representation ρ of G without a nonzero G + -invariant vector and with K -invariant unit vectors v and
UNIFORM POINTWISE BOUNDS
163
w, we have hρ(g)v, wi ≤ ξS (g)
for g ∈ G.
Proof Since both functions are bi-K -invariant, it suffices to consider the case when g ∈ A+ . In fact, we have \ Y A+ (Hα ) · C A0 (Hα ) A+ ⊂ α∈S
α∈S
for k archimedean (see Lemma 5.2). Hence it remains only to apply Theorem 4.9 with Lemma 4.10. Theorem 1.2(2) is a special case of the above theorem.
5. Uniform pointwise bound ξS 5.1 Let k be any local field with char(k) 6 = 2. Unless stated otherwise, G denotes the group of k-rational points of a connected reductive linear algebraic group over k with k-ss rank(G) ≥ 2 and G/Z (G) almost k-simple. We also assume that G/Z (G) Sp2n (C) (locally). We state some more structure theory of algebraic groups. We continue the same notation from Sections 2.1, 3.1, and 4.1. In particular, recall that for each α ∈ 8, U˜ α ˜ denotes the 1-dimensional root subgroup (resp., the maximal k-split torus) (resp., A) ˜ of G 0 such that U˜ α ∩ G 0 = Uα (resp., A˜ ∩ G 0 = A). In the following discussion we freely use some facts about algebraic groups from [BT, Secs. 3.8 – 3.11]. Let S be a strongly orthogonal system of 8. Then the set ±S = {α, −α | α ∈ S } is a closed subset of 8. Denote by G˜ S the subgroup of G˜ 0 generated by A˜ and U˜ α , α ∈ ±S . Then G˜ S is a connected reductive algebraic subgroup of G˜ 0 defined over k. We also denote by H˜ S the subgroup of G˜ 0 generated by U˜ α , α ∈ ±S . Then H˜S is a connected semisimple algebraic subgroup of G˜ 0 defined over k. We set G S = G˜ S (k) and HS = H˜ S (k). Then G S = HS A. It follows from Proposition 2.1 that HS and G S admit Cartan decomposition and Iwasawa decomposition compatible with those of G: HS = (K ∩ HS )A+ (HS )(K ∩ HS ) = (K ∩ HS )A0 (HS )(Ru (B) ∩ HS ), G S = (K ∩ G S )A+ (K ∩ G S ) = (K ∩ G S )A0 (Ru (B) ∩ G S ).
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5.2 The subgroup H˜ S is an almost direct product of H˜ α , α ∈ S , and hence the subgroup generated by Hα , α ∈ S , has a finite index in HS . Since the centralizer of any T subset is algebraic, C A0 (HS ) = α∈S C A0 (Hα ) and A+ (Hα ) ⊂ A+ (HS ) for each α ∈ S. LEMMA
Let S be a strongly orthogonal system of 8. Then we have ( T A+ ⊂ A+ (HS ) · for k archimedean, α∈S C A0 (Hα ) T + + 2A ⊂ A (HS ) · for k nonarchimedean, α∈S C A0 (Hα ) where 2A+ = {a 2 | a ∈ A+ }. In fact, for any a ∈ A+ (resp., for a ∈ 2A+ for the latter case), there exist elements aα ∈ A+ (Hα ) (unique up to mod K ∩ Hα ), α ∈ S , Q and c ∈ C A0 (HS ) such that a = α∈S aα · c. Proof Consider the character map α : A0 → k 0 for each α ∈ S . For k archimedean, α must be surjective since both α(A0 ) and k 0 are 1-dimensional connected groups. Hence A0 = (A0 ∩ Hα ) ker(α). When k is nonarchimedean, we claim that 2A0 ⊂ (A0 ∩ Hα )2(ker(α)). Denote by π the natural projection A0 → A0 / ker α. Consider the map α˜ : A0 / ker(α) → k 0 which is induced by α. Then α(π(a)) ˜ = α(a) for all a ∈ A0 ∩ Hα . Denote by a (resp., b) the generator of A0 ∩ Hα (resp., A0 / ker(α)) as a Zmodule. We may assume that π(a) = bm for some positive integer m. Define a character β : A0 ∩ Hα → k 0 by setting β(a) = α(b). ˜ Then β m (a) = α(b ˜ m ) = απ(a) ˜ = m 0 α(a). Hence β = α on Hα ∩ A . Since Hα is locally isomorphic to SL2 (k), the Z-module generated by the root α of Hα is of index 2 in the Z-module generated by all characters of A(Hα ) defined over k. Hence m = 1 or m = 2. Therefore 2A0 / ker(α) ⊂ A0 ∩ Hα ; hence (2A0 / ker(α))(2 ker(α)) ⊂ (A0 ∩ Hα )(2 ker(α)). This shows that 2A0 ⊂ (A0 ∩ Hα )2(ker(α)) for k nonarchimedean. For the rest of the proof, let r = 1 or 2 depending on whether k is archimedean or not. Note that ker(α) = C A0 (Hα ) and hence that A0 ∩ Hβ ⊂ ker(α) for all β ∈ S \{α}. If β ∈ S \{α}, by the same argument as before, we have r ker(α)/(ker(α) ∩ ker(β)) ⊂ (A0 ∩ Hβ ).
UNIFORM POINTWISE BOUNDS
165
By an inductive argument, we obtain Y \ 0 0 rA ⊂ A (Hα ) · ker(α) . α∈S
α∈S
Since α∈S A0 (Hα ) ⊂ A0 (Hα ), α∈S ker(α) = C A0 (HS ), and α(a) ∈ k + for any a ∈ A+ and α ∈ 8+ , this proves the inclusion relation. To show the Q Q uniqueness, assume that α∈S aα c = α∈S aα0 c0 , where aα , aα0 ∈ Hα and c, c0 ∈ T −1 0 α∈S C A0 (Hα ). Then for each α ∈ S , aα aα ∈ C A0 (Hα ) since Hβ ⊂ C A0 (Hα ) for any β 6= α in S . Therefore aα−1 aα0 ∈ Z (Hα ) ⊂ K ∩ Hα . This finishes the proof. Q
T
Example For G = SL4 (k), let a = diag(a1 , a2 , a3 , a4 ) ∈ A+ . Consider the simple roots α1 , α2 , and α3 such that αi (a) = ai /(ai+1 ) for each 1 ≤ i ≤ 3. Set γ1 (a) =
a1 a4
and
γ2 (a) =
a2 . a3
Then {γ1 , γ2 } is a (maximal) strongly orthogonal system of 8. Observe that a is decomposed into q a 1 1 q a4 a2 1 a3 , , q a 3 1 q a2
a4 a1
√ a1 a4
1
√ a2 a3
√ a2 a3
√ a1 a4
.
Therefore the decomposition of a into A+ (Hγ1 )A+ (Hγ2 )C A0 (Hγ1 Hγ2 ) can be achieved, provided γi (a) ∈ kˆ 2 . For k archimedean, this is always the case. For k nonarchimedean, γi (a) = q n for some positive integer n. Hence γi (a) ∈ kˆ 2 if and only if n is even. Therefore A+ cannot be contained in A+ (Hγ1 )A+ (Hγ2 )C A0 ·(Hγ1 Hγ2 ). 5.3 We also denote by MS the subgroup generated by HS and C A (HS ). Then MS has a finite index in G S since 2A+ ⊂ MS by Lemma 5.2 and A+ has a finite index in A. Observe that HS /Z (HS ) = MS /Z (MS ). Since H˜ S /Z ( H˜ S ) has finite index
166
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Q in G˜ S /Z (G˜ S ) and PGL2 is of adjoint type, the map α∈S ψα (see Section 3.2) factors through G˜ S /Z (G˜ S ) and the following diagram is commutative: Q
α∈S
φα
HS /Z (HS ) ←−−−−−−
Y
SL2 (k)
α∈S
iy
jy Q
α∈S
ψα
G S /Z (G S ) −−−−−−→
Y
PGL2 (k)
α∈S
Q where i and j are canonical maps. We mention that the above map α∈S φα is not in general surjective and that HS /Z (HS ) and G S /Z (G S ) are not isomorphic. Example Let G = SL4 (k), and let S = {γ1 , γ2 }, where a1 a1 a a 2 = 1, γ1 γ1 a2 a3 a4 Then
MS =
xA 0
and GS =
A 0
a3 = . a4
a2 a3 a4
0 A, B ∈ SL (k) , 2 B 0 A, B ∈ SL2 (k), x ∈ k ∗ , x −1 B
HS =
A 0
0 A, B ∈ GL2 (k), det(AB) = 1 . B
Hence HS /Z (HS ) ∼ = MS /Z (MS ) ∼ = PSL2 (k) × PSL2 (k) and G S /Z (G S ) ∼ = PGL2 (k) × PGL2 (k). 5.4 Recall the Cartan decomposition G = K A+ K . Here is a finite subset in the centralizer of A. The bi-K -invariant function ξS of G = K A+ K is defined as follows (see Definition 4.10): Y α(a) 0 4PGL2 (k) for k1 a dk2 ∈ K A+ K . ξS (k1 a dk2 ) = 0 1 α∈S
UNIFORM POINTWISE BOUNDS
167
LEMMA
For a strongly orthogonal system S of 8, we have ξS | HS = 4 HS
and
ξS |G S = 4G S .
Proof If a ∈ A ∩ HS , then 4 HS (a) = 4G S (a). Since, for any a ∈ A, Y 4G S (a) = 4PGL2 (k) ψα (a) , α∈S
it suffices to see that ψα (a) = mas 3.2 and 4.10.
α(a) 0 0 1
, which can be found in the proofs of Lem-
THEOREM 5.5 For any unitary representation ρ of G without a nonzero G + -invariant vector, the restrictions ρ| HS and ρ|G S are tempered.
Proof Denote by H the subgroup generated by Hα , α ∈ S . Theorem 4.9 and the remark following Proposition 2.7 imply that the (K ∩ H )-finite matrix coefficients of ρ| H are bounded by a constant multiple of ξS | H . Since 4 H = ξS | H , ρ| H is tempered. Since H has finite index in HS , an argument similar to the proof of Lemma 4.3 shows that ρ| HS is tempered. Now, to show that ρ|G S is tempered, write ρ|G S as a direct R integral Y σ y of irreducible representations of G S . Without loss of generality, we may assume that for all y ∈ Y , σ y has no nonzero G + S -invariant vector and hence σ y | HS is tempered. Consider the matrix coefficient g 7 → hσ y (g)v, wi, where v and w are (K ∩ G S )-finite vectors of σ y . Recall that the subgroup MS (see Section 5.3) has a finite index in G S . Let h 1 , . . . , h k be a set of representatives in G S /MS . We denote by d[g] the G-invariant measure on G S /Z (G S ). We also denote by d[m] and d[h] the restrictions of d[g] to MS /Z (MS ) and HS /Z (HS ), respectively. Fix > 0. Then Z G S /Z (G S )
k X hσ y ([g])v, wi 2+ d[g] =
=
Z
i=1 MS /Z (MS ) k Z X i=1
HS /Z (HS )
hσ y (h i [m])v, wi 2+ d[m] hσ y ([h])v, h −1 wi 2+ d[h]. i
Since any nonzero matrix coefficient of σ y | HS is L 2+ -integrable (see Theorem 2.5) and hence L 2+ (HS /Z (HS ))-integrable (since Z (HS ) is finite), we have
168
shown that
HEE OH
Z G S /Z (G S )
hσ y ([g])v, wi 2+ d[g] < ∞.
Hence σ y is strongly L 2+ G S /Z (G S ) -integrable. By Theorem 2.4, this implies that ρ|G S is tempered. 5.6 For k archimedean, a translation of a K -finite vector is not necessarily K -finite. However, that is the case for the nonarchimedean field case. LEMMA
Let k be nonarchimedean, and let ρ be any unitary representation of G. Then for any K -finite vector v and for any g ∈ G, the vector gv is K -finite. Furthermore, dimhK (gv)i ≤ [K : gK g −1 ∩ K ] · dimhK vi. Proof Since (gK g −1 )(gv) = gK v, the subspace spanned by (gK g −1 )(gv) has the same dimension as hK vi. Now gK g −1 ∩ K is an open compact subgroup of K and hence has finite index in K . Therefore dimhK (gv)i ≤ [K : gK g −1 ∩ K ] · dimh(gK g −1 ∩ K )(gv)i ≤ [K : gK g −1 ∩ K ] · dimh(gK g −1 )(gv)i = [K : gK g −1 ∩ K ] · dimhK vi. 5.7. Main theorem We are now ready for our main theorem. THEOREM
Let k be a local field with char(k) 6 = 2. Let G be the k-rational points of a connected linear reductive algebraic group with k-ss rank(G) ≥ 2, G/Z (G) almost k-simple, and G/Z (G) Sp2n (C) (locally). Let S be a strongly orthogonal system of 8. Then for any unitary representation ρ of G without a nonzero G + -invariant vector and with K -finite unit vectors v and w, we have hρ(g)v, wi ≤ [K : K ∩ d K d −1 ] · dimhK vi dimhK wi 1/2 ξS (g) for all g = k1 adk2 ∈ K A+ K = G. Proof By Theorem 5.5, ρ|G S is tempered. By the definition of G S , A+ ⊂ G S . For g =
UNIFORM POINTWISE BOUNDS
169
k1 adk2 ∈ K A+ K (recall that = {e} for k = R, C), we have hρ(k1 adk2 )v, wi = hρ(a)(dk2 v), (k −1 w)i 1 1/2 ≤ dimhK (d(k2 v) dimhK (k1−1 w)i 4G S (a) ≤ [K : K ∩ d K d −1 ] · dimhK (k2 v)i 1/2 · dimhK (k1−1 w)i 4G S (a) 1/2 = [K : K ∩ d K d −1 ] · dimhK vi dimhK wi 4G S (a). Since 4G S (a) = ξS (a) = ξS (g) by Lemma 5.4, this finishes the proof. As an immediate corollary, we have the following. COROLLARY 5.8 Keeping the same notation as above, assume, furthermore, that G = K A+ K holds. With the same conditions on ρ, v, and w as above, we have hρ(g)v, wi ≤ dimhK vi dimhK wi 1/2 ξS (g) for all g ∈ G.
5.9 We now summarize. (Properties of ξS ) For any a ∈ A+ , d ∈ , and k1 , k2 ∈ K , 0 < ξS (k1 adk2 ) = ξS (a) ≤ 1. For any a ∈ A+ , ξS (a) = 1 if and only if α(a) = 1 for all α ∈ S . For any > 0, there are positive constants c1 and c2 such that Y −1/2 Y −1/2+ c1 |α(a)| ≤ ξS (a) ≤ c2 |α(a)| for any a ∈ A+ .
THEOREM
(1) (2) (3)
α∈S
α∈S
For instance, in the case when k = Q p , if we set 1 X n S (g) = log p |α(g)|, 2 α∈S
then for any > 0, there exist constants c1 and c2 () such that for any g ∈ G(Q p ), C1 p −n S (g) ≤ ξS (g) ≤ C2 () p −n S (g)(1−) . In inequality (3), using the well-known estimate of 4PGL2 (k) , we may replace Q α∈S |α(a)| with some polynomial of variables |α(a)|; more precisely, for any sufficiently large integer r , there are positive constants c1 and c2 such that Y 1/2 Y r c1 ≤ |α(a)| ξS (a) ≤ c2 1 + |α(a)| α∈S
α∈S
170
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(see [GV] for k archimedean, and see [Si] for k nonarchimedean). For two nonnegative bi-K -invariant functions f 1 and f 2 of K A+ K such that f 1 (g), f 2 (g) ≤ 1 for all g ∈ A+ , we write f 1 ≈ f 2 if for any > 0, there are constants d1 and d2 such that Q d1 f 1 (a) ≤ f 2 (a) ≤ d2 f 1 (a)1− for all a ∈ A+ . Since ξS ≈ α∈S |α|−1/2 , clearly ξS decays fastest when S is a maximal strongly orthogonal system of 8; that is, P among all strongly orthogonal systems of 8 (see Section 3.3). α∈S α is the largest P Set η(8) := (1/2) α∈Q α, where Q is a maximal strongly orthogonal system of 8. Note that η(8) does not depend on the choice of maximal strongly orthogonal systems. In the appendix we present the list of η(8). 6. Representations with the slowest decay In this section we show that for G = SLn (k) or G = Sp2n (k), the pointwise bound function ξQ is an optimal bound for the K matrix coefficients of the class-one part of the unitary dual of G for a maximal strongly orthogonal system Q . We remark that a priori it is not clear whether there should exist one representation whose K matrix 1+ coefficients behave essentially like ξQ in every direction of A+ . This is indeed the case for G = SLn (k) or Sp2n (k). 6.1 Consider the case when G = SLn (k). Let A+ and K be as in Example 2.1.1, so that G = K A+ K holds. Define the characters γi by Pn−i α for i ≤ bn/2c − 1, γi = k=i ( k αbn/2c for n even, γbn/2c = α +α for n odd, bn/2c
bn/2c+1
where αi (a) = ai /(ai + 1) for each 1 ≤ i ≤ n − 1 and for a = diag(a1 , . . . , an ) ∈ A+ . That is, ai . γi (a) = an+1−i Then Q = {γi | 1 ≤ i ≤ bn/2c} is a maximal strongly orthogonal system of 8 (see [Oh1, Prop. 2.3]). The function ξQ (see Section 5.4) is a bi-K -invariant function of G defined by ! bn/2c ai Y 0 a n+1−i 4PGL2 (k) . ξQ (a) = 0 1 i=1
Then by Theorem 5.9, for any > 0, there exist constants d1 and d2 (depending only on ) such that d1 · ξQ (a)1+ ≤ F(a) ≤ d2 · ξQ (a)
for all a ∈ A+ ,
UNIFORM POINTWISE BOUNDS
171
where F is a bi-K -invariant function of G defined by (Qn/2 −1 i=1 |ai | F(a) = Q (n−1)/2 |ai |−1 |a(n+1)/2 |−1/2 i=1
for n even, for n odd.
6.2 We recall the formula for the matrix coefficients of the induced representation IndGP (1) (cf. [Kn]), where P is a parabolic subgroup of G. Consider the Langlands decomposition of P: P = M A P N . Denote by N¯ the unipotent radical of the opposite parabolic subgroup of P with the common Levi subgroup M A P . If g decomposes under the decomposition G = K M A P N , we denote by exp H (g) the A P -component of g. If g decomposes under N¯ M A P N as g = n(g)m(g) ¯ exp a(g)n(g), then the action is given by IndGP (1)(g) f (x) = e−δ0 (a(g
−1 x))
f n(g ¯ −1 x)
for any f ∈ L 2 ( N¯ , d x) and x ∈ N¯ ,
where δ0 is the half-sum of positive N -roots. Define the vector f 0 of IndGP (1) as follows: f 0 (x) = e−δ0 (H (x)) . It is not difficult to see that f 0 is K -invariant and that the matrix coefficient of IndGP (1) with respect to f 0 is as follows: Z
−1 ¯ −1 x))) −δ0 (H (x)) e−δ0 (a(g x)) e−δ0 (H (n(g e d x. IndGP (1)(g) f 0 , f 0 = N¯
Let G = SLn (k), and let P be the maximal parabolic subgroup of G which stabilizes the line k · e1 . We write an element of N¯ as x = (x1 , x2 , . . . , xn )T , where x1 = 1. The realization of the representation IndGP (1) on L 2 ( N¯ , d x) can be formulated as follows (see [Oh1, Sec. 4.4]): for a ∈ A+ , for f ∈ L 2 ( N¯ , d x), and for x = (x1 = 1, x2 , . . . , xn )T ∈ N¯ , IndGP (1)(a) f (x) = |a1 |n/2 f (a1−1 x1 , a2−2 x2 · · · an−1 xn ), and the K -invariant vector f 0 is defined by f 0 (x) = |x|−n/2 . Here the absolute value | · | in k n is defined as follows: qP n 2 for k = R, i=1 x i Pn |x| = for k = C, i=1 x i x¯i max1≤i≤n |xi | for k nonarchimedean.
172
HEE OH
THEOREM 6.3 Let k be any local field, and let G = SLn (k), n ≥ 3. Then for any > 0, there is a constant c > 0 (depending on ) such that for any g ∈ G,
c · ξQ (g)1+ · k f 0 k ≤ hIndGP (1)(g) f 0 , f 0 i ≤ ξQ (g) · k f 0 k. Proof This theorem is shown in [Oh1, Prop. 4.4] for k = R. For k = C, essentially the same proof holds. We briefly go over the case when k is a nonarchimedean local field. The proof given here follows line by line the proof of [Oh1, Prop. 4.4]. For a ∈ A+ , we have Z
|xi | −n/2 G Ind P (1)(a) f 0 , f 0 = max max |xi |−n/2 dm, 1≤i≤n k n−1 1≤i≤n |ai | where dm is a normalized Haar measure on k n−1 . Set r = b(n + 1)/2c, and let T be the following set: n x = (x2 , . . . , xn ) ∈ k n−1 0 ≤ |xi | ≤ 1 for 2 ≤ i ≤ r − 1, o |ai | |xr | for r + 1 ≤ i ≤ n . |ar |
1 ≤ |xr | ≤ 2, |xi | ≤
Note that if x = (x2 , . . . , xn ) ∈ T , then for each 1 ≤ i ≤ n, we have |xi | |xr | ≤ . |ai | |ar |
and
|xi | ≤ |q| Therefore
IndGP (1)(a) ˜ f0, f0
Z
|ar |n/2 dm ≥ C |ar |n/2
≥C T
=C
Y
n Y |ai | |ar |
i=r +1 −1/2
|α(a)|
.
α∈Q
Hence we have
IndGP (1)(a) f 0 , f 0 ≥ d · ξQ (a)1+
for some constant d > 0. 6.4 We now consider the case when G = Sp2n (k). The group Sp2n (k) is defined by the ¯ bilinear form −0I¯ I0n , where I¯n denotes the skew diagonal (n × n) identity matrix. n
We may take the positive Weyl chamber A+ and the maximal compact subgroup K
UNIFORM POINTWISE BOUNDS
173
to be the intersections of those of SLn (k) with G. Then an element a of A+ is of the form a = diag(a1 , . . . , an , an−1 , . . . , a1−1 ). Define the characters γi by ( Pn−1 γi = k=i 2αk + αn
for i ≤ n − 1,
γn = αn , where αi (a) = ai /(ai+1 ), 1 ≤ i ≤ n − 1, and αn (a) = an2 are the simple roots of 8. That is, γi (a) = ai2 for each i = 1, . . . , n. Then Q = {γ1 , . . . , γn } is a maximal strongly orthogonal system of 8 (see [Oh1, Prop. 2.3]). By the definition of ξQ , we have ! 2 0 Q a n i=1 4PGL2 (k) i for k 6= C, 0 1 !!1/2 ξQ (a) = Qn ai2 0 for k = C. i=1 4PGL2 (C) 0 1 (Note that γi is a long root in Cn for each 1 ≤ i ≤ n.) Hence for any > 0, there exist constants d1 and d2 (depending on ) such that d1 · ξQ (a)1+ ≤ F(a) ≤ d2 · ξQ (a)
for all a ∈ A+ ,
where (Q n F(a) =
−1 i=1 |ai | −1/2 i=1 |ai |
Qn
for k 6 = C, =
Qn
i=1 ai
−1
for k = C (all ai positive).
6.5 Let P be the maximal parabolic subgroup of G which stabilizes the line k · e1 . Let P = M A P N denote the Langlands decomposition of P, and let N¯ ∼ = k 2n−1 denote the unipotent radical of the opposite parabolic subgroup to P with the common Levi subgroup M A P . Then an element x of N¯ can be identified with x = (x1 = 1, x2 , . . . , xn , yn , . . . , y1 )T . In the realization of the representation IndGP (1) on L 2 ( N¯ , d x), we have for a ∈ for f ∈ L 2 ( N¯ , d x), and for x ∈ N¯ , T IndGP (1)(a) f (x) = |a1 |n f (a1−1 x1 , a2−2 x2 , . . . , an−1 xn , an−1 yn , . . . , a1−1 y1 ) .
A+ ,
174
HEE OH
Then the vector f 0 given by f 0 (x) = |x|−n is K -invariant, where the absolute value | · | of k 2n is defined as in Section 6.2. THEOREM (k 6 = C and char k 6 = 2) Let G = Sp2n (k), n ≥ 2. Then for any > 0, there is a constant c > 0 (depending on ) such that for any g ∈ G,
c · ξQ (g)1+ · k f 0 k ≤ IndGP (1)(g) f 0 , f 0 ≤ ξQ (g) · k f 0 k.
Proof As in the case of Theorem 6.3, the other inequality is also shown in [Oh1, Prop. 4.4] for k = R. Let k be a nonarchimedean local field. Then for a ∈ A+ , Z o−n n |x | −n
i G , |ai yi | · max |xi |, |yi | dm, Ind P (1)(a) f 0 , f 0 = max 1≤i≤n k 2n−1 1≤i≤n |ai | where x1 = 1. Let T be the following set: n
|an | |yn |, (x1 , . . . , xn , yn , . . . , y1 ) 1 ≤ |yn | ≤ 2, |yi | ≤ |ai | o 0 ≤ |xi | ≤ 1, 0 ≤ |yi | ≤ 2 for 1 ≤ i ≤ n .
Note that if (x1 , . . . , xn , yn , . . . , y1 ) ∈ T , then |xi | ≤ |ai an yn | since |ai | ≥ 1 for all 1 ≤ i ≤ n. Therefore Z −n
G Ind P (1)(a) f 0 , f 0 ≥ C |an yn | dm ≥ C T
1 1+ = d · ξQ (a), |a1 · · · an |
where C and d are some positive constants. 6.6 The inequality in Theorem 6.2 is not true for G = Sp2n (C); in fact, the K matrix coefficient of the representation IndGP (1) satisfies ξQ (a) ≈ hIndGP (1)(a) f 0 , f 0 i
1/2
.
In this case, the minimal pointwise decay can be achieved by the oscillator representation ω of Sp2n (C). (We refer the reader to [Ho] for a detailed description of ω.)
UNIFORM POINTWISE BOUNDS
175
In the realization of ω in L 2 (Cn ), we have the following formula: for a ∈ A+ , for f ∈ L 2 (Cn ), and for (z 1 , . . . , z n ) ∈ C n , ω(a) f (z 1 , . . . , z n ) =
n Y
|ai |−1/2 f (a1−1 z 1 , . . . , an−1 z n ),
i=1
where a = diag (a1 , . . . , an , an−1 , . . . , a1−1 ) ∈ A+ (recall that |ai | = ai a¯i = ai2 ). The representation ω decomposes into two irreducible components, the even part ω+ and the odd part ω− . In the realization of ω in L 2 (Cn ), the space ω+ can be taken as the even functions: functions such that f (−x) = f (x); the space ω− consists of the odd functions: functions such that f (−x) = − f (x). And only the even part ω+ is class one. In fact, the function n X −2 z i z¯ i f 0 (z 1 , . . . , z n ) = exp π i=1
is a K -invariant unit vector in ω+ , where K = SU(2n) ∩ Sp2n (C). Then hω+ (a) f 0 , f 0 i =
n Y i=1
Since
R
R2
(ai )−1
Z Z
exp − π 2 (1 + ai−2 )(xi2 + yi2 ) d xi dyi .
R R
exp(−(x 2 + y 2 )) d x dy = π, hω+ (a) f 0 , f 0 i =
n Y (ai2 + 1)−1/2 . i=1
Since FSp2n (C) (a) = rem.
Qn
−1/2 i=1 |ai |
=
Qn
i=1 ai
−1 ,
we have shown the following theo-
THEOREM
Let G = Sp2n (C), n ≥ 2. For any > 0, there exists a constant C > 0 (depending on ) such that 1+ C · ξQ (g) ≤ hω+ (g) f 0 , f 0 i ≤ ξQ (g) for any g ∈ G. COROLLARY 6.7 Let G = Sp2n (C), n ≥ 2, and let α be a long root in its root system Cn . Then ω+ | Hα is strongly L 4+ but not strongly L 2+ .
Proof By conjugation, we may assume that α = γ1 , where γ1 is defined as in Section 6.4. It
176
HEE OH
then follows from the above theorem that for any > 0, there exists a constant C > 0 such that 1/2+
C · 4 Hγ
1
1/2
(h) ≤ hω+ (h) f 0 , f 0 i ≤ 4 Hγ (h) for any h ∈ Hγ1 . 1
1/2
Note that 4 Hγ is L 4+ -integrable for any > 0. Hence the restriction to Hγ1 of the 1
matrix coefficient hω+ (h) f, f i is L 4+ -integrable for any f ∈ G f 0 . Since ω+ is irreducible, G f 0 spans a dense subset in the Hilbert space attached to ω+ . Hence ω+ | Hγ1 is strongly L 4+ . Now suppose that ω+ | Hγ1 is strongly L 2+ . Then by Theorem 2.4, its (K ∩ Hγ1 ) matrix coefficient hω+ (h) f 0 , f 0 i must be L 2+ -integrable. This is a 1/2+ contradiction since it is bounded from below by the function C · 4 Hγ , which is not L 2+ -integrable.
1
7. Uniform L p -bound for matrix coefficients: p K (G) 7.1 In this section, let G be the group of k-rational points of a connected almost k-simple algebraic group over k, where k is any local field. As before, let K be a good maximal compact subgroup of G. Denote by Gˆ the set of equivalence classes of infinitedimensional irreducible unitary representations of G. Recall that p(G) denotes the ˆ ρ is strongly L q for any smallest real number such that for any nontrivial ρ ∈ G, q > p(G) (cf. [Li], [LZ]). Similarly, we denote by p K (G) the smallest real number ˆ the K -finite matrix coefficients of ρ are L q such that for any nontrivial ρ ∈ G, integrable for any q > p K (G). By the Peter-Weyl theorem, we have p(G) ≤ p K (G). (We point out that in the literature the number p(G) has been implicitly identified with p K (G).) M. Cowling [Co] showed that p(G) < ∞ if and only if G has property (T ). For G = Sp2n (k), n ≥ 2, Howe showed that p K (Sp2n (C)) = 4n and p K (G) = 2n for other local fields k 6= C with char(k) 6 = 2 (see [Ho]). For other real (or complex) classical simple Lie groups, the exact number p K (G) is obtained by combining the known cases of a classification of the unitary dual by D. Vogan and D. Barbasch, and the results of J.-S. Li (see [Li] for references). The precise values of p K (G) are not known in general, but upper bounds have been given in many cases (see [Ho], [Li], [LZ], [Oh1]). 7.2 Let Q be a maximal strongly orthogonal system of 8. Then it follows from Theorem 5.7 that p K (G) ≤ inf q ∈ R ξQ ∈ L q (G) .
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LEMMA 7.3 Let f be a continuous function on G such that f (k1 adk2 ) = f (a) for any k1 adk2 ∈ R K A+ K . If A+ | f (a)| p δ B (a) da < ∞ for some p > 0, then f ∈ L p (G).
Proof For K = R or C, this can be seen using the decomposition of the Haar measure dg on G = K A+ K , dg = 1(a)dk1 dadk2 , and the well-known fact (cf. [Kn, Prop. 5.2.8]) that for any t > 1, there exist constants d1 (t) and d2 such that d1 (t) δ B (a) ≤ 1(a) ≤ d2 δ B (a) for all + + a ∈ A+ t = g ∈ A |α(g)| ≥ t for all α ∈ 8 . For k nonarchimedean, we have Z | f (g)| p dg = G
X
Vol(K ad K )| f (a)| p .
ad∈A+
In fact, there exist positive constants c1 and c2 such that c1 δ B (ad) ≤ Vol(K ad K ) ≤ c2 δ B (ad) for all ad ∈ A+ (see [Si, Lem. 4.1.1]). Since the modular function δ B is a homomorphism on B and is finite, it follows that for some positive constants c10 and c20 , c10 δ B (a) ≤ Vol(K ad K ) ≤ c20 δ B (a) for any ad ∈ A+ . Hence the above claim follows. 7.4 Hence by the above lemma together with the inequality in Theorem 5.9(3), getting an upper bound for p K (G) boils down to a matter of comparing the coefficients of each Q simple root in α∈Q α −1/2 with those in the modular function δ B . THEOREM
Let k be a local field with char(k) 6 = 2. Let G be the k-rational points of a connected linear almost k-simple algebraic group over k with k-rank(G) ≥ 2 and G/Z (G) P Sp2n (C) (locally). Let δ B be the modular function of B, and set η(8) = 1/2 α∈Q α for a maximal strongly orthogonal system Q of 8 (e.g., one in the appendix). Then n the coefficient of α in δ o i B p K (G) ≤ max i = 1, . . . , n , the coefficient of αi in η(8) where {α1 , . . . , αn } is the set of simple roots in 8. For example, if G is split over k with rank greater than or equal to 2, p K (G) is bounded above as shown in Table 1. We remark that for k = R, if we let F(k1 ak2 ) =
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Table 1
8: p K (G) ≤:
An 2n
Bn 2n
Cn 2n
Dn:even 2(n − 1)
Dn:odd 2n
E6 16
E7 18
E8 29
F4 11
G2 6
α −1/2 (a) for any k1 ak2 ∈ K A+ K = G, this function F coincides with FG in [Oh1] except for Dn , n odd, in which case F was improved by replacing two of the Hγ ’s by SO (see Section 3.3; see the remark in [Oh1, Sec. 3.5]), and one obtains a stronger estimate 2n − 2 for p K (G). Since we believe that the novelty of the above corollary lies in the simplicity of our method for giving an upper bound for p K (G) rather than in improving the bound, we do not elaborate on it here. Q
α∈Q
7.5 The results in Section 6 yield the precise number p K (G) in the following cases (in which the classification of the unitary dual is also known). THEOREM
(1) (2) (3)
Let n ≥ 3. Then p K (SLn (k)) = 2(n − 1) for any local field k. Let n ≥ 2. Then p K (Sp2n (k)) = 2n for any nonarchimedean local field k, char k 6 = 2, or k = R. Let n ≥ 2. Then p K (Sp2n (C)) = 4n.
7.6 By the work of Cowling, U. Haggerup, and Howe [CHH] (see Theorem 2.5), we have a passage from a uniform L p -bound to a uniform pointwise bound; that is, we let m be any integer such that 2m ≥ p K (G). Then any K -finite matrix coefficient hρ(g)v, wi 1/2 of an infinite-dimensional ρ ∈ Gˆ is bounded by dimhK vi dimhK wi 4G (g)1/m . We remark that even in the case when the number p K (G) is precisely known, Theorem 5.7 (for a maximal strongly orthogonal system) provides a much sharper pointwise bound in general. 8. Kazhdan constants In this section, we discuss some applications of the above results in terms of a quantitative estimate of Kazhdan property (T ) of the group G, namely, a Kazhdan constant. 8.1 For a locally compact group G, we say that a unitary representation ρ of G almost has an invariant vector if for any > 0 and any compact subset Q of G, there exists a unit vector v that is (Q, )-invariant; that is, kρ(g)v − vk ≤ for all g ∈ Q. Recall
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that G is said to have Kazhdan property (T ) if any unitary representation of G which almost has an invariant vector actually has a nonzero invariant vector. Definition (cf. [HV], [Bu]) For a locally compact group G with a compact subset Q, a positive number is said to be a Kazhdan constant for (G, Q) if for any unitary representation ρ without a nonzero G + -invariant vector and for any unit vector v of ρ, max kρ(s)v − vk ≥ . s∈Q
If there exists such an , we call Q a Kazhdan set for G. In other words, if is a Kazhdan constant for (G, Q), then any unitary representation of G which has a (Q, )-invariant vector actually has a nonzero invariant vector. 8.2 In what follows, we keep the notation from Section 2.1. PROPOSITION
Let H be any subset of G such that K ∩ H is a closed subgroup of K . Let 8 be a bi-(K ∩ H )-invariant function of H with the following properties: (i) 0 < 8(h) ≤ 1 for any h ∈ H ; (ii) for h ∈ H , 8(h) = 1 if and only if h ∈ K ∩ H ; (iii) for any unitary representation ρ of G and any (K ∩ H )-fixed unit vector v, hρ(h)v, vi ≤ 8(h) for any h ∈ H. Set
√ 2(1 − 8(h)) χ (h) = √ 2(1 − 8(h)) + 3
for any h ∈ H.
Then χ is a bi-(K ∩√ H )-invariant function of H satisfying the following: √ (1) 0 ≤ χ (h) < 2/ 2 + 3 for any h ∈ H ; (2) for h ∈ H , χ (h) = 0 if and only if h ∈ K ∩ H ; (3) we have inf
max
s∈{K ∩H,h}
kρ(s)v − vk ≥ χ(h) for any h ∈ H,
where the infimum is taken over all unitary representations ρ of G without a nonzero invariant vector and for all unit vectors v of ρ.
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Proof (1) and (2) are obvious from the definition of χ. Let h be a nontrivial element of H such that h ∈ / K ∩ H . Fix a unit vector v of ρ. Suppose that for all k ∈ K ∩ H , we have kρ(k)v − vk ≤ χ(h). We show that kρ(h)v − vk ≥ χ (h). Let v1 be the average of the (K ∩ H )-transform of v, Z v1 = kv dk, K ∩H
where dk is the normalized Haar measure on K ∩ H . Note that v1 is (K ∩ H )-fixed. We compute kv − v1 k ≤ χ(h), so that kv1 k ≥ 1 − χ(h). Since χ (h) < 1, the inequality implies that v1 is nonzero. Recall that for any unit vector w, kρ(h)w − wk2 = 2 − 2 Rehρ(h)w, wi. Hence
q
v v1 1
−
ρ(h)
≥ 2 1 − 8(h) kv1 k kv1 k
and kρ(h)v1 − v1 k ≥
q
q 2 1 − 8(h) kv1 k ≥ 2 1 − 8(h) 1 − χ(h) .
Therefore
kρ(h)v − vk = ρ(h)v1 − v1 + ρ(h)v − ρ(h)v1 + (v1 − v) is greater than or equal to q 2 1 − 8(h) 1 − χ(h) − 2χ(h) = χ(h). Hence kρ(h)v − vk ≥ χ(h). This shows that for any h ∈ H , max
{s∈K ∩H,h}
kρ(s)v − vk ≥ χ(h),
proving the proposition. The above proposition can be considered as a generalization of a quantitative statement of the fact that the unitary representations of G which are not class one are uniformly bounded away from the trivial representation in the Fell topology: if ρ has an (K , )-invariant vector, say v, for some 0 < < 1, then ρ does have a K R invariant vector. This can be seen by considering the average K ρ(k)v of v along its K -transform. If < 1, then the average is nonzero, which is K -invariant. This is the reason that a pointwise bound for the matrix coefficients (only) with respect to K -invariant vectors yields Kazhdan constants.
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181
8.3 In what follows, let k be a local field with char(k) 6 = 2, and let G be the group of k-rational points of a connected simply connected almost k-simple linear group over k with k-rank at least 2 with the Cartan decomposition G = K A+ K (see the remark in Section 2.1). In particular, G = G + . Recall the bi-K -invariant function ξS on G for a strongly orthogonal system S defined in Definition 4.10 (see Section 5.9 as well). Notation. Define a bi-K -invariant function χS on G by √ 2(1 − ξS (a)) χS (a) = √ for any a ∈ G. 2(1 − ξS (a)) + 3 Since HS (see Section 5.1) and ξS satisfy Proposition 8.2 by Corollary 5.8 and Theorem 5.9, we have the following theorem. THEOREM
Let S be a strongly orthogonal system of 8. Then for any s ∈ HS \K , χS (s) is a Kazhdan constant for ({K ∩ HS , s}, G). 8.4 Note that if S is a large strongly orthogonal system S (see Section 3.3), then K = g ∈ K A+ K ξS (g) = 1 ; hence it follows that χS (g) > 0 for any g ∈ / K. THEOREM
Let S be a large strongly orthogonal system of 8. (1) For any s ∈ / K , χS (s) is a Kazhdan constant for ({K , s}, G). (2) For any compact subset Q properly containing K , maxg∈Q χS (g) is a Kazhdan constant for (Q, G). Proof Statement (1) follows from Corollary 5.8 and Proposition 8.2. To deduce (2) from (1), it suffices to observe that maxg∈Q χS (g) = χS (x) for some x ∈ Q since χS is a continuous function on G and Q is compact. Using an explicit form of ξS (and hence of χS ) in the above theorem provides us with a simple machine that produces a Kazhdan constant for any set {K , s}, s ∈ / K, and hence for any compact subset properly containing K . One of the simplest methods
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is to use S = {the highest root}, but to get an optimal Kazhdan constant from this method, we can use a maximal strongly orthogonal system, as in the appendix. Remark. The assumption that Q contains K can be loosened in many generic cases when k is archimedean. For instance, if Q contains some neighborhood of the identity of G, then there is a positive integer n such that Q n contains K properly, where Q n denotes the set of all elements whose word lengths with respect to Q are at most n. Then maxg∈Q n χS (g) is a Kazhdan constant for (Q n , G) by Theorem 8.4. Observe that n · max kρ(g)v − vk ≥ maxn kρ(g)v − vk g∈Q
g∈Q
for any positive integer n, any unitary representation ρ of G, and a unit vector v of ρ. Therefore (1/n) maxg∈Q n χS is a Kazhdan constant for (Q, G). Remark. For the real group Sp(1, n), n ≥ 2, Theorem 8.4 remains valid in view of Theorem 4.9. 8.5 For any nontrivial irreducible class-one unitary representation ρ of G and a K invariant unit vector v, COROLLARY
kρ(g)v − vk ≥ χS (g)
for all g ∈ G.
Proof Since v is K -invariant, ρ(k)v − v = 0 for any k ∈ K . If g ∈ K , then both sides are zero. For g ∈ / K , by the above theorem, max kρ(s)v − vk ≥ χS (g).
s∈{K ,g}
But maxs∈{K ,g} kρ(s)v − vk = kρ(g)v − vk, proving the claim. 8.6 While the function ξS gives an exact value for k nonarchimedean, it is not the case for k = R or C. In the following we discuss some estimates of ξS for k = R (the process is similar for k = C). We have (cf. [HT, Chap. V, Sec. 3.1]) Z π/2 2 −1/2 cos t 1.09 + log x 2 2 + sin t dt ≤ for any x ≥ 16. 4R (x) = √ √ x2 π x 0 x Hence ξS (s) ≤
Y γ ∈I (s)
h R γ (s)
for any s ∈ / K,
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183
√ where h R (x) = (1.09 + log x)/ x and I (s) = {γ ∈ S | γ (s) ≥ 16}. LEMMA
For any s ∈ HS \K , the set {K ∩ HS , s} is a Kazhdan set with a Kazhdan constant 1 1 Ns χS (s Ns ) ≥ f h R (16)|I (s )| , Ns Ns where Ns denotes the minimum positive integer n such that I (s n ) 6= ∅ and f (x) = √ √ 2(1 − x)/ 2(1 − x) + 3. Proof Observe that for any positive integer n, 1 max kρ(g)v − vk ≤ max kρ(g)v − vk g∈{K ,s} n g∈{K ,s n } for any unitary representation ρ and a unit vector v of ρ. On the other hand, since n f (x) is a strictly decreasing function on [0, 1], we have χS (s n ) ≥ f (h R (16)|I (s )| ). Since maxg∈{K ,s n } kρ(g)v − vk ≥ (1/n)χS (s n ), the lemma is proved. 8.7. Examples of Kazhdan constants We note that any compact subset of G which contains K properly generates the group G in the topological sense. Any compact generating subset of a Kazhdan property (T ) group is a Kazhdan set (see [HV, Chap. 1, Prop. 15]). Furthermore, any compact subset of G which generates a nonamenable subgroup is in fact a Kazhdan set (see [Sh2]). The following theorem yields examples of Kazhdan sets that are contained in a proper closed semisimple subgroup of G. We give examples of Kazhdan constants where Kazhdan sets are taken from SL2 (k) or SL4 (k) embedded into the upper left corner of SLn (k). Example 8.7.1 Let m be any nonzero integer. (1) For any n ≥ 3, the group SLn (R) has a Kazhdan constant (2)
0.08 SO(2), diag(41/m , 4−1/m ) , . |m|
For any n ≥ 4, the group SLn (R) has a Kazhdan constant
0.109 SO(4), diag(41/m , 41/m , 4−1/m , 4−1/m ) , . |m|
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Proof Let m = 1. Denote by s1 and s2 the diagonal elements in (1) and (2), respectively. Set γ1 (a) =
a1 an
and
γ2 (a) =
a2 , an−1
where a = diag(a1 , a2 , . . . , an ) ∈ A+ , that is, where ai ≥ ai+1 > 0 for all 1 ≤ i ≤ n − 1. Note that S1 = {γ1 } and S2 = {γ1 , γ2 } are large strongly orthogonal systems (since γ1 is the highest root). Then γ1 (s1 ) = 16, γ1 (s2 ) = 16, and γ2 (s2 ) = 16. Therefore, by Lemma 8.6, χS1 (s1 ) ≥ f (h R (16)) ≥ 0.08 and χS2 (s2 ) ≥ f (h R (16)2 ) ≥ 0.109. Now the claim for an arbitrary m follows from Lemma 8.6. Example 8.7.2 Let m be any nonzero integer. (1) For any n ≥ 3, the group SLn (R) has a Kazhdan constant 0.104 1 m4 SO(2), , . 0 1 |m| (2)
For any n ≥ 4, the group SLn (R) has a Kazhdan constant 1 m4 0 0 SO(4), 0 1 0 0 , 0.139 . 0 0 1 4 |m| m 0 0 0 1
Proof By Lemma 8.6, it suffices to consider the case of m 1 0 1 4 s1 = , s2 = 0 0 1 0
= 1. Let 4 1 0 0
0 0 1 0
0 0 . 4 1
For g ∈ G, denote by A+ (g) the A+ -part of the decomposition of g in the Cartan decomposition G = K A+ K . It is not difficult to see that √ 1/2 √ −1/2 A+ (s1 ) = diag (9 + 4 5) , 1, . . . , 1, (9 + 4 5) and that for n ≥ 4, A+ (s2 ) √ 1/2 √ 1/2 √ −1/2 √ −1/2 = diag (9 + 4 5) , (9 + 4 5) , 1, . . . , 1, (9 + 4 5) , (9 + 4 5) .
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185
Let γ1 , γ2 , S1 , and S2 √ be as in the proof of Example 8.7.1. Therefore γ1 (s1 ) = γ1 (s2 ) = γ2 (s2 ) = 9 + 4 5 ≥ 16. By Lemma 8.6, χ S1 (s1 ) ≥ f h R (γ1 (s1 )) ≥ 0.104 and χ S2 (s2 ) ≥ f h R (γ1 (s2 ))h R (γ2 (s2 )) ≥ 0.139. The following two examples are direct applications of Theorem 8.4. Let √ 2(1 − x) n( p − 1) + ( p + 1) . and f (x) = cn = √ √ n p ( p + 1) 2(1 − x) + 3 Example 8.7.3 For n ≥ 3, SLn (Q p ) has Kazhdan constants SL2 (Z p ), diag ( p m , p −m ) , f (c2|m| ) for any m ∈ Z. Example 8.7.4 For n ≥ 3, the group SLn (Q p ) has Kazhdan constants SL3 (Z p ), diag ( p n 1 , p n 2 , p n 3 ) , max f (c|n i −n j | ) 1≤i, j≤3
for any n 1 , n 2 , n 3 ∈ Z such that
P3
i=1 n i
= 0.
So for SLn (Q2 ), SLn (Q3 ), and SLn (Q5 ), respectively, the following are Kazhdan constants: SL4 (Z2 ), diag(2, 2−1 , 22 , 2−2 ) , 0.25) , SL4 (Z3 ), diag(3, 3−1 , 32 , 3−2 ) , 0.29) , and
SL4 (Z5 ), diag(5, 5−1 , 52 , 5−2 ) , 0.31) .
8.8 Recall the definition of κ(G, Q) from the introduction. PROPOSITION
Let k be any local field with char(k) 6 = 2, and let G be the group of k-rational points of a connected simply connected almost k-simple algebraic group over k with k-rank(G) ≥ 2.
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(1)
(2)
HEE OH
Let k be nonarchimedean, and let G be the k-split. Then 2√ p , inf κ G, {K , s} ≥ f s∈G\K p+1 where p is the cardinality of the residue field of k. We have inf
inf
inf
p=prime n≥3 s ∈SL / n (Z p )
(3)
κ SLn (Q p ), {SLn (Z p ), s} ≥ f
2√2 3
> 0.10.
Let k = R or C. Then inf κ G, {K , s} = 0.
s∈G\K
(4)
For any sequence gi ∈ G going to infinity, √ 2
lim inf κ G, {K , gi } ≥ √ . i→∞ ( 2 + 3)
Proof (1) Denote by γ the highest root in 8. Then S = {γ } forms a large strongly orthogonal system of 8. Let s ∈ G\K . By replacing s by its A+ -component in the Cartan decomposition G = K A+ K , we may assume that s ∈ A+ . Hence |γ (s)| = p m for some positive integer m. By the definition of ξS , −1 γ (s) 0 p 0 ξS (s) = 4PGL2 Q p ≤ 4PGL2 Q p . 0 1 0 1 Hence we have κ G, {K , s} ≥ f ξS (s) ≥ f
−1 p 4PGL2 Q p 0
2√ p 0 ≥ f . 1 p+1
(2) Since p ≥ 2 for k = Q p , we have √ √ 2 p 2 2 ≥ . p+1 3 Hence by claim (1), we have 2√2 κ SLn (Q p ), {SLn (Z p ), s} ≥ f > 0.10. 3 Since this is independent of n and p, claim (2) follows.
UNIFORM POINTWISE BOUNDS
187
(3) Let ρ be any class-one unitary representation of G without any invariant vector (whose existence we know of), and let v be a K -invariant unit vector of ρ. Then the matrix coefficient hρ(g)v, vi is a continuous function that has value 1 for all g ∈ K . Hence for a sequence gi tending to an element of K with gi ∈ / K, we have limi→∞ hρ(gi )v, vi = 1. By the well-known equality kρ(g)v − vk2 = 2 − 2 Rehρ(g)v, vi, we obtain limi→∞ kρ(gi )v − vk = 0. + (4) Without √ √ loss of generality, we may assume that gi ∈ A . Then f (ξS (gi )) tends to 2/( 2 + 3). Thus the claim follows from Theorem 8.4. 8.9 Let 0 be a lattice in G. If (Q, ) is a Kazhdan constant for G, then in principle one can find some positive real number R (at least bigger than the radius of Q) such that 0 ∩ B R yields a Kazhdan set for 0, where B R denotes a ball of radius R of the identity in a suitable metric in G/K (see [Sh2, Th. B]; see also [HV, Lem. 3.3]). For instance, our results imply that if 0 is a cocompact lattice in the group G = PGL3 (Q p ) such that 0 acts simply transitively on G/ PGL3 (Z p ), then 0.10 is a Kazhdan constant for (0, B1 ∩ 0), where B1 = {g ∈ PGL3 (Q p ) | g ∈ PGL3 (Z p ) diag( p, 1, 1) PGL3 (Z p )}. However, obtaining Kazhdan constants for a lattice 0 of G using this method involves understanding the size of the fundamental domain of 0 in G, which seems highly nontrivial in general. Appendix: Maximal strongly orthogonal systems Let 8 be a reduced irreducible root system with a basis α1 , . . . , αn . The subscripts of the αi ’s are determined by the following choice of the highest root (see [Bo]): 8
the highest root,
An
α1 + α2 + · · · + αn ,
Bn
α1 + 2α2 + · · · + 2αn ,
Cn
2α1 + 2α2 + · · · + 2αn−1 + αn ,
Dn
α1 + 2α2 + · · · + 2αn−2 + αn−1 + αn ,
E6
α1 + 2α2 + 2α3 + 3α4 + 2α5 + α6 ,
E7
2α1 + 2α2 + 3α3 + 4α4 + 3α5 + 2α6 + α7 ,
E8
2α1 + 3α2 + 4α3 + 6α4 + 5α5 + 4α6 + 3α7 + 2α8 ,
F4
2α1 + 3α2 + 4α3 + 2α4 ,
G2
3α1 + 2α2 .
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The notation bxc denotes the largest integer not bigger than x. Set b n+1 for 8 = An , 2 c 2b n c for 8 = Dn , 2 N (8) = N = 4 for 8 = E 6 , rank(8) for 8 = Bn , Cn , F4 , G 2 , E 7 , E 8 . In the following we list maximal strongly orthogonal systems Q constructed in [Oh1]. We correct a typo for γ3 in E 8 from [Oh1]. We have the following: 8 An
Bn (n ≥ 2)
Cn (n ≥ 2)
Dn (n ≥ 4)
Q (8), a maximal strongly orthogonal system of 8, γi = α(i + · · · + αn−i+1 for 1 ≤ i ≤ N − 1, αN for n odd, γ N = α + α for n even, N N +1 γ2i−1 = αi + · · · + αn−i + 2αn−i+1 + · · · + 2αn , γ2i = αi + · · · + αn−i for 1 ≤ i ≤ b n2 c, γ = α for n odd, n (n+1)/2 + · · · + αn ( γi = 2αi + · · · + 2αn−1 + αn for 1 ≤ i ≤ N − 1,
γ N = αn , γ1 = α1 + · · · + αn−2 + αn , γ2 = α1 + · · · + αn−1 , γ2i−1 = αi + · · · + αn−i + 2αn−i+1 + · · · + 2αn−2 + αn−1 + αn , γ = α + · · · + α for 2 ≤ i ≤ b n c, 2i
E6
γ1 γ 2 γ 3 γ4
i
n−i
2
= α1 + α2 + α3 + 2α4 + 2α5 + α6 , = α1 + α2 + 2α3 + 2α4 + α5 + α6 , = α2 + α3 + 2α4 + α5 , = α2 ,
UNIFORM POINTWISE BOUNDS
E7
E8
F4
G2
189
γ1 = α1 + α2 + α3 + 2α4 + 2α5 + α6 + α7 , γ2 = α1 + α2 + 2α3 + 2α4 + α5 + α6 + α7 , γ3 = α1 + α2 + 2α3 + 2α4 + 2α5 + α6 , γ4 = α1 + α2 + α3 + 2α4 + α5 + α6 , γ5 = α2 + α3 + 2α4 + α5 , γ6 = α2 + α3 + 2α4 + 2α5 + 2α6 + α7 , γ7 = α2 , γ1 = α1 + 2α2 + 3α3 + 5α4 + 4α5 + 3α6 + 2α7 + α8 , γ2 = α1 + 2α2 + 2α3 + 3α4 + 3α5 + 2α6 + α7 + α8 , γ3 = α1 + 2α2 + 2α3 + 3α4 + 2α5 + 2α6 + α7 , γ = α + α + α + 2α + 2α + 2α + 2α + α , 7 4 1 2 3 4 5 6 8 γ = α + α + 2α + 2α + α + α + α + α , 5 7 1 2 3 4 5 6 8 γ = α + α + 2α + 2α + 2α + α + α , 7 6 1 2 3 4 5 6 γ = α + α + α + 2α + α , 7 1 2 3 4 5 γ = α + α + α + α + α , 8 1 3 4 5 6 γ1 = α1 + 2α2 + 4α3 + 2α4 , γ = α + 2α + 2α + 2α , 2 1 2 3 4 γ = α + 2α + 2α , 3 1 2 3 γ4 = α1 , ( γ1 = 3α1 + 2α2 , γ2 = α1 .
We set η(8) to be the half-sum of the roots in a maximal strongly orthogonal system of 8. Recall that η(8) does not depend on the choice of a maximal strongly orthogonal system. We have the following: 8 An Bn (n ≥ 2)
η(8), (P(n−1)/2
Pn (n−i+1) i αi i=(n+1)/2 i=1 2 αi + 2 Pn/2 i P n (n−i+1) n αi i=n/2+2 i=1 2 αi + 4 αn/2+1 + 2 bn/2c n X X iαi +
i=1
Cn (n ≥ 2)
n−1 X i=1
i=bn/2+1c
iαi +
n αn , 2
n αi , 2
for n odd, for n even,
190
HEE OH
Dn (n ≥ 4)
bn/2c X
iαi +
i=1
n−2 X i=bn/2c+1
jn k 1jn k αi + (αn−1 + αn ), 2 2 2
E8
α1 + 2α2 + 2α3 + 3α4 + 2α5 + α6 , 7 9 3 2α1 + α2 + 4α3 + 6α4 + α5 + 3α6 + α7 , 2 2 2 4α1 + 5α2 + 7α3 + 10α4 + 8α5 + 6α6 + 4α7 + 2α8 ,
F4
2α1 + 3α2 + 4α3 + 2α4 ,
G2
2α1 + α2 .
E6 E7
Acknowledgments. I would like to thank Hillel Furstenberg and Shahar Mozes for their support and encouragement during the year (1998 – 1999) I spent at Hebrew University in Jerusalem. I would also like to thank Amos Nevo for his interest in this work and for helpful comments. Thanks also are due to Wee Teck Gan for valuable comments on the preprint. References [Be]
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138 Mathematics Department, Fine Hall, Princeton University, Princeton, New Jersey 08544, USA;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 2,
ON THE p-ADIC REALIZATION OF ELLIPTIC POLYLOGARITHMS FOR CM-ELLIPTIC CURVES KENICHI BANNAI
Abstract Let E be a CM-elliptic curve over Q with good ordinary reduction at a prime p ≥ 5. The purpose of this paper is to construct the p-adic elliptic polylogarithm of E, following the method of A. Be˘ılinson and A. Levin. Our main result is that the specializations of this object at torsion points give the special values of the one-variable p-adic L-function of the Gr¨ossencharakter associated to E. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . 1.1. Overview . . . . . . . . . . . . . . . . . . . . 1.2. Notation . . . . . . . . . . . . . . . . . . . . . 2. Review of rigid syntomic cohomology with coefficients . 2.1. Review of geometric cohomology . . . . . . . . 2.2. Review of syntomic cohomology . . . . . . . . . 2.3. The Gysin exact sequence . . . . . . . . . . . . 3. Construction of the elliptic polylogarithm . . . . . . . . 3.1. The elliptic logarithmic sheaf . . . . . . . . . . 3.2. The pullback of the elliptic logarithmic sheaf . . . 3.3. The elliptic polylogarithm . . . . . . . . . . . . 4. On the p-adic elliptic polylogarithmic function . . . . . 4.1. Some preliminaries . . . . . . . . . . . . . . . 4.2. The p-adic elliptic polylogarithmic function . . . 4.3. The relation to the one-variable p-adic L-function 5. The partial logarithmic sheaf . . . . . . . . . . . . . . 5.1. The definition of the partial logarithmic sheaf . . 5.2. The calculation of cohomology . . . . . . . . . . 6. The partial elliptic polylogarithm . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
194 196 196 196 197 199 203 206 206 210 212 214 215 215 218 220 220 223 225
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 2, Received 6 December 2000. Revision received 28 June 2001. 2000 Mathematics Subject Classification. Primary 14G10, 14F30; Secondary 11G40, 14G20, 11R23. Author’s work supported by the Japan Society for the Promotion of Science (JSPS) Research Fellowship for Young Scientists. 193
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7. The main result References . . . . .
KENICHI BANNAI
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
231 234
1. Introduction In the paper [BL], Be˘ılinson and Levin constructed the elliptic polylogarithm as an element in the absolute Hodge and l-adic cohomology of elliptic curves minus the identity. The significance of this element is that, as in the classical case, it is motivic in nature. The aim of this paper is to study the p-adic analogue of the elliptic polylogarithm. Let E be a CM-elliptic curve over Q with good reduction at a prime p ≥ 5. Using the tools developed in [Ban1], we construct the p-adic elliptic polylogarithm in rigid syntomic cohomology following the method of Be˘ılinson and Levin. Our main result is that this element, when specialized to the torsion points of E, gives the special values of the one-variable p-adic L-function of the Gr¨ossencharakter associated to E. This result is a generalization to the case of elliptic curves of our previous work [Ban1]. Hence, philosophically, this result is a generalization to the case of elliptic curves of the calculations by M. Gros and M. Kurihara [G1], Gros [G2], and M. Somekawa [So] on the calculation of the image of the cyclotomic element by the syntomic regulator. This may also be viewed as a generalization to the case of higher twists of the result of R. Coleman and E. de Shalit [CdS] (see also [Bes2]) on the calculation of the p-adic regulator for CM-elliptic curves. We follow [HK, Appendix A] for the construction of the elliptic polylogarithm. Suppose that E has complex multiplication in the integer ring OK of an imaginary quadratic field K. Since we have assumed that E has good ordinary reduction at p, we have a splitting p = pp∗ in OK . We denote by ψ = ψE/K the Gr¨ossencharakter associated to EK = E ⊗Q K, with conductor f prime to p. Let g be an ideal of OK prime to p such that f|g, and denote by K(g) the ray class field of K modulo g. We let K be the completion of K(g) at a prime above p, and we let O K be its ring of integers. We fix a smooth model E over O K of E K = E ⊗Q K . We define H ∨ = H 1 (E, K (0)) to be the geometric syntomic cohomology of E with coefficients in K (0), which in this case is simply the crystalline cohomology of E with the Frobenius and the Hodge filtration. It is a direct sum H ∨ = K (ω) ⊕ K (η) of two one-dimensional filtered Frobenius modules with fixed basis ω and η. We denote by H the dual of H ∨ , with dual basis ω∨ and η∨ . Let S(E) be the category of syntomic coefficients on E. It is a rough p-adic analogue of the category of variations of mixed Hodge structures on E. The elliptic logarithmic sheaf L og is a pro-object in S(E). One of its main features is the splitting principle.
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PROPOSITION (Cor. 3.7) Let u ∈ E(O K ) be a torsion point of E of order prime to p. Then we have Y u ∗ L og ∼ Sym j H = j≥0
in S(O K ), the category of filtered Frobenius modules. We again denote by H ∨ the pullback of H ∨ to E with respect to the structure morphism. Let U = E \ {e}, where e is the identity element of E. The elliptic polylogarithm is an element 1 pol ∈ Hsyn U, H ∨ ⊗ L og(1) , 1 (U, H ∨ ⊗ L og(1)) is the absolute syntomic cohomology. The signifiwhere Hsyn cance of this element is that it is conjectured to be motivic in origin. In other words, it is conjectured to be the image, by the syntomic regulator, of the motivic elliptic polylogarithm in motivic cohomology. Let u ∈ E(O K ) be a torsion point as above. By the splitting principle, the pullback of pol to u gives an element Y 1 u ∗ pol ∈ Hsyn O K , H ∨ ⊗ Sym j H (1) . j≥0
Q By abuse of notation, we denote again by u ∗ pol the element in j≥0 K ω∨ j defined to be the image of u ∗ pol with respect to the isomorphism 1 Hsyn O K , H ∨ ⊗ Sym j H (1) = H ∨ ⊗ Sym j H and the projection to the direct factor Y Y H ∨ ⊗ Sym j H . K ω∨ j ⊂ j≥1
j≥0
Our main result is as follows. THEOREM (Th. 7.1) Let v ∈ E(O K ) be a primitive g-torsion point of E, and denote by ψp the p-adic character of G = Gal(K(gp∞ )/K) associated to ψ. Then we have the equality X j j τ (v)∗ pol = (−1) j p L p,g (ψp )ω∨ j j≥0 , τ ∈Gal(K(g)/K) j
j
where L p,g (ψp ) is the value at ψp of the p-adic L-function of modulus g, and p is the p-adic period.
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The result for j = 1 is essentially the result of Coleman and de Shalit on the calculation of the p-adic regulator for E (see [CdS, Par. 5.11]; see also [Bes2]). We remark that, recently, G. Kings calculated the p-adic elliptic polylogarithm in global e´ tale cohomology (see [Ki]), using a different construction. If there exist adequate comparison theorems between smooth p-adic sheaves on the generic fiber and syntomic coefficients, then our polylogarithm should be a local case of his. This paper is a revised version of the author’s thesis [Ban2]. 1.1. Overview In Section 2, we briefly review the theory of rigid syntomic cohomology with coefficients developed in [Ban1]. The only new result is the construction of the Gysin exact sequence of geometric syntomic cohomology for curves (see Th. 2.19). Then, in Section 3, following the method of Be˘ılinson and Levin, we construct the elliptic logarithmic sheaf L og (see Def. 3.3) and the p-adic elliptic polylogarithm pol (see Def. 3.11). In Section 4, we review some notions in the construction of the one-variable p-adic L-function. We then define the p-adic polylogarithmic function Da, j (see Def. 4.4) and give its relation to the special values of the p-adic L-function (see Prop. 4.12). We then begin the explicit calculation of the p-adic elliptic polylogarithm. In Section 5, we construct the partial logarithmic sheaf L og (see Def. 5.3), which is a quotient of L og. This sheaf is easier to handle than L og, but it is sufficient to prove our main result. In Section 6, we define the modified partial polylogarithm gpola in absolute syntomic cohomology with coefficients in L og (see Def. 6.2). As in the cyclotomic case, both pol and pola have interpretations as extension classes in the category of syntomic coefficients. Unfortunately, the author was unable to give the explicit shape of pol. The crucial step in the proof of our main result is Theorem 6.6, which gives the explicit shape of pola as an extension in the category of syntomic coefficients. Our main result (see Th. 7.1) is given in Section 7. 1.2. Notation Throughout this paper, we fix a prime p. In what follows, let K be a finite unramified extension of Q p with ring of integers O K and residue field k. We denote by σ the lifting of the Frobenius of k to O K and K . 2. Review of rigid syntomic cohomology with coefficients The purpose of this section is to review the theory of rigid syntomic cohomology with coefficients developed in [Ban1].
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Definition 2.1 We define the category of syntomic data D K as follows: The object in this category is a triple X = (X, X , φ X ), where (i) X is a smooth scheme, separated, and of finite type over O K ; (ii) X is a smooth compactification of X , separated, and of finite type over O K , such that the complement D is a simple normal crossing divisor relative to OK ; (iii) φ X : X → X is a lifting of the absolute Frobenius of X k = X ⊗O K k, when we let X be the formal completion of X with respect to the special fiber, such that the diagram X y
φX
−−−−→
X y
σ∗
Spf O K −−−−→ Spf O K is commutative. 2.1. Review of geometric cohomology In this subsection, we review the construction of geometric syntomic cohomology with coefficients associated to a syntomic datum X in D K . Fix a syntomic datum X in D K . We denote by X K = X ⊗ K the generic fiber of X , by X k the special fiber of X , by X an K the rigid analytic space associate to X K , by X the formal completion of X with respect to the special fiber, and by X K the rigid analytic space associated to X . We use the same notation for X . By [Ber2, an Prop. 0.3.5], there is an isomorphism X K ∼ = XK . an Let j : X K ,→ X an K be the natural inclusion. We say that a subset V ⊂ X K is a strict neighborhood of X K in X an K if an X an K = V ∪ (X K \ X K )
is a covering of X an K for the rigid topology. Definition 2.2 † 0 For any abelian sheaf M 0 on X an K , we define j M by j † M 0 = α∗ ( lim αV ∗ αV∗ M 0 ), − →an V ⊂X K
where α : X an K ,→ X K is the inclusion, and the limit is taken with respect to strict an neighborhoods αV : V ,→ X an K of X K in X K .
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We let j † OX K = j † (α ∗ OX K ). If M 0 is a O X an -module, then j † M 0 has a structure K † of a j OX K -module. Next, let M be a coherent O X K -module. We define Mrig to be the coherent j † OX K -module j † (M| X K )an . If there exists an integrable logarithmic connection ∇ : M → M ⊗ 1X (log D) K
on M, then this induces an integrable connection ∇rig : Mrig → Mrig ⊗ 1X
K
on Mrig . In this case, a Frobenius structure on Mrig is a horizontal isomorphism 8M : φ ∗X Mrig ∼ = Mrig , where φ ∗X Mrig is the inverse image of Mrig by the Frobenius. Definition 2.3 We define the category of syntomic coefficients on X to be the category S(X) defined as follows: The objects of S(X) consist of the 4-tuple M = (M, ∇, F • , 8M ), where (i) M is a coherent O X K -module of finite rank; (ii) ∇ is an integrable connection on M with logarithmic poles along D; (iii) F • is a descending exhaustive separated filtration by sub O X K -modules on M, called the Hodge filtration, satisfying Griffiths transversality ∇(F m M) ⊂ F m−1 M ⊗ 1X (log D); K
φ ∗X Mrig
∼ (iv) 8M : = Mrig is a Frobenius structure on Mrig . The morphisms in this category are homomorphisms of underlying O X K -modules which are compatible with the above structure. Definition 2.4 For each integer j, we define the Tate object K ( j) to be the object K ( j) = (M, ∇, F • , 8M ) in S(X) defined as follows. (i) M = O X K e j is a free O X K -module of rank one with generator e j . (ii) The connection ∇ is defined by ∇(e j ) = 0. (iii) The Hodge filtration F • is defined by F − j M = M and F − j+1 M = 0. (iv) We have φ ∗X Mrig = Mrig . The Frobenius morphism is defined to be 8M (e j ) = p− j e j .
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Let M be an object in S(X). As in [Ban1, Sec. 1], we define the de Rham and rigid cohomologies with coefficients in M by i HdR (X, M ) := Ri 0 X K , DR•dR (M ) , i Hrig (X, M ) := Ri 0 X K , DR•rig (M ) , where DR•dR (M ) := M ⊗ •X (log D), K
DR•rig (M )
:= Mrig ⊗ •X
K
are the de Rham complexes associated to M and Mrig . The de Rham cohomology i (X, M ) has a filtration induced from the spectral sequence HdR i, j i E 1 = Ri+ j 0 X K , Gri DR•dR (M ) ⇒ HdR (X, M ),
(2.1)
i (X, M ) has a σ -linear automorphism φ, called the and the rigid cohomology Hrig Frobenius. There is a natural homomorphism i i (X, M ) → Hrig (X, M ) θ : HdR
(2.2)
of K -vector spaces (see [Ban1, Def. 1.12]). Definition 2.5 Let M be an object in S(X) such that θ is an isomorphism. We define the geometric syntomic cohomology, denoted by H i (X, M ), to be the filtered Frobenius module i (X, M ), where the action of the Frobenius φ is as above and the filtration is Hrig i (X, M ). To ease induced through the isomorphism θ from the Hodge filtration on HdR the terminology, we refer to geometric syntomic cohomology simply as geometric cohomology. Remark 2.6 The map θ is an isomorphism for any object in S(X) which we use in the calculation of the elliptic polylogarithm. This follows from the result of F. Baldassarri and B. Chiarellotto [BC, Cor. 2.6]. 2.2. Review of syntomic cohomology The purpose of this subsection is to review the definition of absolute syntomic cohomology, which we refer to simply as syntomic cohomology, with coefficients in the category of admissible syntomic coefficients. This cohomology is a p-adic analogue of absolute Hodge cohomology with coefficients.
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Definition 2.7 We define the category S ad (X) of admissible syntomic coefficients on X to be the full subcategory of S(X) consisting of objects M satisfying the following: (i) the spectral sequence (2.1) degenerates at E 1 ; (ii) the homomorphism θ is an isomorphism; (iii) the geometric cohomology H i (X, M ) is a weakly admissible filtered Frobenius module in the sense of J.-M. Fontaine [F1, Def. 4.1.4]. Remark 2.8 (1) For the syntomic datum O K = (Spec O K , Spec O K , σ ),
(2)
the category S(O K ) is equivalent to the category of filtered Frobenius modules of finite rank over K . The category S ad (O K ) is equivalent to the category of weakly admissible filtered Frobenius modules over K . Suppose that M is a syntomic coefficient in S ad (X). Then, by definition, the morphism θ of (2.2) is an isomorphism. Hence geometric cohomology H i (X, M ) is defined (see Def. 2.5). Again, by definition, this filtered Frobenius module is weakly admissible and hence is an object in S ad (O K ).
Remark 2.9 The syntomic coefficients that we use for the calculation of the elliptic polylogarithm are all admissible. The first condition follows either from the fact that the coefficient comes from a filtered module with connection which underlies a variation of mixed Hodge structures, or simply from direct calculation. The second condition comes from Remark 2.6. The last condition comes from explicit calculation of geometric cohomology. LEMMA 2.10 ([Ban1, Prop. 1.19]) For any exact sequence
0 → M 0 → M → M 00 → 0 in S ad (X), there is an associated long exact sequence · · · → H i (X, M 0 ) → H i (X, M ) → H i (X, M 00 ) → · · · in S ad (O K ). Remark 2.11 Occasionally in this paper, we claim that a morphism of a filtered Frobenius module
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is strict with respect to the filtration without the weakly admissible hypothesis. In the special cases considered in this paper, this follows either from the fact that the morphism underlies a morphism of mixed Hodge structures, or from direct computation. Let X = (X, X , φ X ) be a syntomic datum, and let M be an object in S ad (X). Let I be a finite set, and let U = {U i }i∈I be a covering of X by Zariski open sets. We put T U i0 ···in K = 0≤ j≤n U i j K . Next, let Ui = U i ∩ X , and let Ui K be the rigid analytic space over K associated to the formal completion Ui of Ui with respect to the special T fiber. For Ui0 ···in K = 0≤ j≤n Ui j K , we denote by ji0 ···in the inclusion ji0 ···in : Ui0 ···in K ,→ X K . • (U , M ) be the simple complex associated to the Cech ˇ We let RdR complex
Y i
Y 0 U i K , DR•dR (M ) → 0 U i0 i1 K , DR•dR (M ) → · · · , i 0 ,i 1
• (U , M ) be the simple complex associated to and we let Rrig
Y i
Y 0 X K , ji† DR•rig (M ) → 0 X K , ji†0 i1 DR•rig (M ) → · · · . i 0 ,i 1
• (U , M ) has a filtration induced from the Hodge filtration, and there The complex RdR are canonical homomorphisms • • φU : K ⊗σ,K Rrig (U , M ) → Rrig (U , M ), • • θU : RdR (U , M ) → Rrig (U , M )
induced, respectively, from the Frobenius and θ . We let • • • (U , M ) → Rrig (U , M ) [−1], (U , M ) := Cone F 0 RdR Rsyn where the morphism is (1 − φU ) ◦ θU . Definition 2.12 We define the absolute syntomic cohomology of X with coefficients in M by i • Hsyn (X, M ) := lim H i Rsyn (U , M ) , − → U
where the limit is taken with respect to coverings U as above ordered by refinements. Again, to ease the terminology, we refer to absolute syntomic cohomology simply as syntomic cohomology.
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Remark 2.13 Let M = (M, F • , 8) be an admissible object in S ad (O K ). Then we have 1−8 i Hsyn (O K , M ) = H i [F 0 M −−→ M] ,
(2.3)
1−8
where [F 0 M −−→ M] is the complex with F 0 M in degree zero, whose terms are zero for degrees not equal to 0, 1. We have the following (see [Ban1, Prop. 2.7]). 2.14 Let the notation be as above. We have a short exact sequence 1 i+1 0 → Hsyn O K , H i (X, M ) → Hsyn (X, M ) 0 → Hsyn O K , H i+1 (X, M ) → 0. LEMMA
We use the following result (see [Ban1, Th. 1]). 2.15 Suppose that X is a syntomic datum in D K , and let M be an object in S ad (X). Then there exists a canonical and functorial isomorphism THEOREM
∼ = 1 4 : Ext1S(X) K (0), M − → Hsyn (X, M ). Definition 2.16 We define the category M(X) to be the category consisting of the pair (M, ∇), where (i) M is a coherent O X K -module; (ii) ∇ is an integrable connection on M with logarithmic poles along D. There is a natural forgetful functor For : S(X) → M(X), defined by For(M ) := (M, ∇). For any object M in S(X), we say that For(M ) is the underlying coherent module with logarithmic connection of M . We have the following (see [Ban1, Props. 4.3, 4.4]). 2.17 Suppose that X is a syntomic datum in D K , and let M be an object in S ad (X). Then there exists a canonical and functorial isomorphism PROPOSITION
∼ = 4 : Ext1M(X) K (0), M − → H 1 (X, M )
(2.4)
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such that the diagram Ext1S(X) K (0), M ∼ 4y= 1 (X, M ) Hsyn
For
−−−−→ Ext1M(X) K (0), M ∼ 4y= −−−−→
(2.5)
H 1 (X, M )
is commutative. Here the bottom arrow is defined to be the composition 1 0 O K , H 1 (X, M ) ,→ H 1 (X, M ), Hsyn (X, M ) → Hsyn where the first arrow is the surjection of the exact sequence in Lemma 2.14. Remark 2.18 The map 4 of Proposition 2.17 is given as follows: Let U = {U i }i∈I be a covering of X by affine open sets, and let M and N be objects in S ad (X) with underlying coherent module with connection (M, ∇) and (N , ∇) which fits into the exact sequence 0 → M → N → K (0) → 0 in M(X). Let Ai K = 0(U i K , OU i K ). Then the above sequence gives rise to the exact sequence 0 → Mi → Ni → Ai K e0 → 0 of Ai K -modules, where Mi = 0(U i K , M) and similarly for Ni . For a lifting e˜i of e0 in Ni , we have a splitting Ni = Mi ⊕ Ai K e˜i of Ai K -modules. Let ai = ∇i (e˜i ) ∈ Mi ⊗ 1Ai K (log D) and bi j = e˜ j − e˜i ∈ Mi j , where ∇i is the connection induced from ∇ and Mi j = 0(U i j K , M). Then the pair • (U , M )). (ai , bi j ) defines an element in Z 1 (RdR • 1 Since H (RdR (U , M )) is isomorphic to H 1 (X, M ), the above correspondence gives the desired map. 2.3. The Gysin exact sequence The purpose of this subsection is to construct the Gysin exact sequence for the case of curves which is needed in this paper. Let X = (X, X , φ X ) be a syntomic datum in D K such that X is a curve over O K . Assume in addition that for D = X \ X , the Frobenius φ X induces a morphism φ D on D such that for the syntomic datum D = (D, D, φ D ), the natural inclusion induces a morphism of syntomic data i : D ,→ X. Let X = (X , X , φ X ), and let j : X ,→ X be the inclusion. We have the following. THEOREM 2.19 (Gysin exact sequence) Let M be an admissible syntomic coefficient on X, such that the pullbacks of M to X
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and D are also admissible. Then there exist an isomorphism H 0 (X, j ∗ M ) ∼ = H 0 (X, M ) and an exact sequence 0 → H 1 (X, M ) → H 1 (X, j ∗ M ) → H 0 (D, i ∗ M )(−1) → H 2 (X, M ) → 0. Before proving the theorem, we prepare some notation. Let U = {U i }i∈I be a finite covering of X by Zariski open subschemes. We take sufficiently small subschemes such that U i is affine, and there exist local parameters ti in 0(U i , OU i ) such that for D ∩ U i 6= ∅, the subscheme D ∩ U i is defined by ti = 0. We let • • RdR U , M (log D) := RdR (U , j ∗ M ). i (X, j ∗ M ). Next, let A = By definition, the cohomology of this complex is HdR i 0(U i K , OU i K ), Bi = 0(Ui K , OUi K ), and
1Ai (log D) = Ai dti /ti ⊂ 1Bi . For Mi†0 ···in = 0(X K , ji†0 ···in Mrig ), we let Mi†0 ···in (log D) := Mi†0 ···in ⊗ Ai j 1Ai (log D). j
This does not depend on the choice of 0 ≤ j ≤ n. We define the complex • (U , M (log D)) to be the simple complex associated to the double complex Rrig Q
Mi† y
i
Q
i
Q
−−−−→
Mi† (log D) −−−−→
i 0 ,i 1
Mi†0 i1
−−−−→ · · ·
y Q
i 0 ,i 1
Mi†0 i1 (log D) −−−−→ · · ·
This complex has a Frobenius • • φU : K ⊗σ,K Rrig U , M (log D) → Rrig U , M (log D) induced from φ X and φM . There is a natural morphism of complexes • • Rrig U , M (log D) → Rrig (U , j ∗ M ). By [T, Th. 4.2.2], this morphism is a quasi-isomorphism compatible up to homotopy with the Frobenius.
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There is a natural quasi-isomorphism • • θ : RdR U , M (log D) → Rrig U , M (log D) which gives the commutative diagram ∗ jdR • U , M (log D) • (U , M ) − −−−→ RdR RdR θ y∼ θ y∼ = =
(2.6)
∗ jrig
• (U , M ) − • U , M (log D) Rrig −−−→ Rrig PROPOSITION 2.20 Define morphisms
• • ResD/X : RdR U , M (log D) → RdR (U , i ∗ M )(−1)[−1], • • ResD/X : Rrig U , M (log D) → Rrig (U , i ∗ M )(−1)[−1] of complexes of K -vector spaces by f (ti ) dti /ti 7 → f (0) in degree 1 and by the zeromap otherwise. Then this induces quasi-isomorphisms ∼ =
∗ • ResD/X : Cone( jdR )[−1] − → RdR (U , i ∗ M )(−1)[−2], ∼ =
∗ • ResD/X : Cone( jrig )[−1] − → Rrig (U , i ∗ M )(−1)[−2],
where the first quasi-isomorphism is compatible with the Hodge filtration, and the second is compatible up to homotopy with the Frobenius. Proof The case for de Rham cohomology is standard. The case for rigid cohomology follows from [T, Prop. 4.3.1]. Proof of Theorem 2.19 • (U , M ), Since U is a covering by affine open sets, the complexes RdR • • • • ∗ ∗ RdR (U , j M ), and RdR (U , i M ) (resp., Rrig (U , M ), Rrig (U , j ∗ M ), and • (U , i ∗ M )) calculate the de Rham cohomology (resp., rigid cohomology) of M , Rrig ∗ j M , and i ∗ M . Since the maps defined above are all compatible with the isomorphism θ, Proposition 2.20 and the definition of a cone give an exact sequence · · · → H i (X, M ) → H i (X, j ∗ M ) → H i−1 (D, i ∗ M )(−1) → · · · of geometric cohomology. The statement of the theorem follows from the fact that H i (D, i ∗ M ) = 0 for i 6= 0, and H i (X, j ∗ M ) = 0 for i 6 = 0, 1.
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3. Construction of the elliptic polylogarithm The purpose of this section is to construct the elliptic logarithmic sheaf and the elliptic polylogarithm for an elliptic curve with complex multiplication defined over Q. The construction is done following the method of Be˘ılinson and Levin [BL] (see also [HK, Appendix A]). We prepare some notation. Let E = EQ be an elliptic curve over Q with complex multiplication by the integer ring OK of an imaginary quadratic field K. In this case, K is of class number 1. Let IK be the group of id`eles of K, and let ψ = ψE/K : IK → K× be the Gr¨ossencharakter of EK = E ⊗Q K. We denote by f the conductor of ψ. We often view ψ as a character on the group of fractional ideals of K prime to f. In addition, we assume that p splits in OK , in the form p = pp∗ . This implies that E has good ordinary reduction at p. Let K be a finite unramified extension of Q p . Throughout this paper, we fix an embedding i p : Q ,→ C p
(3.1)
induced from the prime above p. In particular, we view K as a subfield of K through the inclusion K ,→ Kp ∼ = Q p ,→ K . Let E K be the base extension of E by K . Since E has good reduction at p, there exists a smooth model E of E K over O K . Following the convention of Section 2.1, we let E k be the special fiber of E, E the formal completion of E with respect to the special fiber, and E K the associated rigid analytic space over K . Let ω be an invariant differential of E. We fix an isomorphism [−] : OK ∼ = End(E K ) such that [a]∗ ω = aω for a ∈ OK . Let π = ψ(p), which is a generator of p in OK . The endomorphism [π ] : E → E over O K is a lifting of the relative Frobenius of E k (see [Si2, Cor. 5.4]). We denote by φ E the lifting of the absolute Frobenius of E k induced from [π] on the formal completion E . The triple E := (E, E, φ E ) is a syntomic datum in D K . 3.1. The elliptic logarithmic sheaf In this subsection, we define the elliptic logarithmic sheaf L og (n) in S(E) (see Def. 3.3). Then we calculate its geometric cohomology (see Lem. 3.4). We follow the presentation of [HK, App. A.1]. Since E is a syntomic datum that is associated to a projective variety defined over a global field, the Tate objects K ( j) of S(E) are admissible. We let H ∨ = H 1 (E, K (0)), which is a filtered Frobenius module of rank two in S ad (O K ). We fix a
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basis {ω, η} of H ∨ such that ω ∈ F 1 H ∨ is represented by the invariant differential ω, and the action of φ ∗E is given by φ ∗E ω = πω, φ ∗E η = π ∗ η. Here π ∗ = ψ(p∗ ) is a generator of p∗ . We denote by K (e) any one-dimensional filtered Frobenius module with basis e. With this notation, H ∨ = K (ω) ⊕ K (η). For any object M in S(X), we denote by M ( j) the object M ⊗ K ( j), where K ( j) is the Tate object defined in Definition 2.4. We denote the base e ⊗ e j of K (e)( j) by e( j). We let K (e j ) = K (e)⊗ j . Next, let H = H 1 (E, K (1)). This object is dual to H ∨ in S ad (O K ). We denote by ω∨ and η∨ the basis of H dual to ω and η. For ease of calculation, we identify K (η) with K (ω∨ )(−1) by taking η = ω∨ (−1). We denote by the same symbols H and H ∨ the objects in S ad (E) defined to be the pullback of H and H ∨ with respect to the structure morphism E → Spec O K . Since H i (E, H ) = H i (E, K (0)) ⊗ H (see [Ban1, Lem. 1.22]), the geometric cohomology of H is given as follows: H 0 (E, H ) = H , H 1 (E, H ) = H ⊗ H ∨ ,
(3.2)
H (E, H ) = H (1). 2
We also have a canonical isomorphism 0 Hsyn (O K , H ⊗ H ∨ ) = F 0 (H ⊗ H ∨ )8=1 = End(H ),
(3.3)
where End(H ) denotes the endomorphisms of H as a filtered Frobenius module. The short exact sequence in Lemma 2.14 and the calculations above give the following exact sequence: 1 1 0 → Hsyn (O K , H ) → Hsyn (E, H ) → End(H ) → 0.
(3.4)
Let e : Spec O K → E be the identity element of E. The sequence above is split by 1 (E, H ) → H 1 (O , H ). e∗ : Hsyn K syn Definition 3.1 1 (E, H ) characterized by the properties that We define l to be the element in Hsyn (i) the image of l in End(H ) is the identity map; (ii) the pullback e∗ (l) is zero.
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Definition 3.2 Let L og (1) be the extension 0 → H → L og (1) → K (0) → 0 in S(E), whose extension class in Ext1S(E) (K (0), H ) corresponds to l through the canonical and functorial isomorphism 1 (E, H ) 4 : Ext1S(E) K (0), H ∼ = Hsyn given in Theorem 2.15. The extension L og (1) is determined up to unique isomorphism. Since the morphism 4 is functorial, by the definition of l, the pullback of L og (1) by e∗ is split. Definition 3.3 We define the elliptic logarithmic sheaf L og (n) in S(E) by L og (n) = Symn L og (1) .
The splitting of e∗ L og (1) induces e∗ L og (n) =
n Y
Sym j H .
j=0
There is a natural projection map pr : L og (n+1) → L og (n) defined to be the composition Symn+1 L og (1) → Symn+1 L og (1) ⊕ K (0) → Symn L og (1) . Here the first map is induced from the sum of the identity map and the projection L og (1) → K (0). The second map is the canonical projection in the symmetric algebra of a direct sum. By definition, we have an exact sequence 0 → Symn+1 H → L og (n+1) → L og (n) → 0. The same proof as in [HK, Lem. A.1.4] gives the following. 3.4 The geometric cohomology of L og (n) is as follows:
LEMMA
(a)
H 0 (E, L og (n) ) = Symn H , H 1 (E, L og (n) ) = (Symn+1 H )(−1), H 2 (E, L og (n) ) = K (−1).
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(b)
209
The morphisms H 0 (E, L og (n+1) ) → H 0 (E, L og (n) ), H 1 (E, L og (n+1) ) → H 1 (E, L og (n) ) induced from the projection L og (n+1) → L og (n) are zero-maps.
Proof (a) For the proof, we let L og (0) = K (0). The proof is by induction on n. When n = 0, then the statement is the definition. We first prove the lemma for H 0 , H 2 , and the dimension of H 1 . Assume that the statement is true for an integer n ≥ 0. Consider the commutative diagram 0 −−−−− → H ⊗ Symn H −−−−− → L og (1) ⊗ Symn H −−−−− → Symn H − −−−− → 0 y y y 0 −−−−− →
Symn+1 H
L og (n+1)
−−−−− →
−−−−−→ L og (n) − −−−− → 0
where the first two vertical arrows are the multiplication maps, and the last vertical arrow is the natural inclusion. The horizontal sequences are exact. The first boundary morphism of the long exact sequence for geometric cohomology gives the commutative diagram Symn H −−−−→ H ∨ ⊗ H ⊗ Symn H ∼ y= yid ⊗ mult γ
H 0 (E, L og (n) ) −−−−→
H ∨ ⊗ Symn+1 H
where the left vertical arrow is an isomorphism from the induction hypothesis. From the definition of L og (1) and explicit calculation, the diagonal map is given on the basis by ω∨r η∨n−r 7→ (ω ⊗ ω∨r +1 η∨n−r + η ⊗ ω∨r η∨n−r +1 ) (3.5) and hence is injective. This shows that γ is injective; hence ∼ =
− H 0 (E, Symn+1 H ) = Symn+1 H . H 0 (E, L og (n+1) ) ← The second boundary morphism of the long exact sequence of geometric cohomology gives the commutative diagram ∼ =
H (−1) ⊗ Symn H −−−−→ H (−1) ⊗ Symn H y ymult
H 1 (E, L og (n) )
β
−−−−→ (Symn+1 H )(−1)
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Since the diagonal map is surjective, β is surjective. Hence from the induction hypothesis, β is an isomorphism by reason of dimension. This implies that the surjection H 2 (E, L og (n+1) ) → H 2 (E, L og (n) ) = K (−1) is an isomorphism. Furthermore, we have a short exact sequence γ
0 → H 0 (E, L og (n) ) − → H ∨ ⊗ Symn+1 H → H 1 (E, L og (n+1) ) → 0. Hence the dimension of H 1 (E, L og (n+1) ) is n + 3, which is the dimension of (Symn+2 H )(−1). The assertion is now proved for H 0 , H 2 , and the dimension of H 1. In particular, the proof above is valid for any integer n ≥ 1. The assertion of the lemma for H 1 follows from the fact that β is an isomorphism for any n ≥ 0. (b) The statement follows from the fact that both γ and β above are injective. 3.2. The pullback of the elliptic logarithmic sheaf Let the notation be as in the beginning of Section 3. Let a be an element of OK such that the ideal a = (a) is prime to p, and let [a] : E → E be the endomorphism induced from a. Since End(E) is commutative, this induces a morphism [a] : E → E of syntomic data. The purpose of this subsection is to prove the following. PROPOSITION 3.5 There exists a natural isomorphism ∼ =
%a : L og (n) −−−−→ [a]∗ L og (n) .
(3.6)
Since L og (n) = Symn L og (1) , in order to prove the proposition, it is sufficient to prove the following. 3.6 There exists a natural isomorphism LEMMA
∼ =
L og (1) −−−−→ [a]∗ L og (1) .
Proof It is possible to prove this lemma by calculating the object L og (1) explicitly (see [Ban2, Prop. 2.9]). In this paper, we give a more functorial proof. The morphism [a] : E → E of syntomic data induces an isomorphism [a]∗ : H → H , given on the basis by ω∨ 7 → a −1 ω∨ and η∨ 7 → (a ∗ )−1 η∨ . This induces an isomorphism ∼ = 1 (E, H ) − 1 (E, H ) ρa : Hsyn −−−→ Hsyn
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211
on syntomic cohomology. Since [a] is compatible with the structure morphism of E, we have [a]∗ H = H on S(E). This gives the following commutative diagram with exact rows: 1 (O , H ) − 1 (E, H ) − Hsyn −−−→ Hsyn −−−→ End(H ) K [a]∗ y [a]∗ y [a]∗ y 1 (O , H ) − 1 (E, H ) − Hsyn −−−→ Hsyn −−−→ End(H ) K ρa y ρa y ρa y 1 (O , H ) − 1 (E, H ) − Hsyn −−−→ Hsyn −−−→ End(H ) K
where the vertical arrows on the top are the pullback [a]∗ , and the vertical arrows on the bottom are induced from ρa . By definition, the composition of the right vertical arrows End(H ) → End(H ) maps the identity to identity. The commutativity of the above diagram combined with the definition of L og (1) shows that the extension class of [a]∗ L og (1) maps to the extension class of L og (1) through the isomorphism ρa . Therefore, there exists an isomorphism ∼ =
%a : L og (1) −−−−→ [a]∗ L og (1) which fits into the commutative diagram 0 −−−−→ H −−−−→ ρa−1 y
L og (1) %a y
−−−−→ K (0) −−−−→ 0 idy
0 −−−−→ H −−−−→ [a]∗ L og (1) −−−−→ K (0) −−−−→ 0
COROLLARY 3.7 (Splitting principle) Let u : Spec O K → E be a torsion point of E, whose order is prime to p. Then we have n Y u ∗ L og (n) ∼ Sym j H . (3.7) = j=0
Proof Take a ∈ OK such that [a] : E → E is an isogeny that maps u to the identity element e. Since u ∗ ◦ [a]∗ = e∗ , we have ρa
u ∗ L og (n) ∼ = u ∗ [a]∗ L og (n) = e∗ L og (n) =
n Y j=0
as desired.
Sym j H ,
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3.3. The elliptic polylogarithm Let the notation be as before, and let U = E \ (e). Then the triple U = (U, E, φ E ) is a syntomic datum over K . The purpose of this subsection is to give the definition of the elliptic polylogarithm pol(n) in S(U) (see Def. 3.11), following the method of [HK, Sec. A.3]. We let D be the syntomic datum (D, D, φ D ) for D = (e). Then the inclusions j : U ,→ E and e : D → E are morphisms of syntomic data. We start by constructing the map 1 δ (n) : Hsyn U, H ∨ ⊗ L og (n) (1) → End(H ) for each integer n ≥ 1 (see Def. 3.8). First, by Theorem 2.19 and Lemma 3.4, we have an isomorphism H 0 U, H ∨ ⊗ L og (n) (1) = H ∨ ⊗ Symn H (1). Hence Lemma 2.14 gives the short exact sequence 1 1 0 → Hsyn O K , H ∨ ⊗ Symn H (1) → Hsyn U, H ∨ ⊗ L og (n) (1) 0 → Hsyn O K , H 1 (U, H ∨ ⊗ L og (n) (1)) → 0.
(3.8)
Next, the Gysin exact sequence in Theorem 2.19 gives an exact sequence 0 → H 1 E, H ∨ ⊗ L og (n) (1) → H 1 U, H ∨ ⊗ L og (n) (1) u → H 0 (D, H ∨ ⊗ e∗ L og (n) ) − → H 2 E, H ∨ ⊗ L og (n) (1) → 0.
(3.9)
By (3.7) and the calculation of geometric cohomology on O K , we have H 0 (D, H ∨ ⊗ e∗ L og (n) ) =
n Y
H ∨ ⊗ Sym j H .
j=0
By Lemma 3.4, we have H 2 E, H ∨ ⊗ L og (n) (1) = H ∨ ; Q hence the kernel of u in (3.9) is nj=1 H ∨ ⊗ Sym j H . Since H 1 E, H ∨ ⊗ L og (n) (1) = H ∨ ⊗ Symn+1 H (see Lem. 3.4), sequence (3.9) is reduced to 0 → H ∨ ⊗ Symn+1 H → H 1 U, H ∨ ⊗ L og (n) (1) →
n Y j=1
H ∨ ⊗ Sym j H → 0.
(3.10)
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By (2.3) of Remark 2.13, we have 0 Hsyn (O K , H ∨ ⊗ Symn+1 H ) = 0 (n ≥ 1), n Y 0 Hsyn OK , H ∨ ⊗ Sym j H = F 0 (H ∨ ⊗ H )8=1 . j=1 0 (O , −) of (3.10), we have an exact sequence Hence, by taking Hsyn K
0 0 → Hsyn O K , H 1 (U, H ∨ ⊗ L og (n) (1)) → F 0 (H ∨ ⊗ H )8=1 1 → Hsyn (O K , H ∨ ⊗ Symn+1 H ) → · · · .
(3.11)
Definition 3.8 For each integer n ≥ 1, we define the map 1 δ (n) : Hsyn U, H ∨ ⊗ L og (n) (1) → End(H ) to be the composition of the maps in (3.8), (3.11), and the canonical isomorphism F 0 (H ∨ ⊗ H )8=1 = End(H ).
(3.12)
PROPOSITION 3.9 The maps δ (n) give rise to an isomorphism
1 δ : lim Hsyn U, H ∨ ⊗ L og (n) (1) ∼ = End(H ). ← − n Proof By the definition of the map δ (n) , we have an exact sequence 1 1 0 → Hsyn O K , H ∨ ⊗ Symn H (1) → Hsyn U, H ∨ ⊗ L og (n) (1) δ (n)
1 −−→ End(H ) → Hsyn (O K , H ∨ ⊗ Symn+1 H ) → · · · .
By Lemma 3.4(b), the maps 1 1 Hsyn O K , H ∨ ⊗ Symn+1 H (1) → Hsyn O K , H ∨ ⊗ Symn H (1) , 1 1 Hsyn (O K , H ∨ ⊗ Symn+2 H ) → Hsyn (O K , H ∨ ⊗ Symn+1 H )
induced from the projection pr : L og (n+1) → L og (n) are zero-maps. Hence the assertion follows by taking the limit.
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Definition 3.10 We define the elliptic polylogarithm to be the system of elements pol(n) ∈ 1 (U, H ∨ ⊗ L og (n) (1)) such that Hsyn δ(lim pol(n) ) = id ∈ End(H ). ← − Let 4 be the isomorphism 1 Ext1S(U) K (0), H ∨ ⊗ L og (n) (1) ∼ U, H ∨ ⊗ L og (n) (1) = Hsyn given in Theorem 2.15. Definition 3.11 We define the elliptic polylogarithmic extension to be any object in S(U) representing the extension class 4−1 (pol(n) ) ∈ Ext1S(U) K (0), H ∨ ⊗ L og (n) (1) . By abuse of notation, we denote this also by pol(n) . This extension is determined uniquely up to unique isomorphism. Remark 3.12 This is a p-adic version of the element referred to as a cohomological polylogarithm in [HK, Def. A.3.3]. 4. On the p-adic elliptic polylogarithmic function In this section, we construct the p-adic elliptic polylogarithmic functions Da, j (see Def. 4.4), and we give their relation to special values of the one-variable p-adic Lfunction (see Prop. 4.12). We keep the notation of Section 3, and we assume in addition that p ≥ 5. We fix a Weierstrass model E : y 2 + a1 x y + a3 y = x 3 + a2 x 2 + a4 x + a6
(4.1)
of E over Z p = OKp with good reduction at p, and we take for ω the associated invariant differential dx . ω= 2y + a1 + a3 We denote by L = L (E, ω) the lattice of periods, and by ξ(z, L ) : C/L → E(C) the analytic uniformization given in standard affine coordinates by x = ℘ (z, L ) − (a12 + 4a2 )/12, y = ℘ 0 (z, L ) − a1 x − a3 /2.
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4.1. Some preliminaries b be the formal group law of E In this subsection, we prepare some preliminaries. Let E b and the formal multiplicative over Z p with parameter t = −x/y. The formal group E b group Gm are isomorphic over W (k), and an isomorphism bm ∼ b ρ:G = E,
t = ρ(S) = p S + · · ·
(4.2)
is uniquely determined by p ∈ W (k)× satisfying σp −1 = ψ(p) p −1 , where σ is the Frobenius automorphism of W (k). In this paper, we fix once and for all an p as above. Note that since ρ is an isomorphism of formal groups, we have ω ◦ ρ(S) = p (1 + S)−1 d S. an . We have the Let E Canp be the extension to C p of the rigid analytic space E K reduction morphism red : E Canp → E F p . The inverse image of a point in E F p under the reduction morphism is called the residue class, which is an admissible open in E Canp (see [BGR, Sec. 9.1.4, Prop. 5]). Each residue class has a “Teichm¨uller representative,” which is a torsion point of order relatively prime to p. For a Teichm¨uller representative u in E Canp , the points ρu (s) = u + ρ(s) for |s| < 1 parametrize the residue class represented by u. In other words, (4.2) gives rise to an open immersion of rigid analytic spaces ρu : B − (0, 1) ,→ E Canp , where B − (0, 1) is the rigid analytic open unit disk B − (0, 1) = {s ∈ C p | |s| < 1} in C p , such that the image is the residue class represented by u. Let D be the translation-invariant derivation satisfying d f = D( f )ω for any function f in 0(E Canp , O E Can ). Then D restricts to the differential p
D = −1 p (1 + S)
d dS
on B − (0, 1). 4.2. The p-adic elliptic polylogarithmic function Let a be an integral ideal of OK prime to fp, and let a = ψ(a), which is a generator of a. Let Ua = E \ E[a], let Ua be the formal completion of Ua with respect to the special fiber, and let UaK be the associated rigid analytic space over K . The purpose of this subsection is to define for j ≥ 0 the p-adic elliptic polylogarithmic functions Da, j in 0(UaK , OUaK ). We use the tools developed in [CdS, b C p is the rigid analytic space over C p denoted Sec. 5]. Note that UaC p = UaK ⊗ X (a) in [CdS].
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KENICHI BANNAI
Definition 4.1 We define a rational function 2E,a in K(E)× by Y 2E,a = a −12 1(E)Na−1
−6 x − x(u) .
u∈E[a]\(e)
Here 1(E) is the discriminant of E associated to the Weierstrass equation (4.1). Note that this is the function denoted f (P) in [CdS, (51)]. Its divisor is given by X div(2E,a ) = 12Na(e) − 12 (u). u∈E[a]
Definition 4.2 As in [CdS, (73)], we define the function Log(2E,a )] in 0(UaK , OUaK ) by φ∗ Log(2E,a )] = 1 − E log 2E,a . p As in [CdS, Lem. 5.6], for any torsion point u of UaC p of order prime to p, there exists a unique W (k)-valued measure on Z× p , which we denote by µa,u , such that Z (4.3) Log(2E,a )] ◦ ρu (S) = (1 + S)x dµa,u (x) ∈ W (k)[[S]]. Z× p
A proof similar to [CdS, Lem. 5.6] gives the following. PROPOSITION 4.3 Let j ≥ 0, and let m be an integer such that p m −1
p
≡1
Then the sequence of functions D( p uniformly. In particular, the limit
m −1) pr − j
Da, j := lim D( p r →∞
mod pW (k) . (Log(2E,a )] ) for r → ∞ converges
m −1) pr − j
Log(2E,a )]
is a rigid analytic function on UaK . Proof Since 0(UaK , OUaK ) ⊂ 0(UaC p , OUaC p ) is a strict monomorphism (see [BGR, Sec. 6.1.1, Cor. 9]), it is sufficient to check the m r convergence after base change to C p . Applying D( p −1) p − j , as r goes to infinity,
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has the same effect as applying p (p D)( p −1) p − j . This operation applied to (4.3) gives the following on each residue class: Z m r j p (1 + S)x x ( p −1) p − j dµa,u (x). j
m
r
Z× p
Hence limr →∞ D( p
(Log(2E,a )] ) converges uniformly to Z j p (1 + S)x x − j dµa,u (x).
m −1) pr − j
Z× p
Since Da, j is a uniform limit of algebraic functions, it is a rigid analytic function on UaK , as desired. Definition 4.4 For j ≥ 0, we define the p-adic polylogarithmic function to be the rigid analytic function Da, j on UaK of Proposition 4.3. The proof of Proposition 4.3 gives the following. 4.5 For any j ≥ 0 and torsion point u of UaK of order prime to p, we have Z j Da, j (u) = p x − j dµa,u (x). COROLLARY
Z× p
4.6 The functions Da, j for j ≥ 0 are the unique system of functions in 0(UaK , OUaK ) satisfying φ∗ d Da,0 = 1 − E d log 2E,a , p THEOREM
D(Da, j+1 ) = Da, j
( j ≥ 1).
(4.4)
Proof Since D is a continuous operator, the fact that Da, j satisfies (4.4) follows from the definition. It is sufficient to prove uniqueness. 0 0 Let Da, j be another set of functions satisfying (4.4), and assume that Da,m = Da,m for any integer m such that j > m ≥ 0. Then, since the kernel of the differential 0 is the constant function, we have Da, j = Da, j + c for some constant c ∈ K . This 0 implies that d(Da, j − Da, j ) = cω. By Lemma 4.7, nonzero constant multiples of ω are not integrable by rigid analytic functions on UaK . Hence we have c = 0 and 0 Da, j = Da, j , as desired.
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KENICHI BANNAI
The author learned of the following lemma from Nobuo Tsuzuki. His original proof used p-adic analysis. We give a different proof. LEMMA 4.7 The differential equation d f = ω has no rigid analytic solutions f on UaK .
Proof Let b be an integral ideal of O K of order prime to afp. Without loss of generality, we may replace K by a finite extension. Hence assume that K(ab) ⊂ K with respect to the inclusion i p of (3.1). Fix a torsion point Q in E(O K ) with exact annihilator b. Let [+Q] : E → E be the translation by Q map, and let V Q = [+Q]−1 (Ua ) be its inverse image. Since E = Ua ∪ V Q is a covering of E for the Zariski topology, E K = UaK ∪ V Q K is an admissible covering of E K for the rigid topology (see [Ber2, Prop. 0.2.3(i)]). Suppose that there exists a rigid analytic function f on UaK such that d f = ω. Then, since ω is the invariant differential, g = [+Q]∗ f is a rigid analytic function on V Q K satisfying dg = ω. Then f and g restricted to the admissible open set UaK ∩ V Q K satisfy f = g + c for some constant c ∈ K . This implies that f extends to a rigid analytic function on E K , again denoted f , such that d f = ω. Since E K is proper, we have an isomorphism 1 HdR (E K /K ) ∼ = R1 0(E K , •E K ). 1 (E /K ), The existence of f as above would imply that the class of ω is zero in HdR K which is a contradiction.
4.3. The relation to the one-variable p-adic L-function In this subsection, we give the relation between the p-adic elliptic polylogarithmic functions Da, j defined in Section 4.2 and the special values of the one-variable padic L-function (see Prop. 4.12). Fix an integral ideal g of K such that f|g and (g, ap) = 1. Since the class number of K is one, there exists an in C× such that L (E, ω) = g.
We let v = ξ(, L ). Then v is an element in E[g]. Let G = Gal K(gp∞ )/K , G = Gal K(gp∞ )/K(g) . The Gr¨ossencharakter ψ of EK gives rise to a p-adic character ψp of G defined by ψp (σc ) = ψ(c), (c, fp) = 1, σc = c, K(gp∞ )/K .
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219
This induces an isomorphism ψp : G ∼ = Z× p . For each torsion point u of UaK prime to p, we denote again by µa,u the measure on G induced through ψp from the measure µa,u on Z× p. Since G = Gal(K(g)/K) × G, we extend each τ ∈ Gal(K(g)/K) to G by taking τ = (τ, id). In particular, we have ψp (τ ) = 1. Definition 4.8 We define a measure µa,g on G as follows: on G, we define µa,g to be µa,v . We extend this measure to G by setting dµa,g (τ −1 g) = dµa,τ (v) (g)
(g ∈ G)
for each τ ∈ Gal(K(g)/K). By construction, the measure µa,g is the measure denoted µe in [CdS, Th. 5.9]. By [CdS, Th. 5.9], we have the following. THEOREM 4.9 As a measure on G , we have
µa,g = 12(σa − Na)µg , where µg is the measure that gives rise to the p-adic L-function with conductor g. Remark 4.10 The p-adic L-function of K with modulus g is a function L p,g (−) on the continuous p-adic characters of G given by Z L p,g (χ) = χ −1 dµg (4.5) G
for χ : G → where µg is a certain C p -valued measure on G . This function satisfies the interpolation property j j p L p,g (ψp ) = j 1 − p −1 ψ(p)− j L ∞,g (ψ j , 0) ( j < 0), C× p,
where L ∞,g (ψ j , 0) is the value at zero of the Hecke L-function of ψ j . The main j result of this paper (see Th. 7.1) expresses the values L p,g (ψp ) for j ≥ 0 in terms of the specializations of the p-adic elliptic polylogarithm. COROLLARY 4.11 For a continuous character χ : G → C× p of G , we have Z χ −1 dµa,g = 12 χ(σa ) − Na L p,g (χ ). G
(4.6)
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KENICHI BANNAI
Proof This follows directly from Theorem 4.9 and the construction of the p-adic L-function (4.5). PROPOSITION 4.12 For j ≥ 0, we have X
j j Da, j τ (v) = 12 ψ − j (a) − Na p L p,g (ψp ).
τ ∈Gal(K(g)/K)
Proof By (4.6) and Theorem 4.9, we have 12 ψ
−j
(a) − Na
j L p,g (ψp )
Z
−j
= G
=
ψp (g) dµa,g (g) Z X −j ψp (g) dµa,g (τ −1 g)
τ ∈Gal(K(g)/K) G
X
=
Z
τ ∈Gal(K(g)/K) G
−j
ψp (g) dµa,τ (v) (g).
The assertion follows from the definition of µa,τ (v) and Corollary 4.5. 5. The partial logarithmic sheaf In this section, we define and study the partial logarithmic sheaf, which is a quotient of the elliptic logarithmic sheaf. The calculation of the partial logarithmic sheaf is important for the proof of our main theorem. 5.1. The definition of the partial logarithmic sheaf In this subsection, we define the partial logarithmic sheaf (see Def. 5.3); then we explicitly determine its shape (see Prop. 5.4). Definition 5.1 1 (E, K (ω∨ )) to be the image of l with respect to the map We define l in Hsyn 1 1 Hsyn (E, H ) → Hsyn E, K (ω∨ ) induced from the projection H → K (ω∨ ). As in the case of (3.2), we have by [Ban1, Lem. 1.22] the isomorphisms H 0 (E, K (ω∨ )) = K (ω∨ ) and H 1 (E, K (ω∨ )) = K (ω∨ ) ⊗ H ∨ . By (2.3), we have 8=1 0 O K , K (ω∨ ) ⊗ H ∨ = F 0 K (ω∨ ) ⊗ H ∨ = End K (ω∨ ) , Hsyn
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where End(K (ω∨ )) denotes the endomorphisms of K (ω∨ ) as a filtered Frobenius module. Hence the short exact sequence in Lemma 2.14 gives the short exact sequence 1 1 0 → Hsyn O K , K (ω∨ ) → Hsyn E, K (ω∨ ) → End K (ω∨ ) → 0. 1 (E, K (ω∨ )) satisfying the following: The element l is the unique element in Hsyn ∨ (i) the image of l in End(K (ω )) is the identity map; (ii) the pullback e∗ (l) is zero.
Definition 5.2 (1) We let L og be an extension 0 → K (ω∨ ) → L og
(1)
→ K (0) → 0 (1)
in S(E), whose extension class corresponds to l through the isomorphism ∼ = 1 E, K (ω∨ ) . 4 : Ext1S(E) K (0), K (ω∨ ) −−−−→ Hsyn By definition, the sheaf L og 0 −−−−→
H y
(1)
fits into the commutative diagram
−−−−→ L og (1) −−−−→ K (0) −−−−→ 0 y y
0 −−−−→ K (ω∨ ) −−−−→ L og
(1)
−−−−→ K (0) −−−−→ 0
Definition 5.3 We define the partial logarithmic sheaf in S(E) by L og
(n)
= Symn L og
(1)
.
(1)
(n)
The projection L og (1) → L og induces a map L og (n) → L og on the symmetric power. As in the case of the elliptic logarithmic sheaf, there are natural projections pr such that the diagram pr
L og (n+1) −−−−→ L og (n) y y L og
(n+1)
pr
−−−−→ L og
(n)
is commutative. 5.4 is isomorphic to the object L (n) = (L (n) , ∇, F • , 8) in S(E) such that
PROPOSITION (n)
L og
(5.1)
222
(i)
KENICHI BANNAI
L (n) is the free O E K -module L (n) =
Y
O E K ω∨ j ;
0≤ j≤n
(ii) (iii) (iv)
the connection is given by ∇(ω∨ j ) = ω∨ j+1 ⊗ω if 0 ≤ j < n, and ∇(ω∨n ) = 0 when j = n; Q the filtration is given by F m L (n) = 0≤ j≤−m O E K ω∨ j ; the Frobenius 8 on L (n) is given by 8(ω∨ j ) = π − j ω∨ j .
Proof (1) Since L (n) = Symn L (1) , it is sufficient to prove that L og is isomorphic to L (1) . By considering ω∨0 to be the lifting of the base e0 of K (0), we may regard L (1) as an extension 0 → K (ω∨ ) → L (1) → K (0) → 0. Since ∇(ω∨0 ) = ω∨ ⊗ ω, the explicit calculation of the isomorphism 4 shows that 1 (E, K (ω∨ )) corresponding to the extension class of L (1) maps to the element in Hsyn ∨ the identity ω ⊗ ω in End(K (ω∨ )). Furthermore, since the only structure preventing L (1) from splitting is the connection and since the connection on a module over a point is zero, we have e∗ L (1) = K (0) ⊕ K (ω∨ ). We conclude that the extension (1) class of L (1) is l; hence L og and L (1) are isomorphic as extensions of K (0) by K (ω∨ ). (n)
For the rest of this paper, we take L og = L (n) . A proof similar to that of the elliptic polylogarithm gives the following. LEMMA 5.5 The partial logarithmic sheaf satisfies the following properties. (a) For any isogeny [a] : E → E of degree prime to p, there is an isomorphism
%a : L og (b)
(n)
∼ =
−−−−→ [a]∗ L og
(n)
.
For any torsion point u : Spec O K → E of order prime to p, we have the splitting principle n (n) ∼ Y ∗ u L og = K (ω∨ j ). (5.2) j=0
Remark 5.6 The isomorphism %a above is given explicitly on the basis by %a (ω∨ j ) = a j ω∨ j .
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223
5.2. The calculation of cohomology In this subsection, we calculate the geometric cohomology of the partial logarithmic sheaf. LEMMA 5.7 Let n ≥ 1 be an integer. We have
H 0 (E, L og H 1 (E, L og H 2 (E, L og
(n) (n) (n)
) = K (ω∨n ), ) = K (ω) ⊕ K (ω∨n+1 )(−1),
(5.3)
) = K (−1).
Proof (0) The proof is similar to that of Lemma 3.4. We let L og = K (0), and we prove the lemma by induction for n ≥ 0. If n = 0, then the statement is the definition. We first prove the statement for H 0 , H 2 , and that H 1 is an extension 0 → K (ω∨n+1 )(−1) → H 1 (E, L og
(n)
) → K (ω) → 0.
(5.4)
Suppose that the lemma holds for n ≥ 1. Consider the diagram 0 −−−−→ K (ω∨n+1 ) −−−−→ L og yid 0 −−−−→ K (ω∨n+1 ) −−−−→
(1)
⊗ K (ω∨n ) −−−−→ K (ω∨n ) −−−−→ 0 y y
L og
(n+1)
−−−−→ L og
(n)
−−−−→ 0
where the center vertical arrow is induced from the multiplication, and the last vertical arrow is the canonical inclusion. The horizontal sequences are exact in S(E). The first boundary morphism of the long exact sequence for geometric cohomology gives the following commutative diagram: K (ω∨n ) ∼ = y H 0 (E, L og
−−−−→ H ∨ ⊗ K (ω∨n+1 ) yid (n)
γ
) −−−−→ H ∨ ⊗ K (ω∨n+1 )
where the left vertical arrow is an isomorphism from the induction hypothesis. The diagonal map is given by ω∨n 7 → ω ⊗ ω∨n+1 and hence is injective. This shows that γ is injective; hence H 0 (E, L og
(n+1)
∼ = )← − H 0 E, K (ω∨n+1 ) = K (ω∨n+1 ).
This also implies that Coker γ = K (η) ⊗ K (ω∨n+1 ) = K (ω∨n+2 )(−1).
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KENICHI BANNAI
Next, the second boundary morphism of the long exact sequence gives the commutative diagram H (−1) ⊗ K (ω∨n ) −−−−→ K (ω∨n+1 )(−1) y yid
H 1 (E, L og
(n)
)
β
−−−−→ K (ω∨n+1 )(−1)
Since the top horizontal arrow is surjective, β is also surjective. This implies that the surjection (n+1) (n) H 2 (E, L og ) → H 2 (E, L og ) = K (−1) is an isomorphism. The composition of the injection of (5.4) with β is nonzero and hence an isomorphism. This implies that we have a splitting H 1 (E, L og
(n)
) = K (ω) ⊕ K (ω∨n+1 )(−1)
(5.5)
and that ker β = K (ω). Hence the short exact sequence 0 → Coker γ → H 1 (E, L og
(n+1)
) → ker β → 0
gives the exact sequence (5.4) for n + 1. The assertion is now proved for H 0 , H 2 , and that H 1 is of the form (5.4). In particular, the proof above is valid for any integer n ≥ 1. The assertion of the lemma for H 1 follows from the fact that (5.5) holds for any integer n ≥ 1. COROLLARY
5.8
We have lim H 0 E, K (ω) ⊗ L og (n) (1) = 0, ← − lim H 1 E, K (ω) ⊗ L og (n) (1) = K ω ⊗ ω(1). ← − COROLLARY
(5.6)
5.9
The map
K ω ⊗ ω(1) → lim H 1 E, K (ω) ⊗ L og (n) (1) , ← − defined by associating to cω ⊗ ω(1) the element represented by cω ⊗ ω∨0 (1) ⊗ ω ∈ 0 E K , O E K ω ⊗ L (n) (1) ⊗ 1E K (log Da ) in H 1 (E, K (ω) ⊗ L og (n) (1)), is an isomorphism.
(5.7)
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225
Proof Suppose that the class of cω ⊗ ω∨0 (1) is zero in H 1 (E, K (ω) ⊗ L og (n) (1)). This, by definition, implies that there exists an element f =
n X
f j ω ⊗ ω∨ j (1) ∈ 0 E K , O E K ω ⊗ L og (n) (1)
j=0
with f j ∈ 0(E K , O E K ) such that ∇( f ) = cω ⊗ ω∨0 (1) ⊗ ω. The explicit calculation of the connection on L og (n) (see Def. 5.4) implies that d f 0 = cω. Since ω is not integrable by functions in 0(E K , O E K ), we have c = 0. Hence the map K ω ⊗ ω(1) → H 1 E, K (ω) ⊗ L og (n) (1) is injective. By passing to the limit, we see that (5.7) is injective. By Corollary 5.8, lim H 1 (E, K (ω) ⊗ L og (n) (1)) is of dimension one. Hence we conclude that (5.7) is ← − an isomorphism. 6. The partial elliptic polylogarithm In this section, we first define the partial elliptic polylogarithm and the modified partial polylogarithm (see Def. 6.2). The main purpose of this section is to determine the explicit shape of the modified partial polylogarithm (see Th. 6.6). This is essential for the proof of our main result. Definition 6.1 We define the partial elliptic polylogarithm pol
(n)
1 ∈ Hsyn U, K (ω) ⊗ L og
(n)
(1)
to be the image of pol(n) by the map (n) 1 1 Hsyn U, H ∨ ⊗ L og (n) (1) → Hsyn U, K (ω) ⊗ L og (1) induced from the projection. Let a be an integral ideal of OK prime to fp. We assume that K(a) ⊂ K through the inclusion i p of (3.1). Let Ua := E \ E[a]. Then the set Ua := (Ua , E, φ E ) is a syntomic datum with natural inclusion ja : Ua ,→ E. The morphism [a] : E → E for a = ψ(a) induces a morphism [a] : Ua → U of syntomic data. Definition 6.2 We define the modified partial polylogarithm by (n)
pola = 12(Na − [a]∗ )pol
(n)
1 ∈ Hsyn Ua , K (ω) ⊗ L og
(n)
(1) .
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KENICHI BANNAI (n)
By abuse of notation, we denote by the same symbol pola the system of objects in S(Ua ) representing the isomorphism class of the above system of extensions. Let M Y (n) H 1 Ua , K (ω) ⊗ L og (1) → K (ω) ⊗ K (ω∨ j ) (6.1) u∈E[a]
0≤ j≤n
be the residue map of the Gysin exact sequence (see Th. 2.19), combined with (5.2). 0 (O , −), we have a map By taking Hsyn K 0 Hsyn O K , H 1 (Ua , K (ω) ⊗ L og
(n)
M (1)) → End(K ω∨ ) .
(6.2)
u∈E[a]
Definition 6.3 Following the definition of δ (n) in Definition 3.8, we define the map M (n) 1 δa(n) : Hsyn Ua , K (ω) ⊗ L og (1) → End K (ω∨ ) u∈E[a]
to be the composition of the surjection of the short exact sequence in Lemma 2.14 for M = K (ω) ⊗ L og
(n)
(1)
with the residue map (6.2). LEMMA 6.4 (n) The system pola satisfies (n)
δa(n) (pola ) = (Nu id)u∈E[a] ∈
M
End K (ω∨ ) ,
(6.3)
u∈E[a]
where Nu = 12(Na − 1) if u = e and Nu = −12 otherwise. Proof Let δ
(n)
1 : Hsyn U, H ∨ ⊗ L og
(n)
(1) → End K (ω∨ )
be the map defined as in Definition 3.8. By functoriality and the characterization of pol(n) (see Def. 3.10), we have δ
(n)
pol
(n)
= id ∈ End K (ω∨ ) .
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227
By functoriality, we have the commutative diagram 1 U, K (ω) ⊗ L og Hsyn [a]∗ y
(n)
1 U , K (ω) ⊗ L og Hsyn a
(n) δ (1) −−−−→
(n)
End K (ω∨ ) 1y
δa(n) L ∨ (1) −−−−→ E[a] End K (ω ) (n)
where 1 is the diagonal. This, together with the definition of pola (see Def. 6.2), proves the desired assertion. PROPOSITION 6.5 Let p (n) be a system
of elements lim p (n) ∈ lim H 1 Ua , K (ω) ⊗ L og ← − ← −
(n)
(1)
(6.4)
such that the image by the residue map (6.1) is M M Y (Nu id) ∈ End K (ω∨ ) ⊂ K (ω) ⊗ K (ω∨ j ) , u∈E[a]
u∈E[a]
0≤ j≤n
where Nu is as in Lemma 6.4. Then the system of objects in M(Ua ) corresponding to (n) p (n) through 4 of Proposition 2.17 is isomorphic to objects of the form (Pc , ∇a,c ), where c is an element in K , and (n) (i) Pc is the free O E K -module Pc(n) = O E K e0c ⊕ O E K ω ⊗ L (n) (1); (ii)
∇a,c is a connection given by ∇a,c ω ⊗ ω∨ j (1) = ω ⊗ ω∨ j+1 (1) ⊗ ω, ∇a,c (e0c ) = cω ⊗ ω∨0 (1) ⊗ ω + ω ⊗ ω∨ (1) ⊗ d log 2E,a .
Proof (n) For any c ∈ K , by construction, (Pc , ∇a,c ) is a system of extensions 0 → K (ω) ⊗ L (n) (1) → Pc(n) → K (0) → 0 (n)
in M(Ua ). Here the surjection is defined by mapping e0c to e0 . Since the module Pc (n) (n) (n) is free, the element pc = 4(Pc , ∇a,c ) corresponding to (Pc , ∇a,c ) through (2.4) is the element represented by cω ⊗ ω∨0 (1) ⊗ ω + ω ⊗ ω∨ (1) ⊗ d log 2E,a
(6.5)
228
KENICHI BANNAI
in 0(E K , O E K ω ⊗ L (n) (1) ⊗ 1E K (log Da )) (see Rem. 2.18). By definition of the residue map, since ω is regular on E K and X div(2E,a ) = 12Na(e) − 12 (u), u∈E[a] (n)
the extension class of (Pc , ∇a,c ) maps by (6.1) to (Nu id). The element (n)
( pc(n) − p0 ) ∈ lim H 1 E, K (ω) ⊗ L og ← − n
(n)
(1)
is represented by cω ⊗ ω∨0 (1) ⊗ ω ∈ 0 E K , O E K ω ⊗ L (n) (1) ⊗ 1E K (log Da ) . (n)
If ( p (n) ) is a system of elements as in (6.4), then ( p (n) − p0 ) is an element in limn H 1 (E, K (ω) ⊗ L og ← −
(n)
(1)). By Corollary 5.9, we have (n)
(n)
p (n) − p0 = pc(n) − p0 (n)
for some c in K . Hence p (n) corresponds to (Pc , ∇a,c ), as desired. We now explicitly determine the element (n)
pola = (P (n) , ∇a , F • , 8). (n)
Since pola is an extension 0 → K (ω) ⊗ L og
(n)
(n)
(1) → pola → K (0) → 0,
the strictness of the Hodge filtration implies that we have an exact sequence of coherent O E K -modules 0 → F 0 O E K ω ⊗ L (n) (1) → F 0 P (n) → O E K e0 → 0. Let e˜0 be a lifting of e0 to F 0 P (n) . Since O E K e0 is free, we have a splitting P (n) = O E K e˜0 ⊕ O E K ω ⊗ L (n) (1) as an O E K -module. The connection, the Hodge filtration, and the Frobenius structure on O E K ω ⊗ L (n) (1) are the tensor products of the structure on K (ω) with the structure on L (n) given in Proposition 5.4 tensored by K (1).
ON THE p-ADIC ELLIPTIC POLYLOGARITHMS
229 ∼ =
(n)
By Proposition 6.5, there exist c in K and an isomorphism P (n) − → Pc , compatible with the connection, which fit into the commutative diagram 0 −−−−→ O E K ω ⊗ L (n) (1) −−−−→ P (n) −−−−→ O E K e0 −−−−→ 0 ∼ idy idy =y (n)
0 −−−−→ O E K ω ⊗ L (n) (1) −−−−→ Pc −−−−→ O E K e0 −−−−→ 0 P Let e˜0 = e0c + nj=0 f j ω ⊗ ω∨ j (1) through the above isomorphism. The compatibility with the connection gives ∇a (e˜0 ) = cω ⊗ ω∨0 (1) ⊗ ω + ω ⊗ ω∨ (1) ⊗ d log 2E,a +
n X j=0
ω ⊗ ω∨ j (1) ⊗ d f j +
n−1 X
f j ω ⊗ ω∨ j+1 (1) ⊗ ω.
j=0
The connection satisfies Griffiths transversality ∇a (e˜0 ) ⊂ F −1 P (n) . Since Y F −1 O E K ω ⊗ L (n) (1) = O E K ω ⊗ O E K ω∨ j (1) 0≤ j≤1
and since differential equations of the form d f j + f j+1 ω = 0 have solutions if and only if f j+1 = 0, we have f j = 0 for 1 ≤ j ≤ n. Hence e˜0 maps to e0c + f 0 ω⊗ω∨0 (1) through the above isomorphism. Let e0 = e˜0 − f 0 ω ⊗ ω∨0 (1) in P (n) . Then e0 is again a lifting of the basis e0 of K (0) to F 0 P (n) . By taking e0 instead of e˜0 , we conclude that for P (n) = O E K e0 ⊕ O E K ω ⊗ L (n) (1), we have the following: (1) the connection ∇a on P (n) is given by ∇a (e0 ) = cω ⊗ ω∨0 (1) ⊗ ω + ω ⊗ ω∨ (1) ⊗ d log 2E,a ; (2)
the Hodge filtration on P (n) is the direct sum ( O E K e0 ⊕ F m O E K ω ⊗ L (n) (1) , m ≤ 0, m (n) F P = 0, m > 0.
The remaining problem is to calculate the Frobenius structure 8. We calculate 8 with respect to the lifting e0 chosen above. Let A = 0(Ua , OUa ). The Frobenius φ E of E induces a Frobenius φ ∗E : A†K → † an , j † O an ). By [Ber2, Prop. 2.5.2(ii)], the functor 0(E an , −) inA K on A†K = 0(E K a EK K
duces an equivalence of categories between the category of coherent ja† O E Kan -modules
with integrable connection and the category of projective A†K -modules of finite type with integrable connection.
230
KENICHI BANNAI
Suppose that Mrig is a coherent ja† O E Kan -module with integrable connection. To
give a Frobenius structure on Mrig is equivalent to defining an A†K -homomorphism an an 8 : 0(E K , φ ∗E Mrig ) → 0(E K , Mrig )
compatible with the connection. (n) (n) In the case above, we have φ ∗E Prig = Prig as a ja† O E Kan -module. The Frobenius is given on n Y (n) an 0(E K , Prig ) = A†K e0 ⊕ A†K ω ⊗ ω∨ j (1) j=0
by 8(ω∨ j ) = π − j ω∨ j (see Prop. 5.4) and 8(e0 ) = e0 +
Pn
j=0 C a, j−1 ω
⊗ ω j∨ (1)
for some functions Ca, j in A†K . Then we have n X ∇a,c ◦ 8(e0 ) = ∇a,c e0 + Ca, j−1 ω ⊗ ω∨ j (1) j=0
= cω ⊗ ω (1) ⊗ ω + ω ⊗ ω∨ (1) ⊗ d log 2E,a ∨0
+
n X
ω ⊗ ω∨ j (1) ⊗ dCa, j−1 + Ca, j−1 ω ⊗ ω∨ j+1 (1) ⊗ ω
j=0
and 8 ◦ ∇a,c (e0 ) = 8 cω ⊗ ω∨0 (1) ⊗ ω + ω ⊗ ω∨ (1) ⊗ d log 2E,a =
φ∗ cπ 2 ω ⊗ ω∨0 (1) ⊗ ω + ω ⊗ ω∨ (1) ⊗ E (d log 2E,a ). p p
Hence the compatibility of the Frobenius with the connection gives the differential equations dCa,−1 = c (π 2 / p) − 1) ω, φ∗ E − 1 d log 2E,a − Ca,−1 ω, dCa,0 = p dCa, j+1 = −Ca, j ω
( j ≥ 0).
bK = Since nonzero constant multiples of ω are not integrable in A†K ⊂ A 0(UaK , OUaK ) (see Lem. 4.7), we have c = 0 and dCa,−1 = 0. Since the function (φ ∗E / p − 1) log 2E,a is an element in A†K , the differential (φ ∗E / p − 1)d log 2E,a is integrable in A†K . Again, since nonzero constant multiples of ω are not integrable in A†K , we have Ca,−1 = 0.
ON THE p-ADIC ELLIPTIC POLYLOGARITHMS
231
The above argument shows that c = 0 and Ca, j for j ≥ 0 are functions in A†K such that φ∗ E dCa,0 = − 1 d log 2E,a , p D(Ca, j+1 ) = −Ca, j
( j ≥ 0).
From the fact that A†K ⊂ 0(UaK , OUaK ) and Theorem 4.6, we have the following. 6.6 The modified partial polylogarithm is given explicitly by THEOREM
(n)
pola = (P (n) , ∇a , F • , 8), where (i) P (n) is the free O E K -module P (n) = O E K e0 ⊕ O E K ω ⊗ L (n) (1); (ii)
∇a is a connection given by ∇a ω ⊗ ω∨ j (1) = ω ⊗ ω∨ j+1 (1) ⊗ ω, ∇a (e0 ) = ω ⊗ ω∨ (1) ⊗ d log 2E,a ;
(iii)
F • is the Hodge filtration given by ( O E K e0 ⊕ F m O E K ω ⊗ L (n) (1) , m ≤ 0, m (n) F P = 0, m > 0;
(iv)
8 is the Frobenius given by 8 ω ⊗ ω∨ j (1) = (1/ pπ j−1 ) ω ⊗ ω∨ j (1), 8(e0 ) = e0 +
n X
(−1) j Da, j−1 ω ⊗ ω∨ j (1).
j=1
7. The main result The purpose of this section is to prove our main result, Theorem 7.1. Let the notation be as in the previous sections, and assume that p ≥ 5. In particular, let f be the conductor of ψ, and fix an integral ideal g of OK such that f|g and g is prime to p. Let be an element such that L (E, ω) = g. We assume that K(E[g]) ⊂ K through the
232
KENICHI BANNAI
inclusion i p of (3.1). Then v = ξ(, ω) is a torsion point in U K with exact annihilator g. For any torsion point u in U K of order prime to p, we define Y 1 h u : Hsyn U, H ∨ ⊗ L og (n) (1) → K ω∨ j (7.1) 0≤ j≤n−1
to be the composition of the restriction map 1 1 U, H ∨ ⊗ L og (n) (1) → Hsyn O K , H ∨ ⊗ L og (n) (1) , u ∗ : Hsyn the projection to 1 Hsyn O K , K (ω) ⊗ L og
(n)
Y 1 (1) = Hsyn O K , K (ω∨ j−1 )(1) , 0≤ j≤n
and the canonical isomorphism of Lemma 7.2. Our main result is the following. THEOREM 7.1 Assume that p is a prime greater than or equal to 5. We have X j j h τ (v) (pol(n) ) = (−1) j p L p,g (ψp ) ω∨ j 0≤ j≤n−1 . τ ∈Gal(K(g)/K)
The proof is given at the end of this section. We first prepare a lemma. LEMMA 7.2 We have a canonical isomorphism
( 1 Hsyn
O K , K (ω∨ j )(1) =
K ω∨ j ,
j ≥ 0,
0,
j < 0.
(7.2)
Proof By Remark 2.13, for any M = (M, F • , 8) in S ad (O K ), the first syntomic cohomology of M is the cokernel of the map (1 − 8) : F 0 M → M. Since F 0 K (ω∨ j )(1) = 0 for j ≥ 0, the assertion follows for this case. We have removed the (1) from the notation since the twist is no longer relevant. If j < 0, then 8 acts on F 0 K (ω∨ j )(1) = K (ω∨ j )(1) by multiplication by 1/ pπ j . This implies that (1 − 8) is bijective and hence the assertion. Remark 7.3 For an integer j ≥ 0, if M = (M, F • , 8) is an extension 0 → K (ω∨ j )(1) → M → K (0) → 0
ON THE p-ADIC ELLIPTIC POLYLOGARITHMS
233
in S(O K ), then the element corresponding through the isomorphism 4 to the class of M in Ext1S(O K ) (K (0), K (ω∨ j )(1)) is (8 − 1)(e0 ) ∈ K (ω∨ j )(1), where e0 is any lifting to M of the base e0 of K (0). We fix an integral ideal a of OK prime to p, and we define the map Y (n) 1 h a,u : Hsyn Ua , K (ω) ⊗ L og (1) → K ω∨ j 0≤ j≤n−1
for u 6 ∈ E[a] similarly to (7.1). PROPOSITION
7.4
We have X
(n)
h a,τ (v) (pola )
τ ∈Gal(K(g)/K) j
j
= 12(ψ − j (a) − Na)(−1) j−1 p L p,g (ψp )ω∨ j
0≤ j≤n−1
.
Proof (n) The proposition follows from the explicit shape of pola proved in Theorem 6.6, the calculation in Proposition 4.12, and the definition of the isomorphism (7.2) (see also Rem. 7.3). Proof of Theorem 7.1 Since the order of v is prime to a, the map h τ (v) factors h a,τ (v) . Hence it is sufficient to prove that X (n) j j h a,τ (v) (pol ) = (−1) j p L p,g (ψp ) ω∨ j 0≤ j≤n−1 . τ ∈Gal(K(g)/K)
(n)
This follows from Proposition 7.4, the definition of pola , and the fact that %a maps ω∨ j (1) to ψ(a) j ω∨ j (1) (see Rem. 5.6). Acknowledgments. The author thanks Seidai Yasuda for discussion on the proof of Lemma 4.7. The author is sincerely grateful to Nobuo Tsuzuki for taking time to explain in detail some points in p-adic analysis. The author also thanks Kazuya Kato, Shigeki Matsuda, Atsushi Shiho, and Takeshi Tsuji for discussion and comments. The author is very grateful to the referees for their comments. Finally, the author sincerely thanks his advisor Takeshi Saito for discussion and helpful comments.
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Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan; current: Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 2,
ALMOST UNRAMIFIED DISCRETE SPECTRUM FOR SPLIT GROUPS OVER Fq (t) AMRITANSHU PRASAD
Abstract Let G be a split adjoint group defined over Fq , let F = Fq (t), and let A be the ad`eles of F. We describe the local constituents at two points of automorphic representations of G in the discrete part of L 2 (G(F)\G(A)) which have vectors invariant under Iwahori subgroups at these two points and are unramified at all other points.
1. Introduction 1.a. Overview Let G be a split reductive group with an irreducible root system defined over the finite field Fq . We take F to be the function field Fq (t) in one variable, and we denote the ad`eles of F by A. A fundamental problem in the theory of automorphic forms is to determine the irreducible constituents of the right regular representation L 2 G(F)\G(A) of G(A). The unramified automorphic representations are described by I. Efrat in [E] for G = PGL2 , and the unramified representations for G = PSp4 appearing in the discrete part are described by P. Anspach in [A]. The aforementioned works are closest in spirit to our own. When F is a number field and G = GLn , C. Moeglin and J.-L. Waldspurger [MW] have described the discrete spectrum in terms of cuspidal unitary representations of GLd for d|n, using R. Langlands’s theory of Eisenstein series and the analysis of certain normalized intertwining operators. In [M1], Moeglin has proved that for a classical group, the representations in the unramified discrete spectrum have multiplicity one and are parameterized by unipotent orbits in the dual group which do not DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 2, Received 18 October 2000. Revision received 25 May 2001. 2000 Mathematics Subject Classification. Primary 11F70, 22E55. 237
238
AMRITANSHU PRASAD
intersect any proper Levi subgroup. In [M2], when G is either orthogonal or symplectic, she characterizes the square-integrable automorphic forms with cuspidal data consisting of a split torus and the trivial character as residues of certain Eisenstein series. These include forms in irreducible representations that have vectors invariant under Iwahori subgroups at an arbitrary finite set of places and are unramified everywhere else. The techniques that we use here are uniform for all groups. We obtain, for the first time, a nice description of a part of the discrete spectrum for all exceptional groups. 1.b. The main results For simplicity we assume that G is semisimple. The results hold for non-semisimple groups after standard modifications. For each place v of F, we denote the corresponding local field by Fv and denote its ring of integers by Ov . Fix a Borel subgroup B containing a maximal split torus T . Let Iv denote the Iwahori subgroup in G(Fv ) which is the preimage of B(Fq ) under the natural map G(Ov ) → G(Fq ). We fix uniformizing elements t and t −1 at the places v = 0 and v = ∞, respectively. Our main results describe the local constituents at 0 and ∞ of the irreducible representations that occur in L 2 (G(F)\G(A)) that have vectors invariant under the compact and open subgroup Y K := I∞ × I0 × G(Ov ). v6=0,∞
Let Hv denote the Iwahori-Hecke algebra of compactly supported measures on G(Fv ) which are bi-invariant under Iv . We begin by studying the (H∞ ⊗ H0 )-module K M = Cc G(F)\G(A) which has a basis consisting of the characteristic functions of double cosets in G(F)\G(A)/K . In §2.a, we define a map φt −1 from the extended affine Weyl group W˜ to G(F∞ ) (which is not necessarily a group homomorphism). Composing this map with the canonical inclusion G(F∞ ) ,→ G(A), we get a map φtA−1 : W˜ → G(A). The function w 7 → G(F)φtA−1 (w)K
(1.1)
from W˜ to G(F)\G(A)/K is a bijection. (The double cosets above are in bijective correspondence with the set of principal G-bundles on P1 (Fq ) and complete flags specified at 0 and ∞.) A proof of this result, which is widely known, may be found in [P]. Thus M has a basis {tw } indexed by w ∈ W˜ , where tw is the characteristic function of G(F)φtA−1 (w)K . For each rational place v of F, the Iwahori-Hecke algebra Hv has generators T0v , . . . , Trv , where r is the rank of G (see the paragraph following
ALMOST UNRAMIFIED DISCRETE SPECTRUM
239
the proof of Claim 3.2). We compute the action of the Hecke algebras in terms of these basis elements and find that they are related to the formulae (see (4.1), (4.2)) that N. Iwahori and H. Matsumoto obtained in [IM] for the convolution product in the Hecke algebra. We exploit this relation by introducing the local field F• = Fq ((π)) of Laurent series in one variable with ring of integers O• . Let I• be the preimage of B(Fq ) under the canonical map G(O• ) → G(Fq ). We denote by H the Iwahori-Hecke algebra of G(F• ) consisting of compactly supported measures that are bi-invariant under I• , with generators T0 , . . . , Tr (see §4.a). We consider the right (H ⊗ H )-module Mloc = Cc G(F• )\(G(F• ) × G(F• ))/(I• × I• ) , where the left action of G(F• ) is the diagonal action. The double coset space in the definition of Mloc is essentially equivalent to I• \G(F• )/I• , which by the affine Bruhat decomposition is also in bijective correspondence with the extended affine Weyl group. The formulae of Iwahori and Matsumoto allow us to write down this action in terms of basis elements {τw }, w ∈ W˜ , defined in (4.3). We define a vector space isomorphism I : Mloc → M (see (5.4)). We denote by ιv the composition of the natural isomorphism H → Hv with the Iwahori-Matsumoto involution ι (see (5.1)). We use another useful involution κ, defined in §5.a. We obtain the following formula (proved in §5.c). THEOREM 1.2 For any m ∈ Mloc , h, h 0 ∈ H ,
I m · (h ⊗ h 0 ) = I (m) · ι∞ (h) ⊗ ι0 ◦ κ(h 0 ) . 2 → We find (see Proposition 5.5) that I extends to an isometry of Hilbert spaces Mloc 2 M , where 2 Mloc = L 2 G(F• )\(G(F• ) × G(F• ))/(I• × I• ) . 2 in terms of the In §6.a, we describe the representations in the discrete part of Mloc irreducible square-integrable representations of an abstract Iwahori-Hecke algebra associated to G. This enables us to describe the representations of H0 ⊗ H∞ which occur in the discrete part of M. Let Tˇ denote the Langlands dual torus Hom(X ∗ (T ), C× ), where X ∗ (T ) denotes the group of algebraic homomorphisms Gm → T . The complex torus ˇ There is a natural isomorphism Tˇ is a maximal torus in the Langlands dual group G. ∼ X ∗ (T ) = (T (Fv ))/(T (Ov )). Recall from [L, Proposition 3.11] that the center of Hv is naturally isomorphic to C[X ∗ (T )]W , where W denotes the finite Weyl group of G. ˇ being conjugate to an element in Tˇ , determines Thus any semisimple element s ∈ G,
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AMRITANSHU PRASAD
a central character χv of Hv at each place v. We show that a nonzero vector in the χ0 -eigenspace of H0 is also in the χ∞ -eigenspace of H∞ . Let Ms2 denote the vectors in M 2 on which H∞ ⊗ H0 acts by central character χ∞ ⊗ χ0 . In general, if f : A → B is a ring homomorphism and W is a B-module, then the B-module structure B → End(W ) restricts to an A-module structure, A → B → End(W ), which we denote by W f . THEOREM 1.3 (Main theorem) The Ms2 ’s span a dense subspace of the discrete part of the (H∞ × H0 )-module M 2 . Moreover, there is an isomorphism of right (H∞ ⊗ H0 )-modules M V ι∞ ⊗ V˜ ι0 ◦κ −→ g Ms2−1 ,
where the direct sum is taken over the isomorphism classes of irreducible squareintegrable right modules V of H with central character corresponding to s. Here V˜ denotes the contragredient of V . When G is of adjoint type, then the classification of representations of an abstract Iwahori-Hecke algebra by D. Kazhdan and G. Lusztig in [KL] allows us to parameterize the irreducible representations that appear on the left-hand side of the equation in Theorem 1.3. THEOREM 1.4 (Kazhdan-Lusztig) Let Gˇ denote the Langlands dual group of G. Let (s, u) be a pair of elements in Gˇ with the following properties: • u is unipotent; • s is semisimple; • sus −1 = u q ; • there is no proper parabolic subgroup of Gˇ with Levi subgroup L, with L containing both u and s. Let ρ be an irreducible representation of the group of components of the simultaneous centralizer of s and u, such that ρ appears in H∗ (Bs,u , Q), where
Bs,u = {Borel subgroups B ⊂ Gˇ | s, u ∈ B }.
Then conjugacy classes of pairs (s, u) and ρ as above parameterize the set of all square-integrable irreducible representations of H . Moreover, the central character by which H acts on the module is the one corresponding to s. A referee has pointed out to the author that M. Reeder has generalized the above theorem, eliminating the hypothesis that G is of adjoint type (see [R, Theorem 2]).
ALMOST UNRAMIFIED DISCRETE SPECTRUM
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Moreover, when the derived group of G is of adjoint type, κ takes a representation to its contragredient (see Proposition 6.3). Thus we have the following. THEOREM 1.5 Let Xs be the collection of data consisting of conjugacy classes of (s, u, ρ) as in Theorem 1.4, with s ∈ Tˇ fixed. Let Vs,u,ρ denote the irreducible square-integrable representation corresponding to the datum s, u, ρ. We have an isomorphism of right (H∞ × H0 )-modules: M ι∞ ι0 Vs,u,ρ ⊗ Vs,u,ρ −→ g Ms2−1 . (s,u,ρ)∈Xs
1.c. Further remarks At places other than ∞ and 0, M 2 is a module over spherical Hecke algebras, which are commutative and have actions that commute with those of H∞ and H0 . This means that they act by scalars on each irreducible (H∞ ⊗ H0 )-module. Therefore, these irreducible (H∞ ⊗ H0 )-modules correspond to the K -invariant vectors of irreducible G(A)-modules. One of the conditions for the cuspidality of an automorphic function f is the R vanishing of the integral N (F)\N (A) f (nx) dn almost everywhere in x, where N is the unipotent radical of the Borel subgroup B of G. By writing such an integral as a sum over the K 0 -orbits of G(F)\G(A), one may see that this condition implies that f is identically zero. Therefore, there are no cuspidal representations in the discrete part of M 2 . For G = PGL2 , there are two one-dimensional representations in M 2 , one of which is trivial (coming from twisting the Steinberg representation by the IwahoriMatsumoto involution). For G = PSp4 , there is one nontrivial representation in M 2 coming from very nontrivial Kazhdan-Lusztig data (see [A, Chapter 5]). S. Evens and I. Mirkovi´c point out in [EM] that the Iwahori-Matsumoto involution takes many tempered representations to nontempered ones, and thus Theorem 1.5 gives us many automorphic representations with nontempered local constituents. 2. Reduction to local calculations For simplicity we assume that G is simply connected through §5.c. In §5.d, we extend the results to general G. 2.a. Notation The set of right cosets G(F• )/I• is in bijective correspondence with the maximal simplices of the Bruhat-Tits building B associated to G(F• ). There is an identification of X ∗ := X ∗ (T ) ⊗ R with an apartment A0 ⊂ B. The group G(F• ) acts on the building. The Iwahori subgroup I• fixes a fundamental alcove C0 ⊂ A0 , whose closure is a
242
AMRITANSHU PRASAD
fundamental domain for the action of the affine Weyl group Wa on A0 . This group is a semidirect product W n X ∗ (T ), with respect to the usual action of the Weyl group W of G on X ∗ (T ). Let X ∗ := X ∗ (T ) ⊗ R, where X ∗ (T ) is the lattice of algebraic characters T → Gm . The vector space X ∗ contains the root system 8 of G with respect to T . The Borel subgroup B determines a subset 8+ of positive roots, which may be expressed as linear combinations of the set 1 = {α1 , . . . , αr } of simple roots. Let si denote the reflection about the hyperplane αi = 0 in A0 for i = 1, . . . , r . These are generators for W . The relations are that si2 = 1 for each i, and for each pair (i, j), we have a braid relation of the form (si s j )m = (s j si )m
or
(si s j )m si = (s j si )m s j ,
(2.1)
where m depends on i and j and is possibly infinite (which is to be interpreted as the absence of a relation). Let wl be the unique element in W of greatest length with respect to the generators si . Then conjugation by wl permutes the si ’s. We denote by s i the reflection wl si wl (wl is its own inverse). Denote by 8a the affine root system {α + n | α ∈ 8 and n ∈ Z}. Denote the affine root α i + 0 by αi . Let α0 denote the highest positive root in 8. Let α0 be the affine root 1 − α0 , and let s0 be the affine reflection corresponding to α0 . Let s 0 denote the reflection corresponding to the affine root 1 + α0 . (Thus si = wl si wl for i = 0, 1, . . . , r .) Then s0 , . . . , sr as well as s 0 , . . . , s r are generating sets for the Coxeter group Wa . Let l(w) and l(w) denote the lengths of an element w ∈ Wa with respect to the first and second generating sets, respectively. Let U α denote the root subgroup in G corresponding to α. Fix an isomorphism u α : Ga → U α for each root subgroup. For each affine root α = α + n, our choice of a uniformizing element π gives a map u α : Fq → U α (F• ) defined by u α (x) = u α (π −n x). Let Uα denote the image of u α . Fix an inclusion j : W → G(Fq ) (which may not be a homomorphism), taking an element to a representative in N G (T )(Fq ). Then (since we have chosen a uniformizer π) we get an inclusion Wa = W n X ∗ (T ) → G(F• ) by wµ 7→ j (w)µ(π) for w ∈ W and µ ∈ Hom(Gm , T ). If v is a rational place of F, then the choice of a uniformizing element πv allows us to identify F• and Fv . In particular, we have an inclusion map φπv : Wa → G(Fv ). If πv ∈ F, then the image of Wa under φπv is contained in G(F). In order to carry out our calculations, we find it convenient to replace the Iwahori subgroup I0 with I 0 := wl I0 wl , so that certain subgroups eventually involved in the calculations have a very simple description in terms of affine roots (see ProposiQ tion 3.1). Let K = I∞ × I 0 × v6=0,∞ G(Ov ). We work with M = Cc G(F)\G(A)/K
instead of M. M is a module over H∞ and H 0 , where H 0 is the algebra of com-
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243
pactly supported measures on G(F0 ) bi-invariant under I 0 . We interpret these calculations in terms of M in §3.c. 2.b. Comparison of φt and φt −1 Since t and t −1 are also elements of F, the images of Wa under φt −1 and φt lie in G(F). Consider the automorphism of the root system (8, X ∗ (T )) of G defined by x 7→ −x, for all x ∈ X ∗ (T ). It induces an automorphism κ of the affine Weyl group Wa which leaves the finite Weyl group W fixed but takes the affine reflection s0 to the reflection s 0 = wl s0 wl about a parallel hyperplane. Moreover, if we write Wa = W n X ∗ (T ), then κ(w0 eλ ) = w0 e−λ . (Notice that λ 7 → −λ is a W -invariant automorphism of X ∗ (T ).) 2.2 The following diagram commutes: PROPOSITION
κ
/ Wa Wa F FF x x FF xx F xx φt φt −1 FF" x |x G(F)
Proof Indeed, φt κ(w0 eλ ) = φt (w0 e−λ ) = j (w) − λ(t) = j (w) λ(t −1 ) = φt −1 (w0 eλ ). 2.c. Local representatives of double cosets Since the map φtA−1 in (1.1) is a bijection, every double coset in G(F)\G(A)/K also has a representative in G(F∞ ). Moreover, we may modify an element in G(F∞ ) on the left by the group 0 consisting of those elements in G(F) whose images in G(Fv ) lie in G(Ov ) for all v different from 0 or ∞ and whose images in G(F0 ) lie in I 0 while remaining in the same double coset. It is not difficult to see the following. PROPOSITION 2.3 There is an isomorphism of H∞ -modules Cc G(F)\G(A)/K → Cc 0\G(F∞ )/I∞
such that 1G(F)φ A
t −1
(w)K
7 → 10φt −1 (w)I∞
for all w ∈ Wa .
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AMRITANSHU PRASAD
Similarly, to understand the action of H 0 , we take 0 to be the subgroup of G(F) consisting of elements whose image in G(Fv ) is in G(Ov ) for v 6= 0, ∞, and whose image in G(F∞ ) is in I∞ . PROPOSITION 2.4 There is an isomorphism of H 0 -modules Cc G(F)\G(A)/K → Cc 0\G(F0 )/I 0
such that 1G(F)φ A
t −1
7→ 10φ −1 (κ(w−1 ))I 0
(w)K
t
for all w ∈ Wa .
The point is that φt −1 (w) can be thought of as an element of G(F) (which acts di−1 agonally on G(A)). Call this element wglob . The component at 0 of wglob φt −1 (w) is −1 κ(w ) (by Proposition 2.2). We denote both 1G(F)φ A (w)K and its images in Cc (0\G(F∞ )/I∞ ) and Cc (0\G(F0 )/I 0 ) by Tw .
t −1
3. Formulae for convolutions 3.a. The action of H∞ We now calculate the action of H∞ on M∞ = Cc 0\G(F∞ )/I∞ . Our choice of uniformizer t −1 for F∞ gives us an isomorphism G(F• ) ∼ = G(F∞ ). We abuse notation, and we let Uα refer to the image in G(F∞ ) of Uα and w refer to the image φt −1 (w) of w in G(F∞ ). It is easy to see the following. 3.1 Uα ⊂ I∞ if and only if α > 0. Uα ⊂ 0 if and only if α < 0.
PROPOSITION
(1) (2)
Proof Note that affine root α = α + n, where α is an ordinary root and n is an integer, is positive if and only if either (a) α is positive and n is nonnegative or (b) α is negative and n is positive.
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It follows that Uα is contained in G(O∞ ) and that its image under the natural map G(O∞ ) → G(Fq ) is contained in B(Fq ) if and only if the affine root α is positive, proving (1). From the definitions of Uα and 0, we see that Uα ⊂ 0 if and only if u α ((t −1 )−n ), viewed as an element of G(F), lies in I 0 ∩ G(F). This is equivalent to the assertion that the affine root wl (α) − n is a positive affine root. This, in turn, is equivalent to the assertion that α = α + n is a negative affine root, proving (2). Right cosets in I∞ si I∞ We now describe the right I∞ -cosets in G(F∞ ) which are contained in I∞ si I∞ . CLAIM 3.2 The right cosets contained in the double coset I∞ si I∞ may be enumerated in two ways: a a I∞ si I∞ = si I∞ u αi (ξ )si I∞ ξ ∈Fq×
= si I∞
a a
si u αi (ξ )si I∞
ξ ∈Fq×
Proof Since [I∞ : I∞ ∩ si I∞ si ] = q, I∞ si I∞ consists of q right I∞ -cosets. Clearly, a a si I∞ u αi (ξ )si I∞ ⊂ I∞ si I∞ . ξ ∈Fq×
Moreover, no nontrivial element of Uαi fixes si C0 . Hence the right cosets appearing above are distinct. Since Uαi fixes the hyperplane αi (x) = 0 in A , each of the chambers u αi (ξ )C0 shares a face contained in this hyperplane with C0 . Now si u αi (ξ )si fixes si C0 , but not C0 for ξ ∈ Fq× . Therefore, for nonzero ξ , si u αi (ξ )si C0 are the alcoves sharing a face with C0 which is contained in αi (x) = 0. Let d x denote the Haar measure on G(F∞ ) which assigns unit measure to I∞ . This determines an identification of H∞ with the convolution algebra of functions Cc (I∞ \G(F∞ )/I∞ ). For φ ∈ M∞ , and f (x) in H∞ , Z (φ ∗ f )(t) = φ(t x −1 ) f (x) d x. (3.3) G(F∞ )
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AMRITANSHU PRASAD
Specifically, taking f = Ti∞ to be the characteristic function 1 I∞ si I∞ and φ = Tw to be 10w I∞ (recall that w here represents φt −1 (w)), Z ∞ (Tw ∗ Ti )(t) = Tw (t x −1 )Ti∞ (x) d x G(F∞ ) Z X = Tw (t x −1 ) d x g I∞ ∈(I∞ si I∞ )/I∞
= Tw (tsi ) +
X
g I∞
Tw tu αi (ξ )si
(3.4)
ξ ∈Fq×
= Tw (tsi ) +
X
Tw tsi u αi (ξ )si .
(3.5)
ξ ∈Fq×
The last two steps use Claim 3.2. Relation to lengths From the theory of Tits systems (see, e.g., [Bou, Chapitre IV]), we know the following. LEMMA 3.6 If w ∈ W˜ , then l(wsi ) = l(w) + 1 if and only if wαi > 0. LEMMA 3.7 Let w ∈ W˜ , with w ∈ Wa . Then we have the following: (1) wsi u αi (ξ )si w−1 ∈ 0 for all ξ ∈ Fq× if and only if wαi > 0, that is, if and only if l(wsi ) = l(w) + 1; (2) wu αi (ξ )w−1 ∈ 0 for all ξ ∈ Fq× if and only if wαi < 0, that is, if and only if l(wsi ) = l(w) − 1.
Proof The lemma follows from Lemma 3.6, Proposition 3.1, and the fact that si Uαi si = U−αi . Evaluation of Tw ∗ Ti∞ Case 1: l(wsi ) = l(w) + 1. We have Tw wsi u αi (ξ )si = Tw wsi u αi (ξ )si w −1 w = 1 since wsi u αi (ξ )si w−1 ∈ 0 by Lemma 3.7(1). Substitute t = wsi in equation (3.4) to
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get Tw ∗ Ti∞ (wsi ) = Tw (w) +
X
Tw wsi u αi (ξ )si
Tw wsi u αi (ξ )si
ξ ∈Fq×
= q − 1. Substitute t = w in equation (3.5) to get Tw ∗ Ti∞ (w) = Tw (wsi ) +
X ξ ∈Fq×
= q − 1. Case 2: l(wsi ) = l(w) − 1. We have Tw wu αi (ξ )si = Tw wu αi (ξ )w −1 wsi = 0
for ξ 6= 0
since wu αi (ξ )w −1 ∈ 0 by Lemma 3.7(2). Proceeding as in Case 1, we see that Tw ∗ Ti∞ = 0 and Tw ∗ Ti∞ (wsi ) = 1. This proves the following. THEOREM 3.8 The action of H∞ on M∞ is given by ( (q − 1)Tw + q Twsi Tw ∗ Ti∞ = Twsi
if l(wsi ) = l(w) + 1, if l(wsi ) = l(w) − 1.
3.b. The action of H 0 Now we describe the action of H 0 on M0 = Cc 0\G(F0 )/I 0 . 0
Let T i = 1 I 0 φt (s i )I 0 . The calculations are analogous to the ones for H∞ , giving us the following. THEOREM 3.9 The action of H 0 on M0 is given by ( (q − 1)Tw + q Tκ(s i )w 0 Tw ∗ T i = Tκ(s i )w
if l s i κ(w) = l κ(w) + 1, if l s i κ(w) = l κ(w) − 1.
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AMRITANSHU PRASAD
3.c. Translation from M to M In this section, (x ∞ , x 0 ) denotes an element of the adelic group G(A), whose components at all places other than ∞ and 0 are the identity element and whose components at ∞ and 0 are φt −1 (x ∞ ) and φt (x 0 ), respectively. Let tw = 1G(F)(w,1)K = 1G(F)(w,wl )K (1,wl ) = 1G(F)(wl w,1)K (1,wl ) = Twl w ∗ δ(1,wl ) ,
(3.10)
where δ(1,wl ) is a delta measure at (1, wl ) ∈ G(A). Also, 0
0
Ti0 (x) = T i (wl xwl ) = δ(1,wl ) ∗ T i ∗ δ(1,wl ) (x). Therefore, 0
tw · Ti0 = Twl w ∗ δ(1,wl ) ∗ δ(1,wl ) ∗ T i ∗ δ(1,wl ) 0
= Twl w ∗ T i ∗ δ(1,wl ) (q − 1)Twl w ∗ δ(1,wl ) + q Tκ(si )wl w ∗ δ(1,wl ) if l s κ(wl w) = l κ(wl w) + 1, i = T κ(si )wl w ∗ δ(1,wl ) if l si κ(wl w) = l κ(wl w) − 1 (q − 1)tw + qtκ(si )w if l s κ(wwl ) = l κ(wwl ) + 1, i = tκ(si )w if l si κ(wwl ) = l κ(wwl ) − 1. In the last step we used the following facts: si wl w = wl si w, l(si wl w) = l(si wwl ), and l(wl w) = l(wwl ). On the other hand, δ(1,wl ) commutes with H∞ ; therefore, tw ·
Ti∞
=
( (q − 1)tw + qtwsi twsi
if l(wl wsi ) = l(wl w) + 1, if l(wl wsi ) = l(wl w) − 1.
ALMOST UNRAMIFIED DISCRETE SPECTRUM
249
4. The local module 4.a. Algebraic structure Recall (from [IM]) that the Iwahori-Hecke algebra H has generators Ti , i = 0, 1, . . . , r , with relations Ti2 = q + (q − 1)Ti
for all i ∈ {0, 1, . . . , r }
(4.1)
and, corresponding to each braid relation (2.1), a relation (Ti T j )m = (T j Ti )m
or
(Ti T j )m Ti = (T j Ti )m T j .
(4.2)
For each rational place v of F, we have a canonical algebra isomorphism θv : H → Hv . This isomorphism may be defined by choosing a uniformizer at v to identify F• and Fv . Since the double coset of Iv φπv (w)Iv is independent of the choice of the uniformizer, θv does not depend on this choice. Recall that Mloc = Cc G(F• )\(G(F• ) × G(F• ))/(I• × I• ) , where the left action of G(F• ) is the diagonal action. Let τw = 1G(F• )(wl w,1)(I• ×I• ) .
(4.3)
The space of double cosets used to define Mloc and the double coset space I• \G(F• )/I• , which is used to define the Iwahori-Hecke algebra, are the (I• × I• )orbits of G(F• ) and G(F• )\(G(F• ) × G(F• )), respectively. Since these are isomorphic as right (G(F• ) × G(F• ))-spaces with measure, it is easy to write down the convolutions defining the action: ( τwsi if l(wl wsi ) = l(wl w) + 1, τw · (Ti , 1) = qτwsi + (q − 1)τw if l(wl wsi ) = l(wl w) − 1 and ( τw · (1, Ti ) =
τs i w
if l(si wl w) = l(wl w) + 1,
qτs i w + (q − 1)τw
if l(si wl w) = l(wl w) − 1.
4.b. L 2 -norm on the local module Since we use a similar technique in §5.b, let us recall the (well-known) computation of the norm kτw k = meas(G(F• )(wl w, 1)(I• × I• )) (normalizing so that I• × I• has unit measure). This measure is simply the number of right (I• × I• )-double cosets in the double coset under consideration, which equals the index [I• : wl w I• w−1 wl ].
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AMRITANSHU PRASAD
LEMMA 4.4 Let α be an affine root. Then Uα ⊂ w I• w−1 if and only if w−1 α > 0.
Proof Uα ⊂ w I• w−1 is the same as w−1 Uα w ⊂ I• or Uw−1 α ⊂ I• , which, by Proposition 3.1, is equivalent to w−1 α > 0. As a set, I• = T (F• ) ×
Y
Uα .
α>0
And by Lemma 4.4, Y
I• ∩ w I• w−1 = T (F• ) ×
Uα .
α>0,w−1 α>0
Consequently, the index [I• : I• ∩ w I• w−1 ] = q |α<0 such that w Therefore,
−1 α>0|
= q l(w) .
kτw k = q l(w w) . l
5. Comparison of module structures 5.a. Involutions on H The Iwahori-Matsumoto involution on Iwahori-Hecke algebras is related to Fourier transforms (see [EM]). It is a C-algebra automorphism ι : H → H of order two defined by ι(Tsi ) = −qTs−1 = (q − 1) − Tsi i
for i = 0, . . . , r.
(5.1)
We also define an involution κ : H → H in the following manner. Conjugation by wl is an automorphism of the finite Weyl group W which permutes the set of generators {s1 , . . . , sr }. Let i be such that si = wl si wl . Then κ : Ti 7 → Ti κ : T0 7→ T0 .
for i = 1, . . . , r,
(5.2) (5.3)
Since κ simply permutes the generators, it preserves the relations (4.1). Moreover, κ comes from an automorphism w 7 → wl κ(w)wl of the affine Weyl group. Hence it must preserve the braid relations (4.2), which come from the affine Weyl group. We give a nice description of κ for G of adjoint type in Proposition 6.3.
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251
5.b. L 2 -norm on M Now we compute measures of double cosets in G(F)\G(A)/K via those in G(F)\G(A)/K (normalizing the measure on G(F)\G(A) so that G(F)\G(F)K has unit measure). We use the identification of G(F)\G(A)/K with 0\G(F∞ )/I∞ (see Proposition 2.3). 0 acts transitively on right cosets in 0w I∞ with finite stabilizers. Therefore, the double coset 0w I∞ has measure inversely proportional to the cardinality of this stabilizer. Again, as a set, Y Y 0=A F∩ Ov × Uα . v6 =∞
α<0
Therefore, by Lemma 4.4, |0 ∩ w I∞ w−1 | ∝ q |α<0 such that w
−1 α>0|
= q l(w) .
Therefore, meas(0w I∞ ) =
1 q l(w)
,
and by (3.10), meas(G(F)wK ) = 1/q l(wl w) , so that ktw k =
1 q l(wl w)
.
5.c. The relation between M and Mloc Define a vector space homomorphism I : Mloc → M by I (τw ) = (−q)l(w w) tw . l
(5.4)
5.5 I is an isometry.
PROPOSITION
Proof Indeed,
kI (tw )k2 = k(−q)−l(w w) τw k2 = q −l(w w) = ktw k. l
l
Proof of Theorem 1.2 To prove the theorem we verify, for all w ∈ W˜ and i = 0, . . . , r , the equations I τw · (Ti , 1) = I (τw ) · ι∞ (Ti ), (5.6) I τw · (1, Ti ) = I (τw ) · ι0 ◦ κ(Ti ). (5.7)
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Using the formula in Section 3.c, we have l I (τw ) · ι∞ (Ti ) = (−q)l(w w) tw · (q − 1) − Ti∞ = (q − 1)(−q)l(w w) tw l (−q)l(w w) (q − 1)tw + qtwsi if l(wl ws ) = l(wl w) + 1, i − l w) l(w (−q) twsi l if l(w wsi ) = l(wl w) − 1 l (−q)l(w wsi ) twsi if l(wl ws ) = l(wl w) + 1, i = (q − 1)(−q)l(wl w) tw + q(−q)l(wl wsi ) twsi if l(wl wsi ) = l(wl w) − 1 = I τw · (Ti , 1) . l
This proves (5.6). Also, l I (τw ) · ι0 ◦ κ(Ti ) = (−q)l(w w) tw · (q − 1) − Tκi0 = (q − 1)(−q)l(w w) tw l (−q)l(w w) (q − 1)tw + qts i w if l κ(s )κ(wwl ) = l κ(wwl ) + 1, i − (−q)l(wl w) ts i w if l κ(si )κ(wwl ) = l κ(wwl ) − 1 l (−q)l(w s i w) ts i w if l(s wl w) = l(wl w) + 1, i = l(wl s i w) t l(wl w) t q(−q) w s i w + (q − 1)(−q) if l(si wl w) = l(wl w) − 1 = I τw · (1, Ti ) , l
which proves (5.7). 5.d. The case of a nonsimply connected group So far, we have assumed that G is simply connected. Propositions 5.5 and 1.2 hold with minor modifications when this condition is dropped. As a vector space, a local Hecke algebra Hv has a basis indexed by the extended affine Weyl group W˜ , where W˜ = Wa o , where is the quotient of the lattice of cocharacters by the coweight
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lattice. Each ρ ∈ acts on {s0 , . . . , sr } by a permutation si 7→ sρ(i) . For ρ ∈ , let Tρ = 1 Iv ρ Iv . Then W˜ is generated by Ti , i = 0, . . . , r , and Tρ . Since ρ normalizes Iv , it is easy to compute convolutions with Tρ : Tρ · Ti · Tρ−1 = Tρ(i) . The modules M and M also have basis vectors indexed by elements of W˜ . For w ∈ W˜ , let tw = 1G(F)wK . Once again, since ρ normalizes I , it is easy to see that tw · Tρ∞ · Tρ00 = tρ 0−1 wρ . Since acts on Wa by length-preserving automorphisms, the length function can be extended uniquely to W˜ by requiring that l(wρ) = l(w) for all w ∈ Wa and all ρ ∈ . Mloc is defined exactly as before; its basis is enlarged to be indexed by W˜ instead of Wa . The action of Tρ ∈ H is given by τw · (Tρ , Tρ 0 ) = τρ 0−1 wρ . We may also extend ι and I by ι(Tρ ) = Tρ
and
I (τwρ ) = (−1)l(w w) twρ l
for w ∈ Wa and ρ ∈ . With these modifications to our conventions, Propositions 5.5 and 1.2 continue to hold.
6. Decomposition of the modules 2 6.a. Decomposition of Mloc 2 Given a central character χ of H corresponding to s ∈ Tˇ , let Mloc,s denote that part 2 −1 of Mloc on which the center of H ⊗ H acts by χ ⊗ χ , where χ is the central character of H corresponding to s. Let h 7 → h op be the anti-involution of H which op fixes the generators; Ti = Ti . Given a finite-dimensional right H -module V , define the contragredient to be the vector space V˜ = HomC (V, C), with a right H -action given by hξ · h, vi = hξ, v · h op i for all ξ ∈ V˜ , v ∈ V, h ∈ H .
We call the closed (H ⊗ H )-submodule generated by the irreducible ones the dis2 . crete part of Mloc
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PROPOSITION 6.1 2 ’s span the discrete part of M 2 , and there is an isomorphism The Mloc,s loc
8:
M
2 V ⊗ V˜ → Mloc,s ,
where the direct sum is taken over all irreducible square-integrable right modules V of H with central character χ . (This set is finite (see [KL]).) Proof Define 8 as the matrix coefficient 8(v, ξ )(g1 , g2 ) = hξ g2−1 , vg1−1 i; then
8(v, ξ )(gg1 , gg2 ) = ξ(gg2 )−1 , v(gg1 )−1 = hξ, vg2−1 g −1 gg1 i = hξ, vg2−1 g1 i = 8(v, ξ )(g1 , g2 ). Therefore, 8(x, ξ ) is invariant under the diagonal action of G(Fv ) on G(Fv )×G(Fv ). Since V consists of Iwahori-invariant vectors, 8(v, ξ ) is invariant under Iwahori subgroups in each variable. Moreover, if V has central character χ , then the image of 2 . V ⊗ V˜ is contained in Mloc,s 2 , the center of On each irreducible (H ⊗ H )-representation occurring in Mloc H ⊗ {1} acts by scalars. Let us assume that H ⊗ {1} acts on f by a central character χ. We show that f lies in the image of 8. Consider the right (H ⊗ {1})-module V f = f · (H ⊗ {1}) generated by f . 6.2 V f is a finite-dimensional C-vector space.
CLAIM
Proof H is a finitely generated module over its center, Z (see [Be]). Let h 1 , . . . , h k generP ate H over Z . Let h = ai h i , ai ∈ Z . Since Z acts on f by χ, X X X h· f = ai h i · f = h i ai · f = h i · χ (h i ) f . Therefore, V f is generated by h i · f as a C-vector space. This proves the claim. H is a C∗ -algebra under h ∗ (x) = h(x −1 ). V f inherits an inner product (·, ·) from Mloc . V f is a unitary representation of H in the sense that
for f 1 , f 2 ∈ V f and h ∈ H ,
( f 1 · h, f 2 ) = ( f 1 , f 2 · h ∗ ).
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⊕m As a consequence, V f is semisimple. Therefore, we may assume that V f ∼ = V1 1 ⊕ · · · ⊕ Vn⊕m n , where each Vi is irreducible and nonzero, and the Vi ’s are pairwise nonisomorphic. Clearly, Z acts on each Vi by χ. Let V f = V1 ⊕ · · · ⊕ Vn . The natural map Vi⊕m i → Vi defined by addition induces a map p : V f → V f . Similarly, we ff → Vff . This map preserves matrix coefficients have a map on contragredients p˜ : V
hv, v˜ i = p(v), p(˜ ˜ v)
ff . for all v ∈ V f , v˜ ∈ V
ff be the vector defined by φ(m) = m(1) for m ∈ V f . Then hφ · g −1 , p( f ) · Let φ ∈ V 2 g1−1 i = f (g1 , g2 ). Therefore, f lies in the image of 8. From the above proof, we may also conclude that the center of H ⊗ H acts on f by χ ⊗ χ −1 . We have shown 2 ’s span the discrete part of M 2 and that 8 is surjective. that the Mloc,s loc To see that 8 is injective, observe that it is an (H × H )-module homomorphism. The V ⊗ V˜ ’s are irreducible and pairwise nonisomorphic modules. Therefore, the kernel must consist of a direct sum of a subset of the constituent modules. But no nontrivial representation could lie in the kernel. Therefore, the morphism must be injective. 6.b. Decomposition of M 2 Applying Theorem 1.2 to the above decomposition gives us the following commutative diagram (we must also keep track of central characters; ι takes s to s −1 ): 2 Mloc,s ×H ⊗H I ×ι∞ ×ι0 ◦κ
Ms2−1
× H∞ ⊗ H0
/ M2 o loc,s
L
V ⊗ V˜
I
/ M 2−1 o s
L
V ι∞ ⊗ V˜ ι0 ◦κ
Here the sums in the right-hand column are over the set of isomorphism classes of irreducible square-integrable right H -modules V . This proves Theorem 1.3. 6.c. Groups of adjoint type When the derived group of G is of adjoint type, then the involution κ has a simple interpretation. PROPOSITION 6.3 Suppose that G is of adjoint type and that V is an irreducible H -module. Then V κ is isomorphic to V˜ .
Proof Let R be the commutative subring of H which coincides with the algebra generated
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by the characteristic functions of the double cosets of elements in T which are dominant and their inverses in H . R is isomorphic to the group ring C[X ∗ (T )]. By [EM, Theorem 5.5], the isomorphism class of an irreducible finite-dimensional H -module is determined by the weights of R on it. The weights of R on V coincide with the weights of R on the normalized Jacquet module VN , where N ⊂ B is the maximal unipotent subgroup (see [C, Section 3]). By [Bor, Lemma 4.7], it suffices to show that the weights of (V κ ) N are the same as those of (V˜ ) N . By [C, Corollary 4.2.5], (V˜ ) N is isomorphic to the contragredient of VN , where N is the maximal unipotent subgroup of B. But µ is a weight of VN if and only if wl (µ) is a weight of VN . Therefore, the weights of (V˜ ) N are of the form −wl (µ). However, κ induces the involution µ 7 → −wl (µ) on X ∗ (T ). Finally, we may deduce Theorem 1.5 from Theorem 1.3 by using Proposition 6.3 to replace V˜ ι0 ◦κ by V ι0 and by using Theorem 1.4 to get a parameterization of irreducible square-integrable representations of H with a given central character. Acknowledgments. We thank Robert Kottwitz, who suggested the basic approach that we use here and was extremely helpful at every stage of its execution. We are grateful to Stephen DeBacker, Sam Evens, and Dipendra Prasad for many useful ideas and suggestions. References [A]
P. ANSPACH, Unramified discrete spectrum of PSp4 , Ph.D. dissertation, University of
[Be]
J. N. BERNSTEIN, “Le ‘centre’ de Bernstein” in Representations of Reductive Groups
Chicago, 1995. 237, 241
[Bor]
[Bou] [C]
[E] [EM]
over a Local Field, ed. P. Deligne, Travaux en Cours, Hermann, Paris, 1984, 1 – 32. MR 86e:22028 254 A. BOREL, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 233 – 259. MR 56:3196 256 ´ ements de math´ematique, fasc. 34: Groupes et alg`ebres de Lie, N. BOURBAKI, El´ chapitres 4 – 6, Actualit´es Sci. Indust. 1337, Hermann, Paris, 1968. 246 W. CASSELMAN, Introduction to the theory of admissible representations of p-adic reductive groups, unpublished notes, 1974 – 1993, http://www.math.ubc.ca/˜cass/research.html 256 I. EFRAT, Automorphic spectra on the tree of PGL2 , Enseign. Math. (2) 37 (1991), 31 – 43. MR 92e:11054 237 S. EVENS and I. MIRKOVIC´ , Fourier transform and the Iwahori-Matsumoto involution, Duke Math. J. 86 (1997), 435 – 464. MR 98m:22022 241, 250, 256
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[IM]
257
N. IWAHORI and H. MATSUMOTO, On some Bruhat decomposition and the structure of
[KL]
D.
[L]
G.
[M1]
C.
[M2]
´ the Hecke rings of p-adic Chevalley groups, Inst. Hautes Etudes Sci. Publ. Math. 25 (1965), 5 – 48. MR 32:2486 239, 249 KAZHDAN and G. LUSZTIG, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), 153 – 215. MR 88d:11121 240, 254 LUSZTIG, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599 – 635. MR 90e:16049 239 MOEGLIN, Orbites unipotentes et spectre discret non ramifi´e: Le cas des groupes classiques d´eploy´es, Compositio Math. 77 (1991), 1 – 54. MR 92d:11054 237 , Repr´esentations unipotentes et formes automorphes de carr´e int´egrable, Forum Math. 6 (1994), 651 – 744. MR 95k:22024 238 ´ MOEGLIN and J.-L. WALDSPURGER, Le spectre r´esiduel de GL(n), Ann. Sci. Ecole
[MW]
C.
[P]
A. PRASAD, Almost unramified discrete spectrum of split reductive groups over a
[R]
rational function field, Ph.D. dissertation, University of Chicago, 2001, http://aprasad.bravepages.com 238 M. REEDER, Isogenies of Hecke algebras and a Langlands correspondence for ramified principal series representations, preprint, 2000, http://www2.bc.edu/˜reederma/papers.html 240
Norm. Sup. (4) 22 (1989), 605 – 674. MR 91b:22028 237
Centre de recherches math´ematiques, Universit´e de Montr´eal, Box 6128, Centre-ville Station, Montr´eal, Quebec H3C 3J7, Canada;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 2,
ACCRETIVE SYSTEM T b-THEOREMS ON NONHOMOGENEOUS SPACES F. NAZAROV, S. TREIL, and A. VOLBERG
Abstract We prove that the existence of an accretive system in the sense of M. Christ is equivalent to the boundedness of a Calder´on-Zygmund operator on L 2 (µ). We do not assume any kind of doubling condition on the measure µ, so we are in the nonhomogeneous situation. Another interesting difference from the theorem of Christ is that we allow the operator to send the functions of our accretive system into the space bounded mean oscillation (BMO) rather than L ∞ . Thus we answer positively a question of Christ as to whether the L ∞ -assumption can be replaced by a BMO assumption. We believe that nonhomogeneous analysis is useful in many questions at the junction of analysis and geometry. In fact, it allows one to get rid of all superfluous regularity conditions for rectifiable sets. The nonhomogeneous accretive system theorem represents a flexible tool for dealing with Calder´on-Zygmund operators with respect to very bad measures. 0. Introduction: Main objects and results In what follows, the symbol K (x, y) stands for a Calder´on-Zygmund kernel defined for x, y ∈ Rn of order d: |K (x, y)| ≤
C1 , |x − y|d
|K (x1 , y) − K (x2 , y)| + |K (y, x1 ) − K (y, x2 )| ≤
C2 |x1 − x2 |α |x1 − y|d+α
for a positive α and any points x1 , x2 , y satisfying |x1 − x2 | ≤ (1/2)|x1 − y|. We always assume that µ is adapted to K via the Ahlfors condition: µ B(x, r ) ≤ C3r d . However, we assume no estimate from below. DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 2, Received 25 May 2000. Revision received 9 July 2001. 2000 Mathematics Subject Classification. Primary 47B37; Secondary 30E20. Authors’ work supported by National Science Foundation grant number DMS 9970395 and United States–Israel Binational Science Foundation grant number 00030. 259
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By a Calder´on-Zygmund operator with kernel K we mean any bounded operator on L 2 (µ) such that Z T f (x) =
K (x, y) f (y) dµ(y)
(0.1)
for all x outside the support of f dµ. Sometimes we use the phrase an operator with Calder´on-Zygmund kernel. By that we mean any operator with this property for f ’s from its domain of definition (which can consist of, say, all bounded functions with compact support, or all smooth functions, or anything like that). Usually we are dealing with boundedness of an operator with Calder´on-Zygmund kernel on L 2 (µ) (then it is bounded on all L p (µ), 1 < p < ∞; see [24]). Very often the kernel K is continuous (and the measure has a compact support); thus (0.1) makes sense for any x and defines a Calder´on-Zygmund operator. But we are not interested in this obvious L 2 -boundedness. Our goal is to obtain the bound that depends only on C1 , C2 , C3 , α from the definition of K and maybe on some other parameters, but not on the norm of K as a continuous function. Notice also that kernel K does not define T uniquely. In fact, we have the following. 0.1 Let T1 , T2 be Calder´on-Zygmund operators with kernel K . Then T1 f − T2 f = m · f , where m is a bounded µ measurable function. THEOREM
One can find an easy proof in [27, Chap. 1]. The next definition differs essentially from the classical one. Given a 3 > 1, we introduce the space BMO23 , Z n o 1 | f − h f i Q |2 dµ < ∞ , BMO23 (µ) := f ∈ L 2loc (µ) : k f k2∗ := sup Q µ(3Q) Q R where Q stands for an arbitrary cube in Rn . Here h f i Q := (1/µ(Q)) Q f dµ. We remark that the space BMO23 (µ) depends essentially on 3 (see [25]). However, in what follows, any 3 > 1 works. These doubling BMO spaces turn out to be natural when one works with nondoubling measures. They are obviously larger than the usual BMO space. The main players in what follows are accretive systems. A system of functions {b S } is called an accretive system if there exists δ > 0 such that for any cube S in Rn
ACCRETIVE SYSTEMS ON NONHOMOGENEOUS SPACES
261
there exists b S from the system having the properties kb S k∞ ≤ 1, Z b S dµ ≥ δ · µ(S). S
As always, T is a Calder´on-Zygmund operator. We denote the set of Calder´onZygmund parameters of the kernel by z. To formulate our results, let us introduce three sets of assumptions on T and an accretive system. Each set of assumptions is weaker than the previous one. By an almost cube, we mean a rectangle in Rn with sides parallel to the axis and with ratio of any two sides between 1/2 and 2. In all three sets of assumptions, we are talking about the action of T on bounded functions. In the first two sets of assumptions, T acts on bounded functions with compact support. In the third set of assumptions, we are working with T b, where b is, generally, a bounded function but not with compact support. In this case, the meaning of T b is not readily clear even if T is assumed to be bounded in L 2 (µ). How to interpret condition T b ∈ BMO23 . To interpret this condition in our situation, we assume only that T maps bounded functions with compact support into L 2 (µ). In particular, this is so if T is a priori bounded in L 2 (µ). Fix a cube Q. We define T b as a functional on L 20 (Q, µ), where L 20 (Q, µ) := R { f ∈ L 2 (Q, µ) : Q f dµ = 0}. Fix a function a from L 20 (Q, µ), and let R = 3Q with 3 > 1. Let b1 := bχ R , b2 := b − b1 . We have (T b)(a) := (T b, a) := (T b1 , a) L 2 (µ) + (b2 , T ∗ a).
(0.2)
Notice that this definition does not depend on 3 > 1. Notice also that the last term is an absolutely convergent integral: Z Z (b2 , T ∗ a) = [K (x, y) − K (x0 , y)]a(x)b(y) dµ(x) dµ(y). (0.3) Q
Rn \R
Rn \R [K (x, y) − K (x 0 , y)]b(y) dµ(y). 2 BMO3 if the thus-defined functional satisfies
We introduce T ! bχRn \R := We say that T b ∈
R
|(T b)(a)| ≤ Cµ(3Q)1/2 kak L 2 (µ) with the same finite C for all Q. The absolute convergence of the last integral is standard (see, e.g., [25, Lem. 2.1]. Here are our three sets of assumptions.
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NAZAROV, TREIL, and VOLBERG
L ∞ -system of accretive functions supported on almost cubes. For every almost cube Q, there is a function b Q with the following properties: supp b Q ⊂ Q, 1 Z b Q dµ ≥ δ, |hb Q i Q | := µ(Q) Q kb Q k∞ ≤ 1, kT (b Q )k∞ ≤ B. BMO23 -system of accretive functions supported on almost cubes. We have the following: supp b Q ⊂ Q, 1 Z |hb Q i Q | := b Q dµ ≥ δ, µ(Q) Q kb Q k∞ ≤ 1, kT (b Q )kBMO2 ≤ B. 3
BMO23 -system of accretive functions assigned to almost cubes. We call the attention of the reader to the fact that in this set of assumptions we do not assume that b Q are supported on Q: 1 Z |hb Q i Q | := b Q dµ ≥ δ, µ(Q) Q kb Q k∞ ≤ 1, kT (b Q )kBMO2 ≤ B. 3
Our main results are the following. THEOREM 0.2 Let T be a Calder´on-Zygmund operator with Calder´on-Zygmund parameters z and norm kT k. Then there exists an L ∞ -system of accretive functions supported on almost cubes {b S } such that kT b S k∞ ≤ B(z, kT k) < ∞ and such that the constant of accretivity δ ≥ c(z, kT k) > 0.
Conversely (notice that we need accretive functions only on cubes rather than on almost cubes), we have the following.
ACCRETIVE SYSTEMS ON NONHOMOGENEOUS SPACES
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THEOREM 0.3 Let T be a Calder´on-Zygmund operator with Calder´on-Zygmund parameters z and norm kT k. Suppose that for T there exists an L ∞ -system of accretive functions supported on cubes {b1S }. Suppose that for T ∗ there exists an L ∞ -system of accretive functions supported on cubes {b2S }. Let B := sup{kT b1S k∞ , kT ∗ b2S k∞ }. Then kT k ≤ A(z, δ, B) < ∞.
These results are the generalization of the results of Christ [1] to nonhomogeneous spaces. Theorem 0.4 allows one to replace an L ∞ -assumption with a BMO assumption, answering a question of Christ which seemed to be open even in the homogeneous situation. THEOREM 0.4 Let T be a Calder´on-Zygmund operator with Calder´on-Zygmund parameters z and norm kT k. Suppose also that T has an antisymmetric kernel. Suppose that there exists a BMO23 -system of accretive functions assigned to cubes {b S } such that B := sup kT b S kBMO2 . Then kT k ≤ A(z, δ, B) < ∞. 3
The “classical” T b-theorem for Calder´on-Zygmund operators with antisymmetric kernels on nonhomogeneous spaces (see [25]) corresponds to the case b S = b with b being weakly accretive. Notice that in all the theorems above there is no assumption of weak boundedness. But weak boundedness is usually essential in the theory. The explanation is simple: in Theorem 0.4 antisymmetry is a sort of weak boundedness. In Theorem 0.3 the existence of a local L ∞ accretive system turns out to be such a powerful assumption that it allows us to forget about weak boundedness. This effect was found earlier by Christ in [1] for the homogeneous situation. In general, nonhomogeneous harmonic analysis has received considerable attention recently. We refer the reader to [5], [6], [22]–[25], and [29]–[31].∗ Some applications One of the corollaries of Theorems 0.2 and 0.3 is the following result, in which we use the notions of analytic capacity γ and the Cauchy integral operator. Recall that for a measure µ on the complex plane C (the canonical value of) the Cauchy integral ∗ While
preparing the galley proofs, the authors found out about the preprint of P. Auscher, S. Hoffman, C. Muscalu, T. Tao, and C. Thiele, “Carleson measures, extrapolation, and T (b) theorems,” which contains close results about the local T (b) theorems (but in a homogeneous setting) along with interesting connections with the theory of phase space analysis from the point of view of wave packets on tiles.
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operator Tµ is defined by its bilinear form: Z Z ϕ(z)ψ(ζ ) − ϕ(ζ )ψ(z) (Tµ ϕ, ψ) := dµ(ζ ) dµ(z) ζ −z for ϕ, ψ ∈ C0∞ . The analytic capacity γ of a compact subset K of the plane is γ (K ) := sup
lim |z f (z)| : f is bounded and analytic in C \ K ,
z→∞
k f k∞ ≤ 1, f (∞) = 0 . By the symbol H 1 we denote the 1-Hausdorff measure. It is well known that γ (K ) ≤ H 1 (K ). THEOREM 0.5 Let E be a compact subset of C, 0 < H 1 (E) < ∞. Then the Cauchy integral operator T with respect to H 1 |E is bounded on L 2 (H 1 |E) if and only if there exists δ > 0 such that δ · H 1 (S ∩ E) ≤ γ (S ∩ E)
for any square S. This result has been proved by T. Murai [21] under the extra assumption that E is Ahlfors regular, namely, when there exist 0 < c1 , c2 < ∞ such that c1l(S) ≤ H 1 (S ∩ E) ≤ c2l(S) for every square S centered on E, where l(S) denotes the length of the side of S. The meaning of Theorem 0.5 is that the Cauchy integral operator acts boundedly on L 2 with respect to H 1 |E if and only if the analytic capacity on portions of E is equivalent to the measure. One can deduce from Theorem 0.5 another proof of G. David’s celebrated characterization of Ahlfors-David curves (originally obtained in [4]). We refer the reader to the preprint [26], where this new proof is explained together with the above mentioned generalization of Murai’s theorem. The proof has a sinful feature: there is absolutely no geometry involved in it. But sometimes this may become useful. The proof also illustrates that the assumptions of our accretive system theorems are actually verifiable in practice. Finally, let us remark that we think that accretive system T b-theorems are the most flexible tools for dealing with Calder´on-Zygmund operators.
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Plan In Section 1 we list the basic results on Calder´on-Zygmund operators on nonhomogeneous spaces. We use them in what follows. Then we prove Theorem 0.2 in Section 2. In Section 3 we start to prove Theorems 0.3 and 0.4. Section 7 finishes the proofs. In the appendix we prove some technicalities needed in the estimate of the diagonal part of T . Sections 4 and 5 are the most difficult and involved. They use the strictly technical results from the appendix (surgery by means of accretive systems). 1. Basic facts on Calder´on-Zygmund operators on nonhomogeneous spaces We use the notation T ∈ C Z(K ) to say that T is a Calder´on-Zygmund operator with kernel K . We also need the notion of the cut-off of T : Z Tε f (x) := K (x, y) f (y) dµ(y), y:|y−x|≥ε Z |x − y| f (y) dµ(y), Tεψ f (x) := K (x, y)ψ ε where ψ is a C ∞ function, which vanishes on B(0, 1/2) and equals 1 on Rn \ B(0, 1). ψ Notice that Tε are operators with Calder´on-Zygmund kernels, while Tε are not. The right maximal operator is now Z 1 e f (x) := sup M | f (y)| dµ(y). r µ(B(x, 3r )) B(x,r ) Obviously,
(Tε − T ψ )( f )(x) ≤ A M e f (x). ε
We also introduce the singular maximal function T ] f (x) := sup |Tε f (x)|. ε
The next theorem is important in what follows (see [24] for the proof). 1.1 ∈ C Z(K ), then any of the following equivalent assertions hold: {Tε }ε>0 are uniformly bounded in L 2 (µ), ψ {Tε }ε>0 are uniformly bounded in L 2 (µ), T ] is bounded in L 2 (µ).
THEOREM
If T (1) (2) (3)
Let M0 (Rn ) stand for all complex measures with compact support in Rn . 1.2 If T ∈ C Z(K ), then THEOREM
(1.1)
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(1) (2)
T is a bounded operator from L 1 (µ) to L 1,∞ (µ); if, in addition, the kernel K is continuous, then T is a bounded operator from M0 (Rn ) to L 1,∞ (µ) and its norm depends only on the Calder´on-Zygmund parameters z and the norm of T in L 2 (µ).
The next lemma is well known and widely used (see [1]). LEMMA 1.3 Let X be a compact Hausdorff space, and let T be a bounded linear operator from M(X ) to C(X ), where M(X ) is the space of complex measures on X . Also, assume that the adjoint operator T ∗ acts from M(X ) to C(X ), and assume that for a finite positive measure µ the following holds:
µ{x ∈ X : |T ∗ ν(x)| > t} ≤
Akνk , t
∀ν ∈ M(X ).
(1.2)
Then for any Borel set E ⊂ X , 0 < µ(E), there exists a function h on E, 0 ≤ h ≤ 1, such that Z µ(E) , (1.3) h dµ ≥ 2 E kT (h dµ)k∞ < 4 A.
(1.4)
THEOREM 1.4 Let T ∈ C Z(K ) with kT k denoting its norm as an operator in L 2 (µ). Let δ ∈ (0, 1). Then there exist constants C1 , C2 depending on δ and kT k only such that
e f |δ )1/δ (x) + C2 ( M e f )(x), T ] f (x) ≤ C1 ( M|T
∀ f ∈ L 1 (µ).
(1.5)
This is called Cotlar’s inequality. Notice that |T f |δ is summable by Theorem 1.2. This makes the right-hand side almost everywhere finite. 2. The proof of Theorem 0.2 We prove a slightly more general result. Our operator is assumed to be bounded in L 2 (E, dµ). (Here E denotes just the support of µ.) Then its norm kT k bounds it as an operator in L 2 (E 0 , µ) for any Borel subset E 0 of E. Thus Theorem 0.2 follows from the next theorem. 2.1 Let µ be supported by a compact set E, and let an operator T with Calder´onZygmund kernel be bounded in L 2 (µ). Then there exist δ > 0 and B < ∞ which THEOREM
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267
depend only on kT k, and a function b such that kbk∞ ≤ 1, Z
(2.1)
b dµ ≥ δ µ(E),
(2.2)
E
kT (b dµ)k∞ ≤ B.
(2.3)
Proof We deduce from Theorems 1.1 and 1.2 that (Tεψ )] : M(E) → L 1,∞ (µ)
(2.4)
uniformly in ε. ψ Obviously, Tε satisfies all the assumptions of Lemma 1.3. Applying Lemma 1.3, we obtain the family of functions {h ε } having the following properties: 0 ≤ h ε ≤ 1, Z
h ε dµ ≥ δµ(E), E
kTεψ (h ε dµ)k∞ ≤ B. Let us consider a decreasing sequence εn → 0 such that h εn → h weakly in L 2 (µ) and Tεn → T0 in the weak operator topology. (We can do that because the Tε are uniformly bounded operators from a separable to a reflexive space.) It is clear that T0 ∈ C Z(K ). Let h n := h εn , Tn := Tεn . LEMMA 2.2 The functions Tn h m , m ≥ n, are uniformly bounded.
Proof ψ We know that kTε (h ε dµ)k∞ ≤ B. Using (1.1) and the uniform boundedness of h m , we conclude that kTm h m k ≤ B1 . Similarly to (1.1), e f (x). |Tδ f (x) − T2δ f (x)| ≤ A M ψ
Thus if εn ∈ [εm , 2εm ], then kTn h m k∞ ≤ B2 . If εn > 2εm , then (Tεm )εn = Tεn . Thus in this case, kTn h m k∞ = k(Tεψm )εn h εm k∞ ≤ k(Tεψm )] h εm k∞ .
(2.5)
By Cotlar’s inequality (see Th. 1.4), we get e εψ h εm |δ )1/δ (x) + C2 ( Mh e εm )(x). (Tεψm )] h εm (x) ≤ C1 ( M|T m
(2.6)
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NAZAROV, TREIL, and VOLBERG ψ
But both h εm and Tεm h εm are uniformly bounded. Thus (2.5) and (2.6) give us, finally, the uniform boundedness of Tn h m for all m ≥ n. P mn To continue the proof of Theorem 2.1, let us choose {akn }k=0 , akn ≥ 0, k akn = 1, Pm n n 2 in such a way that gn := j=0 a j h n+ j converges in L (µ)-norm to an h. Consider Tn gn . Then mn X kTn gn k∞ ≤ a nj sup kTn h m k∞ ≤ A < ∞. m≥n
j=0
In particular, let n j be a sequence such that weakly in L 2 (µ).
Tn j gn j → f
(2.7)
Clearly, f is a bounded function and k f k∞ ≤ A := lim sup kTn gn k∞ < ∞.
(2.8)
n→∞
Let us prove that T h is bounded. This finishes the proof of the theorem because Z
Z h dµ = lim E
n
gn dµ = lim E
n
mn X j=0
a nj
Z
h n+ j dµ ≥ δµ(E).
(2.9)
E
To show that T h is bounded, we first show that T0 h = f . In fact, T0 h = T0 h − Tn j gn j + Tn j gn j = Tn j (h − gn j ) + (T0 − Tn j ) h + Tn j gn j . The first term tends to zero in L 2 (µ) because the Tn j are uniformly bounded and gn j → h in L 2 (µ). The second term tends weakly to zero because we have chosen Tn converging in the weak operator topology to T0 . By (2.7), the third term tends weakly to f . So T0 h = f . But T and T0 are both from C Z(K ), and they are both bounded in L 2 (µ). Thus T h − T0 h = m h, where m ∈ L ∞ (µ). Thus T h = m h + f , and so it is a bounded function. We also guaranteed (2.9). The construction implies that khk∞ ≤ 1, obviously. Theorem 2.1 is completely proved. 3. Decomposition of L 2 (µ) with respect to an accretive system We are starting to prove Theorems 0.3 and 0.4. The idea is quite natural. We fix dyadic lattices D1 = {Q}, D2 = {R}. We decompose the functions f, g from L 2 (µ)
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269
into pieces {1 Q f }, {1 R g}, and we want X k1 Q f k2L 2 (µ) ≤ C1 k f k2L 2 (µ) ,
(3.1)
Q
X
k1 R gk2L 2 (µ) ≤ C2 kgk2L 2 (µ) ,
(3.2)
k(1 Q )∗ f k2L 2 (µ) ≤ C1 k f k2L 2 (µ) ,
(3.3)
R
X Q
X
k(1 R )∗ gk2L 2 (µ) ≤ C2 kgk2L 2 (µ) .
(3.4)
R
Then we estimate carefully the “matrix elements” (T 1 Q f, 1 R g)/ k1 Q f k2L 2 (µ) k1 R gk2L 2 (µ) , and we prove that the corresponding infinite matrix
maps l 2 to itself. There is a very subtle point when we see that some “matrix elements” cannot be well estimated. Exactly as in [23] and [25], we cope with this difficulty by using probability: “bad” Q’s and R’s are rare and can be discarded by choosing lattices D1 = {Q}, D2 = {R} randomly. For convenience we work only with compactly supported µ. Let D = {Q} be a lattice of dyadic cubes in Rn . We are given an accretive system {b Q } Q∈D . We may think that E = supp µ ⊂ Q 0 , Q 0 ∈ D , and that Z b Q 0 dµ ≥ δ µ(Q 0 ), kb Q 0 k∞ ≤ 1. Q0
Notice that our dyadic lattice D is going to be random. So Q 0 is also random. But this randomness is (for Q 0 at least) in shifting a bit. So if we fix a big cube such that supp µ is deep inside it, we can shuffle the cube a little randomly, and supp µ is still inside. This large cube is called Q 0 . j j j Consider cubes {Q 1 }, Q 1 ∈ D , Q 1 ⊂ Q 0 , and such that these are maximal cubes satisfying Z j j b Q 0 dµ < δ 2 µ(Q 1 ). Q1
Then Z δ µ(Q 0 ) ≤
Q0
Z b Q 0 dµ ≤
S j Q0\ Q1
X Z b Q 0 dµ + j≥1
j
Q1
b Q 0 dµ .
S j The first term is at most µ(Q 0 ) − µ Q 1 . The second term is at most S j S j 2 Q 1 . We obtain δ µ Q 1 . So the right-hand side is at most µ(Q 0 ) − (1 − δ 2 )µ the following estimate for stopping cubes: [ 1−δ j µ(Q 0 ) ≤ (1 − τ )µ(Q 0 ), (3.5) µ Q1 ≤ 1 − δ2
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NAZAROV, TREIL, and VOLBERG
where τ is positive and depends only on δ. j One cube Q 0 forms the family D 0 of cubes of the zero layer; cubes {Q 1 } form the 0 family of cubes of the first layer. We assign to each Q ∈ D (only Q 0 is such) {b Q } and to each Q in D 1 the function b Q from our accretive system. We call corresponding families of functions the system for the zero layer, and we denote it by the symbol B 0 ; we denote the system for the first layer, B 1 , correspondingly. We want to proceed with the constructions of D k , B k . And this is straightforward. Let D k−1 , B k−1 be already constructed. Fix Q ∈ D k , and let {Q j } be maximal cubes from D , Q j ⊂ Q, such that Z b Q dµ < δ 2 µ(Q j ). Qj
We collect all cubes Q j for all cubes Q ∈ D k−1 and call them D k . The corresponding functions from our accretive system form the B k —the system for the kth layer. Obviously, similarly to (3.5), [ ∀m, ∀Q ∈ D m , µ q ≤ (1 − τ )µ(Q). (3.6) q∈D m+1 ,q⊂Q
Let us now construct 1 Q f , which we mentioned above. Let us consider an arbitrary cube Q and the dyadic grid of its dyadic (with respect to Q) subcubes of sides that are half the sides of Q. Let us call a certain collection T of those subcubes the collection of terminal cubes. Let us also be given a bounded function h ∈ L ∞ (µ) such that Z h dµ ≥ δ µ(Q). (3.7) Q
prep
Now we first introduce a preparatory 1 Q f . This function is defined to be zero outside of Q, and on Q we define it on each son S of Q by the following formulae: prep
1Q
f :=
h f i
S
hhi S
−
h f iQ h hhi Q
and prep
1Q
f := f −
on S if S is a son of Q and is not from T
h f iQ h hhi Q
on S if it is a son of Q S ∈ T .
(3.8)
(3.9)
Notice that the collection of terminal cubes T may very well be empty. To determine 1 Q f , let us suppose that for a terminal cube T from T we are given a function h T which is bounded and which satisfies the accretivity condition Z h T dµ ≥ δ µ(T ). T
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271
Very often (but not always) in what follows, the functions h T are just equal to h. But prep in any case we change 1 Q f to 1 Q f on T according to the following rule: 1 Q f := 1hQ f :=
h f iQ h f iT hT − h. hhiT hhi Q
(3.10)
We make this change for every terminal T for which we are given an accretive h T . prep We do not change 1 Q f elsewhere. The resulting function on Q is called 1 Q f := 1hQ f . Notice that always Z 1 Q f dµ = 0, which means that (1 Q )∗ (1) = 0. (3.11) Q
Suppose that Q does not have any terminal sons. Let h be an accretive function with the help of which 1 Q = 1hQ was constructed. Then Z
(1 Q f )∗ h dµ = 0,
which means that 1 Q (h) = 0.
(3.12)
Q
Notice that if all h T are equal to h (e.g., if we do not have any terminal sons of Q), then 1 Q is the projection; that is, (1hQ )2 = 1hQ .
(3.13)
More generally, when we have new h T on all terminal sons T of Q, we have 1 Q f − (1 Q )2 f =
X T is terminal son of Q
o h f i Q n hhiT h T − h χT . hhi Q hh T iT
(3.14)
Definition P We have ω Q f := T is terminal son of Q (h f i Q /hhi Q ) {(hhiT /hh T iT ) h T − h} χT . Clearly, ω Q f is orthogonal to constants, and |ω Q f | ≤ C(δ)h| f |i Q if h is δ-accretive on Q and the h T are δ-accretive on the T ’s. Now the decomposition of f ∈ L 2 (µ) can be performed as follows. We are given D j for j = 0, 1, 2, . . . , and we are given the corresponding systems B j , j = 0, 1, 2, . . . , of accretive functions b Q , Q ∈ B j . Starting with Q 0 , we take h = b Q 0 and construct 1 Q 0 by declaring the cubes from D 1 terminal. It might very well happen that no son of Q 0 is terminal. Then we are able to use the same h = b Q 0 for the sons of Q 0 . If no grandson is terminal, we do this again, and so on. When at a certain stage we meet Q ∈ D 1 , we use its accretive function b Q as h Q to construct the corresponding 1. After this is done, all 1 Q f are constructed
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NAZAROV, TREIL, and VOLBERG
except for Q ∈ D 1 . To construct 1 Q f for Q ∈ D 1 , we just repeat the procedure with h = b Q , b Q ∈ B 1 . Then all 1 Q f are constructed except for Q ∈ D 2 . To construct 1 Q f for Q ∈ D 2 , we just repeat the procedure with h = b Q , b Q ∈ B 2 . We continue, and 1 Q f is eventually constructed for every Q ∈ D , Q ⊂ Q 0 . Also, denote h f iQ0 E Q 0 f := bQ . hb Q 0 i Q 0 0 We would like to see that f = E Q0 f +
X
1Q f
(3.15)
Q∈D
in L 2 (µ). Let us call a collection of finitely many subcubes of Q 0 from D a nest N if the fact that Q ∈ N implies that the father of Q belongs to the nest. First notice that by construction we have for any bounded f ,
X
1 Q f ∞ ≤ A δ −1 k f k L ∞ . (3.16)
E Q0 f + L
Q∈N
P Also, the partial sums of E Q 0 f + Q∈D 1 Q f converge to f , µ-a.e. This follows from (3.6) and from the construction of 1 Q f . Thus we have L 2 (µ)-convergence as in (3.15) for bounded functions. But the bounded functions are dense in L 2 (µ). Thus, to prove the convergence in L 2 (µ) mentioned in (3.15), it is enough to prove the following. LEMMA
We have
3.1 P
Q∈D
k1 Q f k2L 2 (µ) ≤ Ck f k2L 2 (µ) .
Proof We need to estimate
XZ S
|1 Q f |2 dµ,
(3.17)
S
where S ∈ D , S ⊂ Q ⊂ Q 0 , and Q denotes the father of S. Let us put ( S µ(S) if S ∈ j≥0 D j , αS = 0 otherwise. The sequence {α S } S∈D satisfies the Carleson condition X α S ≤ C µ(R). S⊆R
(3.18)
ACCRETIVE SYSTEMS ON NONHOMOGENEOUS SPACES
In fact, X
αS =
S⊆R
∞ X
X
j=0 S∈D j , S⊆R
µ(S) ≤
273
1 µ(R), τ
where the last inequality follows easily from (3.6). R P Let us first estimate the part of the sum in (3.17): S∈S D j S |1 Q f |2 dµ. j≥0 S Recall that, by our construction on each S ∈ j≥0 D j , the function 1 Q f has the form nhfi o h f iQ S 1 Q f |S = bS − b Qa , hb S i S hb Q a i Q where Q a is a certain ancestor of Q. Thus hfi hfi h f i Q S S − |1 Q f | S | ≤ |b Q a − b S | χ S + χS hb S i S hb S i S hb Q a i Q 2 |hb S i S − hb Q a i Q | ≤ |h f i S | χ S + |h f i S | χS δ |hb S i S | |hb Q a i Q | 1 + |h f i Q − h f i S | χS |hb Q a i Q | 2 2 1 ≤ + 2 |h f i S | + |h f i Q − h f i S |. δ δ δ R P S 2 To finish the estimate of S∈ j S |1 Q f | dµ, it is enough to validate the j≥0 D following two inequalities (below Q means the father of S): X |h f i S |2 α S ≤ C1 k f k22 , (3.19) S∈D
X
|h f i S − h f i Q |2 µ(S) ≤ C2 k f k22 .
(3.20)
S∈D
We know by (3.18) that {α S } S possesses the Carleson property. Thus inequality (3.19) follows from the Carleson embedding theorem (see [12], [15], or [22]). As for (3.20), it is obvious. In fact, consider the functions ( h f i S − h f i Q if S is a son of Q, δ Q f := 0 otherwise. The operators δ Q form a family of mutually orthogonal projections in L 2 (µ). And |h f i S − h f i Q |2 µ(S) = kδ Q f k2L 2 (µ) . Thus (3.20) follows. R P Now we need to estimate S ∈/ S D j S |1 Q f |2 dµ. For such S we have j≥0
1 Q f |S =
hfi h f iQ S − b Qa . hb Q a i S hb Q a i Q
(3.21)
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NAZAROV, TREIL, and VOLBERG
Here S is a son of Q, and Q a is an ancestor of S. This ancestor is defined uniquely by S by our construction. Put ( S |hb Q a i S − hb Q a i Q |2 µ(S) if S ∈ / j≥0 D j , β := 0 otherwise. Let us prove that the sequence {β S } S∈D satisfies the Carleson condition X β S ≤ C µ(R), ∀ R ∈ D .
(3.22)
S⊆R
In fact, let R j be the family of subcubes of R such that they are proper subcubes of a certain Q j ∈ D j but are not subcubes of any cube from D j+1 contained in R. S j Thus we are now looking at the collection j≥0 R j . Let {Q n }n denote the collection of cubes from D j which are inside R. Then X X X X X βS = βS = |hb Q j i S − hb Q j i Q |2 µ(S) n
S∈R j
≤
X n
kb Q j χ Q j k2L 2 (µ) ≤ n
n
n
j
S∈R j , S⊆Q n
X
n
n
j
S∈R j , S⊆Q n
j µ(Q n ) = µ
[
Q .
Q∈D j , Q⊆R
n
Summing on j and taking (3.6) into account, we get (3.22): X X [ βS ≤ µ Q ≤ C(τ ) µ(R). S⊆R
Q∈D j , Q⊆R
j
We are ready to estimate Z
X S∈ /
S
j≥0 D
j
|1 Q f |2 dµ. S
From (3.21), |1 Q f | S | ≤ |h f i S | Thus X Z S∈ /
S
j≥0
Dj
|hb Q a i S − hb Q a i Q | 1 χ S + |h f i S − h f i Q | χ S . δ δ2
|1 Q f |2 dµ ≤ C1 (δ) S
X S
|h f i S − h f i Q |2 µ(S) + C2 (δ)
X
|h f i S |2 β S .
S
To see that the first sum is bounded by C1 k f k2L 2 (µ) , look at (3.20). The second
sum is bounded by C2 k f k2L 2 (µ) because of Carleson property (3.22) of the family of numbers {β S }. This completes the proof of Lemma 3.1. The decomposition with respect to a system of accretive functions is completed.
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4. The estimate of the matrix of the operator from Theorem 0.3 The idea of the proof of the main results is pretty simple. We would like to estimate (T f, g). To do that, let us decompose f and g in the martingale difference decomposition given by (3.1) and (3.2), and then let us estimate the matrix (T 1 Q f, 1 R g) and conclude that the operator T is bounded. LEMMA 4.1 Let Q, R be two cubes, `(Q) ≤ `(R), and let dist(Q, R) ≥ `(Q). Let ϕ Q , ψ R ∈ L 2 (µ) be functions supported by the cubes Q and R, respectively. Suppose also that ϕ Q is orthogonal to constants. Then
|(T ϕ Q , ψ R )| ≤ C
`(Q)α µ(Q)1/2 µ(R)1/2 kϕ Q k 2 kψ R k L 2 (µ) . L (µ) dist(Q, R)d+α
Proof Let s0 be the center of the cube Q. Then we get ZZ |(T ϕ Q , ψ R )| = K (t, s)ϕ Q (s)ψ R (t) dµ(s) dµ(t) ZZ = [K (t, s) − K (t, s0 )]ϕ Q (s)ψ R (t) dµ(s)dµ(t) ZZ |s − s0 |α ≤C |ϕ (s)| · |ψ R (t)| dµ(s) dµ(t) |t − s0 |d+α Q `(Q)α ≤C kϕ k · kψ R k 1 L (µ) dist(Q, R)d+α Q L 1 (µ) α `(Q) ≤C µ(Q)1/2 µ(R)1/2 kϕ Q k 2 kψ R k L 2 (µ) . L (µ) dist(Q, R)d+α
Definition Let γ = α/(2α+2d) so that γ d+γ α = α/2. Let r be some positive integer to be fixed later. Consider a pair of cubes Q and R. Suppose for definiteness that `(Q) ≤ `(R). We call this pair singular if dist(Q, R) ≤ `(Q)γ · `(R)1−γ , and essentially singular if, in addition, `(Q) ≤ 2−r `(R). Definition Let D(Q, R) denote the so-called long distance between cubes: D(Q, R) := dist(Q, R) + `(Q) + `(R).
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NAZAROV, TREIL, and VOLBERG
LEMMA 4.2 Let T be a Calder´on-Zygmund operator, and let ϕ Q , ψ R ∈ L 2 (µ) be functions supported by the cubes Q and R, respectively, and normalized by kϕ Q k L 2 (µ) = µ(Q)−1/2 , kψ R k L 2 (µ) = µ(R)−1/2 . Suppose also that `(Q) ≤ `(R) and that ϕ Q is orthogonal to constants. Then
|(T ϕ Q , ψ R )| ≤ C
`(Q)α/2 `(R)α/2 , D(Q, R)d+α
provided that dist(Q, R) ≥ min(`(Q), `(R)) and that the pair Q, R is not essentially singular. Proof Without loss of generality, one can assume that `(Q) ≤ `(R). If dist(Q, R) ≥ `(R), then D(Q, R) ≤ 3 dist(Q, R); thus the estimate from Lemma 4.1 implies |(T ϕ Q , ψ R )| ≤ C
`(Q)α `(Q)α/2 `(R)α/2 ≤ C · . D(Q, R)d+α D(Q, R)d+α
Now let us suppose that dist(Q, R) ≤ `(R), but that the pair Q, R is not singular. That means dist(Q, R) ≥ `(Q)γ `(R)1−γ . The estimate of Lemma 4.1 and the identity γ d + γ α = α/2 imply C · `(Q)α C · `(Q)α/2 `(R)α/2 = α/2 d+α/2 `(R)d+α `(Q) `(R) `(Q)α/2 `(R)α/2 ≤C . D(Q, R)d+α
|(T ϕ Q , ψ R )| ≤
Note that if we do not normalize the functions ϕ Q and ψ R , the estimate from Lemma 4.2 can be rewritten as |(T ϕ Q , ψ R )| ≤ C
`(Q)α/2 `(R)α/2 µ(Q)1/2 µ(R)1/2 kϕ Q k L 2 (µ) kψ R k L 2 (µ) . D(Q, R)d+α
The following theorem shows that the matrix {TQ,R } Q∈D ,R∈D 0 , defined by TQ,R :=
`(Q)α/2 `(R)α/2 µ(Q)1/2 µ(R)1/2 , D(Q, R)d+α
generates a bounded operator in `2 .
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277
THEOREM 4.3 Let the measure µ satisfy µ(Q) ≤ C`(Q)d for all squares Q. Then for the matrix {TQ,R } Q,R∈D defined above, one has
X Q∈D ,R∈D 0
TQ,R x Q · y R ≤ C
X Q∈D
2 xQ
1/2 X R∈D 0
y 2R
1/2
for any sequences of nonnegative numbers {x Q } Q∈D , {y R } R∈D 0 ∈ `2 . The proof can be found in [23]. 5. “Paraproducts” and the estimate of (T ϕ Q , ψ R ) when Q ⊂ R As usual in the theory of singular integral operators, to estimate (T ϕ Q , ψ R ) when Q ⊂ R, one can use the so-called paraproducts. The classical construction does not work in our case, and we slightly modify it. Paraproducts Let b S be accretive functions from the statement of our T b-theorems. Let r be a positive integer to be defined later. (It is the same number we used in the definition of Q Q essentially singular pairs.) We define a paraproduct = T ∗ by (we recall that R a S means the ancestor of R closest to R which lies in j D j ) Y X f := (E R b R a )−1 · E R f · (1 Q )∗ T ∗ b R a . R∈D 0 , Q∈D :`(Q)=2−r `(R) dist(Q,∂ R)≥λ`(Q)
Notice that T ∗ b R a are well defined as functionals on compactly supported functions from L 2 (µ) which are orthogonal to constants. So we can define (1 Q )∗ T ∗ b R a by duality as the function f satisfying ( f, g) = (T ∗ b R a , 1 Q g), ∀g ∈ L 2 (µ). Q Actually, we are not using directly because in the case of the system of accretive functions this seems to be very difficult if possible at all. Instead, we split (T ϕ Q , ψ R ) into pieces, and then we sum up all the first pieces for all Q, R such that Q ⊂ R, and the pair Q, R is not essentially singular. This summation provides a lot of cancellation. The “leftover” pieces of (T ϕ Q , ψ R ) are estimated by absolute value with no cancellation at all. Q Let us explain why we are not using as such. Definition S We call a cube Q unusual if it has a son from j D j . Otherwise, we call it usual. This is the same for cubes R from the lattice D 0 .
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NAZAROV, TREIL, and VOLBERG
Q b a In the above formula for , by 1 Q we understand 1 QQ . Let Q ∈ D , R ∈ D 0 , let Q, R be usual cubes, and let ϕ Q and ψ R be functions of the form ϕ Q (x) = ψ R (x) =
X Q 0 ∈D :Q 0 ⊂Q,`(Q 0 )=`(Q)/2
X R 0 ∈D 0 :R 0 ⊂R,`(R 0 )=`(R)/2
A Q 0 · χ Q 0 (x) · b Q a (x),
(5.1)
B R 0 · χ R 0 (x) · b R a (x),
(5.2)
where A Q 0 , B R 0 are some constants. Suppose also that the functions ϕ Q , ψ R are R R orthogonal to constants; that is, ϕ Q dµ = 0, ψ R dµ = 0. This is how 1 Q f , 1 R g should look. Q Notice that computing (ϕ Q , ψ R ) and wishing to get a simple formula like [25, (7.3)] for this scalar product, we are obliged to consider hϕ Q , (1qbq a )∗ T ∗ b R a i for certain small cubes q which lie in a son of Q. If we had b Q a = bq a (as we have in [25] b
a
because all b’s are the same there), we would conclude that hϕ Q , (1qq )∗ T ∗ b R a i = 0. This is just because of relationships (3.12) and (5.1). In fact, let h = b Q a = bq a . Then (5.1) shows that we come to calculating (1qh )(h), which is zero by (3.12). UnfortuQ nately, we cannot assume b Q a = bq a . And this is why the matrix of is not so nice as in [25]. Instead of using paraproducts as such, we consider the following scheme. We always have Q ∈ D , R ∈ D 0 , Q ⊂ R, and `(Q) ≤ 2−r −1 `(R). We always consider only good Q. By sk(R) we mean the union of all boundaries for all sons of R. Definition Q ∈ D is called good if for any R ∈ D 0 the pair Q, R is not essentially singular. In other words, Q is good means that for any R either `(Q) and `(R) are 2−x , x ∈ (−r − 1, r + 1) comparable, or (if, say, `(Q) ≤ 2−r −1 `(R)) dist Q, sk(R) ≥ `(Q)γ `(R)1−γ . (5.3) We always choose r large enough to satisfy 2r (1−γ ) ≥ 4 3,
(5.4)
where 3 is from the definition of our BMO. For such a good Q, we consider all R’s as above, and we see that, in particular, (5.3) is always satisfied. There exists a smallest R as above. Let us call it S = S(Q), and let S1 be its son containing Q; then `(Q) = 2−r `(S1 ). Let us decompose (T 1 Q f, 1 R g) into pieces.
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279
Case 1: R is usual. Let Rk denote sons of R, and let R1 be its son containing Q. Then hgi hgi R Rk 1 R g| Rk = − b R a χ R a =: B Rk b R a χ Rk . hb R a i Rk hb R a i R Then d
(T 1 Q f, 1 R gi = (T 1 Q f, B R1 b R a χ R1 ) +
2 X
(T 1 Q f, B Rk b R a χ Rk )
k=2
= (T 1 Q f, B R1 b R a ) − T 1 Q f, B R1 b R a (1 − χ R1 ) d
+
2 X
(T 1 Q f, B Rk b R a χ Rk )
k=2
=: t (Q, R) − r (Q, R) + s(Q, R). Let us estimate s(Q, R), r (Q, R). First, |r (Q, R)| ≤ |B R1 | T 1 Q f, B R1 b R a (1 − χ R1 ) Z ≤ |B R1 | |T 1 Q f | dµ ≤ C |B R1 | Rd \R1
`(Q)α k1 Q f k2 µ(Q)1/2 . dist(Q, ∂ R1 )α
On the other hand, Z δ|B R1 | µ(R1 ) ≤ |B R1 | b R a dµ R1 Z Z ≤ 1 R g dµ ≤ R1
|1 R g| dµ ≤ µ(R1 )1/2 k1 R gk2 . R1
So |B R1 | ≤ C(δ)µ(R1 )−1/2 k1 R gk2 . Thus
µ(Q) 1/2 `(Q)α k1 Q f k2 k1 R gk2 . dist(Q, ∂ R1 )α µ(R1 ) But Q is good, and thus the pair Q, R is not essentially singular with Q being much smaller than R. So (5.3) holds. Thus dist(Q, ∂ R1 ) ≥ `(Q)1/2 `(R)1/2 . Finally, `(Q) α/2 µ(Q) 1/2 |r (Q, R)| ≤ C k1 Q f k2 k1 R gk2 . (5.5) `(R) µ(R1 ) |r (Q, R)| ≤ C
To estimate s = s(Q, R), we use Lemma 4.2. It gives (we again use (5.3)) `(Q) α/2 µ(Q) 1/2 |(T 1 Q f, B Rk b R a χ Rk )| ≤ C k1 Q f k2 k1 R gk2 , `(R) `(R)d `(Q) α/2 µ(Q) 1/2 |s| ≤ C k1 Q f k2 k1 R gk2 (5.6) `(R) µ(R1 ) because our measure satisfies the estimate from above: µ(R1 ) ≤ µ(R) ≤ C `(R)d .
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Case 2: Cube R is unusual. If R1 (the son of R in which Q is contained) is not S from j D j , then the estimates of r, s are exactly the same as in (5.5) and (5.6). S a j We are left to consider the case when R1 ∈ j D . In this case R1 = R1 . Let us introduce σ R1 (g) := (hgi R1 /hb R1 i R1 )b R1 χ R1 =: B R1 b R1 χ R1 and σ R (g) := (hgi R /hb R a i R )b R a χ R =: C R b R a χ R . Notice that d
1 R g = σ R1 (g) − σ R (g) χ R1 +
2 X
1 R gχ Rk
k=2 d
= B R1 b R1 χ R1 − C R b R a χ R1 +
2 X
1 R g χ Rk
k=2 d
= B R1 b R1 − C R b R a − B R1 b R1 (1 − χ R1 ) + C R b R a (1 − χ R1 ) +
2 X
1 R g χ Rk .
k=2
And so (T 1 Q f, 1 R g) = (T 1 Q f, B R1 b R1 ) − (T 1 Q f, C R b R a ) − B R1 T 1 Q f, b R1 (1 − χ R1 ) d
+ C R T 1Q
2 X (T 1 Q f, 1 R g χ Rk ) f, b R a (1 − χ R1 ) + k=2
=: t1 − t2 − r1 + r2 + s. The estimate of s repeats the notation of the one in Case 1: |s| ≤ C
`(Q) α/2 µ(Q) 1/2 `(R)
µ(R1 )
k1 Q f k2 k1 R gk2 .
(5.7)
To estimate r1 , r2 , we need to know how to estimate B R1 , C R . But b R1 is δaccretive on R1 , and b R a is δ-accretive on R. This shows that |B R1 | ≤ C(δ) h|g|i R1 ,
|C R | ≤ C(δ) h|g|i R .
So, exactly as in Case 1, we have `(Q) α/2 µ(Q) 1/2
k1 Q f k2 h|g|i R1 µ(R1 )1/2 , `(R) µ(R1 ) `(Q) α/2 µ(Q) 1/2 k1 Q f k2 h|g|i R µ(R1 )1/2 . |r2 | ≤ C `(R) µ(R1 )
|r1 | ≤ C
(5.8) (5.9)
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Let us introduce the notation: t (Q, R) := t1 − t2 and X
T Q :=
t (Q, R).
R: S(Q)⊂R, R∈D 0 ,`(Q)≤2−r −1 `(R)
As the reader can guess, we are not going to estimate t’s by absolute value. Instead, we are going to see that their sum has a very nice form. 5.1 Let Q ∈ D be good. Let S1 be a cube from D 0 such that Q ⊂ S1 , `(Q) = 2−r `(S1 ). Then LEMMA
hgi S1 (T 1 Q f, b S1a ), hb S1a i S1 X X X TQ =
TQ =
(1) (2)
Q good
J ∈D 0 Q⊂J, Q∈D , `(Q)=2−r `(J )
1 Q f,
hgi J T ∗b J a . hb J a i J
Proof The proof of (1) is obvious by construction. We just sum up all “martingale” differences of g for all sufficiently large R containing Q. The proof of (2) then is just the change of order of summation. 5.2 P
THEOREM
We have |
Q good T Q |
≤ Ck f k2 kgk2 .
We need some additional lemmata. Fix λ > 1. We fix r large enough so that 2r ≥ 4λ. For a function ϕ and a dyadic cube R ∈ D 0 , define X a R = a R (ϕ) = k(1 Q )∗ ϕk2 2 . Q∈D :Q⊂R, `(Q)=2−r `(R) dist(Q,∂ R)≥λ`(Q)
L (µ)
Notice that R ∈ D 0 and that the smaller cubes Q are taken from another dyadic lattice D . LEMMA 5.3 If ϕ ∈ BMO2λ (µ), then the family {a R (ϕ)} R∈D 0 defined above satisfies the Carleson measure condition X a R 0 ≤ Cµ(R), ∀R ∈ D 0 . R 0 ⊂R
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Proof It is sufficient to prove that for any dyadic cube R ∈ D 0 , X k(1 Q )∗ ϕk2 2 ≤ Cµ(R). L (µ)
Q∈D : Q⊂R, `(Q)≤2−r `(R) dist(Q,∂ R)≥λ`(Q)
(5.10)
Consider the following Whitney-type covering of the cube R by cubes Q ∈ D : take all cubes Q ⊂ R of size 2−r `(R) such that dist(Q, ∂ R) ≥ λ`(Q) (the assumption 2r ≥ 4λ guarantees that there exists at least one such Q), then take the layer around them consisting of all cubes of size 2−r −1 `(R) such that dist(Q, ∂ R) ≥ λ`(Q), then the layer of cubes of size 2−r −2 , and so on. Let us call the collection of such Whitney cubes W . Pick a cube Q ∈ W . By the definition of BMO2λ (µ), Z |ϕ − ϕ Q |2 dµ ≤ Cµ(λQ). Q
Now (3.3) implies X Q 0 ∈D ,Q 0 ⊂Q
Estimate (5.11) implies X X Q∈W Q 0 ∈D ,Q 0 ⊂Q
k(1 Q 0 )∗ ϕk2 2
≤ Cµ(λQ).
k(1 Q 0 )∗ ϕk2 2
≤C
L (µ)
L (µ)
X
µ(λQ).
(5.11)
(5.12)
Q∈W
Since for any cube Q from the Whitney-type decomposition W we have dist(Q, ∂ R) ≥ λ`(Q), any point in R is covered by at most M = M(N , λ) cubes P λQ, Q ∈ W . Therefore Q µ(λQ) ≤ Mµ(R). To complete the proof of the lemma, it is enough to notice that the sum in the left-hand side of (5.12) coincides with the sum in (5.10). Suppose now that we have the generations of terminal cubes D 0 j with the Carleson property X µ(R 0 ) ≤ A · µ(R). (5.13) S R 0 ⊂R,R 0 ∈ j D 0 j
As before, for any R we define the closest ancestor from D 0 j and call it R a . For a system of functions ϕ J , J ∈ D 0 , and a dyadic cube R ∈ D 0 , define X a aR = k(1 Q )∗ ϕ R k2 2 . Q∈D : Q⊂R, `(Q)=2−r `(R) dist(Q,∂ R)≥λ`(Q)
L (µ)
Notice that R ∈ D 0 and that the smaller cubes Q are taken from another dyadic lattice D .
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283
LEMMA 5.4 If ϕ J ∈ BMO2λ (µ) uniformly, then the family {a R } R∈D 0 defined above satisfies the Carleson measure condition X a R 0 ≤ Cµ(R), ∀R ∈ D 0 . R 0 ⊂R
Proof a Fix R ∈ D 0 . First we are summing a R 0 for which ϕ R is involved. Their sum is bounded by Cµ(R) as in Lemma 5.3. Suppose R a ∈ D 0 i . Consider cubes Rki+1 which i+1
are from D 0 i+1 and located inside R. Fix a k. Collect and sum a R 0 for which ϕ Rk is i+2 involved. Their sum is bounded by Cµ(Rki+1 ) as in Lemma 5.3. Consider cubes Rkm i+2 i+1 which are from D 0 and located inside Rk . Fix an m. Collect and sum a R 0 for i+2
i+2 which ϕ Rkm is involved. Their sum is bounded by Cµ(Rkm ) as in Lemma 5.3. And so on. P Finally, we apply (5.13) to estimate J ⊂R,J ∈S D ; j µ(J ) by Cµ(R). j≥i+1
For a function ϕ and a dyadic cube R ∈ D 0 , define X X d R = d R (ϕ) = Q∈D : `(Q)=2−r `(R) T term. son of Q dist(Q,∂ R)≥λ`(Q)
infh|ϕ − c|i2T µ(T ). c
Notice that R ∈ D 0 and that the smaller cubes Q are taken from another dyadic lattice D. 5.5 If ϕ ∈ BMO2λ (µ), then the family {d R (ϕ)} R∈D 0 defined above satisfies the Carleson measure condition X d R 0 ≤ Cµ(R), ∀R ∈ D 0 . LEMMA
R 0 ⊂R
Proof It is sufficient to prove that for any dyadic cube R ∈ D 0 , X X infh|ϕ − c|i2T µ(T ) ≤ Cµ(R). Q∈D : dist(Q,∂ R)≥λ·`(Q)
Q⊂R, `(Q)≤2−r `(R)
T term. son of Q
c
(5.14)
Consider the following Whitney-type covering of the cube R by cubes Q ∈ D : take all cubes Q ⊂ R of size 2−r `(R) such that dist(Q, ∂ R) ≥ λ`(Q) (the assumption 2r ≥ 4λ guarantees that there exists at least one such Q), then take the layer around them consisting of all cubes of size 2−r −1 `(R) such that dist(Q, ∂ R) ≥
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NAZAROV, TREIL, and VOLBERG
λ`(Q), then the layer of cubes of size 2−r −2 , and so on. Let us call the collection of such Whitney cubes W . Pick a cube Q ∈ W . Denote by ϕ Q the average hϕi. By the definition of BMO2λ (µ),
Z Q
S
|ϕ − ϕ Q |2 dµ ≤ Cµ(λQ).
We know that the following is a Carleson sequence: αT := 0 if T is not from j , and it equals µ(T ) otherwise. This and the previous estimate imply D j X Q 0 ∈D ,Q 0 ⊂Q
Estimate (5.15) implies X X
X
h|ϕ − ϕ Q |i2T µ(T ) ≤ Cµ(λQ).
T term. son of
X
Q∈W Q 0 ∈D ,Q 0 ⊂Q T term. son of Q 0
(5.15)
Q0
infh|ϕ − c|i2T µ(T ) ≤ C c
X
µ(λQ).
(5.16)
Q
Since for any cube Q from the Whitney-type decomposition W we have dist(Q, ∂ R) ≥ λ`(Q), any point in R is covered by at most M = M(N , λ) cubes P λQ, Q ∈ W . Therefore Q µ(λQ) ≤ Mµ(R). To complete the proof of the lemma, it is enough to notice that the sum in the left-hand side of (5.16) coincides with the sum in (5.14). For a system of functions ϕ J , J ∈ D 0 , and a dyadic cube R ∈ D 0 , define X X a dR = infh|ϕ R − c|i2T µ(T ). Q∈D : `(Q)=2−r `(R) T term. son of Q dist(Q,∂ R)≥λ`(Q)
c
Notice that R ∈ D 0 and that the smaller cubes Q are taken from another dyadic lattice D . 5.6 BMO2λ (µ) uniformly, then the family {d R } R∈D 0 defined above satisfies the Carleson measure condition X d R 0 ≤ Cµ(R), ∀R ∈ D 0 . LEMMA If ϕ J ∈
R 0 ⊂R
Proof The proof of this lemma follows from Lemma 5.5 in exactly the same way as the proof of Lemma 5.4 follows from Lemma 5.3.
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285
Proof of Theorem 5.2 First, for any J we have |hb J a i J | ≥ δ 2 , and thus X X X T Q ≤ C(δ)
|(1 Q f, hg) J T ∗ b J a )|.
J ∈D 0 Q⊂J, Q∈D , `(Q)=2−r `(J )
Q good
Recall that 1 Q f = (1 Q )2 f + ω Q f (see the definition after (3.14)). We have X X |(1 Q f, hgi J T ∗ b J a )| J ∈D 0 Q⊂J, Q∈D , `(Q)=2−r `(J )
X
≤
X
··· +
J ∈D 0 Q⊂J, Q∈D , Q usual...
X
=
X
X
X
···
J ∈D 0 Q⊂J, Q∈D , Q unusual...
|(1 Q f, hgi J (1 Q )∗ T ∗ b J a )|
J ∈D 0 Q⊂J, Q∈D ,...
+
X
X
|(ω Q f, hgi J χ Q (T ∗ b J a − c))|
J ∈D 0 Q⊂J, Q∈D , Q unusual...
=: T 0 + T 00 . Here constants c can be chosen arbitrarily because ω Q f is orthogonal to constants. There is only one J for every Q. And so 1/2 X 1/2 X X k(1 Q )∗ T ∗ b J a k22 . |T 0 | ≤ C(δ) k1 Q f k22 · |hgi J |2 J ∈D 0
Q
Q⊂J, Q∈D ,...
On the other hand, the second factor above can be estimated by the Carleson embedding theorem by C kgk2 because Lemma 5.4 shows that numbers a J := P 2 ∗ ∗ Q⊂J, Q∈D ,... k(1 Q ) T b J a k2 form a Carleson sequence. S For any unusual Q, we call terminal those sons of Q which are from j D j . Now we want to estimate T 00 : XX X |T 00 | ≤ kω Q f k∞ infh|T ∗ (b J a ) − c|iT µ(T ). J ... Q...
T term. son of Q
c
There is only one J for every Q. And so X 1/2 [ 00 2 T |T | ≤ C(δ) |ω Q f | µ Q
·
X J ∈D 0
|hgi J |2
T term son of Q
X
X
Q⊂J, Q∈D ,... T term. son of Q
1/2 infh|T ∗ (b J a ) − c|i2T µ(T ) . c
But |ω Q f | ≤ C(δ) h| f |i Q (see the definition after (3.14)). This (and the fact S that the following is a Carleson sequence: αT := 0 if T is not from j D j , and it
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equals µ(T ) otherwise) allows us to conclude that the first factor above is bounded by C k f k2 . As for the second factor, we use Lemma 5.6 to conclude that the numbers X X d J := infh|T ∗ (b J a ) − c|i2T µ(T ) Q⊂J, Q∈D ,... T term. son of Q
c
form a Carleson sequence. So the second factor is bounded by C kgk2 . Theorem 5.2 is completely proved. Let us introduce
P
Reg
:= |
P
R∈D 0
P
Q good,Q⊂R, `(Q)≤2−r −1 `(R) (T 1 Q
f, 1 R g)|.
P 6. The estimate of Reg Recall that Q is good. In Case 1 above we introduced s(Q, R) and r (Q, R), and in Case 2 we had s(Q, R), r1 (Q, R), r2 (Q, R). Let us put r (Q, R) := r1 − r2 for Case 2 (see above). And let us introduce X X S := s(Q, R), R∈D 0 Q⊂R,`(Q)≤2−r −1 `(R),Q∈D
R :=
X
X
R∈D 0
Q⊂R,`(Q)≤2−r −1 `(R),Q∈D
r (Q, R).
Now the reader should look at (5.5) – (5.9) to be convinced that we need to consider the matrix defined by `(Q) α/2 µ(Q) 1/2 if Q ⊂ R, `(Q) < 2−r `(R), TQ,R = `(R) µ(R1 ) 0 otherwise, where R1 is the subcube of R of the first generation (`(R1 ) = `(R)/2) containing Q. 6.1 The matrix {TQ,R } Q∈D ,R∈D 0 defined above generates a bounded operator in `2 ; that is, X 1/2 X 1/2 X 2 TQ,R x Q · y R ≤ C xQ y 2R LEMMA
Q∈D ,R∈D 0
Q∈D
R∈D 0
for any sequences of nonnegative numbers {x Q } Q∈D , {y R } R∈D 0 ∈ `2 . One can find the proof in [23]. Let us again use the following facts: S (1) the following is a Carleson sequence: αT := 0 if T is not from j D j , and it equals µ(T ) otherwise;
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(2)
287
Lemma 6.1. Then we get |S | ≤ C k f k2 kgk2 ,
(6.1)
|R | ≤ C k f k2 kgk2 .
(6.2)
Estimates (6.1) and (6.2) combined with Theorem 5.2 give X X (T 1 Q f, 1 R g) ≤ C k f k2 kgk2 .
(6.3)
R∈D 0 Q good,Q⊂R,`(Q)≤2−r −1 `(R)
7. Bad and good cubes; well-intersected and badly intersected cubes We fix a large natural number r . The reader needs to recall the definition of essential singularity. It involves parameter r . Let dyadic lattices D and D 0 be fixed. We recall that a dyadic square Q in one lattice (say, in D ) is called bad if there exists a bigger square R in the other lattice (in D 0 in this case), such that the pair Q, R is essentially singular; otherwise the square is called good. Let a function f ∈ L 2 (µ) be supported by a cube of size 2n . We call the function f good (D -good) if 1 Q f = 0 for any bad square Q ∈ D , `(Q) < 2n . If one replaces D by D 0 , one gets the definition of D 0 -good functions. Here and in the sequel, we assume that n is fixed. We call a dyadic square Q in one lattice (say, in D ) well intersected if (1) for any cube R from D 0 such that 2−r `(Q) ≤ `(R) ≤ 2r `(Q), the “rectangle” of intersection has the ratio of any two of its sides not exceeding 2r ; (2) for any son Q 0 of Q and any son R 0 of R, the rectangle of intersection of Q 0 and R 0 has the ratio of any two of its sides not exceeding 2r . Otherwise, it is called badly intersected. 7.1. Random dyadic lattice Our random lattice contains the dyadic cubes of standard size 2k , k ∈ Z, but it is randomly shifted with respect to the standard dyadic lattice D0 . The simplest idea would be to pick up a random variable ξ uniformly distributed over Rd and to define the random lattice as ξ + D0 . Unfortunately, there exists no such ξ , and we have to act in a slightly more sophisticated way. Let us construct a random lattice of dyadic intervals on the real line R, and then let us define a random lattice in Rd as the product of the lattices of intervals. Let 1 be some probability space, and let x(ω) be a random variable uniformly distributed over the interval [0, 1)d . Let ξ j (ω) be random variables satisfying P{ξ j = +1} = P{ξ j = −1} = 1/2. Assume also that x(ω), ξ j (ω) are independent. Define the random lattice D (ω) as follows.
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NAZAROV, TREIL, and VOLBERG
Let I0 (ω) = [x(ω) − 1, x(ω)] ∈ D (ω). This uniquely determines all intervals in D (ω) of length 2k , where k ≤ 0. (2) The intervals Ik (ω) ∈ D (ω) of length 2k with k > 0 are determined inductively. If Ik−1 (ω) ∈ D is already chosen, Ik (ω) is determined by the following rule: (Ik (ω))+ = Ik−1 (ω) if ξk (ω) = +1 and (Ik (ω))− = Ik−1 (ω) if ξk (ω) = −1. In other words, at every step we extend the interval Ik−1 (ω) to the left if ξk (ω) = +1 and to the right otherwise. Clearly, to know one interval of length 2k in the lattice is enough to determine all of them. To get a random dyadic lattice in Rd , we just take a product of N independent random lattices in R. It is easy to check that the random lattice D (ω) in Rd constructed in this way is uniformly distributed over Rd and satisfies the following. Equidistribution property: For x ∈ Rd , k ∈ Z, the probability that dist(x, ∂ Q) ≥ ε`(Q) for some cube of size 2k is exactly (1 − 2ε)d . (1)
7.2. Bad cubes Let D (ω) and D 0 (ω0 ), (ω, ω0 ) ∈ × , be two independent random dyadic lattices, as constructed above. We call a cube Q ∈ D (ω) bad if there exists a cube R ∈ D 0 (ω0 ) of length `(R) ≥ `(Q) such that the pair Q, R is essentially singular. Otherwise, we call the cube Q good. The definition of bad cubes in D 0 (ω0 ) is the same. (Now we look for a bigger cube in D (ω).) LEMMA 7.1 Let r , γ be from the definition of essentially singular pairs. Then for any fixed ω and a cube Q ∈ D (ω), we have
P := Pω0 {Q is bad} ≤ 2d
2−rγ . 1 − 2−γ
Proof Given a cube Q ∈ D (ω) (ω is fixed), the probability P k that there exists a cube R ∈ D 0 (ω0 ), Q ⊂ R, of size 2k `(Q) such that dist(Q, ∂ R) ≤ `(Q)γ `(R)1−γ can be estimated by (see Fig. 1) d P k ≤ 1 − 1 − (2−k + 2−γ k ) ≤ 2d2−γ k . So, the probability P can be estimated: X X 2−r γ P= Pk ≤ 2d 2−γ k = 2d . 1 − 2−γ k≥r
k≥r
ACCRETIVE SYSTEMS ON NONHOMOGENEOUS SPACES
6 −γ k 2 ?
289
6 −k 6 ?2
1
?
Figure 1. Estimate of probability Pk
LEMMA 7.2 Let the integer r be from the definition of badly intersected cubes. Then for any fixed ω and a cube Q ∈ D (ω), we have
P := Pω0 {Q is badly intersected} ≤ (r), where (r ) → 0 when r tends to ∞. Proof The cube Q is badly intersected. By pl , denote the probability that there exists a cube R from D 0 such that `(R)/`(Q) = 2l , l = −r, . . . , 0, . . . , r . Probability pl can be uniformly (independently of l) estimated by C(d)2−r . Summing up over r, . . . , 0, . . . , r , we get 2C(d)r 2−r .
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NAZAROV, TREIL, and VOLBERG
7.3. With large probability bad parts are small Consider functions f and g supported by some cube of size 2n . One can write down the decomposition X X f = 1Q f + E Q f, Q∈D ,`(Q)≤2n
Q∈D ,`(Q)=2n
where the series converges in L 2 (µ) (see (3.1)). Let us split f = f good + f bad , where X f bad := Q∈D , `(Q)≤2n
1 Q f.
Q is bad or badly intersected
Here bad means where D 0 = D 0 (ω0 ) is the other random dyadic lattice. Similarly, one can decompose g = ggood + gbad , where X gbad := 1 Q g; D 0 -bad,
Q∈D 0 , `(Q)≤2n Q is bad or badly intersected
here bad means D -bad. Let us estimate the mathematical expectation E k f bad k2L 2 (µ) (taken over the random dyadic lattices constructed above). To do that, let us consider (for a fixed dyadic lattice D ) the so-called square function S(x) defined for x ∈ Rn by X S f (x) = SD f := k1 Q f k2L 2 (µ) µ(Q)−1 χ Q Q∈D :Q3x `(Q)≤2n
X
+
Q∈D :Q3x
`(Q)=2n
kE Q f k2L 2 (µ) µ(Q)−1 χ Q .
Clearly, Z RN
X
S f (x) dµ(x) =
Q∈D :Q3x `(Q)≤2n
+
k1 Q f k2L 2 (µ)
X Q∈D :Q3x `(Q)=2n
kE Q f k2L 2 (µ) k f k2L 2 (µ) ,
where means equivalence in the sense of a two-sided estimate (see (3.1), (3.3)). Consider the average square function Eω S f (x). (For each x ∈ Rd , take the mathematical expectation over all dyadic lattices D = D (ω).) Changing the order of inteR gration, one can see that Rd Eω S f (x) dµ(x) ≤ Ck f k2L 2 (µ) . The (conditional, ω is fixed) probability Pω0 that a square Q is bad, or badly intersected, is at most (2d(2−r γ /1 − 2−γ )) + (r ) (see Lemma 7.1), so Eω0 S f bad (z) ≤ 0 S f (z).
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291
Since 2
E k f bad k ≤ AE ω0
ω0
≤ A 0
Z
Z RN
Z S f bad dµ = A Eω0 S f bad dµ S f dµ ≤ 2−8 2−4d k f k2L 2 (µ) ,
we get Eω,ω0 k f bad k2 = Eω Eω0 k f bad k2 ≤ 2−8 2−4d k f k2L 2 (µ) if 0 is sufficiently small (ensured by large r ). The probability that k f bad k2L 2 (µ) ≥ 4 · 2−8 2−4d k f k2L 2 (µ) cannot be more than 1/4; therefore with probability 3/4 we have k f bad k L 2 (µ) ≤ 2 · 2−4 2−2d k f k L 2 (µ) . So, if we have two functions f and g and two random dyadic lattices D (ω) and D 0 (ω0 ), then with probability at least 1/2 we have simultaneously k f bad k L 2 (µ) ≤ 2−3 2−2d k f k L 2 (µ) ,
kgbad k L 2 (µ) ≤ 2−3 2−2d kgk L 2 (µ) .
7.4. First pulling ourselves up by the hair Let us now almost prove Theorem 0.4. Let us pick functions f, g ∈ L 2 (µ), k f k = kgk = 1 such that |(T f, g)| ≥ (3/4)kT k. Since compactly supported functions are dense in L 2 (µ), we can always assume that both functions are supported by some cube of size 2n . Pick dyadic lattices D , D 0 such that k f bad k2 ≤ 2−3 2−2d k f k2
and
kgbad k2 ≤ 2−3 2−2d kgk2 .
We can always pick such a lattice because, as we have shown above, a random pair of lattices fits with probability at least 1/2. First let us assume temporarily that we have the estimate for most of the pairs of lattices (it would be nice to have this for all pairs of lattices, but we will have proved it for most pairs): 1 |(T f good , ggood )| ≤ Ck f good k2 kggood k2 + kT kk f good k2 kggood k2 . (7.1) 4 We can write |(T f, g)| ≤ |(T f good , g)| + |(T f bad , g)| ≤ |(T f good , ggood )| + |(T f good , gbad )| + |(T f bad , g)|. We have 1 1 |(T f good , ggood )| ≤ Ck f k2 kgk2 + kT kk f k2 kgk2 ≤ C + kT k, 4 4 |(T f good , gbad )| ≤ 2−3 2−2d kT k, |(T f bad , g)| ≤ 2−3 2−2d kT k
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because k f bad k2 ≤ 2−3 2−2d , k f good k2 ≤ k f k2 ≤ 1, and the same is true for g. Therefore, since |(T f, g)| ≥ (3/4)kT k and 2−2d ≤ 1, 3 1 kT k ≤ C + kT k + 2 · 2−3 kT k. 4 4 So kT k ≤ 4C and we are done. 7.5. The proof of estimate (7.1) for |(T f good , ggood )| P P If in |(T f good , ggood )| we decompose f good = Q 1 Q f , ggood = R 1 R g, we should remember that there are nonzero terms only for Q, R which are not bad cubes and which are well intersected at the same time. We split the sum over Q, R such that P P 2−r `(Q) ≤ `(R) ≤ 2r `(Q); call it 1 and the rest 2 . In this second sum the sizes of cubes Q, R are quite (2r ) different, and because there are no essentially singular pairs here, the smaller is either well inside the larger one or quite relatively separate from the larger one. Such sums we know how to estimate. Just gather together the estimates of Lemma 4.2, Theorem 4.3, and (6.3) with the symmetric estimate, where R is inside Q, and we get X ≤ Ck f k2 kgk2 . (7.2) 2
We are left to prove X
≤ Ck f k2 kgk2 .
(7.3)
1
7.6. Finitely many diagonals P In the sum 1 there is a part that can be easily estimated. This is the P dist(Q,R)≥10 max(`(Q),`(R)) . . . . This sum has a necessary estimate just by Lemma 4.2 and Theorem 4.3. Let us fix a very small τ . (Its smallness depends on r ; fixing a very large r , we need to fix then a very small τ .) If we consider now another part of 61 , namely, P 10 max(`(Q),`(R))≥dist(Q,R)≥τ min(`(Q),`(R)) . . . , we can readily estimate it as X µ(Q)1/2 µ(R)1/2 ...
dist(Q, R)d
k1 Q f k2 k1 R gk2 ≤ C(r )τ −d
X
k1 Q f k22
1/2 X
Q∈D
which is bounded by C 0 (r )τ −d k f k2 kgk2 by (3.1) and (3.2).
R∈D 0
k1 R gk22
1/2
,
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293
2ε`(Q)
Figure 2. The set δ Q (the shaded area)
All of the previous was necessary just to say that we only need to estimate τ X 1
:=
X
|(T 1 Q f, 1 R g)|.
2−r `(Q)≤`(R)≤2r `(Q) dist(Q,R)<τ min(`(Q),`(R))
For a cube Q, let δ Q := (1 + 2τ )Q \ (1 − 2τ )Q (see Fig. 2). For a fixed point x ∈ let pτ be the probability that x ∈ δ R for some cube R ∈ D 0 (ω0 ), 2−r −1 `(Q) ≤ `(R) ≤ 2r +1 `(Q), where D 0 (ω0 ) is the random dyadic lattice constructed above. Note that pτ does not depend on `(Q) and that pτ → 0 as τ → 0 when r is fixed. Of course, if we consider the random dyadic lattice D (ω), we get the same probability pτ . Note that one can compute the probability pτ , but we only need the fact that it can be arbitrarily small. For a cube Q ∈ D , let Q b be its bad part, [ Qb = Q ∩ δR . Rn ,
R∈D 0 2−r −1 `(Q)≤`(R)≤2r +1 `(Q)
For a function f ∈ L 2 (Q, µ), define bad parts f b of f as follows: f b := χ Q f ; b
here we use subscript “b” instead of “bad” to avoid the confusion with f bad .
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Let us estimate the mathematical expectation Eω0 (k f b k2 2 lattices
D 0 (ω0 ).
L (µ)
) over all random
(The lattice D = D (ω) is fixed.) First notice that for a fixed x ∈ Rn , Eω0 | f b (x)|2 ≤ pτ | f (x)|2 .
Therefore, changing the order of integration, we get Z Eω0 k f k2 2 = E | f b (x)|2 dµ(x) L (µ) d R Z ≤ pτ | f (x)|2 dµ(x) Rd
= p τ k f k2 2
L (µ)
.
Since the above inequality holds for any dyadic grid D = D (ω), we get for the mathematical expectation E = Eω,ω0 , E k f b k2 2 ≤ pτ k f k2 2 . L (µ)
L (µ)
In particular, E k(1 Q f )b k22 ≤ pτ k1 Q f k22 . P Now let us consider the decomposition f = Q∈D 1 Q f : X X E k (1 Q f )b k22 ≤ pτ k1 Q f k22 ≤ C pτ k f k22 Q
Q
because of (3.1). In particular, given f, g, we may think that for a set of lattices D , D 0 of probability at least 1/2, the inequality
X
2
(1 Q f )b ≤ C pτ k f k22 (7.4)
2
Q
holds along with the same inequality for g. Consider two squares Q ∈ D and R ∈ D 0 such that 2−r `(Q) ≤ `(R) ≤ 2r `(Q),
dist(Q, R) < τ min `(Q), `(R)
(see Fig. 3). We would like to estimate |(T 1 Q f, 1 R g)|. Consider first the case when the cubes Q and R are in general position (as on Fig. 3); the estimate for cases when Q ∩ R = ∅ or one of the cubes contains the other can be done similarly.
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Let us consider a son R 0 of R. We introduced above δ R 0 = (1+2τ )R 0 \(1−2τ )R 0 . Q 0 −mod Let Q 0 denote any son of Q intersecting R 0 . Let us consider δ R 0 , which is of the form (1 + 2τ 0 )R 0 \ (1 − 2τ 0 )R 0 , where τ 0 ∈ (τ/2, τ ) is such that its boundary hyperplanes are L(τ )-good for Q 0 and where L(τ ) := (1/τ 2 )µ(Q 0 ). In other words, Q 0 −mod if Q 0 is given and if P is a boundary hyperplane of δ R 0 (recall that we consider only cubes R close to Q of 2r approximately the same size as Q), then t . (7.5) µ{x ∈ Q 0 : dist(x, P) ≤ t} ≤ L diam(Q 0 ) We can always guarantee that such a τ 0 ∈ (τ/2, τ ) exists because of the following lemma. The set of all hyperplanes parallel to a fixed one has natural Lebesgue measure on it. 7.3 All hyperplanes in Q parallel to a fixed one are L-good in the sense of (7.5) except for those in the set of A(diam(Q) µ(Q)/L) Lebesgue measure, where A is an absolute constant. LEMMA
Proof Let ` be a line orthogonal to our fixed hyperplane. Project µ|Q on this `. We get the measure µ` . If we apply the weak-type maximal theorem to this measure and to number λ = (L/diam(Q)), we obtain a small set on which the maximal function of µ` is large. And on the rest of hyperplanes orthogonal to `, we have inequality (7.5), which says exactly that our maximal function of µ` is smaller than λ = (L/diam(Q)). S Q 0 −mod Put δ RQ−mod := δ R 0 , where the union is taken over intersecting sons of Q, R. Q−mod Let Q ∂ = Q ∩ δ R . The symbol ∂ here means boundary; that is, this set is a fuzzy set around that part of the boundary of R which lies inside Q (actually around that part of the boundaries of all sons R 0 of R which lie inside Q). Note that Q ∂ ⊂ S Q ∩ R 0 is a son of R δ R 0 . Let Q sep = (Q \ δ RQ−mod ) \ (R ∩ Q) (the square Q without the intersection of Q and R and without the shaded part on Fig. 3); sep means separated (from R). Let 1 Q = (Q \ Q ∂ ) \ Q sep . Similarly, let us split R as R = Rsep ∪ R∂ ∪1 R , where all sets are disjoint. We use Q−mod R−mod here δ Q to construct Rsep and R∂ as we have just been using δ R to construct Q sep and Q ∂ . Finally, let 1 = 1 Q ∩ 1 R . Then we use repeatedly Q∂ ⊂ Qb,
R∂ ⊂ Rb .
(7.6)
We have (T 1 Q f, 1 R g) = T 1 Q f, (1 R g)sep + T 1 Q f, (1 R g)∂ + (T 1 Q f, χ1 1 R g).
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NAZAROV, TREIL, and VOLBERG
Rsep R
Q Q sep
Figure 3. Cutting out the bad part. The sets Q ∂ \ R and R∂ \ Q are the shaded parts of the squares Q and R, respectively. The set Q ∂ ∩ R is the image of Q ∂ \ R under reflections in boundary hyperplanes of R adjusted to this shaded Q ∂ \ R. (This is similar for R∂ ∩ Q.) Actually, the picture is slightly more complicated because we have to draw it for every pair of intersecting sons of Q and R.
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The first two terms are easy to estimate. Since Q and Rsep are separated, T 1 Q f, (1 R g)sep ≤ C(τ )k1 Q f k2 k1 R gk2 . The second term can be estimated as T 1 Q f, (1 R g)∂ ≤ kT k · k1 Q f k2 k(1 R f )b k2 because R∂ ⊂ Rb . To estimate the last term, let us write it as (T 1 Q f, χ1 1 R g) = (T χ1 1 Q f, χ1 1 R g) + T (1 Q f )∂ , χ1 1 R g + T (1 Q f )sep , χ1 1 R g .
(7.7)
Case 1. Let Q, R be usual cubes. To avoid tedious considerations and to make the explanation more clear, let us first assume that there is only one son of Q, say, Q 0 , and only one son of R, say, R 0 , such that their intersection is nonempty. Then 1, which was introduced above, consists of one rectangle. Also, there are constants c Q , c R such that χ1 1 Q f =: c Q b Q a χ1 , χ1 1 R f =: c R b R a χ1 . Remember that Q a , R a are some ancestors of Q, R, respectively. Also, |c Q |µ(Q 0 )1/2 ≤ C(δ)k1 Q f k2 . The same is true for R. We use Theorem A.1, Theorem A.2, or Theorem A.3 (depending on what we assumed) to conclude that the first term is bounded by C(τ )|c Q ||c R |µ(1)1/2 (µ(Q 0 )1/2 + µ(R 0 )1/2 ), which is bounded by C(τ )|c Q ||c R | min µ(Q 0 )1/2 , µ(R 0 )1/2 µ(Q 0 )1/2 + µ(R 0 )1/2 ≤ C(τ, δ)k1 Q f k2 k1 R gk2 . When several sons of Q intersect several sons of R, the reasoning is the same but more tedious. Let us indicate it. Let Q i be all sons of Q, and let R j be all sons of R. Then χ Q i 1 Q f =: c Q i b Q a χ Q i , χ R j 1 R g =: c R j b R a χ R j . For brevity, denote ci = c Q i , c j = c R j . Then |ci |µ(Q i )1/2 ≤ C(δ)k1 Q f k2 . The same is true for R. Let p denote a pair (i, j) such that Q i ∩ R j 6 = ∅. Let P be the set of all such pairs. Let 1 p = 1 ∩ Q i ∩ R j . We use Theorem A.1, Theorem A.2, or Theorem A.3 (depending on what we assumed) to conclude that the first term of (7.7) is bounded by C(τ )6 p=(i, j)∈P |ci ||c j |µ(1 p )1/2 (µ(Q i )1/2 + µ(R j )1/2 ) + 6 p, p0 ∈P , p6= p0 (T χ1 p 1 Q f, χ1 p0 1 R g). We denote the first sum by F, and we denote the second sum by S. Then F is bounded by C(τ )
X
|ci ||c j | min µ(Q i )1/2 , µ(R j )1/2 µ(Q i )1/2 + µ(R j )1/2 ≤ C(τ, δ)k1 Q f k2 k1 R gk2 .
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The other two terms in (7.7) and S can be estimated as above, so summarizing all, we get |(T 1 Q f, 1 R g)| ≤ Ck1 Q f k2 k1 R gk2 + kT k k1 Q f k2 k(1 R g)sep k2 + k(1 Q f )b k2 k1 R gk2 . Case 2. One of cubes Q, R is an unusual cube. Let it be Q. Only the estimate of (T 1 Q f, χ1 1 R g) differs from the previous ones. Again, let us assume as before that only one son of Q, say, Q 0 , intersects with sons of R, and, actually, with only one son of R, say, R 0 . The estimate of this term remains the same if Q 0 , R 0 are not terminal sons. So suppose that Q 0 is a terminal son. In our case, 1 is just one “rectangle.” Then χ1 1 Q f = (c0Q b Q 0 − c Q b Q a )χ1 . We can notice that always |c0Q | ≤ C(δ)h| f |i Q 0 and |c Q | ≤ C(δ)h| f |i Q . To estimate |(T 1 Q f, χ1 1 R g)|, we use Theorem A.1, Theorem A.2, or Theorem A.3 (depending on what we assumed) to conclude that the first term is bounded by C(τ )(|c Q | + |c0Q |)|c R |µ(1)1/2 (µ(Q 0 )1/2 + µ(R 0 )1/2 ), which is bounded by C(τ, δ)(|c Q | + |c0Q |)|c R | min µ(Q 0 )1/2 , µ(R 0 )1/2 µ(Q 0 )1/2 + µ(R 0 )1/2 , and this is bounded by C(τ, δ)(h| f |i Q + h| f |i Q 0 )µ(Q 0 )1/2 k1 R gk2 . In the general case (many sons of Q intersect with many sons of R), we repeat the reasoning indicated in Case 1. Case 3. Both cubes Q, R are unusual cubes. Again, let us assume as before that only one son of Q, say, Q 0 , intersects with sons of R, and, actually, with only one son of R, say, R 0 . And Q 0 , R 0 are both terminal. Then, similarly to the above estimate, we estimate |(T 1 Q f, χ1 1 R g)| by using Theorem A.1, Theorem A.2, or Theorem A.3 (depending on what we assumed) to conclude that the first term is bounded by C(τ, δ)(h| f |i Q + h| f |i Q 0 )µ(Q 0 )1/2 (h|g|i R + h| f |i R 0 )µ(R 0 )1/2 . For brevity we denote by N the family of pairs Q, R such that 2−r `(Q) ≤ `(R) ≤ 2r `(Q), dist(Q, R) ≤ τ min `(Q), `(R) . We have X X |(T 1 Q f, 1 R g)| ≤ C |(T χ1 1 Q f, χ1 1 R g)| N
N
+ kT k
X
k1 Q f k2 k(1 R g)b k2 + k(1 Q f )b k2 k1 R gk2 .
N
Now we can finish the estimate of
Pτ
1.
ˆ Rˆ denote the fathers In what follows, Q,
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299
of Q, R, respectively: X |(T χ1 1 Q f, χ1 1 R g)| N
≤ C(r, τ, δ)
X
k1 Q f k22
1/2 X
Q∈D
+ C(r, τ, δ)
+ C(r, τ, δ)
+ C(r, τ, δ)
k1 R gk22
1/2
R∈D 0
X
1/2 X 1/2 (h| f |i Qˆ + h| f |i Q )2 µ(Q) k1 R gk22
Q∈D , Q terminal
R∈D 0
1/2 X 1/2 (h| f |i Rˆ + h| f |i R )2 µ(R) k1 Q Fk22
X
R∈D 0 , R terminal
X
Q∈D
1/2 (h| f |i Qˆ + h| f |i Q )2 µ(Q)
Q∈D , Q terminal
×
1/2 (h| f |i Rˆ + h| f |i R )2 µ(R) .
X
R∈D 0 , R terminal
Using the Carleson measure property and (3.1), (3.2), we conclude that X |(T χ1 1 Q f, χ1 1 R g)| ≤ Ck f k2 kgk2 .
(7.8)
On the other hand, X k1 Q f k2 k(1 R g)b k2 + k(1 Q f )b k2 k1 R gk2 ≤ C ∗ τ k f k2 kgk2
(7.9)
N
N
because of the four following estimates: (7.4), a similar estimate for g, and (3.1), (3.2). Notice that all these estimates hold for at least 1/2 of all pairs of lattices if τ is extremely small. Choose τ maybe even smaller by requiring that C ∗τ <
1 . 4
Combining this with (7.8) and (7.9), we conclude that X N
1 |(T 1 Q f, 1 R g)| ≤ C(r, τ, δ)k f k2 kgk2 + kT kk f k2 kgk2 . 4
The estimate of the diagonal part is finished. 7.7. Second pulling ourselves up by the hair If we gather now our diagonal and off-diagonal estimates, we conclude that for any f, g one can find at least 7/8 of pairs of lattices for which we are able to go through
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our procedure and to prove 1 |(T f good , ggood )| ≤ C(r, τ, δ)k f good k2 kggood k2 + kT kk f good k2 kggood k2 . (7.10) 4 So we did prove (7.1) for many lattices. If we choose f, g such that |(T f, g)| ≥ (3/4)k f k2 kgk2 , we conclude the estimate for kT k. 7.8. From almost cubes to cubes In Theorems A.1 – A.3 one uses the existence of an accretive system for all almost cubes. But actually one can apply the same considerations if such a system exists Q−mod R−mod only for cubes. This can be achieved by choosing δ Q and δ R more carefully. Just move them slightly. It is possible because good hyperplanes are so abundant. By moving slightly, we achieve that all almost cubes in Theorems A.1 – A.3 are cubes. Let us provide slightly more detail. The rectangle S in Theorems A.1 – A.3 may be equal to 1 from Section 7.6 (1 may consist of several rectangles, but let us now think only about one of them). In principle, the ratio of different sides of 1 can be an arbitrary number between K (τ, r )−1 and K (τ, r ). (We recall that our cubes are well intersected and that τ = τ (r ) can be chosen very small for large r .) So, in principle, it may happen that s = 1 cannot be covered by the grid of equal cubes. However, let us recall the construction of 1: its boundary was formed by good hyperplanes, and we had the freedom to move them a bit. So we can always think that the ratio of any two sides of 1 is a rational number with denominator not exceeding n(τ, r ). This allows us to use actual cubes rather than almost cubes in Theorems A.1 – A.3. Appendix. Surgery by means of accretive functions Under our sets of assumptions (see Sec. 0), we perform the surgery expressed in the following theorems. Notice that the first theorem has absolutely no requirement of weak boundedness type. In contrast with that we have a weak boundedness requirement in the next two theorems. Here it takes the form of the antisymmetric property of the kernel of T . In the theorems below, S is always a special rectangle. We call the rectangle special or, more precisely, L-special if for every face F of our rectangle the following holds: µ x ∈ S : dist(x, F) ≤ t ≤ L
t . diam(S)
THEOREM A.1 Let S be an L-special rectangle with sides parallel to the axis and with the ratio of any two sidelengths between 2−r and 2r . Let φ1 and φ2 be two bounded functions,
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kφi k∞ ≤ B, such that kT φi k∞ ≤ K , i = 1, 2. Suppose that we also have an L ∞ system of accretive functions supported on almost cubes (our first set of assumptions). Then T (φ1 χ S ), φ2 χ S ≤ C1 (B, K , r, z) max µ(S), L 1/2 µ(S)1/2 . A.2 Let T have an antisymmetric kernel. Let S be an L-special rectangle with sides parallel to the axis and with the ratio of any two sidelengths between 2−r and 2r . Let φ1 and φ2 be two bounded functions, kφi k∞ ≤ B, such that kT φi kBMO2 ≤ K , i = 1, 2. THEOREM
3
Suppose that we also have a BMO23 -system of accretive functions supported on almost cubes (our second set of assumptions). Then |(T (φ1 χ S ), φ2 χ S )| ≤ C2 (B, K , r, z, 3, ε) max µ((1 + ε)S)1/2 µ(S)1/2 , L 1/2 µ(S)1/2 . THEOREM A.3 Let T have an antisymmetric kernel. Let S be an L-special rectangle with sides parallel to the axis and with the ratio of any two sidelengths between 2−r and 2r . Let φ1 and φ2 be two bounded functions, kφi k∞ ≤ B, such that kT φi kBMO2 ≤ K , i = 1, 2. 3
Suppose that we also have a BMO23 -system of accretive functions assigned to almost cubes (our third set of assumptions). Then |(T (φ1 χ S ), φ2 χ S )| ≤ C3 (B, K , r, z, 3, ε) max µ((1 + ε)S)1/2 µ(S)1/2 , L 1/2 µ(S)1/2 . Proof of Theorem A.1 Let ` be a smallest sidelength of S. Let d := `/10, and let us consider the cubic grids G with cubes of sidelength d. The number of cubes intersecting S of any such grid is bounded by C(r ). Now we want to choose such a grid in a special way. The hyperplane P is called S-good if µ x ∈ S : dist(x, P) ≤ t ≤ Aµ(S)
t . diam(S)
We call a grid S-good if all hyperplanes forming G and intersecting S are S-good. Notice that we can always choose an S-good grid with A = A(r ). Now we want the intersection of every cube of the grid with S to be an almost cube. The problem is with cubes intersecting the boundary of S. Let us delete the hyperplanes that are closest to faces of S. Then the rest of the hyperplanes give us the partition of S into almost cubes qi (almost cubes strictly inside S are just cubes).
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S We have a disjoint union S = i qi ∪ P , where P denotes the union of the faces of all our almost cubes except the boundary of S. Notice that P consists of (pieces of) S-good hyperplanes. Denote φ i := φ1 χqi , ψ i := φ2 χqi . Let i ◦ j mean that qi and q j are disjoint but have common boundary points (one can call them neighbors): X X X T (φ1 χ S ), φ2 χ S = (T φ i , ψ j ) + 6i (T φ i , ψ i ) =: + . i6 = j
1
2
Now (T φ i , ψ j ) when i 6= j but i, j are not neighbors is easy to estimate. In fact, by the first Calder´on-Zygmund property of T , we see that |(T (φ i ), ψ j )| ≤ C(z)`−m ||φ i || L 1 (qi ) ||ψ j || L 1 (q j ) ≤ C(B, z)µ(S). (A.1) P To estimate 1 , we are left to estimate (T φ i , ψ j ) when i ◦ j. We need the following lemma. LEMMA A.4 Let q1 , q2 be two disjoint almost cubes with ratio of any two sidelengths of both cubes between 1/2 and 2 and of comparable diameters, let P be a hyperplane containing their common boundary, and suppose that measure µ is such that τ , i = 1, 2. µ{x ∈ qi : dist(x, P) ≤ τ } ≤ M diam qi
Let φ, ψ be bounded functions supported by q1 , q2 with norms at most B. Then 1/2 |(T φ, ψ)| ≤ C(B, z)M 1/2 min µ(q1 ), µ(q2 ) . Proof Suppose that µ(q1 ) ≤ µ(q2 ). Let x ∈ q1 , and let d(x) := dist(x, P). By the rough estimate of measure and kernel of T ∗ , we conclude that |T ∗ ψ(x)| ≤ C(B, z) log
diam q2 . d(x)
Thus |(T φ, ψ)| ≤ ||T ∗ ψ|| L 2 (q1 ) ||φ|| L 2 (q1 ) Z 1/2 1/2 diam q2 ≤ C(B, z) log2 d µ(x) · µ(q1 ) d(x) q1 Z diam q1 1/2 diam q2 Mt 1/2 ≤ C(B, z) log2 d · µ(q1 ) t diam q1 0 ≤ C(B, z)M 1/2 µ(q1 )1/2 . We used here the fact that the diameters of our almost cubes are comparable.
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To estimate (T φ i , ψ j ) when i ◦ j, we use Lemma A.4 with M = µ(S) because all our hyperplanes are S-good. Using this and (A.1), we get X ≤ C(B, r, z)µ(S). 1
Now we estimate
P
2.
It is enough to get the estimate
|(T φ i , ψ i )| ≤ C(B, z)L 1/2 µ(S)1/2
(A.2)
to get X
≤ C(B, r, z)L 1/2 µ(S)1/2 .
2
P P Estimates of 1 , 2 finish the proof of Theorem A.1. So we are left to prove (A.2). Fix q = qi . Let us first consider the case when q is a boundary cube of S, that is, when it has a common face with S. We need the following lemma, which basically repeats the wording of Theorem A.1, and whose meaning is that we can always reduce everything to almost cubes. (Everything is reduced even to usual cubes in the applications of our theorems.) A.5 Let q be an almost cube, let φ1 , φ2 be two bounded functions with norms at most B, and let T be an operator with Calder´on-Zygmund kernel with parameters z. Suppose that T φ1 is a bounded function with norm at most B. Suppose that the measure µ on q is such that for every face P of q, LEMMA
µ x ∈ qi : dist(x, P) ≤ t ≤ L
t . diam q
Suppose that we also have an L ∞ -system of accretive functions supported on almost cubes (our first set of assumptions). Then |(T φ1 χq , φ2 χq )| ≤ C(B, z) max µ(q), L 1/2 µ(q)1/2 . Proof Consider 2q: (T φ1 χq , φ2 χq ) = (T φ1 , φ2 χq ) − (T φ1 χRn \2q , φ2 χq ) − (T φ1 χ2q\q , φ2 χq ) =: t1 − t2 − t3 . We have the following estimates: Estimate of t3 : |t3 | = |(T φ1 χ2q\q , φ2 χq )| ≤ C(B, z)L 1/2 µ(q)1/2
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by Lemma A.4; Estimate of t1 : |t1 | = |(T φ1 , φ2 χq )| ≤ C(B)µ(q) by the boundedness of T φ1 ; Estimate of t2 : let us denote f := T (φ1 χRn \2q ) and notice that ∀x, y ∈ q,
| f (x) − f (y)| ≤ C(B, z).
Now we use the fact that we have an L ∞ -system of accretive functions supported on almost cubes. In particular, let bq be a function bounded by K together with T ∗ bq , supported on q, and such that its average over q is 1. Let us denote βq := ( f, bq /µ(q)). Our last inequality implies that ∀x ∈ q,
| f (x) − βq | ≤ C(B, z).
(A.3)
Now |t2 | ≤ |( f − βq , φ2 χq )| + |βq |µ(q) =: t21 + t22 . Estimate (A.3) immediately gives t21 ≤ C(B, z)µ(q). To estimate t22 , we are left to prove L 1/2 |βq | ≤ C(B, K , z) + C(B, z) . (A.4) µ(q)1/2 We write βq :=
bq bq bq = T (φ1 χRn \2q ), = T φ1 , µ(q) µ(q) µ(q) bq bq − T φ1 χ2q\q , − T φ1 χq , =: b1 − b2 − b3 . µ(q) µ(q)
f,
Obviously, |b1 | ≤ C(B, z) because T φ1 is bounded. The estimate for b2 again follows from Lemma A.4: |b2 | ≤ C(B, z)
L 1/2 . µ(q)1/2
To estimate b1 , we write bq bq |b1 | = T φ1 χq , = φ1 χq , T ∗ µ(q) µ(q) −1 ≤ ||T ∗ bq ||∞ ||φ1 χq ||1 µ(q) ≤ C(B, K , z). Estimate (A.4) is proved, and the lemma is finished.
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Thus we proved (A.2), and this finishes the proof of Theorem A.1 because we can apply our estimates not only to almost cubes adjusted to the boundary of S but to cubes qi strictly inside S as well, just by noticing that they are L-special with L ≤ Aµ(S). Taking into account that the number of cubes qi is bounded by C(r ), we are done. Proof of Theorem A.2 We keep the same notation. In particular, we subdivide S into qi . (Inside S, these are cubes; near ∂ S, these are almost cubes.) The subdivision is made by S-good hyperplanes. But here is a hitch. We now have two more parameters: a large 3 > 1 and a small ε > 0. We want to have diam qi so small that all dilated almost cubes 3qi are in (1 + ε)S. This is not a problem, but the number of qi ’s now depends on 3, ε, r ; we call it C(3, ε, r ). Also, the measure of the neighborhoods of S-good hyperplanes of the subdivision now satisfies τ µ x ∈ S : dist(x, P) ≤ τ ≤ A(3, ε, r ) . (A.5) diam S It is easy to see that estimates of |(T φ i , ψ j )| when i 6= j follow exactly along the same lines as in the proof of Theorem A.1. We just need to estimate |(T φ i , ψ i )|, which brings us to the following lemma. A.6 Suppose that q is an almost cube, that φ1 , φ2 are two bounded functions with norms at most B, and that T is an operator with Calder´on-Zygmund kernel with parameters z. Suppose that T φ1 is a BMO23 -function with norm at most B. Suppose that the measure µ on q is such that for every face P of q, τ µ x ∈ q : dist(x, P) ≤ τ ≤ L . diam q LEMMA
Suppose that we also have a BMO23 -system of accretive functions supported on almost cubes (our second set of assumptions). Then |(T φ1 χq , φ2 χq )| ≤ C(B, z) max µ(3q)1/2 µ(q)1/2 , L 1/2 µ(q)1/2 . Proof Suppose first that φ1 has compact support. We are given that T φ1 ∈ BMO23 , which we can understand in the usual sense. Let cq denote a constant such that Z |T φ1 (x) − cq |2 dµ(x) ≤ Bµ(3q). (A.6) q
Let us write (T φ1 χq , φ2 χq ) = (T φ1 − cq , φ2 χq ) − (T φ1 χRn \3q − cq , φ2 χq ) − (T φ1 χ3q\q , φ2 χq ) =: t1 − t2 − t3 . Then we have the following estimates:
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Estimate of t3 : |t3 | = |(T φ1 χ3q\q , φ2 χq )| ≤ C(B, z)L 1/2 µ(q)1/2 by Lemma A.4; Estimate of t1 : |t1 | = |(T φ1 − cq , φ2 χq )| ≤ C(B)µ(3q)1/2 µ(q)1/2 by (A.6); Estimate of t2 : let us denote by f := T (φ1 χRn \3q ), and let us notice that ∀x, y ∈ q,
| f (x) − f (y)| ≤ C(B, z).
Now we use that we have a BMO23 -system of accretive functions supported on almost cubes. In particular, let bq be a function with ||bq ||∞ ≤ K for which BMO23 norm of T ∗ bq is bounded by K , supported on q, and such that its average over q is 1. Let us denote βq := ( f, bq /µ(q)). Our last inequality implies that ∀x ∈ q,
| f (x) − βq | ≤ C(B, z).
(A.7)
Now to estimate t2 , we start with the following decomposition: bq bq − T φ1 , T φ1 χRn \3q − cq = ( f − βq ) + T φ1 χRn \3q , µ(q) µ(q) bq + T φ1 − cq , =: T1 + T2 + T3 . µ(q) We have the following estimates: Estimate of T1 : |T1 | ≤ C(B, z) by (A.7); Estimate of T3 : |T3 | ≤ C(B)
µ(3q)1/2 ; µ(q)1/2
Estimate of T2 : bq bq −T2 = T φ1 χ3q\q , + T φ1 χq , =: T21 + T22 ; µ(q) µ(q) Estimate of T21 : |T21 | ≤
L 1/2 µ(q)1/2
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by Lemma A.4; Estimate of T22 : We are given that T bq ∈ BMO23 (µ). Function bq has compact support, so we understand this in the usual sense. In particular, let kq be a constant such that kT (bq ) − kq k2L 2 (q,µ) ≤ K µ(3q). Then bq bq T22 = T φ1 χq , = φ1 χq , T µ(q) µ(q) b −1 kq q = φ1 χq , T − + kq φ1 χq , χq )(µ(q) . µ(q) µ(q) Then
T (bq ) − kq µ(3q)1/2 ≤K φ1 χq , µ(q) µ(q)1/2
because the BMO23 -norm of bq is bounded by K . Also, |kq | = |(T (bq ) − kq , bq /µ(q))|. We use here the fact that T has an antisymmetric kernel; in fact, this gives (T (bq ), bq ) = 0. (Recall also that bq is chosen to have average 1 over q.) From this we deduce now an estimate of kq : |kq | ≤ K (µ(3q)1/2 /µ(q)1/2 ). Thus |T22 | ≤ K
−1 µ(3q)1/2 µ(3q)1/2 + |k ||(φ χ , χ ) µ(q) | ≤ C(K ) . q q 1 q µ(q)1/2 µ(q)1/2
Now t2 = (T1 + T2 + T3 , φ2 χq ), and our previous estimates together give |t2 | ≤ C(K , B, z) max L 1/2 µ(q)1/2 , µ(3q)1/2 µ(q)1/2 , which proves Lemma A.6 in the case when φ1 has compact support. We now have to modify the proof for the case when φ1 does not have compact support. Then we interpret T φ1 as a functional on L 20 (q, µ). First, introduce φ˜2 := R φ2 − ( q φ2 dµ)(bq /µ(q)). Then we write Z −1 φ2 dµ (T φ1 χq , bq ) µ(q) . (T φ1 χq , φ2 χq ) = (T φ1 χq , φ˜2 χq ) + q
K µ(3q)1/2 (µ(q))1/2 .
The second term is bounded by Just return to the estimate of T22 to see this. As for the first term, we first write the definition of the functional T φ1 using (0.2), (0.3): (T φ1 χ3q , φ˜2 χq ) = (T φ1 )(φ˜2 χq ) + (T ! φ1 χRn \3q , φ˜2 χq ), which implies (T φ1 χq , φ˜2 χq ) = (T φ1 )(φ˜2 χq ) − (T ! φ1 χRn \3q , φ˜2 χq ) − (T φ1 χ3q\q , φ˜2 χq ) =: t10 − t20 − t30 .
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Now |t30 | ≤ C(B, z)L 1/2 µ(q)1/2 by Lemma A.4. The estimate is the same as that of t3 at the beginning of our proof. Similarly, |t10 | ≤ C(B)µ(3q)1/2 µ(q)1/2 , exactly as t1 because of our assumption that T φ1 belongs to BMO23 (µ). Turning to t20 , we notice that (0.3) contains a formula for T ! φ1 χRn \3q , from which it follows that |T ! φ1 χRn \3q | ≤ C(B, z) on q. Thus, clearly, |t20 | ≤ C(B, z)µ(q). Gathering everything together, we get the required estimate of |(T φ1 χq , φ2 χq )|. This proves Lemma A.6. Theorem A.2 is also completely proved. Proof of Theorem A.3 First, let us consider the case when φ1 and all bq have compact support. We repeat the proof of Theorem A.2 word for word until the function bq appears. Then we replace all entries of bq by bq χq . Here is the difficulty: in the estimate of T22 , the function T (bq χq ) − kq cannot be estimated trivially in L 2 (q). We start from this place: b χ b kq q q q T22 = φ1 χq , T = φ1 χq , T − µ(q) µ(q) µ(q) b χ n −1 k q R \3q q − φ1 χq , T − φ1 χq , T (bq χ3q\q ) µ(q) − µ(q) µ(q) =: T221 − T222 − T223 . Clearly, we can now estimate |T221 | ≤ K (µ(3q)1/2 /µ(q)1/2 ), and T223 can be estimated by Lemma A.4. To estimate T222 , we need to write bq χq n n n T (bq χR \3q ) − kq = T (bq χR \3q ) − T (bq χR \3q ), µ(q) bq χq bq χq + T (bq χRn \3q ), − T (bq ), µ(q) µ(q) bq χq + T (bq ) − kq , µ(q) =: B1 + B2 + B3 . Clearly, Calder´on-Zygmund properties of the kernel show that |B1 | ≤ C(B, z). By the BMO property of T bq , we get |B3 | ≤ K (µ(3q)1/2 /µ(q)1/2 ). On the other hand, bq χq bq χq |B2 | = T (bq χ3q ), = T (bq χ3q\q ), µ(q) µ(q) max(L 1/2 , µ(S)1/2 ) ≤B . µ(q)1/2
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Finally, all this together proves that |T22 | ≤ C(K , B, z)
max(L 1/2 , µ(S)1/2 ) . µ(q)1/2
After that, we finish the proof in exactly the same fashion as the proof of Theorem A.2. To get rid of the compactness assumption, let us consider E := (T φ1 χq , bq χq ), and let us prove the same estimate that we have already proved under the assumption of compactness of supports: |E| ≤ C(K , B, z)µ(3q)1/2 µ(q)1/2 .
(A.8)
Suppose that this is done. Then (exactly as at the end of the proof of Theorem A.2) we interpret T φ1 as a functional on L 20 (q, µ). Again, introduce φ˜2 := R φ2 − ( q φ2 dµ)(bq χq /µ(q)). Then we write Z −1 (T φ1 χq , φ2 χq ) = (T φ1 χq , φ˜2 χq ) + φ2 dµ (T φ1 χq , bq χq ) µ(q) . q
The second term is bounded by K µ(3q)1/2 (µ(q))1/2 because of (A.8). The estimate of the first term is verbatim the one at the end of the proof of Theorem A.2. To prove (A.8), let us use the antisymmetry of T . We write φ˜1 := φ1 − R ( q φ1 dµ)(bq χq /µ(q)). Then E = (T φ1 χq , bq χq ) = −(φ1 χq , T (bq χq )) = R −(φ˜1 χq , T (bq χq )) − ( φ1 dµ)(bq χq /µ(q), T (bq χq )). The last term vanishes beq
cause T is antisymmetric. Thus (using the definition of T ! right after (0.3)) −E = (φ˜1 χq , T (bq χq )) = (φ˜1 χq , T bq ) − (φ˜1 χq , T ! bq χRn \3q ) − (φ˜1 χq , T bq χ3q\q ) =: E 1 − E 2 − E 3 . Now E 1 can be estimated by K µ(3q)1/2 µ(q)1/2 by the assumption that T bq ∈ BMO23 (µ). The term E 3 can be estimated by Lemma A.4. As for E 2 , we can notice that |T ! bq χRn \3q | (see the definition of T ! right after (0.3)) is bounded by B on q. Thus |E 2 | ≤ C(B)µ(q). We are done with Theorem A.3. Acknowledgments. We are very grateful to the referee for many valuable remarks. We express our sincere gratitude to Michael Frazier and Jacob Plotkin for reading the manuscript and for their numerous remarks. References [1]
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estimates for singular integral operators and to other classical problems of harmonic analysis, St. Petersburg Math. J. 8 (1997), 721 – 824. MR 99d:42026 263, 273 F. NAZAROV, S. TREIL, and A. VOLBERG, Cauchy integral and Calder´on-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 1997, 703 – 726. MR 99e:42028 269, 277, 286 F. NAZAROV, S. TREIL, and A. VOLBERG, Weak type estimates and Cotlar inequalities for Calder´on-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 1998, 463 – 487. MR 99f:42035 260, 265 F. NAZAROV, S. TREIL, and A. VOLBERG, T b-theorem on non-homogeneous spaces, preprint, 1999, http://math.msu.edu/˜volberg 260, 261, 263, 269, 278 F. NAZAROV and A. VOLBERG, On analytic capacity of portions of continuum and a theorem of Guy David, preprint, 1999, Erwin Schr¨odinger International Institute for Mathematical Physics, ESI 718, http://esi.ac.at/preprints/ESI-Preprints.html 264 E. M. STEIN, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, with assistance of Timothy S. Murphy, Princeton Math. Ser. 43, Monogr. Harmon. Anal. 3, Princeton Univ. Press, Princeton, 1993. MR 95c:42002 260 X. TOLSA, Cotlar’s inequality without the doubling condition and existence of principal values for the Cauchy integral of measures, J. Reine Angew. Math. 502 (1998), 199 – 235. MR 2000a:42030 , L 2 -boundedness of the Cauchy integral operator for continuous measures, Duke Math. J. 98 (1999), 269 – 304. MR 2000d:31001 263 , Curvature of measures, Cauchy singular integral, and analytic capacity, Ph.D. dissertation, Universitat Aut`onoma de Barcelona, 1998. , BMO, H 1 , and Calder´on-Zygmund operators for non doubling measures, preprint, arXiv:math.CA/0002152 263 J. VERDERA, On the T (1)-theorem for the Cauchy integral, Ark. Mat. 38 (2000), 183 – 199. MR 2001e:30074
Nazarov Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA;
[email protected] Treil Department of Mathematics, Brown University, 151 Thayer Street, Box 1917, Providence, Rhode Island 02912, USA;
[email protected]
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Volberg Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA;
[email protected]; Equipe d’Analyse, Universit´e Paris VI, 4 Place Jussieu, Tour 46, 4i`eme e´ tage, 75252 Paris CEDEX 05, France
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 2,
THE PRIME NUMBER RACE AND ZEROS OF L-FUNCTIONS OFF THE CRITICAL LINE KEVIN FORD and SERGEI KONYAGIN
Abstract We examine the effects of certain hypothetical configurations of zeros of Dirichlet L-functions lying off the critical line on the distribution of primes in arithmetic progressions. 1. Introduction Let πq,a (x) denote the number of primes p 6 x with p ≡ a (mod q). The study of the relative magnitudes of the functions πq,a (x) for a fixed q and varying a is known colloquially as the “prime race problem” or the “Shanks-R´enyi prime race problem.” Fix q and distinct residues a1 , . . . , ar with (ai , q) = 1 for each i. Consider a game with r players called “1” through “r ” as colorfully described in the first paper of [KT1]. At time t, each player “ j” has a score of πq,a j (t) (i.e., player “ j” scores 1 point whenever t reaches a prime ≡ a j (mod q)). As t → ∞, will each player take the lead infinitely often? More generally, will all r ! orderings of the players occur for infinitely many integers t? It is generally believed that the answer to both questions is yes for all q, a1 , . . . , ar . As first noted by P. Chebyshev [Ch] in 1853, some orderings may occur far less frequently than others (e.g., if q = 3, a1 = 1, a2 = 2, then player “1” takes the lead for the first time when t = 608, 981, 813, 029; see [BH]). More generally, when r = 2, a1 is a quadratic residue modulo q, a2 is a quadratic nonresidue modulo q, and πq,a2 (x) − πq,a1 (x) tends to be positive more often than it is negative (this phenomenon is now called Chebyshev’s bias). In 1914, J. Littlewood [L] proved that the functions π4,3 (x) − π4,1 (x) and π3,2 (x) − π3,1 (x) both change sign infinitely often. Later S. Knapowski and P. Tur`an [KT1], [KT2] proved for many q, a, b that πq,b (x) − πq,a (x) changes sign infinitely often. The distribution of the functions πq,a (x) is closely linked with the distribution of the zeros in the critical DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 2, Received 30 January 2001. 2000 Mathematics Subject Classification. Primary 11N13, 11M26. Ford’s work supported in part by National Science Foundation grant number DMS-0070618. Konyagin’s work supported by grant number 99-01080 from the International Association for the Promotion of Co-operation with Scientists from the New Independent States of the Former Soviet Union. 313
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strip 0 < <s < 1 of the Dirichlet L-functions L(s, χ) for the characters χ modulo q. Some of the results of Knapowski and Tur`an are proved under the assumption that the functions L(s, χ) have no real zeros in (0, 1), or that for some number K q , the zeros of the functions L(s, χ) with |=s| 6 K q all have real part equal to 1/2. Theoretical results for r > 2 are more scant, all depending on the unproven extended Riemann hypothesis for q (ERHq ), which states that all these zeros lie on the critical line <s = 1/2. J. Kaczorowski [K1], [K2], [K3] has shown that the truth of ERHq implies that for many r -tuples (q, a1 , . . . , ar ), πq,a1 (x) > · · · > πq,ar (x) for arbitrarily large x. If, in addition to ERHq , one assumes that the collection of nontrivial zeros of the L-functions for characters modulo q are linearly independent over the rationals (grand simplicity hypothesis (GSHq )), M. Rubinstein and P. Sarnak [RS] have shown that for any r -tuple of coprime residue classes a1 , . . . , ar modulo q, all r ! orderings of the functions πq,ai (x) occur for infinitely many integers x. In fact, they prove more, that the logarithmic density of the set of real x for which any such inequality occurs exists and is positive. In light of the results of Littlewood and of Knapowski and Tur`an, one may ask if such results for r > 2 can be proved without the assumption of ERHq . In particular, can it be shown, for some quadruples (q, a1 , a2 , a3 ), that the six orderings of the functions πq,ai (x) occur for infinitely many integers x without the assumption of ERHq (while still allowing the assumption that zeros with imaginary part < K q lie on the critical line for some constant K q )? In this paper we answer this question in the negative (in a sense) for all quadruples (q, a1 , a2 , a3 ). Thus, in a sense, the hypothesis ERHq is a necessary condition for proving any such results when r > 2. Let Cq be the set of nonprincipal characters modulo q. Let D = (q, a1 , a2 , a3 ), where a1 , a2 , a3 are distinct residues modulo q which are coprime to q. Suppose for each χ ∈ Cq that B(χ) is a sequence of complex numbers with positive imaginary part (possibly empty, duplicates allowed), and denote by B the system of B(χ ) for χ ∈ Cq . Let n(ρ, χ) be the number of occurrences of the number ρ in B(χ ). The system B is called a barrier for D if the following hold: (i) all numbers in each B(χ) have real part in [β2 , β3 ], where 1/2 < β2 < β3 6 1; (ii) for some β1 satisfying 1/2 6 β1 < β2 , if we assume that for each χ ∈ Cq and ρ ∈ B(χ ), L(s, χ) has a zero of multiplicity n(ρ, χ ) at s = ρ, and all other zeros of L(s, χ) in the upper half-plane have real part 6 β1 , then one of the six orderings of the three functions πq,ai (x) does not occur for large x. If each sequence B(χ) is finite, we call B a finite barrier for D and denote by |B | the sum of the number of elements of each sequence B(χ ), counted according to multiplicity.
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THEOREM 1.1 For every real number τ > 0 and σ > 1/2 and for every D = (q, a1 , a2 , a3 ), there is a finite barrier for D, where each sequence B(χ) consists of numbers with real part 6 σ and imaginary part > τ . In fact, for most D, there is a barrier with |B | 6 3.
We do not claim that the falsity of ERHq implies that one of the six orderings does not occur for large x. For example, take σ > 1/2, and suppose that each nonprincipal character modulo q has a unique zero with positive imaginary part to the right of the critical line, at σ + iγχ . If the numbers γχ are linearly independent over the rationals, it follows easily from Lemma 1.1 and the Kronecker-Weyl theorem that in fact all φ(q)! orderings of the functions {πq,a (x) : (a, q) = 1} occur for an unbounded set of x. We now present a general formula for πq,a (x) in terms of the zeros of the functions L(s, χ). Throughout this paper, constants implied by the Landau O and Vinogradov symbols may depend on q, but not on any other variable. 1.2 Let β > 1/2, let x > 10, and for each χ ∈ Cq , let B(χ) be the sequence of zeros (duplicates allowed) of L(s, χ) with <s > β and =s > 0. Suppose further that all L(s, χ ) are zero-free on the real segment 0 < s < 1. If (a, q) = (b, q) = 1 and x is sufficiently large, then LEMMA
φ(q) πq,a (x) − πq,b (x) i h X X = −2< χ(a) − χ(b) f (ρ) + O(x β log2 x), χ∈Cq
ρ∈B(χ ) |=ρ|6x
where f (ρ) :=
xρ 1 + ρ log x ρ
x
Z 2
tρ t log2 t
dt =
x <ρ xρ . +O ρ log x |ρ|2 log2 x
Proof Let 3(n) be the von Mangolt function, and define X X 9q,a (x) = 3(n), 9(x; χ) = 3(n)χ (n). n6x n≡a (mod q)
n6x
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Let Dq be the set of all Dirichlet characters χ modulo q. Then X 3(n) πq,a (x) = + O(x 1/2 ) log n n6x n≡a (mod q) Z x d9q,a (t)
+ O(x 1/2 ) log t Z x 9q,a (x) 9q,a (t) = + dt + O(x 1/2 ) 2 log x 2 t log t 9(x; χ) Z x 9(t; χ) 1 X + dt + O(x 1/2 ). = χ(a) 2 log x φ(q) 2 t log t =
2−
χ∈Dq
Then φ(q) πq,a (x) − πq,b (x) Z x X 9(x; χ) 9(t; χ) χ (a) − χ(b) dt + O(x 1/2 ). (1.1) + = 2 log x t log t 2 χ∈C q
By well-known explicit formulas (see [D, Chap. 19, (7), (8)]), when χ ∈ Cq , X xρ + O(log2 x), 9(x; χ) = − ρ
(1.2)
|=ρ|6x
where the sum is over zeros ρ of L(s, χ) with 0 < <ρ < 1. Since the number of zeros with 0 6 =ρ 6 T is O(T log T ) (see [D, Chap. 16, (1)]), by partial summation we have X 1 X x ρ x β log2 x. 6 xβ ρ |ρ| 0<=ρ6x <ρ6β
0<=ρ6x
The implied constant depends on the character and hence only on q. By (1.2), X xρ 9(x; χ) = − + O(x β log2 x). (1.3) ρ |=ρ|6x <ρ>β
The first part of the lemma follows by inserting (1.3) into (1.1) and combining zeros ρ of L(s, χ ) and ρ of L(s, χ). Lastly, if 1/2 6 σ = <ρ, integration by parts gives Z Z x tρ t ρ x 2 x t ρ−1 dt = dt + 2 ρ log2 t 2 ρ 2 log3 t 2 t log t Z √x Z x i xσ 1 h 1 8 σ −1 σ −1 t dt + t dt + |ρ| log3 2 2 |ρ| log2 x log3 x √x σ x . |ρ| log2 x
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This completes the proof of the lemma. In the next three sections, we show several methods for constructing barriers, which, by Lemma 1.1, boil down to analyzing the two functions <
X χ∈Cq
X xρ χ(a j ) − χ(a3 ) ρ
( j = 1, 2).
ρ∈B(χ)
In Section 2 we construct a barrier using two simple zeros (one of which may be a zero for several characters). Section 3 details a method using a zero for L(s, χ) and a zero for L(s, χ 2 ) (for most D these are simple or double zeros). Lastly, Section 4 presents a more general method with two numbers, which are zeros for each character of certain high multiplicities. Together, the three constructions provide barriers for all quadruples (q, a1 , a2 , a3 ). All of the constructions in Sections 2 – 4 involve two zeros, one with imaginary part t and the other with imaginary part 2t. Thus, we assume that both ERHq and GSHq are false. Answering a question posed by Sarnak, in Section 5 we construct a barrier (with an infinite set B(χ)) where the imaginary parts of the numbers in the sets B(χ ) are linearly independent; in particular, we assume that all zeros of each L(s, χ ) are simple and that L(s, χ1 ) = 0 = L(s, χ2 ) does not occur for χ1 6= χ2 and <s > β2 . We adopt the following notation: e(z) = e2πi z ; bxc is the greatest integer 6 x; dxe is the least integer > x; {x} = x − bxc is the fractional part of x; and kxk is the distance from x to the nearest integer. Also, arg z is the argument of the nonzero complex number z lying in [−π, π). Throughout, q = 5 or q > 7, and (a1 , q) = (a2 , q) = (a3 , q) = 1. 2. First construction LEMMA 2.1 If, for some relabeling of the numbers ai , there is a set S of nonprincipal Dirichlet characters modulo q such that X X X χ(a1 ) = χ(a2 ) 6 = χ(a3 ), χ∈S
χ ∈S
χ ∈S
then there is a barrier B for D = (q, a1 , a2 , a3 ) with |B | 6 |S| + 1. Remark. The hypotheses of Lemma 2.1 are satisfied when, for example, q has a primitive root g, and a3 /a2 is not in the subgroup of (Z/qZ)∗ generated by a2 /a1 . Writing a2 /a1 ≡ g f , we take the character with χ(g) = e 1/( f, φ(q)) and S = {χ}.
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Proof Suppose that 1/2 6 β < σ2 < σ1 6 min(σ, 0.501), and let χ2 be a character with χ2 (a1 ) 6= χ2 (a2 ) (χ2 may or may not be in S). Let Tq be a large number, depending only on q. Let ρ1 = σ1 + it, ρ2 = σ2 + 2it where t > Tq . Suppose L(s, χ ) has a simple zero at s = ρ1 for each χ ∈ S, that L(s, χ2 ) has a simple zero at s = ρ2 , and that no other nontrivial zeros of any L-function in Cq have real part exceeding β. Let D1 (x) := φ(q) πq,a1 (x) − πq,a2 (x) , D2 (x) := φ(q) πq,a3 (x) − πq,a2 (x) . By Lemma 1.1 and our hypotheses, if x is sufficiently large, 1 i 2x σ2 h e2it log x < W +O , W = χ 2 (a2 ) − χ 2 (a1 ), log x σ2 + 2it log x 1 i X 2x σ1 h eit log x D2 (x) = < Z +O , Z= χ(a2 ) − χ(a3 ) . log x σ1 + it log x D1 (x) =
χ∈S
Define
1 eit log x 1
A(x) = arg Z − π σ1 + it 2
1
= t log x + arg Z + tan−1 (σ1 /t) . π If A(x) > (log x)−1/2 , then |D2 (x)| x σ1 / log3/2 x. But D1 (x) = O(x σ2 ), so for such x, πq,a3 (x) is either the largest or the smallest of the three functions. When A(x) < (log x)−1/2 , then C(x) := arg
e2it log x
W
σ2 + 2it σ π 2 ≡ arg W − + tan−1 + 2t log x 2 2t σ 1 σ 1 2 − 2 arg Z − 2 tan−1 +O p ≡ arg W + tan−1 2t t log x
≡ arg W − 2 arg Z − F(x)
(mod π ),
where 1/(2t) < F(x) < 1/t for large x. The number of possibilities for arg W − 2 arg Z depends only on q; hence we may assume that n1 o 1 B= (arg W − 2 arg Z ) − π 2 satisfies either B = 0 or |B| > 2/t > 2F(x) (by taking Tq sufficiently large). We have π C(x) ≡ π B + − F(x) (mod π ). 2
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319
If B = 0, then C(x) is either π/2 − F(x) or 3π/2 − F(x) (mod 2π), whence D1 (x) takes only one sign for such x. Likewise, C(x) ∈ (π/2 + 2/t, π ) if B > 2/t, and C(x) ∈ (−F(x), π/2 − 2/t) if B < −2/t. In all cases, when A(x) < (log x)−1/2 , D1 (x) takes only one sign. Therefore, one of the orderings πq,a1 (x) > πq,a3 (x) > πq,a2 (x) or πq,a2 (x) > πq,a3 (x) > πq,a1 (x) does not occur for large x. Remark. By similar reasoning, for any integer k > 2, one may construct a barrier with one zero having imaginary part t and another zero having imaginary part kt. 3. Second construction The basic idea of this section is to find a character χ such that the values χ (a1 ), χ(a2 ), χ (a3 ) are nicely spaced around the unit circle, but not too well spaced (e.g., cube roots of 1 or translates thereof). In almost all circumstances we can find such a character. LEMMA 3.1 Let s1 = ordq (a2 /a1 ), s2 = ordq (a3 /a2 ), and s3 = ordq (a1 /a3 ). If one of s1 , s2 , s3 is not in {3, 7, 13, 21}, then for some relabeling of the ai ’s, there is a Dirichlet character χ satisfying either (i) χ (a1 ) = χ (a2 ) 6 = χ(a3 ) or (ii) χ (ai ) = e(ri ) with 0 6 r1 < r2 < r3 < 1, and d1 = r2 − r1 , d2 = r3 − r2 satisfy
1 1 < d1 6 d2 < 3 2
or
(d1 , d2 ) ∈
n 6 9 12 16 o , , , . 19 19 37 37
(3.1)
Remark. In the case when (i) holds, the hypotheses of Lemma 2.1 hold with S = {χ }, and thus there is a finite barrier for D with |B | = 2. Therefore, in this section we confine ourselves to the case when (ii) holds (see Lemma 3.5). Before proving Lemma 3.1, we begin with some simple lemmas about the existence of characters with certain properties. LEMMA 3.2 Suppose q > 3 and (b, q) = 1. Let m be the order of b modulo q. Then there is a Dirichlet character χ modulo q with χ(b) = e(1/m).
Proof f f Suppose that g1 , . . . , gt generate (Z/qZ)∗ and that b = g1 1 · · · gt t . Let si = f ordq gi for each i, and let si0 be the order of gi i . Then si0 = si /( f i , si ) and
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m = lcm[s10 , . . . , st0 ]. Let f i0 = f i /( f i , si ), so that in particular (si0 , f i0 ) = 1. The gcd of the t + 1 numbers m, f i0 m/si0 is 1, so there are integers h 1 , . . . , h t such that P h i ( f i0 m/si0 ) ≡ 1 (mod m). Take the character χ with χ(gi ) = e(h i /si ) for each i; Q then χ (b) = χ(gi ) fi = e(h 1 f 10 /s10 + · · · + h t f t0 /st0 ) = e(1/m). 3.3 Suppose that b, c are distinct residues modulo q with (b, q) = (c, q) = 1. Suppose that r | ordq b and that for every pa kr with a > 1, pa+1 - ordq c. Then there is a Dirichlet character χ modulo q such that LEMMA
χ(b) = e(1/r ),
χ(c)r = 1.
Proof Let s1 = ordq b and s2 = ordq c. By Lemma 3.2, there is a character χ1 with χ1 (b) = e(1/s1 ) and therefore a character χ2 with χ2 (b) = e(1/r ). Since c has order s2 , χ2 (c) = e(g/s2 ) for some integer g. Write s2 = vu, where (u, r ) = 1 and v|r . Define x by xu ≡ 1 (mod r ), and let χ = χ2xu . Then χ(b) = χ2 (b)xu = e(1/r ) and χ (c) = e(gxu/s2 ) = e(gx/v) = e(gx(r/v)/r ). Definition An odd number m is good if for every j, 1 6 j 6 m − 1, there is a number k such that among the points (0, k/m, k j/m) modulo 1, either two are equal (and not equal to the third), or two of the three distances d1 , d2 , d3 (with sum = 1) between the points satisfy (3.1). Remark. To prove that a number m is good, we need only check that 2 6 j 6 (m + 1)/2, since for j = 1 we take k = 1, and if k works for j = j0 , then the same k works for j = m + 1 − j0 . 3.4 Every odd prime p except p ∈ P = {3, 7, 13} is good, and for p ∈ P, p 2 is good. Also, the numbers 39, 91, and 273 are good. LEMMA
Proof A short computation implies that if p ∈ P, then p is not good, but p 2 is good. Also, by a short computation, all other odd primes 6 83 are good, as well as 39, 91, and 273. The following j values have no associated k-value: for m = 3, j = 2; for m = 7, j = 3, 5; for m = 13; j = 3, 5, 6, 8, 9, 11; for m = 21, j = 5, 17. Suppose that m = p > 84 is prime, and write each product k j = `p + r with 0 6 r < p. We prove that for each j ∈ [2, ( p + 1)/2], there is a k such that two
PRIME NUMBER RACE AND ZEROS OF L-FUNCTIONS
321
of the three distances satisfy 1/3 < d1 6 d2 < 1/2. We now divide up the j ∈ [2, ( p + 1)/2] into 9 cases. Case I: j ∈ {3, 5, 7, ( p + 1)/2}. For j = 3, take p/6 < k < 2 p/9, and for j = 5, 7, take any k with p/(2 j) < k < p/(2 j − 2). There is such a k when p > 84. Then p − jk and jk − k both lie in ( p/3, p/2). For j = ( p + 1)/2, take k = 2d p/3e; then r = d p/3e, so both r and k − r lie in ( p/3, p/2) for p > 6. Case II: 9 6 j < p/6 + 1. Take m = b(5/12)( j − 1)c. Then m + 1/2 5 1/2 1 6 + < , j −1 12 j −1 2
m + 1/3 5 7/12 > − > 1/3. j −1 12 j −1
Therefore, if p(m + 1/3) p(m + 1/2)
1, so such a k exists. Case III: 2 6 j < p/3 + 1, j even. Take k = ( p − 1)/2. Then r = p − j/2 and both k and r − k lie in ( p/3, p/2). Case IV: p/3 + 1 < j < 3 p/7, j even. Take h such that 1 6 h < ( p − 3)/18 and 2h + 2/3 2h + 1 p< j< p. 6h + 1 6h + 1 The largest admissible h is at least ( p − 19)/18, so the above intervals cover p( p − 13)/(3( p − 16)), 3 p/7 , which contains [( p + 4)/3, 3 p/7) for p > 64. Then take k = (( p − 1)/2) − 3h, so that r ∈ ( p/2, 2 p/3). Case V: 2 p/5 + 1 < j 6 ( p − 1)/2, j even. We take h so that 1 6 h < ( p − 3)/12 and 2h 2h + 1/3 p < j −1< p. 4h + 1 4h + 1 The largest admissible h is at least ( p − 13)/12, so these intervals cover 2 p/5, ( p( p − 11))/(2( p − 10)) , which includes (2 p/5, ( p − 3)/2] for p > 13. Then take k = (( p − 1)/2) − 2h, so r − k ∈ ( p/3, p/2). Case VI: p/3 + 1 < j 6 ( p − 1)/2, j odd. Take h, 0 6 h < ( p − 15)/12 such that 2h + 1 2h + 4/3 p < j −1< p. 4h + 3 4h + 3
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Then take k = (( p − 3)/2) − 2h, so that k − r ∈ ( p/3, p/2). The above intervals cover ( p/3, (( p − 3)/2)], provided that p > 24. Case VII: p/3 − 1 < j < p/3 + 1. Write j = ( p + t)/3, where −2 6 t 6 2, t 6 = 0. Here we take k = 3d p/9e + b, where 0 6 b 6 2 and t + 3b ≡ w (mod 9), w ∈ {5, 7}. If p > 28, then k ∈ ( p/3, p/2). If w = 5, then r = 5 p/9 + E, where |E| 6 22/9. Thus, r ∈ ( p/2, 2 p/3) when p > 44. When w = 7, t = 1, b = 2, then r ∈ (7 p/9, 7 p/9 + 14/9]. Case VIII: 5 p/21 < j < p/3 − 1, j odd. Take 1 6 h < ( p − 3)/18, so that 6h − 1 2h p< j< p. 18h + 3 6h + 1 Take k = (( p − 1)/2)−3h, so that r ∈ ( p/3, p/2). The above intervals cover (5 p/21, p/3 − 1). Case IX: p/6+1 < j < 5 p/21, j odd. If p/5 < j −1 < 4 p/15, take k = ( p − 5)/2, so that r ∈ (5 p/6 − 5/2, p − 5/2). If p/7 < j − 1 < 4 p/21, then k = ( p − 7)/2 works, and if 5 p/27 < j < 2 p/9, then k = ( p − 9)/2 works. Proof of Lemma 3.1 By hypothesis, there are two possibilities: (i) some si (say, s1 ) is divisible by a prime power p w other than 3, 7, or 13; (ii) each si divides 273 and some si (say, s1 ) equals 39, 91, or 273. Say s1 is divisible by p w , with p w+1 - s2 and p w+1 - s3 . By Lemma 3.3, there is a character χ1 with χ1 (a2 /a1 ) = e(1/ p w ) and χ1 (a3 /a2 ) = e(m/ p w ) for some w−1 integer m. If p = 2, let χ = χ12 , so that χ(a2 /a1 ) = −1 and 1 = χ(a2 /a1 )χ(a3 /a2 )χ(a1 /a3 ) = −χ(a3 /a2 )χ (a1 /a3 ). But each character value on the right is either −1 or 1, so either χ (a2 ) = χ (a3 ) or p w−1
χ (a1 ) = χ (a3 ) and (i) is satisfied. If p is odd, let χ2 = χ1 p w−2 χ1
if p 6∈ P and let
χ2 = if p ∈ P. Then χ2 (a2 /a1 ) = e(1/ p u ), where u = 2 if p ∈ P and u = 1 otherwise. Write χ2 (a3 /a2 ) = e( j/ p u ). If j = 0, then χ2 (a2 ) = χ2 (a3 ) and (i) is satisfied. Otherwise, since p u is good by Lemma 3.4, there is a number k such that two of the three distances of the points (0, k/ p u , k j/ p u ) (mod 1) satisfy (3.1). Taking χ = χ2k gives (ii) for some relabeling of the ai ’s. In the case when each si divides 273 and s1 ∈ {39, 91, 273}, by Lemma 3.3 there is a character χ1 with χ1 (a2 /a1 ) = e(1/r ) and χ1 (a3 /a2 ) = e(g/r ) for some integer g (here r = s1 ). Since r is good by Lemma 3.4, there is a k such that two of the three
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323
distances of the points (0, k/r, k j/r ) (mod 1) satisfy (3.1). Taking χ = χ1k gives (ii) for some relabeling of the ai ’s. LEMMA 3.5 Suppose that for some relabeling of a1 , a2 , a3 and some Dirichlet character χ modulo q, χ (ai ) = e(ri ) with 0 6 r1 < r2 < r3 6 2, d1 = r2 − r1 and d2 = r3 − r2 , and (d1 , d2 ) satisfies (3.1). Then there is a finite barrier B for D = (q, a1 , a2 , a3 ) with |B | 6 14. If d1 > 1/3, then |B | 6 3.
Proof For some 1/2 6 β < α 6 σ and large γ , suppose that L(s, χ ) has a zero at s = α+iγ of order c1 and that L(s, χ 2 ) has a zero at s = α + 2iγ of order c2 , where d1 > 13 , (1, 2), 6 (c1 , c2 ) = (5, 9), d1 = 19 , 12 (3, 5), d = . 1
37
Suppose that all other nontrivial zeros of L-functions modulo q have real part 6 β. Let φ(q) log x πq,a2 (x) − πq,a1 (x) , α x φ(q) log x D2 (x) = πq,a3 (x) − πq,a2 (x) . α x
D1 (x) =
Let u = log x. For large x, Lemma 1.1 and the identity c − b b + c sin sin(a − b) − sin(a − c) = 2 cos a − 2 2 give D1 (x) =
2 4 X c` sin(d1 `π ) cos `γ u − (r1 + r2 )π` + O(1/γ 2 ), γ ` `=1
2 4 X c` D2 (x) = sin(d2 `π ) cos `γ u − (r2 + r3 )π ` + O(1/γ 2 ). γ ` `=1
For j = 1, 2 define c2 g j (y) = c1 sin(πd j ) cos y + sin(2πd j ) cos 2y 2 c2 = c1 sin(πd j ) cos y + cos(πd j ) cos 2y . c1
(3.2)
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Because 0 < d j < 1/2, cos πd j and sin πd j are both positive. We claim that min g1 (γ u − (r1 + r2 )π), g2 (γ u − (r2 + r3 )π) < 0 (u > 0),
(3.3)
which is equivalent to showing that min g1 (y), g2 (y − π(d1 + d2 )) < 0 for all real y. Since g1 and g2 are periodic and continuous, in fact the minimum above is 6 −δ for some δ > 0. If γ is large (depending on δ), this implies that one of the two functions on the left in (3.2) is negative for all large x. Thus, for large x, πq,a3 (x) > πq,a2 (x) > πq,a1 (x) does not occur. To prove (3.3), we consider the one-parameter family of functions h(y; λ) = cos y + λ cos(2y) for 0 < λ < 1. These are all even functions, so it suffices to look at 0 6 y 6 π. We have h(y; λ) √ positive for 0 6 y < vλ and negative for vλ < y 6 π, −1 where vλ = cos [(−1 + 8λ2 + 1)/(4λ)]. As a function of λ, vλ decreases from π/2 at λ = 0 to π/3 at λ = 1. For i = 1, 2, let z i = vλi for λi = (c2 /c1 ) cos πdi . Since π(d1 + d2 ) < π , (3.3) follows from z 1 + z 2 < π(d1 + d2 ).
(3.4)
When (d1 , d2 ) ∈ {(6/19, 9/19), (12/37, 16/37)}, (3.4) follows by direct calculation. When 1/3 < d1 , we have c1 = 1, c2 = 2, and λ j = 2 cos πd j ( j = 1, 2). We claim for j = 1, 2 that z j < π d j or, equivalently, that cos z j > cos πd j = (1/2)λ j . Since 0 < λ j < 1, q q 8λ2j + 1 − 1 4λ4j + 4λ2j + 1 − 1 λj > = , cos z j = 4λ j 4λ j 2 which proves (3.4) in this case as well. Combining Lemmas 3.1 and 3.5 gives the following. COROLLARY 3.6 Let s1 = ordq (a2 /a1 ), let s2 = ordq (a3 /a2 ), and let s3 = ordq (a1 /a3 ). If one of s1 , s2 , s3 is not in {3, 7, 13, 21}, then there is a finite barrier B for D with |B | 6 14.
4. Third construction Throughout this section, we assume that a1 , a2 , a3 do not satisfy the conditions of Lemma 2.1. LEMMA 4.1 Let χ be a character modulo q such that there are at least two different values among χ (a1 ), χ (a2 ), χ(a3 ). Then the following hold:
PRIME NUMBER RACE AND ZEROS OF L-FUNCTIONS
(a) (b) (c) (d)
325
χ (a1 ), χ(a2 ), χ(a3 ) are distinct; <χ (a1 ), <χ(a2 ), <χ(a3 ) are distinct; all the values χ(a1 ), χ(a2 ), χ(a3 ) are not ±1; χ has order > 7.
Proof (a) If this does not hold, the conditions of Lemma 2.1 hold with S = {χ}. (b) If χ (a1 ) = χ (a2 ), then, by (a), <χ(a3 ) 6 = <χ(a1 ) and the conditions of Lemma 2.1 hold for S = {χ, χ }. (c) If χ (a3 ) = 1 and k is the order of the character χ , then the conditions of Lemma 2.1 hold for S = {χ, χ 2 , . . . , χ k−1 }. If χ (a3 ) = −1 and none of χ (ai ) = 1, then χ 2 (a3 ) = 1 6 = χ 2 (a1 ) and the conditions of Lemma 2.1 hold for S = {χ 2 , χ 4 , . . . , χ 2h−2 }, where h is the order of χ 2 . (d) This follows directly from (b) and (c).
4.2 There exists a character χ modulo q of order > 7 such that < χ (a3 ) − χ(a2 ) < χ 2 (a2 ) − χ 2 (a1 ) 6 = < χ(a2 ) − χ(a1 ) < χ 2 (a3 ) − χ 2 (a2 ) , (4.1) and for some integers h, k with 1 6 h < k 6 3, = χ h (a3 ) − χ h (a2 ) = χ k (a2 ) − χ k (a1 ) 6 = = χ h (a2 ) − χ h (a1 ) = χ k (a3 ) − χ k (a2 ) . (4.2) LEMMA
Proof Let χ be any character modulo q such that χ(a2 /a1 ) 6 = 1. By Lemma 4.1(a), the values χ (a1 ), χ(a2 ), χ(a3 ) are distinct. Denote χ(a j ) = e2πiϕ j ( j = 1, 2, 3). By Lemma 4.1(b), the values cos(ϕ1 ), cos(ϕ2 ), cos(ϕ3 ) are distinct. Therefore, the maj=1,2,3 trix A = cos` (ϕ j )`=0,1,2 is nonsingular. Since cos(2ϕ) = 2 cos2 (ϕ) − 1, the matrix j=1,2,3
cos(`ϕ j )`=0,1,2 is also nonsingular, and this implies (4.1). Next, by Lemma 4.1(c), sin(ϕ j ) 6 = 0 ( j = 1, 2, 3). Therefore, the maj=1,2,3 trix B = sin(ϕ j ) cos` (ϕ j )`=0,1,2 is nonsingular. Using the identities sin(2ϕ) = 2 sin(ϕ) cos(ϕ), sin(3ϕ) = 2 sin(ϕ)(4 cos2 (ϕ) − 1), it follows that the matrix j=1,2,3 sin(`ϕ j )`=1,2,3 is also nonsingular. This implies (4.2). 4.3 Let z 1 and z 2 be complex numbers. We can associate with each χ ∈ Cq a nonnegative LEMMA
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real number λχ such that z1 =
X
λχ χ(a2 ) − χ(a1 ) ,
χ ∈Cq
z2 =
X
λχ χ(a3 ) − χ(a2 ) .
(4.3)
χ ∈Cq
Proof Write z j = u j + iv j ( j = 1, 2), where u 1 , u 2 , v1 , v2 are real. By Lemma 4.2, there is a character χ = χ0 for which (4.1) and (4.2) hold. Thus, we can find real numbers λ1 and λ2 such that λ1 < χ0 (a2 ) − χ0 (a1 ) + λ2 < χ02 (a2 ) − χ02 (a1 ) = u 1 /2, λ1 < χ0 (a3 ) − χ0 (a2 ) + λ2 < χ02 (a3 ) − χ02 (a2 ) = u 2 /2 and real numbers λ3 and λ4 such that λ3 = χ0h (a2 ) − χ0h (a1 ) + λ4 = χ0k (a2 ) − χ0k (a1 ) = v1 /2, λ3 = χ0h (a3 ) − χ0h (a2 ) + λ4 = χ0k (a3 ) − χ0k (a2 ) = v2 /2. By Lemma 4.1, the six characters χ0 , χ02 , χ03 , χ 0 , χ 20 , χ 30 are distinct. Now set µχ = λ1 for χ ∈ {χ0 , χ 0 }, µχ = λ2 for χ ∈ {χ02 , χ 20 }, and µχ = 0 for other characters. Also, let νχ h = λ3 , νχ h = −λ3 , νχ k = λ4 , νχ k = −λ4 , and νχ = 0 for other 0 0 0 0 characters. Let θχ = µχ + νχ for each χ . Then (4.3) holds with λχ = θχ for each χ , but it may occur that θχ < 0 for some χ . However, by Lemma 4.1, a j 6 ≡ 1 (mod q) P for each j, so χ ∈Cq χ(a j ) = −1 for every j. Thus, for any real y, (4.3) holds with λχ = θχ + y for each χ. LEMMA 4.4 If a1 , a2 , a3 do not satisfy the conditions of Lemma 2.1, then for all τ > 0 and σ > 1/2, there is a finite barrier for D = (q, a1 , a2 , a3 ), with each B(χ ) consisting of numbers ρ with <ρ 6 σ and =ρ > τ .
Proof (1) (2) By Lemma 4.3, we can find nonnegative νχ and νχ such that X i= νχ(1) χ(a2 ) − χ(a1 ) , χ
−i =
X
i=
X
χ χ
νχ(1) χ(a3 ) − χ(a2 ) , νχ(2) χ(a2 ) − χ(a1 ) ,
PRIME NUMBER RACE AND ZEROS OF L-FUNCTIONS
i=
X χ
327
νχ(2) χ(a3 ) − χ(a2 ) .
(4.4) (1)
Fix small positive ε > 0, and take a positive integer Q and nonnegative integers Nχ , (2) (1) (1) (2) (2) Nχ for all characters χ modulo q such that |νχ −Nχ /Q| < ε, |νχ −Nχ /Q| < ε. For some σ1 ∈ (β1 , σ ] and large γ > τ , suppose that for all characters χ ∈ Cq (k) and for k = 1, 2, the function L(s, χ) has a zero at s = σ1 + kiγ of order Nχ . Suppose that all other nontrivial zeros of L-functions modulo q have real part 6 β1 . Let D1 (x) = φ(q)(πq,a1 (x) − πq,a2 (x)), and let D2 (x) = φ(q)(πq,a2 (x) − πq,a3 (x)). By Lemma 1.1 and (4.4), we have log x Q D1 (x) = 2 cos(γ log x) + cos(2γ log x) + ε1 (x) + O(1/γ ) , σ x 1 2γ log x Q D2 (x) = − 2 cos(γ log x) + cos(2γ log x) + ε2 (x) + O(1/γ ) , σ 1 x 2γ where the functions ε1 (x), ε2 (x) are uniformly small if ε is small. Taking into account that min(2 cos u + cos 2u, −2 cos u + cos 2u) ≤ −1 for all u, we obtain that for large x, πq,a1 (x) > πq,a2 (x) > πq,a3 (x) does not occur. 5. A barrier satisfying GSHq The construction of this barrier is modeled on the construction in Section 2. For one character, B(χ ) √ is infinite, the number of elements of B(χ ) with imaginary part 6 T growing like the parameters in the construction, we can create √ T . By altering barriers with T replaced by T for any fixed . Assume that for some relabeling of a1 , a2 , a3 , there are two characters χ1 ,χ2 satisfying χ1 (a1 ) = χ1 (a2 ) 6= χ1 (a3 ),
χ2 (a1 ) 6 = χ2 (a2 ).
(5.1)
Suppose that 1/2 6 β < σ2 < σ1 , that t is large, and that L(s, χ1 ) has a simple zero at s = σ1 + it. Suppose that L(s, χ2 ) has simple zeros at the points s = ρ j ( j = 1, 2, . . .), where ρ j = σ2 − δ j + iγ j , δ j > 0, γ j > 0, δ j → 0, and γ j → ∞ as P j → ∞, and 1/γ j < ∞. Also, suppose that the numbers t, γ1 , γ2 , . . . are linearly independent over Q. Define Z = χ 1 (a2 ) − χ 1 (a3 ),
W = χ 2 (a2 ) − χ 2 (a1 ).
By (5.1), Z 6 = 0 and W 6 = 0. Also define 1 −1 σ1 arg W 1 α=− tan + arg Z , β= − . π t 2π 4 Let H be the set of integers h such that khα + βk 6 1/5. Since the number of possibilities for Z is finite, if t is large, then 1 1 1 6 kαk 6 − . 10t 2 10t
328
FORD and KONYAGIN
It follows that in every set of b10tc + 1 consecutive integers, one of them is in H . As in Section 2, define D1 (x) := φ(q) πq,a1 (x) − πq,a2 (x) , D2 (x) := φ(q) πq,a3 (x) − πq,a2 (x) . Suppose that x is sufficiently large, and for brevity write u = log x. By Lemma 1.1 and our hypotheses, D2 (x) =
1 i 2x σ1 h eitu < Z +O u σ1 + it u
(5.2)
and D1 (x) =
e−δ j u i 2x σ2 X h e(−δ j +iγ j )u < W +O + O(x β log2 x) u σ2 − δ j + iγ j γ j2 u γ j 6x
=
2x σ2
e−δ j u i Xh + O(x β log2 x),
(5.3)
j
where Bj = W By assumption,
P
arg
j
e(−δ j +iγ j )u . iγ j
(5.4)
|B j | 1; thus D1 (x) x σ2 /u. Modulo 2π,
t π eitu Z ≡ tu − tan−1 + arg Z ≡ tu − − πα. σ1 + it σ1 2
By (5.2), when ktu/π − αk > u −0.9 , D2 (x) x σ1 /(log x)1.9 , and thus for these x either πq,a3 (x) is the largest of the three functions or it is the smallest. Next assume that ktu/π − αk 6 u −0.9 . We choose δ j and γ j as follows: 0 < δ j < σ2 − β, j −3 δ j j −3 , γ j = 2th j + O( j −10 ), where for j > 10t we have h j ∈ H , h j+1 > h j and j 2 6 h j 6 j 2 + j. With these choices, ∞ −δ j u X e j=1
γ j2
e−u
and X ju 2/5
1/4
X j6u 1/4
1/j 4 +
X
1/j 4 u −3/4
j>u 1/4
e−δ j u 1/4 e−u + u −2/5 u −2/5 . γj
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329
Thus, by (5.3) and (5.4), D1 (x) =
2x σ2 h u
i
X
(5.5)
u 1/4 6 j6u 2/5
Suppose u 1/4 6 j 6 u 2/5 . Since h j ∈ H , we have
1
1 π
arg B j = arg W + γ j u −
2π 2π 2
ut
= β + h j + O(u −3/2 ) π
= β + h j α + O(u −0.1 ) 6 0.21 for large u. Hence |B j | cos(0.42π ) > (1/5)|B j |. Therefore, X u 1/4 6 j6u 2/5
X
u 1/3 6 j62u 1/3
1 u −1/3 . γj
It follows from (5.5) that for u large and kut/π − αk 6 u −0.9 , D1 (x) >
cx σ2 , (log x)4/3
where c > 0 depends on q, t, and W . This implies that the inequality πq,a2 (x) > πq,a3 (x) > πq,a1 (x) does not occur for large x. References [BH]
C. BAYS and R. H. HUDSON, Details of the first region of integers x with
[Ch]
π3,2 (x) < π3,1 (x), Math. Comp. 32 (1978), 571 – 576. MR 57:16175 313 P. L. CHEBYSHEV, Lettre de M. le professeur Tch´ebychev a` M. Fuss, sur un nouveau
[D] [K1] [K2] [K3]
th´eoreme r´elatif aux nombres premiers contenus dans la formes 4n + 1 et 4n + 3, Bull. de la Classe phys.-math. de l’Acad. Imp. des Sciences St. Petersburg 11 (1853), 208. 313 H. DAVENPORT, Multiplicative Number Theory, 3d ed., Grad. Texts in Math. 74, Springer, New York, 2000. MR 2001f:11001 316 J. KACZOROWSKI, A contribution to the Shanks-R´enyi race problem, Quart. J. Math. Oxford Ser. (2) 449 (1993), 451 – 458. MR 94m:11105 314 , On the Shanks-R´enyi race mod 5, J. Number Theory 50 (1995), 106 – 118. MR 95j:11089 314 , On the Shanks-R´enyi race problem, Acta Arith. 74 (1996), 31 – 46. MR 96k:11113 314
330
[KT1]
[KT2]
[L] [RS] [Ru] [S]
FORD and KONYAGIN ´ , Comparative prime number theory I, Acta. Math. Sci. S. KNAPOWSKI and P. TURAN
Hungar. 13 (1962), 299 – 314, MR 26:3682a; II, 13 (1962), 315 – 342, MR 26:3682b; III, 13 (1962), 343 – 364, MR 26:3682c; IV, 14 (1963), 31 – 42, MR 26:3683a; V, 14 (1963), 43 – 63, MR 26:3883b; VI, 14 (1963), 65 – 78, MR 26:3683c; VII, 14 (1963), 241 – 250, MR 28:70a; VIII., 14 (1963), 251 – 268, MR 28:70b. 313 , Further developments in the comparative prime-number theory. I, Acta Arith. 9 (1964), 23 – 40, MR 29:75; II, 10 (1964), 293 – 313, MR 30:4739; III, 11 (1965), 115 – 127, MR 31:4773; IV, 11 (1965), 147 – 161, MR 32:99; V, 11 (1965), 193 – 202, MR 32:99; VI, 12 (1966), 85 – 96, MR 34:149. 313 J. E. LITTLEWOOD, Sur la distribution des nombres premiers, C. R. Acad. Sci. Paris 158 (1914), 1869 – 1872. 313 M. RUBINSTEIN and P. SARNAK, Chebyshev’s Bias, Experiment. Math. 3 (1994), 173 – 197. MR 96d:11099 314 R. RUMELY, Numerical computations concerning the ERH, Math. Comp. 61 (1993), 415 – 440. MR 94b:11085 D. SHANKS, Quadratic residues and the distribution of primes, Math. Comp. 13 (1959), 272 – 284. MR 21:7186
Ford Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, USA Konyagin Department of Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia; [email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 2,
A LIMIT FORMULA FOR ELLIPTIC ORBITAL INTEGRALS ˇ ´ ˇ CEVI MLADEN BOZI C
Abstract Let O be a nilpotent orbit for a semisimple Lie group which appears as the leading orbit in the wave-front set of an Aq (λ)-module. We establish a limit formula for the computation of the canonical measure on O through differentiation of the canonical measures on elliptic orbits. 0. Introduction Fourier inversion of the nilpotent orbital integrals is closely related to the computation of the canonical measure on a nilpotent orbit through differentiation of the canonical measures on the semisimple orbits. This leads naturally to the so-called limit formulas. The importance of Fourier inversion for the harmonic analysis on a semisimple group is already apparent in the work of Harish-Chandra on the Plancherel formula. Since then, the problem of computing the nilpotent measures has appeared in the work of various authors. In [BV1], [BV2], [HK], and [R2] the problem was solved for complex semisimple groups and in [B] for the real semisimple groups of rank one. For a general real semisimple group, the problem appears to be quite hard. However, in the special case of U ( p, q), the computation for all the orbits was carried out in [BV3], using character theory and combinatorics. In [R1] a conjectural limit formula was proposed, based on Rossmann’s theory of character contours and Weyl group representations (see [R2], [R1], [R3]). The goal of this paper is to establish a limit formula for the class of nilpotent orbits that arise as the leading orbits in the wave-front set of Aq (λ)-modules. The main technical tool in our approach is provided by the theory developed by W. Schmid and K. Vilonen [SV1], [SV2], [SV3] in the course of their work on the Barbasch-Vogan conjecture. To describe the main result of the paper, we have to introduce some notation. Let G R be a linear, connected, semisimple Lie group, let G be the complexification, and let K R be a maximal compact subgroup of G R . Denote by kR ⊂ gR the Lie algebras, and denote by k ⊂ g the complexified Lie algebras of K R ⊂ G R . We choose DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 2, Received 31 August 2000. Revision received 18 June 2001. 2000 Mathematics Subject Classification. Primary 22E46; Secondary 22E30. Author’s work supported by Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. 331
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a parabolic subalgebra p ⊂ g so that the complex conjugate p is opposite to p. Then l = p ∩ p is a Levi factor of p. Denote by c the center of l, and denote by (c ∩ kR )∗ the real linear dual of c ∩ kR . Let C + ⊂ i(c ∩ kR )∗ be the positive chamber defined by p/l. If V ⊂ ig∗R is a G R -orbit, we denote by m V the canonical measure on V , which is up to a constant multiple the (dimR V /2)-power of the Liouville form. If p is a polynomial on c∗ , write ∂( p) for the corresponding differential operator on c. Finally, let O ⊂ ig∗R be the nilpotent G R -orbit associated via the Sekiguchi correspondence with dense nilpotent K -orbit in K · (g/p + k)∗ . (Here (g/p + k)∗ stands for C-linear dual.) Now we are ready to state the main result of the paper. 0.1 There exist a polynomial p on c∗ and a nonzero constant c so that the following limit formula holds: lim ∂( p)m G R ·λ = cm O . THEOREM
λ→0(C + )
To prove the limit formula, we study the asymptotic behavior at λ = 0 of the holomorphic function Z ˆ λm , λ 7→ φσ G R ·λ
where φˆ is the Fourier transform of a test function φ on gR , σλ is the Liouville form on G R · λ, and 2m = dimR G R · λ. This is accomplished in several steps. First, we transfer the problem to the cotangent bundle T ∗ Y of the generalized flag variety Y of Ad(G)-conjugates of p. We use the twisted moment map µλ : T ∗ Y −→ G · λ to write Z Z m ˆ λm ). ˆ φσλ = µ∗λ (φσ G R ·λ
µ−1 λ (G R ·λ)
In [R3] W. Rossmann shows how to obtain the Taylor series expansion at λ = 0 of distributions on the right-hand side of the formula in the case when p is a Borel subalgebra. We present in Section 4 an analogue of Rossmann’s theory for a generalizied flag variety. Next, we compute in Section 3 the characteristic cycle CC(G ) of a standard sheaf G associated with open orbit G R · p ⊂ Y . In fact, we show that µ−1 λ (G R · λ), with natural orientation, is homologous to CC(G ). This result is a slight generalization of [SV2, §7]. Finally, to understand the leading term in the R ˆ m ), we apply the results from [SV3] on the Taylor series expansion of CC(G ) µ∗λ (φσ λ microlocalization of the Matsuki correspondence for sheaves. In Sections 1 and 2, we give a summary of the results from [SV3] needed for our applications in the setting of a generalized flag variety. We hope the powerful theory developed in [SV1], [SV2], and [SV3] will shed more light on the problem of the Fourier inversion of the nilpotent orbits in the future.
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1. Preliminaries Let G R be a real semisimple Lie group. In addition, we assume that G R is connected and linear. We fix a maximal compact subgroup K R of G R and write K and G for the respective complexifications. We denote further by gR and g Lie algebras of G R and G, respectively. Let θ be the Cartan involution on gR determined by the choice of K R . We denote by the same letter the corresponding involution on g. We fix a compact real form UR of G such that K R ⊂ UR . Denote by uR the Lie algebra of UR . Let P be a fixed parabolic subalgebra of G. Denote by p the Lie algebra of P. Let Y = G/P be the generalized flag variety. Alternatively, we may view Y as the variety of parabolic subalgebras Ad(G)-conjugated to p. Let h ⊂ p be a Cartan subalgebra. For an h-invariant subspace m ⊂ g, we write 1(m) for the set of h-weights on m. We use h to define Levi decompositions p=l+n
and
p = l + n.
(1.1)
Here 1(l) = (−1(p)) ∩ 1(p), 1(n) = 1(p) \ 1(l), 1(n) = −1(n). Recall that p is called the opposite parabolic subalgebra of p. Denote by 2ρp the sum of the roots from 1(n). Write B for the Killing form on g. Since g is semisimple, we may use B to identify g with complex linear dual g∗ . Denote by A the group G R or K , and write a for the Lie algebra of A. As a subgroup of G, the group A acts naturally on Y . We use this action to define the equivariant derived category D A (Y ). Recall that the objects of this category are represented by the complexes of A-equivariant sheaves on Y . By the result of T. Matsuki [M], the group A acts on Y with finitely many orbits that determine a semialgebraic Whitney stratification. The image of an object from D A (Y ) under the forgetful functor is constructible with respect to this stratification. Next, we recall the construction of the equivalence of equivariant categories, γ : D K (Y ) −→ DG R (Y ), which was conjectured by M. Kashiwara and proved in [MUV]. Consider the maps a
q
p
Y ←− G R × Y −→ G R /K R × Y −→ Y given by a(g, y) = g −1 y, q(g, y) = (gK R , y), p(gK R , y) = y. These maps are (G R × K R )-equivariant with respect to the following actions: (g, k) · y = k · y, (g, k) · (g1 , y) = (gg1 k −1 , g · y), (g, k) · (g1 K R , y) = (gg1 K R , g · y), (g, k) · y = g · y. By restricting from K to K R , we may view F ∈ D K (Y ) as an object from DG R ×K R (Y ). We have a ! (F ) ∈ DG R ×K R (G R × Y ), and since K R acts freely on G R × Y , there exists G ∈ DG R /K R (G R × Y ) such that a ! (F ) ∼ = q ! (G ). Now we define γ (F ) = Rp! (G ). The functorial notation for an inverse and direct image is as in [KS].
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We proceed to describe the action of γ on standard sheaves. Let S be an A-orbit on Y . Denote by j : S ⊂ Y the inclusion. Let L be an A-equivariant local system on S. The standard sheaves associated with (S, L ) are defined by I A∗ (L ) = R j∗ (L )
and
I A! (L ) = R j! (L ).
There exists a natural bijection between K - and G R -orbits on Y , called the Matsuki correspondence. If the K -orbit Z and the G R -orbit S are in Matsuki correspondence, then Z ∩ S is a K R -orbit. Moreover, we also have a natural correspondence between local systems {L } on Z and {L 0 } on S characterized by L | Z ∩S ∼ = L 0 | Z ∩S .
Denote by CY the constant sheaf on Y with stalk isomorphic to C. The following result generalizes [MUV, Th. 6.6]. The proof from [MUV] applies in our situation without any changes. THEOREM 1.2 The morphism γ : D K (Y ) −→ DG R (Y ) is an equivalence of categories. Suppose that the pairs (Z , L ) and (S, L 0 ) are in Matsuki correspondence. Then the action of γ on standard sheaves is given by γ I K∗ (L ) = IG! R L 0 ⊗ j ! (CY ) .
Use the complex structure to orient Y , and view Y as a real analytic manifold. Write T ∗ Y for the cotangent bundle of Y , and write T A∗ Y for the union of conormal bundles of the A-orbits on Y . For a locally compact space Z , we denote by H∗ (Z , Z) (H∗ (Z , C)) the Borel-Moore homology with integral (complex) coefficients. Set 2m = dimR Y . The characteristic cycle construction from [KS] yields a homomorphism from the Grothendieck group of D A (Y ) to the top homology group of T A∗ Y : CC : K D A (Y ) −→ H2m (T A∗ Y, Z). Define the map µ : T ∗ Y −→ g∗ as the composition of the natural embedding T ∗ Y ,→ Y × g∗ and the second projection Y × g∗ −→ g∗ . We call µ the moment map of Y . Next, we introduce a homomorphism 8 : H2m (TK∗ Y, Z) −→ H2m (TG∗R Y, Z), which makes the following diagram commutative: γ K D K (Y ) −−−−→ K DG R (Y ) CC y CC y 8
H2m (TK∗ Y, Z) −−−−→ H2m (TG∗R Y, Z)
(1.3)
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First, we define a family of bianalytic maps Fs : T ∗ Y −→ T ∗ Y , s ∈ R>0 , by ∗ setting Fs (ξ ) = ` exp(−s −1 Re µ(ξ )) (ξ ), ξ ∈ T ∗ Y . Here we write Re µ(ξ ) for the real part of µ(ξ ) relative to the real form gR ⊂ g, and we write `(g)∗ for the map induced by the left translation `(g) : Y −→ Y , g ∈ G. One can check that for C ∈ H2m (TK∗ Y, Z), we obtain a family of cycles (Fs (C), s > 0) in the sense of [SV1]. We draw attention now to the notion of the limit of a family of cycles from [SV1]. The proof of the next theorem is completely analogous to [SV3, proof of Th. 3.7]. 1.4 For C ∈ H2m (TK∗ Y, Z), the limit 8(C) = lims→0+ Fs (C) exists and is supported in TG∗R Y . The resulting homomorphism 8 makes diagram (1.3) commutative. THEOREM
2. Integrals associated with characteristic cycles In this section we study the convergence properties of integrals of certain differential forms over the cycles from H2m (TG∗R Y, Z). We also establish results that are needed in order to study the asymptotics of such integrals. This material is completely analogous to [SV2, §3] and [SV3, §5]. Recall that Y = G/P, where P is a fixed parabolic subgroup. Write y0 ∈ Y for the point determined by P. Then Y is a homogeneous space for UR and UR ∩ P is the centralizer of a torus in UR . Denote by cR the Lie algebra of this torus. Extend cR to a Cartan subalgebra hR of uR , and write c ⊂ h for the complexifications. Use h to define Levi decompositions (1.1). The choice of l defines a splitting p = c ⊕ [p, p]. Consider the exact sequence 0 −→ (g/p)∗ −→ (g/[p, p])∗ −→ c∗ −→ 0. Using a direct sum decomposition g = c ⊕ ([p, p] + [g, c]), we define a section of (g/[p, p])∗ −→ c∗ . Thus the above sequence splits; that is, (g/[p, p])∗ ∼ = c∗ ⊕ (g/p)∗ . ∗ ⊥ ∗ ∗ In particular, we view c ⊕ p ⊂ g . We say that λ ∈ h is P-regular if centg (λ) = centg (c). Assume in the following definition that λ is P-regular. As in [R2] and [R1], define a twisted moment map µλ : T ∗ Y −→ G · λ by the formula µλ u · (y0 , ν) = u · (λ + ν), u ∈ UR , ν ∈ Ty∗0 Y ∼ = p⊥ · Since λ is P-regular, u 1 · y0 = u 2 · y0 , u 1 , u 2 ∈ UR , implies u 1 · λ = u 2 · λ, and hence µλ is well defined. PROPOSITION 2.1 If λ ∈ h∗ is P-regular, then µλ : T ∗ Y −→ G · λ is a real algebraic isomorphism.
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Proof Consider the decomposition g = l + n + n. Denote by h λ the image of λ under the isomorphism g ∼ = g∗ defined by the form B. The assumption that λ is P-regular implies that ad(h λ ) is invertible on n. Thus we have P · h λ = h λ + n. Applying the isomorphism g ∼ = g∗ to this formula and using p⊥ ∼ = n, we obtain P · λ = λ + p⊥ . We proceed to define the inverse λ : G · λ −→ T ∗ Y of µλ . Using the above formula and G = UR · P, we write any ξ ∈ G · λ0 in the form ξ = u · (λ + ν), where u ∈ UR and ν ∈ p⊥ . Now we set λ (ξ ) = (u · y0 , u · ν). We have to show that λ is well defined. Let ξ = u 1 · (λ + ν1 ), where u 1 ∈ UR and ν1 ∈ p⊥ . Then we may find p, p1 ∈ P such that λ+ν = p · λ and λ+ν1 = p1 ·λ. The condition up ·λ = u 1 p1 ·λ implies (u 1 p1 )−1 up ∈ P. Thus u −1 1 u ∈ UR ∩ P, and therefore u · y0 = u 1 · y0 and u · λ = u 1 · λ. Finally, we conclude that (u · y0 , u · ν) = (u 1 · y0 , u 1 · ν1 ); hence λ is well defined. It follows immediately from the definitions that λ is the inverse of µλ . Suppose that Z is a space with G-action (UR -action), and suppose that ξ ∈ g (ξ ∈ uR ). We denote by l(ξ ) the vector field on Z defined by the group action. Given a map f of smooth manifolds, we write f ∗ for its differential. If u ∈ g and ξ ∈ g∗ , we write u · ξ for the coadjoint action. After these notational preliminaries, we introduce differential forms σO and τλ . Recall that each orbit O ⊂ g∗ carries a canonical G-equivariant complex symplectic form σO . If ξ ∈ O and u 1 · ξ , u 2 · ξ ∈ Tξ∗ O , then σO is defined by σO |ξ (u 1 · ξ, u 2 · ξ ) = ξ([u 1 , u 2 ]). When O = G · λ, we simply write σλ for σO . Furthermore, we define a UR -invariant two-form τλ on Y by the formula τλ | y0 l(u), l(v) = λ([u, v]), u, v ∈ uR . When λ is P-regular, the differential forms τλ , σλ , and σO are related as follows (see [R2, Lem. 7.2], [R3, Lem. 1.3.1]). PROPOSITION 2.2 Suppose that λ ∈ h∗ is P-regular, and suppose that O ⊂ g∗ is a nilpotent orbit. Then
µ∗λ σλ = µ∗ σO + π ∗ τλ at the smooth points of µ−1 (O ).
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Proof Since Y is UR -homogeneous, it suffices to prove the formula at the smooth point (y0 , ν) ∈ µ−1 (O ). The tangent space T(y0 ,ν) T ∗ Y is spanned by ker(π∗ )(y0 ,ν) ∩ T(y0 ,ν) (µ−1 O ) and l(uR ). We have to show that σλ (µλ∗ v1 , µλ∗ v2 ) = τλ (π∗ v1 , π∗ v2 ) + σO (µ∗ v1 , µ∗ v2 ) if v1 , v2 ∈ T(y0 ,ν) (µ−1 O ). We consider three cases. First, suppose vi = l(u i ) ∈ l(uR ) ⊂ T(y0 ,ν) (µ−1 O ), i = 1, 2. Since the maps µλ , π, and µ are UR -equivariant, we have σλ µλ∗l(u 1 ), µλ∗l(u 2 ) = σλ l(u 1 ), l(u 2 ) = (λ + ν)([u 1 , u 2 ]) = τλ π∗l(u 1 ), π∗l(u 2 ) + σO µ∗l(u 1 ), µ∗l(u 2 ) = π ∗ τλ l(u 1 ), l(u 2 ) + µ∗ σO l(u 1 ), l(u 2 ) , as desired. Next, we suppose v1 ∈ ker π∗ ∩ T(y0 ,ν) (µ−1 O ), v2 = l(u 2 ) ∈ l(uR ). In this case, we make the following identifications: ker π∗(y0 ,ν) ∼ = T(y0 ,ν) (Ty∗0 Y ) ∼ = ∗ ⊥ ∗ Ty0 Y ∼ p . The restriction of µ to T Y is affine; hence µ v = µ v . It follows = λ λ∗ 1 ∗ 1 y0 that µλ∗ v1 = u 1 · (λ + ν) = u 01 · ν = µ∗ v1 for some u 1 , u 01 ∈ g, and thus σλ (µλ∗ v1 , µλ∗ v2 ) = σλ u 1 · (λ + ν), l(u 2 ) = (λ + ν)([u 1 , u 2 ]) = ν([u 01 , u 2 ]). On the other hand, τλ (π∗ v1 , π∗ v2 ) = 0 and σO (µ∗ v1 , µ∗ v2 ) = ν([u 01 , u 2 ]), so the desired equality holds again. Finally, we suppose v1 , v2 ∈ ker π∗ ∩ T(y0 ,ν) (µ−1 O ). Let b ⊂ p be a Borel subalgebra of g. In view of our earlier identifications, we have µλ∗ v1 , µλ∗ v2 ∈ λ + p⊥ ⊂ G · λ
and
µ∗ v1 , µ∗ v2 ∈ O ∩ p⊥ ⊂ O ∩ b⊥ .
A direct computation shows that λ + p⊥ ⊂ G · λ is isotropic. On the other hand, O ∩ b⊥ ⊂ O is Lagrangian by the result of A. Joseph [J], and therefore O ∩ p⊥ is isotropic. In other words, σλ (µλ∗ v1 , µλ∗ v2 ) = 0 = τλ (π∗ v1 , π∗ v2 ) + σO (µ∗ v1 , µ∗ v2 ). This completes the proof. Let C be a semialgebraic chain in T ∗ Y . We say that C is R-bounded if Re µ(supp(C)) is a bounded subset of g∗ (see [SV2, §3]). Recall that the Fourier transform of a test function φ ∈ Cc∞ (gR ) is defined by Z ˆ )= φ(ξ eξ(x) φ(x) d x, ξ ∈ g∗ . gR
The proof of the next proposition comes down to the application of the fact that φˆ decays rapidly in the imaginary directions.
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PROPOSITION 2.3 ([R2, Vol. II, §1.2], [SV2, Lem. 3.16]) If C is a semialgebraic, R-bounded 2m-chain in T ∗ Y , then the integral Z ˆ λm ), φ ∈ Cc∞ (gR ), µ∗λ (φσ C
converges absolutely. The integral depends holomorphically on λ. In particular, if C ∈ H2m (TG∗R Y, Z), the above integral converges. In order to compare integrals over various cycles, we recall the notion of restricted homology. Let C1 , C2 be semialgebraic, R-bounded 2m-cycles in T ∗ Y . We say that C1 and C2 are R-homologous if there exists a semialgebraic, R-bounded (2m + 1)chain C in T ∗ Y such that C1 − C2 = ∂C. 2.4 ([SV2, Lem. 3.19]) Suppose that C1 , C2 are semialgebraic, R-bounded 2m-cycles in T ∗ Y . If C1 and C2 are R-homologous, then Z Z ∗ ˆ m ˆ λm ). µλ (φσλ ) = µ∗λ (φσ PROPOSITION
C1
C2
Denote by N the nilpotent cone in g∗ . Observe that a nilpotent orbit has an even complex dimension. Write N˜k , respectively, Nk , for the union of nilpotent orbits O such that dimC O ≤ 2k, respectively, dimC O = 2k. Suppose that O ⊂ N is a nilpotent orbit. Then O ∩ a⊥ is a union of finitely many A-orbits of real dimension 2k (see [KR]). A simple computation gives T A∗ Y = µ−1 (a⊥ ). Suppose Nk ∩ µ(T A∗ Y ) 6= ∅. Then the restriction of the moment map µ : µ−1 (Nk ∩ a⊥ ) −→ Nk ∩ a⊥ is an A-equivariant fibration whose typical fiber µ−1 (ξ ) is a complex projective variety of complex dimension less than or equal to m − k. It follows that the real dimension of µ−1 (N˜k ∩ a⊥ ) is less than or equal to 2m. Observe that the homomorphism H2m (µ−1 (N˜k ∩ a⊥ ), Z) −→ H2m (T A∗ Y, Z), induced by the closed embedding µ−1 (N˜k ∩ a⊥ ) ⊂ T A∗ Y , is injective. Hence, we view H2m (µ−1 (N˜k ∩ a⊥ ), Z) as a subgroup of H2m (T A∗ Y, Z). Since µ−1 (Nk ∩ a⊥ ) is open in µ−1 (N˜k ∩ a⊥ ), the restriction of homology classes induces the homomorphism H2m µ−1 (N˜k ∩ a⊥ ), Z −→ H2m µ−1 (Nk ∩ a⊥ ), Z , C 7 → C|Nk ∩a⊥ . If λ ∈ c∗ , consider the form (1/(2πi))τλ ∈ H 2 (Y, C). Set eτλ /2πi = 1 +
1 1 τλ + (τλ )2 + · · · , 2πi 2!(2πi)2
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and write π ∗ eτλ /2πi for the pullback of eτλ /2πi under the projection π : T ∗ Y −→ Y . We use the form eτλ /2πi to descend the cycles in T A∗ Y to the nilpotent cone N . In fact, suppose C ∈ H2m (µ−1 (Nk ∩ a⊥ ), Z). We take the cap product of C against the component in π ∗ eτλ /2πi of degree 2m − 2k. This produces the class C ∩ π ∗ eτλ /2πi ∈ H2k (µ−1 (Nk ∩ a⊥ ), C). Finally, we define Z π ∗ eτλ /2πi ∈ H2k (Nk ∩ a⊥ , C) C
as the pushforward of the cycle C ∩ π ∗ eτλ /2πi via the map µ : µ−1 (Nk ∩ a⊥ ) −→ Nk ∩ a⊥ . We remark that the cap product followed by the pushforward agrees with geometric operation of integration over the fiber (see [SV3, §5]). This fact is used in Section 4. Now, if H2m (µ−1 (Nk ∩ a⊥ ), Z) 6= 0, then dimR µ−1 (Nk ∩ a⊥ ) = 2m. Hence, dimR µ−1 (ξ ) = 2m − 2k for ξ ∈ Nk ∩ a⊥ , so we obtain the homomorphism R H2m (µ−1 (Nk ∩ a⊥ ), Z) −→ H2k (Nk ∩ a⊥ , C) by assigning C 7→ C π ∗ eτλ /2πi . Choose C 6 = 0 from H2m (T A∗ Y, Z), and denote by k = k(C) the least integer such that supp(C) ⊂ µ−1 (N˜k ∩a⊥ ). Then dimR supp(C) = 2m implies dimR µ−1 (N˜k ∩a⊥ ) = 2m, and therefore H2m (µ−1 (Nk ∩ a⊥ ), Z) 6= 0. Finally, we define M (gr µ∗ )λ : H2m (T A∗ Y, Z) −→ H2k (Nk ∩ a⊥ , C) R by the formula (gr µ∗ )λ (C) = C 0 π ∗ eτλ /2πi . Here C 0 = C|µ−1 (Nk ∩a⊥ ) . L The next goal is to define a homomorphism φ : H2k (Nk ∩ k⊥ , C) −→ L ⊥ H2k (Nk ∩ gR , C), which makes the following diagram commutative: H2m (TK∗ Y, Z) (gr µ∗ )λ y L
8
H2m (TG∗R Y, Z) (gr µ∗ )λ y
−−−−→
φ
H2k (Nk ∩ k⊥ , C) −−−−→
L
(2.5)
H2k (Nk ∩ g⊥ R , C)
Similarly, as before, consider the family of bianalytic maps f s : N −→ N , s ∈ R>0 , defined by the formula f s (η) = Ad(exp(−s −1 Re(η)))η, η ∈ N . For notational simplicity, we denote here the coadjoint action also by Ad. 2.6 ([SV3, Th. 5.10]) Let C be a 2k cycle in Nk ∩ k⊥ . Then the limit of cycles φ(c) = lims→0+ ( f s )∗ (c) exists as a cycle in Nk and is supported in Nk ∩ g⊥ R . The induced homomorphism L L ⊥ ⊥ φ: H2k (Nk ∩ k , C) −→ H2k (Nk ∩ gR , C) makes diagram (2.5) commutative. THEOREM
The homomorphism φ has a particularly nice description of the invariant part of homology. First, we explain how to orient A-orbits. Given a G-orbit O ⊂ Nk , denote
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by σO the complex symplectic form on O . Any K -orbit O K ⊂ O ∩ k⊥ is a complex manifold of dimension k. We use the underlying complex structure to orient O K . On the other hand, any G R -orbit OG R ⊂ O ∩ g⊥ R is a real manifold of dimension 2k. Observe that σO is purely imaginary on OG R and thus the restriction of (1/2πi)σO to OG R determines a symplectic form. We use it to orient OG R . Hence, if V ⊂ O ∩ a⊥ is an A-orbit, we have a well-defined cycle [V ] which determines a homology class in H2k (Nk ∩ a⊥ , C). There exists a natural bijection between the set of K -orbits in N ∩ k⊥ and the set of G R -orbits in N ∩ g⊥ R , called the Sekiguchi correspondence (see [S]). Recall that the orbits associated by the Sekiguchi correspondence lie in the same G-orbit and, in particular, have the same real dimension. The next result is [SV3, Th. 6.3]. THEOREM 2.7 ([SV3, Th. 6.3]) Let O K be a K -orbit in N ∩ k⊥ . Then φ([O K ]) = [OG R ], where OG R is the image of O K under the Sekiguchi correspondence.
3. The case of open orbits The goal of this section is to compute the characteristic cycle of a standard sheaf associated with an open G R -orbit on Y . The results are analogous to those in [SV2, §7]. We begin by fixing the notation. First, we draw attention to the notation already introduced in Section 1. In addition, assume that G R has a compact Cartan subgroup HR . Write hR = Lie(HR ) and h = (hR )C . The results of this section hold in the more general setting when hR is fundamental, but the arguments (most notably the orientation statement in Lem. 3.2) are simpler under the assumption in force. Let y0 ∈ Y . Denote by P ⊂ G the parabolic subalgebra that stabilizes y0 . Set p = Lie(P), and assume that p ∩ gR contains a compact Cartan subalgebra. Without any loss of generality, we may assume hR ⊂ p ∩ gR . Since the roots of h are real valued on ihR , we have l = p ∩ p. This implies further that l is the complexification of the real Lie algebra lR = l ∩ gR = p ∩ gR . Observe that lR is the Lie algebra of the subgroup L R = g ∈ G R : Ad(g)p = p . Set S = G R · y0 , and identify the tangent space of S, respectively, Y , at y0 with gR /lR , respectively, g/p. A simple computation shows dimR gR /lR = dimR g/p; hence, the orbit S is open in Y . Conversely, if the orbit S is open, it is not difficult to show that gR ∩ p contains a fundamental (in particular, compact, under our assumptions) Cartan subalgebra.
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Let cR be the center of lR . The choice of n determines the positive chamber in c∗ : C + = λ ∈ ic∗R : λ(h α ) > 0 if α ∈ 1(n) .
(3.1)
Here c∗R is the R-linear dual and h α ∈ h is the element corresponding to α ∈ h∗ via the isomorphism g ∼ = g∗ . Clearly, any λ ∈ C + is P-regular, so the twisted moment map µλ : T ∗ Y −→ G · λ is a real analytic isomorphism. Let j : S ,→ Y be the inclusion. The goal is to show that the cycles CC(R j∗ C S ) and [µ−1 λ (G R · λ)] are R-homologous for any λ ∈ C + . First, we study the case λ = 2ρp . Let V = V2ρp be the irreducible G-module with highest weight 2ρp . Choose a UR -invariant positive definite hermitian form h u on V . The space V has a natural action of the involution θ . In fact, identify V with the space of regular (algebraic) functions F : G −→ C satisfying the condition F(gp) = e−2ρp ( p)F(g), g ∈ G, p ∈ P. Here P is the parabolic subgroup of G opposite to P. Then we define θ F by the formula θ F(g) = F(θg). One checks immediately that θ F ∈ V . Another hermitian form h r on V is defined by the formula h r (v1 , v2 ) = h u (v1 , θv2 ),
v1 , v2 ∈ V.
A short computation shows that h r is G R -invariant. Write p y = Ad(g)p and n y = Ad(g)n, whenever y = g · y0 . Observe that n y does not depend on the choice of g. The space of invariant vectors V n y is 1-dimensional, so the formula f (y) =
h r (v, v) , h u (v, v)
v ∈ V ny ,
defines a real algebraic function on Y . If necessary, we multiply h r by a constant so as to make f positive on S. In fact, by the G R -invariance of h r , it suffices to examine the sign of h r (v, v), v ∈ V n . Since n is θ -stable, we have θv ∈ V n . Hence, it suffices to multiply h r by −1 if θ v = −v. We denote the modified form also by h r . Write Y R for the underlying real analytic structure on Y , and identify the real cotangent bundle T ∗ Y R with the holomorphic cotangent bundle T ∗ Y via the pairing (v y , ξ y ) 7→ 2 Rehξ y , v y i, v y ∈ Ty Y , ξ y ∈ Ty∗ Y . View the differential d log f as a section of T ∗ S R , and orient it via the isomorphism d log f ∼ = S and the complex structure on S. On the other hand, orient G R · (2ρp ) so that the top exterior power of −iσ2ρp is positive. Set s = dimC n ∩ k. The properties of the function f are given by the following lemma. 3.2 The function f is positive on S and vanishes on the boundary of S. The twisted moment map µ2ρp restricted to d log f defines a real algebraic isomorphism LEMMA
µ2ρp : d log f −→ G R · (2ρp )
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which preserves (resp., reverses) orientation if s is even (resp., odd). Proof Our argument is completely analogous to [SV2, proof of Lem. 7.10]. Let y ∈ Cl(S) \ S. We show h r (v, v) = 0 for v ∈ V n y . The form h r is G R -invariant, so conjugating by an element from G R , we may assume that p y contains a Cartan subalgebra h1 stable for θ and the conjugation with respect to gR . Then h1 is also stable for the conjugation with respect to the compact real form uR . Let V (λ1 ) and V (λ2 ) be h1 -weight spaces for λ1 6 = λ2 . We claim h u (V (λ1 ), V (λ2 )) = 0. Let v1 ∈ V (λ1 ), v2 ∈ V (λ2 ), and choose h ∈ h1 ∩ uR such that λ1 (h) 6= λ2 (h). Since λ2 is the sum of roots, we have λ2 (h) ∈ iR and, consequently, λ1 (h)h u (v1 , v2 ) = h u (h · v1 , v2 ) = −h u (v1 , h · v2 ) = λ2 (h)h u (v1 , v2 ). Hence, h u (v1 , v2 ) = 0, as desired. Observe that V n y = V (2ρ y ) and θ V n y = V (2θρ y ). Here 2ρ y is the sum of roots from n y . To prove h r (v, v) = 0 for v ∈ V n y , it suffices to show ρ y 6= θρ y . Otherwise, ρ y = θρ y would imply ρ y ∈ ik∗R . Then p y ∩ gR would contain a compact Cartan subalgebra. However, this is impossible since G R · y is not open. Next, we show that µ2ρp induces a real algebraic isomorphism between d log f and G R · (2ρp ). Let ν ∈ p⊥ , u ∈ UR . Suppose µ2ρp (uy0 , Ad(u)ν) = Ad(g)(2ρp ) for some g ∈ G R . The definition of µ2ρp implies Ad(u)ν = Ad(g)(2ρp ) − Ad(u)(2ρp ). Rewriting this condition and using the fact that 2ρp is P-regular, we obtain Ad(u −1 g)(2ρp ) = 2ρp + ν = Ad( p)(2ρp ) for some p ∈ P. Then Ad(g −1 up)ρp = ρp , and, consequently, gy0 = uy0 = y. Thus we obtain g ∈ G R , gy0 = uy0 . µ−1 2ρp Ad(g)(2ρp ) = gy0 , Ad(g)(2ρp ) − Ad(u)(2ρp ) , To establish the statement about the isomorphism, it suffices to prove d log f |gy0 = gy0 , Ad(g)(2ρp ) − Ad(u)(2ρp ) , g ∈ G R , gy0 = uy0 . Regarding d log f as a section of T ∗ Y R , we have for g ∈ G R and Z ∈ g,
d log f (exp t Z · gy0 )|t=0 = 2 Re µ(d log f |gy0 ), Z . dt On the other hand, a simple calculation yields d h r (Z v, v) + h r (v, Z v) h u (Z v, v) + h u (v, Z v) log f (exp t Z · gy0 )|t=0 = − dt h r (v, v) h u (v, v)
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for any v ∈ V n y . Since the space V n y is 1-dimensional, we may take v = gv0 in the first term and v = uv0 in the second term on the right-hand side. Here g ∈ G R , u ∈ UR , v0 ∈ V n , and gy0 = uy0 . Consider the decomposition g = c + [p, p] + n. If T ∈ [p, p] + n, then C · T v0 ∩ C · v0 = {0}. Since the distinct weight spaces are orthogonal with respect to h u , we obtain h r (T v0 , v0 ) = 0. On the other hand, if T ∈ c, then T v0 = h2ρp , T iv0 . Using the above decomposition of g, we regard 2ρp ∈ g∗ and apply the preceding discussion to T = Ad(g −1 )Z . We deduce h r (Zgv0 , gv0 ) + h r (gv0 , Zgv0 ) h r (Ad(g −1 )Z v0 , v0 ) + h r (v0 , Ad(g −1 )Z v0 ) = h r (gv0 , gv0 ) h r (v0 , v0 )
−1 = 2 Re 2ρp , Ad(g )Z . The analogous calculation for the second term finally yields µ(d log f |gy0 ) = Ad(g)(2ρp ) − Ad(u)(2ρp ), as desired. Now we turn to the orientation statement. It suffices to compare the orientations of S and G R · (2ρp ) only at points y0 and 2ρp . First, we describe the tangent spaces. L The holomorphic tangent space T2ρp G · (2ρp ) is isomorphic to g/l ' α∈±1(n) gα , and the real tangent space T2ρp G R · (2ρp ) is isomorphic to gR /lR ⊂ C ⊗R gR /lR ' g/l. On the other hand, the real tangent space Ty0 S R is isomorphic to gR /lR . The map I : S −→ G R · (2ρp ), I (gy0 ) = Ad(g)(2ρp ), g ∈ G R , is a real algebraic isomorphism, and we use it to compare the orientations. Via the above identifications of the tangent spaces, the differential of I at y0 becomes the identity on gR /lR . As before, we regard ρp ∈ g∗ . Then the symplectic form σ2ρp at 2ρp is given by the formula
σ2ρp (T1 , T2 ) = 2ρp , [T1 , T2 ] , T1 , T2 ∈ n + n. In particular, the root spaces gα and gβ are orthogonal for σ2ρp unless α + β = 0. For α ∈ 1(n), consider the 3-dimensional subalgebra gα = gα + g−α + [gα , g−α ]. Let G α , respectively, G α,R , be the connected subgroup of G with Lie algebra gα , respectively, gα ∩ gR . Clearly, G α,R ⊂ G α is a real form. Denote by Bα the Borel subgroup of G α with Lie algebra gα + [gα , g−α ]. Enumerate the elements of 1(n) as α1 , . . . , αm , and write yi for the point in the flag variety of gα defined by Bαi , i = 1, . . . , m. Observe that ρp restricts to a regular weight on gα , and consider the maps G α1 × · · · × G αm −→ Y,
(g1 , . . . , gm ) 7 → g1 · · · gm y0
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and G α1 × · · · × G αm −→ G · (2ρp ),
(g1 , . . . , gm ) 7 → Ad(g1 · · · gm )(2ρp ).
They induce further the maps G α1 /Bα1 × · · · × G αm /Bαm −→ Y and G α1 · (2ρp ) × · · · × G αm · (2ρp ) −→ G · (2ρp ). For the real orbits, we obtain the maps G α1 ,R · y1 × · · · × G αm ,R · ym −→ Y and G α1 ,R · (2ρp ) × · · · × G αm ,R · (2ρp ) −→ G R · (2ρp ). By considering the tangent maps, we deduce immediately that all these maps are local isomorphisms, compatible with complex structures and symplectic forms. In this way the problem is reduced to the special case of G R = SU(1, 1) if α is noncompact and G R = SU(2) if α is compact. In these cases the orientation statement is easily established by a direct computation. THEOREM 3.3 ∗ ∞ Set λ = 2ρp , and consider µ−1 λ (G R ·λ) as a 2m-cycle on T Y . Then, for φ ∈ C c (gR ), the following formula holds: Z Z ˆ λm ) = (−1)s ˆ λm ). µ∗λ (φσ µ∗λ (φσ CC(R j∗ C)
µ−1 λ (G R ·λ)
Proof Set C0 = CC(R j∗ C S ), C1 = d log( f |S). View C0 and C1 as 2m-cycles on T ∗ Y . Consider the map l : (0, 1) × T ∗ Y −→ T ∗ Y defined by l(t, (y, ξ )) = (y, tξ ), t ∈ R, y ∈ Y , ξ ∈ Ty∗ Y . Let C˜ be a (2m + 1)-chain on T ∗ Y with support equal to l((0, 1) × supp(C1 )) and the orientation determined by the product orientation of (0, 1) × supp(C1 ). Applying [SV1, Prop. 3.25, Th. 4.2], we obtain ˜ C0 − C1 = ∂ C. ˜ the statement of the theorem follows from PropoSince Re µ is bounded on supp(C), sition 2.4 and Lemma 3.2.
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THEOREM 3.4 Suppose that λ ∈ h∗ lies in the positive chamber C + . Then the formula from Theorem 3.3 holds.
Proof Set λ(t) = (1 − t)(2ρp ) + tλ, t ∈ [0, 1]. Clearly, λ(t) ∈ C + , and in particular, λ(t) is P-regular. Observe that S ∼ = G R · λ and that the same orientation statement as in Lemma 3.2 holds. Arguing similarly as in the proof of Theorem 3.3, we find −1 ˜ µλ(0) G R · (λ(0)) − µ−1 λ(1) G R · (λ(1)) = ∂ C, ˜ We use where C˜ is a (2m + 1)-chain on T ∗ Y such that Re µ is bounded on supp(C). brackets to indicate that a given set is considered as a cycle. The claim now follows from Theorem 3.3 and Proposition 2.4. 4. Elliptic orbital integrals In this section we study the nilpotent orbital integrals which arise as leading terms in the asymptotic expansion of Fourier transforms of elliptic orbital integrals. The material is analogous to [R2, §§7, 8] and [R3, §1]. We begin with a simple observation. We use the notation introduced in Sections 1 – 3. LEMMA 4.1 Let λ ∈ c∗ and k ∈ Z+ . Then
τλk
=
s X
pi (λ)τi ,
i=1
where p1 , . . . , ps are homogeneous polynomials on c∗ of degree k and τ1 , . . . , τs are UR -invariant 2k-forms on Y not depending on λ. The polynomials pi are real valued on cR . Proof Recall that hR ⊂ uR ∩ p is a Cartan subalgebra. For any α ∈ 1(g) we may find a vector xα in the α-weight space gα so that 2 hα . α(h α ) P Set eα = xα − x−α , f α = i(xα + x−α ), and define the spaces r1 = α∈1+ (l) (Reα + P R f α ), r2 = α∈1(n) ¯ (Reα + R f α ). We have the direct sum decompositions xα − x−α , i(xα + x−α ) ∈ uR
uR = h R + r 1 + r2 ,
and
[xα , x−α ] =
hR = hR ∩ [lR , lR ] + cR ,
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and we use them to identify R-linear duals h∗R , c∗R , r∗1 , r∗2 with subspaces of u∗R . Denote by (eα∗ , f α∗ ; α ∈ 1(¯n)) the basis in r∗2 dual to (eα , f α ; α ∈ 1(¯n)). Let e˜α , respectively, f˜α , be a unique UR -invariant form on Y whose value at y0 is eα∗ , respectively, f α∗ . The form τλ can then be written as X τλ = λ(i h α )e˜α ∧ f˜α . ¯ α∈1(n)
All the claims of the lemma are immediate consequences of this formula. S ⊥ −1 Recall that we have the decomposition TG∗R Y = l≥0 µ (Nl ∩ gR ), so that ⊥ −1 dimR µ (Nl ∩ gR ) ≤ 2m. Set I = l ∈ Z : dimR µ−1 (Nl ∩ g⊥ R ) = 2m . Choose C ∈ H2m (TG∗R Y, Z), and let k = k(C) be the minimal k ∈ Z+ such that ∗ C ∈ H2m (µ−1 (N˜k ∩ g⊥ R ), Z). In Section 1, we remarked that k ∈ I . Set TG R YC = S ⊥ −1 l∈I,l≤k µ (Nl ∩ gR ). 4.2 P In the group of 2m-chains on TG∗R Y , we have C = l∈I,l≤k Cl , where Cl is a G R −1 invariant 2m-chain on µ−1 (Nl ∩ g⊥ R ). If l = k, C k is the restriction of C to µ (Nk ∩ g⊥ R ). LEMMA
Proof ∗ The definition of TG∗R YC implies dimR µ−1 (N˜k ∩ g⊥ R ) \ TG R YC < 2m, so the exact sequence 0 −→ H2m (TG∗R YC , Z) −→ H2m µ−1 (N˜k ∩ g⊥ R ), Z ∗ −→ H2m µ−1 (N˜k ∩ g⊥ R ) \ TG R YC , Z −→ · · · determines the isomorphism H2m (TG∗R YC , Z) ∼ = H2m (µ−1 (N˜k ∩ g⊥ R ), Z). Write ∗ ∗ ∗ ∂ TG R YC = TG R YC \ TG R YC , and consider the exact sequence 0 −→ H2m (∂ TG∗R YC , Z) −→ H2m (TG∗R YC , Z) −→ H2m (TG∗R YC , Z) −→ · · · . The homomorphism H2m (TG∗R YC , Z) −→ H2m (TG∗R YC , Z) is injective since dimR ∂ TG∗R YC < 2m. Via this injection, we view C as an element from H2m (TG∗R YC , Z). For l ∈ I , l ≤ k = k(C), denote by Ul the interior of µ−1 (Nl ∩ gR ) in TG∗R YC . Applying repeatedly a Mayer-Vietoris sequence, we construct the injective homomorphism M 0 −→ H2m (TG∗R YC , Z) −→ H2m (Ul , Z). l∈I,l≤k
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L If we identify C with its image in l∈I,l≤k H2m (Ul , Z), we obtain the required deP composition C = l∈I,l≤k Cl . The cycle C is G R -invariant, and all the constructions involved are G R -invariant; so it follows that Cl is also G R -invariant. Finally, we re⊥ −1 mark that Uk = µ−1 (Nk ∩g⊥ R ); hence, C k is the restriction of C to µ (Nk ∩ gR ). Let O ⊂ N ∩ a⊥ be an A-orbit. The moment map restricts to a fibration µ : µ−1 O −→ O . Let 2dO = dimR O , and let 2eO = dimR µ−1 ν (ν ∈ O ). We orient O as in Section 2. The orientation on µ−1 O is defined via the fibration µ : µ−1 O −→ O , using the orientation on O and the orientation on the fibers induced by the underlying complex structure. Fix ν ∈ O , and denote by Z A (ν) the centralizer of ν in A. Then for p ∈ Z+ there exists a natural bijection between Z A (ν)-invariant (2eO − p)-chains on µ−1 ν and A-invariant (2m − p)-chains on µ−1 O . On the level of sets, this bijection is given by C(ν) 7→ C = A · C(ν),
C 7 → C(ν) = C ∩ µ−1 ν.
(4.3)
We recall the Lebesgue-Fubini theorem in the form convenient for our applications. Let V and W be C ∞ -manifolds of dimensions r and s, respectively. Let α and β be top-dimensional differential forms on V and W . Suppose that β is nonzero at any point of W . Let f : V −→ W be a submersion. Then there exists an (r − s)-form γ on V , so that α = γ ∧ f ∗ β. Here we denote by f ∗ β the pullback of β to V along f . Write αw and γw for the restriction of α and γ to the fiber f −1 w, and write β(w) for the value of β at w ∈ W . Then γw is uniquely determined by αw and β(w), so we write γw = αw /β(w). Suppose that V and W are oriented. Then we orient the fibers f −1 w compatibly with f . Further, we assume that β is the orientation form on W and denote by m β the corresponding measure. If the form α is integrable, then for almost all w ∈ W with respect to m β , the form γw is integrable on f −1 w and R (almost everywhere) defined form w 7 → ( f −1 w γw )β(w) is integrable on W . For the corresponding integrals, the following formula holds: Z Z Z α= γw β(w). (4.4) V
w∈W
f −1 w
Next, we describe how (4.4) is applied. Let O ⊂ N ∩ g⊥ R be a G R -orbit, and −1 let U ⊂ µ O be an open G R -invariant set. Let C ∈ H2m (U, Z). Choose ν ∈ O , and denote by µ−1 νr ⊂ µ−1 ν the set of regular points. The set U ∩ G R · µ−1 νr is open dense in U and dimR (U \ U ∩ G R · µ−1 νr ) < 2m. It follows that the restriction homomorphism H2m (U, Z) −→ H2m (U ∩ G R · µ−1 νr , Z) is injective. Any 2m-cycle on U ∩ G R · µ−1 νr is a Z-linear combination of connected components. If we choose such a component V , then µ : V −→ O is a submersion, and we may apply (4.4) to ˆ m σ m )|V and β = cdO σ dO . Here φ ∈ Cc∞ (gR ), and we set for the forms α = µ∗λ (φc λ O
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simplicity c = 1/(2πi). Adding up terms corresponding to the various components of the chain C, we obtain Z Z Z ˆ m σλm ) = µ∗λ (φc γν cdO σOdO . (4.5) ν∈O
C
C(ν)
Since C is G R -invariant, the inner integral converges for any ν ∈ O . We can describe (4.5) more explicitly. Taking into account Proposition 2.3, we have on µ−1 O , m X m ∗ i ∗ m ∗ ∗ m µλ (σλ ) = (π τλ + µ σO ) = π τλ ∧ µ∗ σOm−i . i i=0
Hence, we obtain for the quotient form m ∗ i m−i X m π τλ ∧ µ∗ σO = . µ∗ (σOdO ) i=eO i µ∗ (σOdO )
µ∗λ (σλm )
(4.6)
For (y, ν) ∈ T ∗ Y , set (y, ν) = u(y) · (y0 , ν0 ), where u(y) ∈ UR , ν0 ∈ Ty∗0 Y ∼ = p⊥ . Since φˆ is holomorphic, we have ∞ X 1 j ˆ ˆ µ∗λ φ(y, D φ(ν). (4.7) ν) = φˆ u(y) · λ + u(y) · ν0 = φˆ ν + u(y) · λ = j! u(y)·λ j=0
Here Dξ denotes the derivative in the direction of ξ ∈ g∗ . We set further for ν ∈ O , k X PO ,k (C, λ)φˆ (ν) = i=0
Denote by C[c∗ ](k)
dO !ceO (k − i)!i!(m − i)!
Z C(ν)
k−i ˆ (Du(y)·λ φ)(ν)
the set of homogeneous polynomials on c∗
π ∗ τλi ∧ µ∗ σOm−i µ∗ (σOdO )
.
(4.8) of degree k. Combining
(4.6), (4.7), and (4.8), we obtain the following result. LEMMA 4.9 Let U ⊂ µ−1 (O ) be a G R -invariant open set, let C ∈ H2m (U, Z), and let φ ∈ ˆ Cc∞ (gR ). Then for any ν ∈ O , we have PO ,k (C, ·)φ(ν) ∈ C[c∗ ](k) , and
Z C
∞
µ∗λ φˆ
1 m m X c σλ = m!
Z
k=0 O
PO ,k (C, λ)φˆ
1 dO dO c σO dO !
is a Taylor series expansion at λ = 0 of the holomorphic function λ 7 → R ∗ m m ˆ C µλ (φ(1/m!)c σλ ). The leading term in this expansion is equal to Z Z 1 dO dO pO (λ) φˆ c σO , where pO (λ) = π ∗ eτλ /2πi . O dO ! C(ν)
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Now we turn to the problem of comparing the leading term defined via the map (gr µ∗ )λ and the leading term of the Taylor series expansion for an arbitrary cycle. P Let C ∈ H2m (T A∗ Y, Z), and let k = k(C). Write C = l≤k,l∈I Cl as in Lemma 4.2. In the notation from Section 2, we have Ck = C ◦ . Any A-orbit O ⊂ Nk ∩ a⊥ is open; P hence, there exists a unique Ck,O ∈ H2m (µ−1 O , Z), so that Ck = O ⊂Nk ∩a⊥ Ck,O . R Choose νO ∈ O . Observe that pO (λ) = Ck,O (νO ) π ∗ eτλ /2πi does not depend on the choice of νO . Now we use the fact that the cup product followed by the pushforward agrees with integration over the fiber. The base component of the cycle Ck,O is [O ], and the fiber component over νO ∈ O is Ck,O (νO ), so we deduce X (gr µ∗ )λ (C) = pO (λ)[O ]. (4.10) O ⊂Nk ∩a⊥
On the other hand, if A = G R , we consider the Taylor series expansion of the R ˆ m ) at λ = 0. By Lemma 4.9 the terms holomorphic function (cm /m!) C µ∗λ (φσ λ R m ∗ ˆ Cl µλ (φσλ ) for l < k cannot contribute to the leading term. It follows that the leading term is equal to Z X ck pO (λ) φˆ σOk . (4.11) O k! ⊥ O ⊂Nk ∩gR
Next, our goal is to specialize the above considerations to the case of the characteristic cycle computed in Section 3. As in Section 3, we fix a θ-stable parabolic subalgebra p and denote by y0 the corresponding point in Y. Set Z = K · y0 , S = G R · y0 , and denote by i : Z −→ Y , j : S −→ Y the inclusions. The orbit Z is closed, and the orbit S is open in Y . Let G = Ri ∗ (C Z ) and F = R j! (C S ) be standard sheaves. Then for the Matsuki correspondence for the sheaves γ , we have γ (G ) = F . Recall that the characteristic cycle CC(L ) of a sheaf L is supported on the microsupport SS(L ). In our case, SS(G ) = TZ∗ Y , and, consequently, µ(SS(G )) is an irreducible K -invariant complex variety in N ∩ k⊥ . It follows that µ(SS(G )) = V for a single K -orbit V ⊂ N ∩ k⊥ . Set 2k = dimR V . An application of Theorem 2.6 shows that k is the minimal integer such that SS(F ) ⊂ µ−1 (N˜k ∩ g⊥ R ), and furthermore ⊥ µ(SS(F )) ∩ Nk = O for a unique G R -orbit O ⊂ N ∩ gR . Finally, we observe that CC(F ) = −CC(R j∗ (C S )) by [SV1, Th. 4.2]. The preceding discussion and Theorems 2.6, 2.7, and 3.4 imply immediately the following result analogous to [R3, Th. 1.6.1]. For the convenience of the reader, we give a detailed explanation of the notation. THEOREM 4.12 Let G R be a connected, linear, semisimple group. Assume that G R has a compact Cartan subgroup. Let p be a θ-stable parabolic subalgebra in g, and let p be the
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complex conjugate. Then l = p ∩ p is a Levi factor of p. Denote by c the center of l, and denote by C + the positive chamber in ic∗R defined by the roots from p/l. Let V ⊂ N ∩ k⊥ be the K -orbit such that K · (p⊥ ∩ k⊥ ) = V . Set 2k = dimR V , and denote by O ⊂ N ∩ g⊥ R the G R -orbit associated with V via the Sekiguchi correspondence. Let Z be the K -orbit of the point y0 corresponding to p in the generalized flag variety Y . Denote by C ∈ H2m (TK∗ Y, Z) the fundamental cycle of the conormal bundle TK∗ Y , denote by C 0 the restriction of C to µ−1 V , and denote by C 0 (ν) the restriction of C 0 to µ−1 (ν), ν ∈ V (cf. (4.3)). Let s = dimC n ∩ s. Then R p(λ) = (−1)s+1 C 0 (ν) π ∗ eτλ /2πi is a homogeneous polynomial of degree m − k, and for λ ∈ C + , φ ∈ Cc∞ (gR ), the leading term in the Taylor series expansion at λ = 0 R ˆ m is equal to of the holomorphic function λ 7 → 1/(m!(2πi)m ) G R ·λ φσ λ Z p(λ)
O
φˆ
σOk . k!(2πi)k
In the next proposition we show that the polynomial p is nonzero. PROPOSITION 4.13 Suppose that −λ ∈ C + is integral. Then p(λ) 6 = 0.
Proof Let V be a K -orbit from Theorem 4.12. We show that there exists an irreducible component D of µ−1 ν, ν ∈ V , such that K · D is open dense in TZ∗ Y . Let D1 , . . . , Dt be the irreducible components of µ−1 ν so that µ−1 V = K · D1 ∪ · · · ∪ K · Dt and K · Di are pairwise distinct. Clearly, K · Di , i = 1, . . . , t, are the irreducible components of µ−1 V . Since TZ∗ Y ∩ µ−1 V is a nonempty open set in TZ∗ Y and TZ∗ Y is irreducible, we have TZ∗ Y = TZ∗ Y ∩ µ−1 V ⊂ K · D1 ∪ · · · ∪ K · Dt . Again for the reasons of irreducibility, TZ∗ Y ⊂ K · Di for some i. Set Di = D. Since dimC TZ∗ Y = dimC K · D, we deduce TZ∗ Y = K · D. Observe that K · D is closed in µ−1 V ; so K · D ∩ µ−1 V = K · D. Hence, the restriction of [TZ∗ Y ] to µ−1 V is equal to [K · D]. The bracket here denotes the fundamental cycle. Let z 1 , . . . , zr be a full set of representatives for the component group Z K (ν)/Z K (ν)◦ . Then the fiber of [K · D] over ν ∈ V can be written as [K · D](ν) = [z 1 D] + R · · · + [zr D]; hence, we obtain p(λ) = r D eτλ /2πi . If −λ ∈ C + is integral, let Vλ be the finite-dimensional representation of G with P-lowest weight λ. If necessary, we
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replace G here by an appropriate covering. Denote by i λ the imbedding of Y into the projective space P(Vλ ) as the orbit of a lowest weight vector. A simple computation shows i λ∗ (ω) = (τλ )/(2πi), where ω is the standard K¨ahler form on P(Vλ ). Finally, we obtain Z r p(λ) = ω(m−k) > 0, (m − k)! D as desired. Given a polynomial p ∈ C[c∗ ], denote by p(∂) a constant coefficient differential operator on c determined uniquely by the property p(∂)eλ = p(λ)eλ ,
λ ∈ c∗ .
Consider the Taylor series expansion of the holomorphic function λ 7→ R ˆ m at λ = 0. Differentiating this expansion by p(∂) and then 1/(m!(2πi)m ) G R ·λ φσ λ letting λ → 0, we deduce the following formula: Z Z p(∂) p 1 ˆ λm = ˆ k. φσ φσ (4.14) lim p(∂) m!(2πi)m G R ·λ k!(2πi)k O O λ→0(C + ) Define the measures m λ and m O on the orbits G R · λ and O by the formulas mλ =
1 σm m!(2πi)m λ
and
mO =
1 σk . k!(2πi)k O
Observe that Lemma 4.1 implies p(∂) p 6= 0; hence, taking the Fourier transform in (4.14), we obtain the following theorem. 4.15 With the same assumptions as in Proposition 4.13, the following limit formula for the orbital measures holds: lim p(∂)m λ = cm O . THEOREM
λ→0(C + )
Here c is a nonzero constant. In fact, we may take c = p(∂) p. Finally, we relate p(λ) to the harmonic polynomials. As in Lemma 4.1 we choose a Cartan subalgebra h ⊂ g so that hR ⊂ l ∩ uR . Write h = h1 + c, where h1 ⊂ [l, l] is a Cartan subalgebra. Taking C-linear duals, we have h∗ = h∗1 + c∗ . Using this splitting, we write for λ ∈ h∗ , λ = λ1 + λ2 , where λ1 ∈ h∗1 and λ2 ∈ c∗ . Denote by H (h∗ ) the space of harmonic polynomials on h∗ . Let X be the flag variety of Borel subalgebras of g. Denote by q : X −→ Y the natural fibration, and write l = dimC q −1 y0 . Let µ0 : T ∗ X −→ g∗ be the moment map defined analogously to µ. Let V be the K orbit defined in Proposition 4.13. Choose ν ∈ V , and view µ−1 0 ν as a subset of X ,
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and µ−1 ν as a subset of Y . Observe that dimC µ−1 ν = m − dimC O = e implies dimC µ−1 0 ν = e + l. Hence, the fibration q determines the pullback homomorphism q ∗ : H2e (µ−1 ν, C) −→ H2(e+l) (µ−1 0 ν, C). Denote by C G (ν) the component group of the centralizer of ν in G. Let W be the Weyl group of g. By the Springer theory, the −1 C G (ν) carry a natural structure of W spaces H2(e+l) (µ−1 0 ν, C) and H2(e+l) (µ0 ν, C) C G (ν) is the Springer module. Furthermore, the character of W on H2(e+l) (µ−1 0 ν, C) representation χν corresponding to the orbit G · ν and the trivial character of C G (ν). Homology classes and harmonic polynomials are related via the Borel map Z −1 C G (ν) ∗ H2(e+l) (µ0 ν, C) −→ H (h ), η 7→ τλe+l . η
We remark that the Borel map intertwines W -actions. In the notation from Proposition 4.13, set C = [TZ∗ Y ]◦ (ν), ν ∈ V . Via the homomorphism H2e (µ−1 ν, Z) −→ H2e (µ−1 ν, C), we view C ∈ H2e (µ−1 ν, C). Define P(λ) ∈ H (h∗ ) by the formula Z P(λ) = τλe+l . q∗C
Our goal is to relate polynomials P(λ) and p(λ). We remark that the proof of Lemma 4.1 implies τλ = τλ1 + τλ2 , so we may write τλe+l =
e+l X
pi (λ1 )τi .
(4.16)
i=0
Here pi (λ1 ) is a homogeneous polynomial in λ1 of degree i, and τi is a 2(e + l) l form not depending on λ1 . If i = e, then pl (λ1 )τl = e+l τλ1 ∧ τλe2 . Set ω L (λ1 ) = l Q ⊥ ⊥ α∈1+ (l) λ1 (i h α ). Since ν ∈ K · (p ∩ k ), we may assume without any loss of −1 generality y0 ∈ µ ν. Hence, from (4.4) we deduce Z Z Z l e l τλ1 ∧ τλ2 = τλ1 · τλe2 = c1 ω L (λ1 ) p(λ2 ), q∗C
q −1 y0
C
where c1 is a nonzero constant. After differentiating the integral of (4.16) by ω L (∂) and letting λ1 → 0, we deduce the following proposition. PROPOSITION 4.17 Let V be the open K -orbit in K · (p⊥ ∩ k⊥ ), and let ν ∈ V . Write λ = λ1 + λ2 according to the splitting h∗ = h∗1 + c∗ . Denote by C the fiber of [TZ∗ Y ] at ν. R Then P(λ) = q ∗ C τλe+l is a harmonic polynomial on h∗ which transforms under the W -action according to the Springer character χν . The polynomial p(λ2 ) can be computed from P(λ) by the formula
p(λ2 ) = c lim ω L (∂)P(λ), λ1 →0
where c is a nonzero constant.
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ˇ ´ ˇ CEVI MLADEN BOZI C
University of Zagreb, Department of Geotechnical Engineering, Hallerova aleja 7, 42000 Varaˇzdin, Croatia; [email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 2,
ISOSPECTRAL DEFORMATIONS OF NEGATIVELY CURVED RIEMANNIAN MANIFOLDS WITH BOUNDARY WHICH ARE NOT LOCALLY ISOMETRIC CAROLYN S. GORDON and ZOLTAN I. SZABO
Abstract To what extent does the eigenvalue spectrum of the Laplace-Beltrami operator on a compact Riemannian manifold determine the geometry of the manifold? We present a method for constructing isospectral manifolds with different local geometry, generalizing an earlier technique. Examples include continuous families of isospectral negatively curved manifolds with boundary as well as various pairs of isospectral manifolds. The latter illustrate that the spectrum does not determine whether a manifold with boundary has negative curvature, whether it has constant Ricci curvature, and whether it has parallel curvature tensor, and the spectrum does not determine whether a closed manifold has constant scalar curvature. Introduction A fundamental question in spectral geometry is the extent to which the spectrum of the Laplacian on a Riemannian manifold determines the geometry of the manifold. The only way to identify specific geometric invariants that are not spectrally determined is through explicit constructions of isospectral manifolds, that is, manifolds whose Laplacians, acting on smooth functions, have the same eigenvalue spectrum. In the case of manifolds with boundary, one may consider the spectrum of the Laplacian acting on functions satisfying either Dirichlet or Neumann boundary conditions. We say that two manifolds with boundary are isospectral if they are both Dirichlet and Neumann isospectral. All examples of isospectral manifolds constructed prior to 1992, as well as many of the more recent examples, are locally isometric (see, e.g., [BGG], [Bu], [DG], DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 2, Received 30 October 2000. Revision received 12 June 2001. 2000 Mathematics Subject Classification. Primary 53C20, 35P05; Secondary 11F72, 34L40. Gordon’s work partially supported by National Science Foundation grant number DMS-0072534. Szabo’s work partially supported by National Science Foundation grant number DMS-0104361 and City University of New York grant number 9-91907.
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[GWW], [GW1], [GW2], [Gt1], [Gt2], [I], [M], [Su], [V], or the expository articles [Be], [Br], [G3], or [GGt]). These examples reveal various global invariants that are not spectrally determined, such as the diameter and the fundamental group, but give no information concerning local invariants such as curvature. In the past several years, examples of isospectral manifolds with different local geometry have appeared. The first such examples were pairs of manifolds with boundary constructed by Z. Szabo (preprint, 1992). These examples, together with examples constructed later of closed manifolds, appeared in [Sz]. Among the latter examples are a pair of isospectral closed manifolds, one of which is homogeneous and the other not. The examples in [Sz] were proven isospectral first by explicit computation of the spectra and later by construction of an intertwining operator between the Laplacians. C. Gordon developed a technique for constructing isospectral manifolds with different local geometry in [G1] and [G2]. This technique was further developed in [GW4], resulting in continuous families of isospectral manifolds with boundary having different Ricci curvature, and in [GGS], resulting in continuous families of isospectral closed manifolds whose scalar curvature functions have different maxima. D. Schueth [Sch1], [Sch2] modified this technique to construct the first examples of isospectral simply connected closed manifolds, in fact, isospectral deformations of simply connected closed manifolds. All these examples of isospectral manifolds with different local geometry are principal torus bundles with totally geodesic fibers. In this article, we give new examples of isospectral manifolds with different local geometry. The manifolds are again principal torus bundles, but the fibers are not totally geodesic. The new examples include the following. (i) Continuous isospectral deformations of negatively curved manifolds with boundary. The boundaries are also isospectral but in general nonisometric. Examples can be constructed in which the curvature is bounded above by any prescribed negative constant. The examples include both families of locally homogeneous manifolds and families of locally inhomogeneous manifolds. These examples contrast with the result of V. Guillemin and D. Kazhdan [GuK] in dimension two, generalized by C. Croke and V. Sharafutdinov [CS] to arbitrary dimensions, stating that negatively curved closed manifolds cannot be continuously isospectrally deformed. (ii) A pair of isospectral manifolds with boundary, one of which has negative sectional curvature and the other mixed curvature. (iii) A pair of isospectral manifolds with boundary, one of which has constant Ricci curvature and the other variable Ricci curvature. (iv) A pair of isospectral closed manifolds, one of which has constant scalar curvature and the other variable scalar curvature. In contrast, V. Patodi [P] showed that from the spectra of the Laplacian acting on functions, 1-forms, and 2-forms, one can tell whether the manifold has constant scalar curvature. Our examples show that
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the spectrum on functions alone does not determine this information. (v) Pairs of isospectral manifolds with boundary such that one manifold has parallel curvature tensor (it is a domain in a locally symmetric space) and the other does not. Examples (iii) and (iv) suggest the possibility that one may not be able to tell from the spectrum of the Laplacian acting on functions whether a closed manifold is Einstein, although the spectra of the Laplacian on functions, 1-forms, and 2-forms jointly determine whether the metric is Einstein (see [P]). The underlying differentiable manifold in each of the isospectral families is of the form T × Q, where Q is a suitable submanifold of the upper half-space of Rm+1 and T is a torus. For c ∈ R+ , we let gc be the Riemannian metric induced on Q by the metric of constant curvature −c2 on the upper half-space. For each j in a linear parameter space to be described in Section 2 and for each c ∈ R+ , we define a Riemannian metric g( j, c) on T × Q so that the projection (T × Q, g( j, c)) → (Q, gc ) is a Riemannian submersion. The induced metric on each toral fiber is flat. With c arbitrary but fixed, suitable variations of the parameter j result in families of isospectral, nonisometric Riemannian metrics g( j, c). The parameter c may be adjusted in order to control the curvature of the metrics g( j, c). It is very likely that the construction also allows a negatively curved Riemannian metric on a manifold with boundary to be continuously isospectrally deformed to a metric with curvature of both signs. In the case when the submanifold Q of the upper half-space is either a domain with boundary or a suitably chosen submanifold (with boundary) of codimension one, the metrics g( j, c) have strictly negative curvature when c is chosen sufficiently large. More precisely, there exists for each j a constant λ( j) such that the metric g( j, c) on T × Q has strictly negative curvature when c > λ( j) and mixed curvature when c < λ( j). In order to obtain an isospectral deformation from a negatively curved metric to a metric of mixed curvature by our technique, one must choose the variation jt so that λ( jt ) depends nontrivially on t. One can then choose c so that λ( jt ) > c for some values of t and λ( jt ) < c for other values of t. We expect that λ( jt ) does indeed vary nontrivially with t for the generic deformations that we construct. Unfortunately, explicit computation of λ( j) for given j seems intractable at this point. In the one case in which we were able explicitly to compare the values of λ( j) for two different j (using results of E. Damek [D]), we found that the two values were indeed different. This pair of j-parameters was used to construct the examples referred to in example (ii) above of isospectral metrics, only one of which is negatively curved. The paper is organized as follows. In Section 1, we explain the general technique that is used to prove isospectrality. The first part of Section 2 (see Sections 2.1, 2.2, 2.4; Lemma 2.3; Corollary 2.5) introduces the Riemannian manifolds
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(T × Q, g( j, c)). The main results of this section (see Theorems 2.8, 2.9; Corollaries 2.10, 2.12) give conditions on a pair of parameters j, j 0 in order for the metrics g( j, c) and g( j 0 , c) on T × Q to be isospectral but not isometric (or locally isometric) and assert the existence of nontrivial isospectral deformations. The remainder of Section 2 gives the proofs. In Section 3, we consider the case in which the submanifold Q of the upper half-space is a bounded domain; in this case each Riemannian metric g( j, c) on T × Q is locally homogeneous. We introduce λ( j), as described in the previous paragraph, and then construct examples of locally homogeneous families or pairs of metrics illustrating each of the properties in (i) – (v) above. In Section 4, we consider the case in which Q has codimension one in the upper half-space. In this case the metrics g( j, c) on T × Q are not locally homogeneous in general. Specializing to a particular class of such submanifolds Q, we again obtain the existence of a map λ( j) satisfying the condition introduced previously and consequently obtain isospectral deformations of negatively curved locally inhomogeneous Riemannian metrics on manifolds with boundary, as asserted in (i) above.
1. Technique for constructing isospectral manifolds with different local geometry 1.1. Background and notation Let T be a torus, let π : M → N be a principal T -bundle, and endow M with a Riemannian metric so that the action of T is by isometries. Give N the induced Riemannian metric so that π is a Riemannian submersion. For a ∈ M, we can decompose Ta (M) into the vertical subspace (the tangent space to the fiber at a) and the horizontal subspace (the orthogonal complement to the vertical space). For X ∈ Ta (M), write X = X v + X h , where X v is vertical and X h is horizontal. Let ∇ be the Levi-Civita connection on M. The mean curvature at a of the toral fiber through a in M is given by k X Ha = (∇ Z i Z i )h , i=1
where Z 1 , . . . , Z k is an orthonormal basis for the space of T -invariant vector fields on the fiber. Since T acts by isometries, we have Hz(a) = z ∗ Ha for all a ∈ M and z ∈ T . Hence we may define a π -related vector field H˜ on N . We refer to H˜ as the projected mean curvature vector field of the submersion. L. Berard-Bergery and J.-P. Bourguignon [BB] gave a decomposition of the Laplacian 1 M into vertical and horizontal components 1 M = 1v +1h . In the case of functions f on M which are constant on the fibers of the submersion, so that f = π ∗ f¯ for some function f¯ on N , we have 1v ( f ) = 0 and (1.1) 1 M ( f ) = 1h ( f ) = π ∗ 1 N ( f¯) + H˜ ( f¯) .
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If N has nontrivial boundary, then ∂ M = π −1 (∂ N ). Since π : M → N is a Riemannian submersion, π ∗ : C ∞ (N ) → C ∞ (M) maps functions on N satisfying Neumann boundary conditions to functions on M satisfying Neumann boundary conditions. Of course, it also maps functions satisfying Dirichlet conditions to functions satisfying Dirichlet conditions. Remark. In the situation of Notation 1.1, the space π ∗ (C ∞ (N )) of functions on M which are constant on the fibers of the submersion is precisely the space of T -invariant functions. Since T acts by isometries, it follows that π ∗ (C ∞ (N )) is invariant under 1 M . The map π ∗ : C ∞ (N ) → π ∗ (C ∞ (N )) is a linear isomorphism, but it need not be unitary. The operator 1 N + H˜ on N is not necessarily selfadjoint; however, it does have a discrete spectrum since it is similar to the restriction of 1 M to π ∗ (C ∞ (N )). 1.2 (Abstract method for proving isospectrality) Let T be a torus. Suppose that M1 and M2 are principal T -bundles, either both closed or both having nonempty boundary, endowed with Riemannian metrics so that the T action is by isometries. For each subtorus K of T of codimension at most one, suppose (1) that the operator 1 K \M1 + H˜ K acting on C ∞ (K \M1 ) (or on the subspace of functions satisfying Dirichlet, resp., Neumann, boundary conditions in the case of mani(2) folds with boundary) is isospectral to the operator 1 K \M2 + H˜ K on C ∞ (K \M2 ) (or on the subspace of functions satisfying the respective boundary conditions), where 1 K \Mi is the Laplacian on K \Mi associated with the Riemannian metric induced (i) from Mi and where H˜ K is the projected mean curvature vector field for the submersion Mi → K \Mi as in Section 1.1. Then M1 and M2 are isospectral (Dirichlet, resp., Neumann, isospectral in the case of nonempty boundary). THEOREM
This theorem generalizes a method developed in [G2] and [GW4] in the special case when the fibers are totally geodesic. Remark 1.3 In our applications of Theorem 1.2, the manifolds satisfy the stronger hypothesis that for each subtorus K of codimension at most one, there exists an isometry τ K : (1) (2) K \M1 → K \M2 satisfying τ K ∗ H˜ K = H˜ K . In particular, in the case when M1 and M2 have nontrivial boundary, we can conclude that they are both Dirichlet and Neumann isospectral. Proof Let 1i denote the Laplacian of Mi , and let L2C (Mi ) denote the space of complexvalued square-integrable functions on Mi . The torus T acts on L2C (Mi ) via the action
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z( f )(x) = f (zx), where zx denotes the action of z ∈ T on x ∈ Mi , i = 1, 2. By a Fourier decomposition for this action, we have X L2C (Mi ) = Hi α , α∈Tˆ
where Tˆ consists of all characters on T , that is, all homomorphisms from the group T to the unit complex numbers, and Hi α = f ∈ L2C (Mi ) : z( f ) = α(z) f for all z ∈ T . Since the torus action on Mi is by isometries, the Laplacian leaves each of the subspaces Hi α invariant. If Mi has boundary, then the subspace of smooth functions satisfying either Neumann or Dirichlet boundary conditions decomposes into its intersections with the spaces Hi α . Define an equivalence relation on Tˆ by α ≡ β if ker(α) = ker(β). Let [α] denote the equivalence class of α, and let [Tˆ ] denote the set of equivalence classes. Setting X β Hi [α] = Hi , β∈[α]
then L2C (Mi ) =
X
Hi [α] .
[α]∈[Tˆ ]
For α = 1 the trivial character, we have [1] = {1}, and the space Hi 1 consists of those functions constant on the fibers of the submersion πi : Mi → T \Mi . By equation (1.1) and the remarks following Section 1.1, πi∗ intertwines the restriction (i) of 1i to Hi 1 with 1T \Mi + H˜ T acting on L2C (T \Mi ). In the case when the manifolds have boundary, πi∗ carries Dirichlet, respectively, Neumann, eigenfunctions of (i) 1T \Mi + H˜ T to Dirichlet, respectively, Neumann, eigenfunctions of 1i associated (1) with the same eigenvalues. Thus the hypothesis that 1T \M1 + H˜ T is isospectral to (2) 1T \M2 + H˜ T (possibly with Dirichlet or Neumann boundary conditions) implies that the restrictions of the Laplacians 1i to the spaces Hi 1 are isospectral (with the corresponding boundary conditions in the case when the manifolds have boundary). For nontrivial α ∈ Tˆ , the kernel of α is a subtorus K of T of codimension one. The space of all functions on Mi constant on the fibers of the submersion πi : Mi → K \Mi coincides with Hi [α] ⊕ Hi 1 . We can again use the hypothesis of Theorem 1.2 to conclude that the restrictions of the Laplacians of Mi to the subspaces Hi [α] ⊕ Hi 1 , i = 1, 2, are isospectral. Since we already know that the restrictions to the subspaces Hi 1 are isospectral, we conclude that the restrictions to Hi [α] are also isospectral, and the theorem follows.
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2. Construction of isospectral metrics We now introduce the specific classes of manifolds that are our main objects of study as follows. • We first recall in Section 2.1 that the upper half-space Rm in Rm+1 has a natural structure as a solvable Lie group. • Associated to each linear map j : Rk → so(m), we define in Section 2.2 a solvable Lie group G( j) diffeomorphic to Rk × Rm and such that the canonical projection Rk × Rm → Rm corresponds to a Lie group homomorphism G( j) → Rm . We emphasize, however, that the Lie group structure does not correspond to a direct product structure on Rk × Rm . • For each c ∈ R+ , we define in Section 2.2(ii) a left-invariant Riemannian metric g˜ c on G( j). We denote by g( j, c) the pullback of this metric by the diffeomorphism above to a Riemannian metric on Rk × Rm (see Lemma 2.3). Letting T = L \Rk be a torus, and letting Q be a submanifold of Rm satisfying a certain symmetry property, the metric g( j, c) induces metrics, also denoted g( j, c), both on T × Rm and on T × Q (see Section 2.4). The Riemannian manifolds of the form (T × Q, g( j, c)) are our main objects of study. 2.1. Notation and remarks (i) The upper half-space realized as a Lie group. We recall that the hyperbolic space of dimension m + 1 admits a simply transitive solvable Lie group Rm of isometries and thus may be identified with Rm endowed with a left-invariant metric. The Lie group Rm is of course diffeomorphic to R+ × Rm ; the Lie group multiplication is given by (t, x)(t 0 , x 0 ) = (tt 0 , x + t x 0 ) for t, t 0 ∈ R+ and x, x 0 ∈ Rm . Let A be the left-invariant vector field on Rm whose value at the identity element (1, 0) is given by A(1,0) = ∂/∂t. Then A is given at an arbitrary point (t, x) by A = t (∂/∂t). The Lie algebra rm of Rm is given by rm = a + v, where v = Rm is an abelian ideal, a = RA, and [A, x] = x for all x ∈ v. (ii) Constant curvature metrics gc . We fix once and for all the “standard” inner product on v = Rm and denote by gc the left-invariant Riemannian metric on Rm for which A ⊥ v and kAk = 1/c; this metric has constant curvature κ = −c2 . As an aside, we remark that every left-invariant Riemannian metric on Rm is isometric to gc for some c. (iii) Action of O(m) on hyperbolic space. Each τ in the orthogonal group O(m) defines an automorphism of Rm , also denoted τ , given by (t, x) → (t, τ (x)) for all
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(t, x) ∈ Rm . This automorphism is an isometry with respect to each of the metrics gc . 2.2. Notation (i) The solvable Lie group G( j) and its nilradical H ( j). Starting with an inner product space z, say, of dimension k (we frequently identify z with Rk ), and a nontrivial linear map j : z → so(m, R), a 1-dimensional vector space a, and a fixed choice of nonzero vector A ∈ a, we construct Lie algebras h( j) and g( j) as follows. Let v = Rm . As vector spaces, we set h( j) = v + z and g( j) = a + v + z. Define a Lie bracket on h( j) so that z is central, [v, v] ⊂ z, and
[x, y], z = j (z)x, y for x, y ∈ v and z ∈ z, where h , i denotes both the standard inner product on v = Rm and the inner product on z. This gives h( j) the structure of a two-step nilpotent Lie algebra. Next, give g( j) the unique bracket structure so that h( j) is an ideal in g( j) and [A, x] = x and [A, z] = 2z for x ∈ v and z ∈ z. With these structures, g( j) is a solvable Lie algebra with nilradical h( j), z is an abelian ideal in g( j), and the quotient z\g( j) is isomorphic to the Lie algebra rm constructed in Section 2.1. Let G( j), respectively, H ( j), be the simply connected Lie group with Lie algebra g( j), respectively, h( j). Then G( j) is diffeomorphic to R+ ×v×z with multiplication given by 1 (t, x, z)(t 0 , x 0 , z 0 ) = tt 0 , x + t x 0 , z + t 2 z 0 + t[x, x 0 ] 2 0 + 0 0 for t, t ∈ R , x, x ∈ v, and z, z ∈ z. Note that (1, 0, 0) is the identity element of G( j). The nilpotent Lie group H ( j) is isomorphic to the subgroup obtained by holding t = 1. Thus H ( j) is diffeomorphic to Rm × z with multiplication given by 1 (x, z)(x 0 , z 0 ) = x + x 0 , z + z 0 + [x, x 0 ] . 2 (ii) Riemannian metrics g˜ c on G( j) and g˜ on H ( j). In the notation of (i), the “standard” inner product on v = Rm and the inner product on z define an inner product on h( j) so that the decomposition h( j) = v + z is an orthogonal sum. This inner product defines a left-invariant Riemannian metric g˜ on H ( j). The metric g˜ is the only Riemannian metric we consider on this Lie group.
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Given a real number c > 0, define an inner product on a by requiring that kAk = 1/c. This inner product, together with the inner product defined above on h( j) = v+z, defines an inner product on g( j) so that the decomposition g( j) = a + v + z is an orthogonal sum. We denote the resulting left-invariant Riemannian metric on G( j) by g˜ c . We caution that for different choices of j, say, j and j 0 , the manifolds (H ( j), g) ˜ 0 and (H ( j ), g) ˜ are in general not isometric, even though we are using the same name for the Riemannian metrics. A similar statement holds for (G( j), g˜ c ) and (G( j 0 ), g˜ c ) with c fixed. The nilpotent Lie subgroup H ( j) of G( j) does not play a direct role in our construction of isospectral metrics but is needed in the proof that the metrics we construct are not isometric. 2.3 We use the notation of Sections 2.1 and 2.2. Identify the vector space z (i.e., Rk ) with the abelian normal subgroup {(1, 0, z) : z ∈ z} of G( j). Then we have the following. (i) The quotient z\G( j) is isomorphic to Rm , and the homomorphic projection LEMMA
π j : G( j) → Rm is a Riemannian submersion relative to the Riemannian metrics g˜ c on G( j) and gc on Rm for each choice of c. The induced metric on each fiber is Euclidean. (ii) Define a diffeomorphism (not a group homomorphism) α j : Rk × Rm → G( j) by α j z, (t, x) = (t, x, z) for (t, x) ∈ Rm and z ∈ z = Rk . Then α j intertwines the action of Rk on Rk × Rm , given by translation in the first factor, with the action of Rk = z on G( j), given by left translation (where z is viewed as a subgroup of G( j) as above). The following diagram commutes, where the maps on the top and bottom rows are the canonical projection and the homomorphic projection, respectively: Rk × Rm −−−−→ αj y G( j)
Rm yId
−−−−→ Rm
(iii) Letting g( j, c) = α ∗j (g˜ c ),
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g( j, c) is a homogeneous Riemannian metric on Rk × Rm , invariant under the action of Rk , and the canonical projection π : (Rk × Rm , g˜ c ) → (Rm , gc ) is a Riemannian submersion. (The metric is not a product metric.) Proof Assertions (i) and (ii) are immediate from Sections 2.1 and 2.2, and (iii) is a consequence of (i) and (ii). While the metric g( j, c) on Rk × Rm is homogeneous, it is not the case that the natural action by Rm (translation in the second factor) is by isometries. 2.4. The Riemannian manifold (T × Q, g( j, c)) Let T = L \z (= L \Rk ) be a torus, and let Q be a submanifold of Rm , invariant under the action of O(m) defined in Section 2.1(iii). Let j : z → so(m) be a linear map, and let c ∈ R+ . Since the Riemannian metric g( j, c) on Rk × Rm defined in Lemma 2.3(iii) is invariant under left translation by elements of Rk , it induces a Riemannian metric, also denoted g( j, c), on T × Rm . We use the same notation g( j, c) for the restriction of this metric to the submanifold T × Q. It is sometimes convenient to identify T × Rm with a quotient of G( j). The lattice L may be identified with a discrete subgroup of G( j) by identifying z with a normal ¯ j) = L \G( j). (We caution that G( ¯ j) subgroup of G( j) as in Lemma 2.3. Let G( does not inherit a Lie group structure from G( j) since L is not a normal subgroup of G( j).) The left-invariant Riemannian metric g˜ c on G( j) induces a Riemannian ¯ j), giving G( ¯ j) the structure of a locally homogeneous metric, also denoted g˜ c on G( Riemannian manifold. The map α j induces an isometry α¯ j : (T × Rm , g( j, c)) → ¯ j), g˜ c ). (G( 2.5 We use the notation of Section 2.4. Let c ∈ R+ . Denote by gc the Riemannian metric on Q induced by the metric of the same name on Rm (see Section 2.1). Then for any choice of j, the canonical projection π : (T × Q, g( j, c)) → (Q, gc ) is a Riemannian submersion. Moreover, the natural action of T on T × Q by translations in the first factor is an isometric action relative to g( j, c). COROLLARY
The manifolds (T × Q, g( j, c)) are our main objects of study. Definition 2.6 (i) Let z be an inner product space. A pair j, j 0 of linear maps from z to so(m) is called equivalent, denoted j ' j 0 , if there exist orthogonal transformations α
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of Rm and β of z such that j 0 β(z) = α j (z)α −1 (ii)
(iii)
for all z ∈ z. Let L be a lattice of full rank in z. We say that the pair ( j, L ) is equivalent to the pair ( j 0 , L ), denoted ( j, L ) ' ( j 0 , L ), if j ' j 0 and if the map β in definition (i) can be chosen so that β(L ) = L . The pair j, j 0 is called isospectral, denoted j ∼ j 0 , if for each z ∈ z, the eigenvalue spectra (with multiplicities) of j (z) and j 0 (z) coincide; that is, for each z ∈ z, there exists an orthogonal linear operator αz ∈ O(m) for which αz j (z)αz−1 = j 0 (z).
Observation 2.7 Suppose that j, j 0 : z → so(m) are isospectral linear maps as in Definition 2.6(iii). If 0 are equivalent. u is any 1-dimensional subspace of z, then the restrictions j|u and j|u 0 Moreover, if L is any lattice in u, then ( j|u , L ) is equivalent to ( j |u , L ). The main results of this section are the following criteria for isospectrality and for isometry of the metrics defined in Section 2.4. We state both results and their consequences before giving the proofs. THEOREM 2.8 We use the notation of Section 2.4 and Definition 2.6. Let T = L \Rk be a torus, and let Q be an O(m)-invariant submanifold of Rm . Let j, j 0 : Rk → so(m) be isospectral linear maps. Then for each c ∈ R+ , the Riemannian metrics g( j, c) and g( j 0 , c) on T × Q are isospectral. If the manifold T × Q has boundary, then the metrics are both Dirichlet and Neumann isospectral. Moreover, the induced metrics on the boundary T × ∂(Q) of T × Q are also isospectral.
The final statement of the theorem is an immediate consequence of the first; we simply let ∂(Q) in the final statement play the role of Q in the first statement. The first statement is proven later in this section. 2.9 We use the notation of Section 2.4 and Definition 2.6. Let T = L \Rk be a torus, and let Q be an O(m)-invariant submanifold of Rm with or without boundary. Let j, j 0 : Rk → so(m) be linear maps, and let c ∈ R+ . (i) If ( j, L ) ' ( j 0 , L ), then the Riemannian metrics g( j, c) and g( j 0 , c) on T ×Q are isometric. Moreover, the isometry can be chosen to carry T -orbits to T -orbits, that is, to be a bundle map relative to the projection π : T × Q → Q. THEOREM
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(ii)
Suppose that j satisfies the generic property that there are only finitely many orthogonal maps of Rm which commute with all the transformations j (z), z ∈ Rk , and that Q contains at least one nontrivial O(m)-orbit. Then the metrics g( j, c) and g( j 0 , c) on T × Q are isometric only if ( j, L ) ' ( j 0 , L ). Suppose that Q is a bounded domain in Rm . Then the metrics g( j, c) and g( j 0 , c) on T × Q are locally isometric if and only if j ' j 0 .
(iii)
Remark. When Q is a closed submanifold, we do not compare the local geometry of the metrics g( j, c) and g( j 0 , c) on T × Q since these metrics are not locally homogeneous; that is, the local geometry of each metric varies from point to point. COROLLARY 2.10 Suppose that Q is an O(m)-invariant bounded domain in Rm and that T = L \Rk is a torus. If j, j 0 : Rk → so(m) are isospectral, inequivalent linear maps, then the metrics g( j, c) and g( j 0 , c) on T × Q are isospectral but have different local geometry. Moreover, the induced metrics on the boundary are isospectral, and if at least one of j, j 0 satisfies the genericity condition in Theorem 2.9(ii), then the metrics on the boundary are not isometric.
Before proving Theorems 2.8 and 2.9, we observe that they yield large continuous families of isospectral, nonisometric Riemannian metrics. PROPOSITION 2.11 ([GW4, Theorem 2.2]) Let m be any positive integer other than 1, 2, 3, 4, or 6. Let W be the real vector space consisting of all linear maps j : R2 → so(m). Then there is a Zariski open subset O of W (i.e., O is the complement of the zero locus of some nonzero polynomial function on W ) such that each j ∈ O belongs to a d-parameter family of isospectral, inequivalent elements of W . Here d ≥ m(m −1)/2−[m/2]([m/2]+2) > 1. In particular, d is of order at least O(m 2 ). Moreover, all elements of O satisfy the genericity condition of Theorem 2.9(ii).
The final statement in the proposition is not stated explicitly in [GW4, Theorem 2.2]. However, the genericity condition appears as one of the defining properties of the Zariski open set O , given in the proof of that theorem. 2.12 Let T = L \R2 be a 2-torus. Let m be any positive integer other than 1, 2, 3, 4, or 6, and let Q be either a bounded domain in Rm or a submanifold of Rm of codimension one. Then for each j in the set O defined in Proposition 2.11, the metric g( j, c) lies COROLLARY
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in a continuous d-parameter family of isospectral, nonisometric metrics on T × Q, where d is given as in Proposition 2.11. Moreover, if Q is a bounded domain, then the induced metrics on the boundary of T × Q are also isospectral but not isometric. Although the expression for d gives zero when m = 6, explicit examples of continuous families of isospectral, inequivalent j maps are given in [GW4, Example 2.4] when m = 6. These maps also satisfy the genericity condition and thus give rise to continuous families of isospectral, nonisometric metrics. The remainder of this section is devoted to proving Theorems 2.8 and 2.9. We first prove Theorem 2.9(i) and (iii); these are elementary consequences of known results concerning left-invariant metrics on the Lie groups G( j). Next we prove Theorem 2.8, using the technique developed in Theorem 1.2. Theorem 2.9(i), along with Observation 2.7, plays a central role in the proof of Theorem 2.8. Finally, we give the proof of Theorem 2.9(ii). PROPOSITION 2.13 We use the notation of Section 2.2 and Definition 2.6. Let j and j 0 be linear maps from Rk to so(m), and let c ∈ R + . Then the following are equivalent: (i) j ' j 0; (ii) (G( j), g˜ c ) is isometric to (G( j 0 ), g˜ c ); (iii) (H ( j), g) ˜ is isometric to (H ( j 0 ), g). ˜ In the case when these conditions hold, the map τ : G( j) → G( j 0 ), given in terms of the global coordinates (t, x, z) on both manifolds by τ (t, x, z) = t, α(x), β(z) ,
where α and β are given as in Definition 2.6(i), is an isometry between the metrics g˜ c on G( j) and G( j 0 ), which restricts to an isometry between the metrics g˜ on H ( j) and H ( j 0 ). Proof We first prove the equivalence of (i) and (ii). By [GW3, Theorem 5.2], (G( j), g˜ c ) is isometric to (G( j 0 ), g˜ c ) if and only if g( j) and g( j 0 ), with the inner products associated with the metrics g˜ c on the two Lie groups, are isomorphic as metric Lie algebras, that is, if and only if there exists a map τ : g( j) → g( j 0 ) which is both a Lie algebra isomorphism and an inner product space isometry. This condition is easily seen to be equivalent to the condition that j ' j 0 : any metric Lie algebra isomorphism must be of the form (s A + x + z) → (s A + α(x) + β(z)) for all s ∈ R, x ∈ v, and z ∈ z, with α and β defining an equivalence between j and j 0 . The corresponding isomorphism τ : G( j) → G( j 0 ) is then an isometry and is given as in the statement of the proposition.
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The equivalence of (i) and (iii) is proven similarly (see [GW4, Proposition 1.4]).
Proof of Theorem 2.9(i) Assume ( j, L ) ' ( j 0 , L 0 ). Let τ : G( j) → G( j 0 ) be the isometry defined in k Proposition 2.13. In the notation of Lemma 2.3(ii), the map α −1 j 0 ◦ τ ◦ α j : (R × Rm , g( j, c)) → (Rk × Rm , g( j 0 , c)) is then an isometry and is given by (z, (t, x)) → (β(z), (t, α(x))), where α and β are the orthogonal maps given in Definition 2.6(i) and (ii). Since β(L ) = L , this isometry induces an isometry, which we denote by τ j, j 0 , from (T × Rm , g( j, c)) to (T × Rm , g( j 0 , c)). Since Q is O(m)-invariant, the isometry (t, x) → (t, α(x)) of Rm leaves Q invariant. Consequently, τ j, j 0 restricts to an isometry between the metrics g( j, c) and g( j 0 , c) on T × Q. Proof of Theorem 2.9(iii) By Section 2.4, the local geometry of (T × Rm , g( j, c)) is identical to that of the simply connected homogeneous manifold (G( j), g˜ c ). In the case when Q is a bounded domain in Rm , the same is true for the local geometry of (T × Q, g( j, c)). Thus part (iii) of the theorem is immediate from Proposition 2.13. We now prepare for the proof of Theorem 2.8. 2.14. Notation and remarks (i) Let T = L \z (= L \Rk ) be a torus, let K be a subtorus, and let K \T be the quotient torus endowed with the flat metric induced from the flat metric on T . The torus K is of the form (k ∩ L )\k, where k is a subspace of z spanned by lattice vectors in L . Let k⊥ be the orthogonal complement of k in z, and let Lk⊥ be the image of L under the orthogonal projection z → k⊥ . Endow the torus Lk⊥ \k⊥ with the flat metric induced by the restriction to k⊥ of the inner product on z. Then Lk⊥ \k⊥ is isometric to K \T . Given a linear map j : z → so(m), we may let k⊥ play the role of z, let j|k⊥ play the role of j, and let K \T = Lk⊥ \k⊥ play the role of T in Sections 2.2 and 2.4 and Lemma 2.3 in order to define the Lie group G( j|k⊥ ), Riemannian metrics g( j|k⊥ , c) on K \T × Q, where Q is any O(m)-invariant submanifold of Rm , and so on. The subspace k of z may be viewed as an ideal in the Lie algebra g( j). The quotient k\g( j) is then isomorphic to g( j|k⊥ ). The resulting homomorphism g( j) → g( j|k⊥ ) yields a homomorphism G( j) → G( j|k⊥ ) which is a Riemannian submersion with respect to the metrics g˜ c on the two spaces. This homomorphism induces a Rie¯ j), g˜ c ) → (G( ¯ j|k⊥ ), g˜ c ). Moreover, the following diagram mannian submersion (G(
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commutes, and all vertical arrows are Riemannian submersions: ¯ j), g˜ c T × Q, g( j, c) ⊂ T × Rm , g( j, c) ' G( ↓
↓ ↓ ¯ j|k⊥ ), g˜ c K \T × Q, g( j|k⊥ , c) ⊂ K \T × Rm , g( j|k⊥ , c) ' G( ↓ Q
↓ ⊂
Rm
↓ =
Rm
(ii) For each inner product space z and linear map j : z → so(m), thus defining a Lie group G( j), we denote by A the left-invariant vector field on G( j) defined by the element A ∈ a ⊂ g( j) of the Lie algebra of G( j). In terms of the global coordinates (t, x, z) on G( j), the vector field A is given by t (∂/∂t). We emphasize that the same notation A is being used independently of the choice of j; in particular, A also denotes a left-invariant vector field on G( j|k⊥ ) for k as in (i). The Lie group on which A is defined will be clear from the context. We similarly use the notation A¯ for ¯ j) by the vector field A on G( j), independently of the the vector field induced on G( choice of j. 2.15 Let K be a subtorus of T . Then, in the notation of Sections 1.1 and 2.14, the pro¯ j), g˜ c ) → jected mean curvature vector field of the Riemannian submersion (G( ¯ j|k⊥ ), g˜ c ) at any point is given by 2c2 (dim(K )) A. ¯ Consequently, if Q is a com(G( pact O(m)-invariant submanifold of Rm , then the projected mean curvature vector field for the submersion (T × Q, g( j, c)) → (K \T × Q, g( j|k⊥ , c)) is given by the orthogonal projection of the vector field 2c2 (dim(K )) A¯ to the submanifold K \T × Q ¯ j|k⊥ ) under the identification of G( ¯ j|k⊥ ) with K \T × Rm . of G( LEMMA
Proof It suffices to show that the mean curvature vector field along each fiber of the submersion (G( j), g˜ c ) → (G( j|k⊥ ), g˜ c ) is given by 2c2 (dim(K ))A, where A is the vector field on G( j) defined in Section 2.14(ii). Let ∇ be the Levi-Civita connection on (G( j), g˜ c ). For left-invariant vector fields u, v, w on G( j), we have
2 ∇u (v), w = [u, v], w + [w, u], v + [w, v], u . Using this formula and the fact that kAk = 1/c with respect to the metric g˜ c , we see that ∇z (z) = 2c2 A for any unit vector z ∈ z. Since A is a horizontal vector field with respect to the Riemannian submersion (G( j), g˜ c ) → (G( j|k⊥ ), g˜ c ), the mean curvature vector field
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along the fibers of this submersion is given by 2c2 (dim(K ))A. The first statement of the lemma follows. The second statement then follows from the commutativity of the diagram in Section 2.14. Proof of Theorem 2.8 We apply Theorem 1.2. Let K be a subtorus of T of codimension one, and recall Notation 2.14. Since k⊥ is 1-dimensional, the condition that j be isospectral to j 0 implies that the pair ( j|k⊥ , Lk⊥ ) is equivalent to the pair ( j|k0 ⊥ , Lk⊥ ) by Observation 2.7. Consequently, by Theorem 2.9(i), there exists an isometry σ between the metrics g( j|k⊥ , c) and g( j|k0 ⊥ , c) on K \T × Q. We must show that this isometry carries the projected mean curvature vector field for the submersion (T × Q, g( j, c)) → (K \T × Q, g( j|k⊥ , c)) to that for the submersion (T × Q, g( j 0 , c)) → (K \T × Q, g( j|k0 ⊥ , c)). As seen in the proofs of Theorem 2.9(i) and Proposition 2.13, the isometry σ arises from an isometry and Lie group isomorphism τ : (G( j|k⊥ ), g˜ c ) → (G( j|k0 ⊥ ), g˜ c ) whose differential carries the leftinvariant vector field A on G( j|k⊥ ) to the left-invariant vector field A on G( j|k0 ⊥ ). ¯ j|k⊥ ), g˜ c ) → (G( ¯ j 0 ⊥ ), g˜ c ) carryThe isometry τ induces an isometry τ¯ : (G( |k
¯ j|k⊥ ) to the vector field A¯ on G( ¯ j 0 ⊥ ). The isomeing the vector field A¯ on G( |k try σ : (K \T × Q, g( j|k⊥ , c)) → (K \T × Q, g( j|k0 ⊥ , c)) is the restriction of τ¯ to ¯ j|k⊥ ). UsK \T × Q under the identification of K \T × Q with a submanifold of G( ing Lemma 2.15, we thus see that σ carries the projected mean curvature vector field for the submersion (T × Q, g( j, c)) → (K \T × Q, g( j|k⊥ , c)) to that for the submersion (T × Q, g( j 0 , c)) → (K \T × Q, g( j|k0 ⊥ , c)). Thus the hypothesis of Theorem 1.2 is satisfied for every codimension-one subtorus K of T . By a more elementary argument, the hypothesis is also satisfied when K = T . Thus we conclude from Theorem 1.2 that the metrics g( j, c) and g( j 0 , c) on T × Q are isospectral. We now prepare for the proof of Theorem 2.9(ii). LEMMA 2.16 Let T = L \Rk , and let V0 denote the volume of T with respect to the standard inner product on Rk . Let j : Rk → so(m) be a linear map. Then for (t, x) ∈ Rm , the fiber over (t, x) of the submersion π : (T × Rm , g( j, c)) → (Rm , gc ) is a flat torus of volume (1/t 2k V0 ) with respect to the metric induced by g( j, c). In particular, the volume of the fiber over (t, x) depends nontrivially on t.
Proof ¯ j) = L \G( j) be the projection. The fiber of the submersion Let ρ : G( j) → G(
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π : (T × Rm , g( j, c)) → (Rm , gc ) over the point (t0 , x0 ) is isometric via α¯ j to ¯ j) where F = {(t0 , x0 , z) ∈ G( j) : z ∈ Rk }. Left the submanifold ρ(F) of G( translation L 0 by (1/t0 , −(x0 /t0 ), 0) carries F isometrically to the subgroup z of G( j) given by z = {(1, 0, z) : z ∈ Rk }. This isometry intertwines the isometry action of L on F with the action by translation of (1/t02 )L on the group z ' Rk since L 0 (t0 , x0 , z) = (1, 0, (1/t02 )z). Thus L 0 induces an isometry from the fiber to the torus ((1/t02 )L )\Rk , and the lemma follows. 2.17 Let τ be an isometry between the metrics g( j, c) and g( j 0 , c) on T × Q which carries T -orbits to T -orbits. Then for t ∈ [t1 , t2 ], the (possibly empty) “slice” S(t) of T × Q given by S(t) = (z, (t, x)) ∈ T × Q : z ∈ T, x ∈ Rm COROLLARY
is invariant under τ . The corollary follows from the lemma by noting that S(t) is the union of all T -orbits in T × Q of volume (1/t 2k )V0 . We see that, depending on the choice of Q, either the components or the boundary components of the nontrivial slices in Corollary 2.17 are isometric to manifolds studied in [GGS]. We reduce the proof of Theorem 2.9(ii) to the analogous result (stated as Lemma 2.19) for the latter manifolds, whose construction we now recall. 2.18. Notation As seen in Section 2.2, the Lie group H ( j) is diffeomorphic to Rm × Rk with multiplication given by 1 (x, z)(x 0 , z 0 ) = x + x 0 , z + z 0 + [x, x 0 ] 2 for x, x 0 ∈ Rm and z, z 0 ∈ Rk . Thus {(0, z) : z ∈ Rk } is a central subgroup isomorphic to Rk . A lattice L of full rank in Rk may be viewed as a discrete central subgroup of H ( j). Denote elements of the quotient H¯ ( j) = L \H ( j) by (x, z¯ ) with z¯ ∈ T = L \Rk and x ∈ Rm . Continue to denote by g˜ the left-invariant Riemannian metric on H¯ ( j) induced by the metric g˜ on H ( j), given in Section 2.2. Let Nr ( j, L ) = {(x, z¯ ) ∈ H¯ ( j) : z¯ ∈ T, kxk = r } with the Riemannian metric induced by g. ˜ LEMMA 2.19 ([GGS]) Suppose that j, j 0 : Rk → so(m) are linear maps and that j satisfies the generic
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property of Theorem 2.9(ii). Let r ∈ R+ , and let L be a lattice of full rank in Rk . If Nr ( j, L ) is isometric to Nr ( j 0 , L ), then ( j, L ) ' ( j 0 , L ). The genericity condition on j is stated differently in [GGS]. To clarify, let h( j) be the metric Lie algebra defined in Section 2.2. Any orthogonal linear map α of v = Rm which commutes with all the transformations j (z), z ∈ Rk , extends to an orthogonal automorphism α of h( j) = v + z (i.e., a Lie algebra automorphism preserving the inner product) which restricts to the identity map on z. Conversely, every orthogonal automorphism of h( j) which restricts to the identity on z must be of this form. Thus the genericity condition in Theorem 2.9(ii) is equivalent to the condition that there are only finitely many orthogonal automorphisms of h( j) which restrict to the identity on z. The latter condition is weaker than the genericity condition used in [GGS] in that the word “orthogonal” has been inserted. However, a glance at the arguments in [GGS] shows that only this weaker condition is actually used. LEMMA 2.20 Given t, r ∈ R+ , let S(t, r ) = {(z, (t, x)) ∈ T × Rm : z ∈ T, kxk = r }. Then S(t, r ), with the metric induced by g( j, c), is isometric to Nr/t ( j, (1/t 2 )L ).
Proof ¯ j) be the projection. The slice S(t, r ) is isometric via the diffeoLet ρ : G( j) → G( ¯ j), where N = {(t, x, z) : morphism α¯ j in Section 2.4 to the submanifold ρ(N ) of G( kxk = r, z ∈ z}. The left translation in G( j) by (1/t, 0, 0) carries N isometrically to the submanifold N0 of H ( j) given by N0 = {(1, x, z) : kxk = r/t, z ∈ Rk } and intertwines the action of L on N with the action of (1/t 2 )L on N0 . Thus this left translation induces an isometry from ρ(N ) to Nr/t ( j, (1/t 2 )L ). Proof of Theorem 2.9(ii) Theorem 2.9(ii) is a consequence of the following two propositions. We may assume without loss of generality that Q is connected. PROPOSITION 2.21 Assume, in addition to the hypotheses of Theorem 2.9(ii), that the isometry τ : (T × Q, g( j, c)) → (T × Q, g( j 0 , c)) can be chosen to carry T -orbits to T -orbits. Then ( j, L ) ' ( j 0 , L ). PROPOSITION 2.22 Suppose that j satisfies the genericity hypothesis of Theorem 2.9(ii). If the Riemannian metrics g( j, c) and g( j 0 , c) on T × Q are isometric, then there exists an isometry
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between them which carries T -orbits to T -orbits. Proof of Proposition 2.21 Choose t such that the slice S(t) of T × Q contains a nontrivial O(m)-orbit. Then for some r > 0, the manifold S(t, r ) defined in Lemma 2.20 is contained in S(t). Let r be maximal such that S(t, r ) ⊂ S(t). Then S(t, r ) is either a boundary component of S(t) or a connected component of S(t); in either case, it has maximal volume among all such components. Thus by Corollary 2.17, S(t, r ) is τ -invariant. The proposition now follows from Lemmas 2.19 and 2.20. The following lemma is used to prove Proposition 2.22. LEMMA 2.23 If j satisfies the genericity hypothesis of Theorem 2.9(ii), then T is a maximal torus in the full isometry group Iso(T × Q, g( j, c)).
We first assume the lemma and prove the proposition. Proof of Proposition 2.22 Suppose that there exists an isometry τ between the Riemannian metrics g( j, c) and g( j 0 , c) on T × Q. Then τ induces an isomorphism τˆ : Iso(T × Q, g( j, c)) → Iso(T × Q, g( j 0 , c)) given by τˆ (β) = τβτ −1 . In particular, the maximal tori in Iso(T × Q, g( j, c)) and Iso(T × Q, g( j 0 , c)) must have the same dimension, so T , viewed as a subgroup of Iso(T × Q, g( j 0 , c)), must be a maximal torus. Since all maximal tori in Iso(T × Q, g( j 0 , c)) are conjugate within Iso(T × Q, g( j 0 , c)), we may assume, after composing τ with an isometry of (T × Q, g( j 0 , c)), that τˆ carries T to T . Equivalently, τ carries T -orbits to T -orbits. Proof of Lemma 2.23 It suffices to prove the lemma in the case when Q is connected. Let C(T ) be the centralizer of T in Iso(T × Q, g( j, c)). We need to show that T is a connected component of C(T ). Let τ ∈ C(T ). Since τ is an isometry of (T × Q, g( j, c)) which commutes with the action of T , it must carry T -orbits to T -orbits. By Corollary 2.17, τ restricts to an isometry on each slice S(t). Let S(t) be a nontrivial slice. Choose r as in the proof of Proposition 2.21. As in that proof, we see that S(t, r ) is τ -invariant. Moreover, τ is uniquely determined by its restriction to S(t, r ). (Indeed, let p ∈ S(t, r ). Since an isometry of a connected manifold is uniquely determined by its value and differential at a point, we need only show that τ∗ p is uniquely determined by its restriction to T p (S(t, r )). Since S(t, r ) is either a connected component or a boundary
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component of S(t), the restriction of the orthogonal map τ∗ p to T p (S(t)) is determined by its restriction to T p (S(t, r )). The facts that S(t) has codimension at most one in T × Q and that τ leaves every slice of T × Q invariant show, moreover, that τ∗( p) is uniquely determined by its restriction to T p (S(t)) and hence to T p (S(t, r )).) Thus the map τ → τ|S(t,r ) is an injective homomorphism from C(T ) to the centralizer of T in the isometry group of S(t, r ), where S(t, r ) has the metric induced by g( j, c). Under the hypothesis on j given in Theorem 2.9(ii), it was proven in [GGS] that T is a maximal torus in the isometry group of Nr ( j, L ) for any r and any L . Hence by Lemma 2.20, T is a connected component of its own centralizer in Iso((S(t, r )) and hence also in Iso(T × Q, g( j, c)). This completes the proof of Theorem 2.9(ii). 3. Locally homogeneous examples Throughout this section, we use the notation of Sections 2.2 and 2.4 with Q chosen to be a bounded domain in Rm , for example, a geodesic ball. Recall in this case that (T × Q, g( j, c)) is locally homogeneous and that its local geometry is that of (G( j), g˜ c ). The following proposition is a special case of the more general classification of homogeneous manifolds of nonpositive curvature, carried out in [Hz] for strictly negative curvature and in [AW1] and [AW2] for nonpositive curvature. PROPOSITION 3.1 ([Hz] or [AW2, Proposition 8.5]) In the notation of Section 2.2, for each choice of j, there exists a constant λ( j) such that (G( j), g˜ c ) has strictly negative curvature when c > λ( j), nonpositive curvature when c = λ( j), and mixed curvature when c < λ( j). Moreover, as c approaches ∞, the maximum of the sectional curvature of (G( j), g˜ c ) approaches −∞. THEOREM 3.2 In every dimension n ≥ 8, there exist continuous isospectral deformations of locally homogeneous negatively curved manifolds with boundary which are not locally isometric. Moreover, given any constant κ > 0, we can choose the isospectral metrics so that their curvature is bounded above by −κ.
Proof Let m ≥ 5, and let { jt } be a family of isospectral, inequivalent maps jt : R2 → so(m) as in Proposition 2.11, or as in the comments following Corollary 2.12, with compact parameter space. We may assume that the maps jt , and thus the metrics g( jt , c) and their curvatures, vary smoothly with t. (The fact that we can allow the j’s to vary
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smoothly follows from the proof of Proposition 2.11 given in [GW4].) For any choice of c > 0, the manifolds (T × Q, g( jt , c)) are isospectral but not locally isometric. Proposition 3.1 allows us to adjust the constant c to achieve any desired curvature bound on the metric g( jt , c) for any given t. Since the curvature varies smoothly with the parameter t and since the parameter space was chosen to be compact, we can thus adjust c to achieve any curvature bound on all the metrics g( jt , c). Remark 3.3 If the constant λ( jt ) depends nontrivially on t, then we can choose c so that c > λ( jt ) for some choices of t and c < λ( jt ) for some other choices of t. In this case, we obtain isospectral deformations in which some of the metrics have negative curvature and others mixed curvature. Unfortunately, due to the complexity of the curvature expressions, we have not been able to compute the constants λ( jt ) for any of the deformations. It is very likely, however, that for generic deformations, λ( jt ) does vary with t. The only situation in which we have explicitly compared the constants thus far is for the pair of isospectral j maps in Theorem 3.8(ii). In this case, the constants are indeed different. This pair of maps does not belong to a continuous family of isospectral, inequivalent maps, however. Definition 3.4 Following A. Kaplan [K], we say that a linear map j : Rk → so(m) is of Heisenberg type if j (z)2 = −kzk2 Id for all z ∈ Rk . Damek and F. Ricci [DR] proved that if j is of Heisenberg type, then in the notation of Section 2.2, (G( j), g˜ 1/2 ) is a harmonic manifold. Included among such harmonic spaces are all the rank-one symmetric spaces of noncompact type as well as the first known examples of nonsymmetric harmonic manifolds. The following lemma is immediate from Definitions 2.6 and 3.4. LEMMA 3.5 If j, j 0 : Rk → so(m) are both of Heisenberg type, then they are isospectral in the sense of Definition 2.6.
Example 3.6 Let k = 3, and let m = 4l with l ≥ 2. Identify R3 with the purely imaginary quaternions and Rm with the direct sum of l copies of the quaternions. Choose nonnegative integers a and b with l = a + b. Define the map ja,b : R3 → so(m) by ja,b ( p)(q1 , . . . , qa , q10 , . . . , qb0 ) = ( pq1 , . . . , pqa , q10 p, . . . , qb0 p), where pqi and q 0j p denote quaternionic multiplication. The map j(a,b) is of Heisen-
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berg type. If (a, b) is another pair of positive integers such that a + b = a 0 + b0 , then j(a,b) and j(a 0 ,b0 ) are equivalent in the sense of Definition 2.6 if and only if (a 0 , b0 ) is a permutation of (a, b). Fix any choice of c ∈ R+ , bounded domain Q in Rm , and lattice L in Rk , and let T = L \Rk . Then by Lemma 3.5 and Corollary 2.10, when (a 0 , b0 ) is not a permutation of (a, b) but a + b = a 0 + b0 , the locally homogeneous Riemannian 0 manifolds (T × Q, g( j(a,b) , c)) and (T × Q, g( j(a,b) , c)) are isospectral but have different local geometry. The following lemma is standard. LEMMA 3.7 Let a and b be nonnegative integers, not both zero, and let l = a + b. In the notation of Example 3.6, the Riemannian manifold (G( j(a,b) ), g˜ 1/2 ) is a symmetric space if and only if one of a or b is zero. When this condition holds, the manifold is isometric to the quaternionic hyperbolic space of dimension 4(l + 1).
3.8 For each l ≥ 2, there exists a pair M1 , M2 of isospectral locally homogeneous Riemannian manifolds with boundary such that M1 is locally isometric to quaternionic hyperbolic space of dimension 4(l + 1) and thus has parallel curvature tensor, while M2 has nonparallel curvature tensor. There exists a pair of 12-dimensional isospectral locally homogeneous Riemannian manifolds with boundary, one of which has strictly negative curvature while the other has mixed curvature.
THEOREM
(i)
(ii)
Proof Assertion (i) is immediate from Lemma 3.7 and Example 3.6: M1 is of the form (T × Q, g( j(l,0) , 1/2)), and M2 is of the form (T × Q, g( j(a,b) , 1/2)), where a and b are both nonzero and a + b = l. For (ii), consider the isospectral, inequivalent maps j(2,0) and j(1,1) in Example 3.6. In the notation of Proposition 3.1, we show that λ( j(2,0) ) 6= λ( j(1,1) ). The quaternionic hyperbolic space G( j(2,0) , 1/2) has curvature bounded above by −1/4. Thus λ( j(2,0) ) < 1/2. Damek [D] showed that if j : Rk → so(m) is of Heisenberg type, then the harmonic manifold (G( j), g˜ 1/2 ) always has nonpositive curvature, and it has some two-planes of zero curvature if and only if there exist vectors x, y ∈ Rm such that j (Rk )(x) ⊥ y and j (Rk )(x) ∩ j (Rk )(y) is nonempty, where j (Rk )(x) denotes the image of the map z → j (z)(x). Damek’s condition is satisfied for j(1,1) ; take x = ( p, p) and y = (q, q), where p and q are distinct nonzero
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purely imaginary quaternions. Hence (G( j(1,1) ), g˜ 1/2 ) has some zero curvature but no positive curvature, and so λ( j(1,1) ) = 1/2. Choosing c so that λ( j(2,0) ) < c < 1/2, we find that (G( j(2,0) ), g˜ c ) has strictly negative curvature while (G( j(1,1) ), g˜ c ) has mixed curvature. Thus the isospectral manifolds of the form (T × Q, g( j(2,0) , c)) and (T × Q, g( j(1,1) , c)) satisfy condition (ii) of the theorem. We next consider conditions for constant Ricci curvature and for constant scalar curvature. PROPOSITION 3.9 ([EH]) We use the notation of Section 2.2, and we let c = 1/2. The solvmanifold (G( j), g˜ 1/2 ) is Einstein if and only if both of the following conditions are satisfied. (i) The map j : z → j (z) ⊂ so(m) is a linear isometry relative to the inner product ( , ) on so(m) given by (α, β) = −(1/m) trace(αβ). Pk (ii) Letting {z 1 , . . . , z k } be an orthonormal basis of z, then i=1 j (z i )2 is a scalar operator. (This condition is independent of the choice of orthonormal basis.) PROPOSITION 3.10 In the notation of Section 2.18, the closed manifold N1 ( j, L ) has constant scalar curvature if and only if j satisfies Proposition 3.9(ii).
Proof Since H¯ ( j) is homogeneous, it has constant scalar curvature τ . The scalar curvature of the submanifold N1 ( j, L ) was computed in [GGS]: scal(x, z¯ ) = τ + (m − 1)(m − 2) −
k X
j (z i )2 (x, x)
i=1
for (x, z¯ ) ∈ N1 ( j, L ), where {z 1 , . . . , z k } is an orthonormal basis of z. The proposition now follows. Example 3.11 We define a pair of isospectral linear maps j, j 0 : R3 → so(6) such that j satisfies conditions of Proposition 3.9(i) and (ii) while j 0 does not satisfy condition (ii). To define j, identify R6 with R3 × R3 . For z ∈ R3 and (u, v) ∈ R6 , set r r 3 3 j (z)(u, v) = z × u, z×v , 2 2 where × denotes the cross product in R3 . Then the eigenvalues of j (z) are √ √ ± 3/2kzk −1 and zero, each occurring with multiplicity 2. It is straightforward to verify that j satisfies Proposition 3.9(i) and (ii).
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To define j 0 , view R6 as H × R2 , where H denotes the quaternions, and view R3 as the space of purely imaginary quaternions. For z ∈ R3 and (u, v) ∈ R6 with u ∈ H and v ∈ R2 , set r 3 0 j (z)(u, v) = zu, 0 , 2 where zu denotes quaternionic multiplication. Then the eigenvalues of j 0 (z) are also √ √ ± 3/2kzk −1 and zero, each occurring with multiplicity 2. Thus j and j 0 are isospectral. However, all the j 0 (z), as z varies over R3 , have the same zero eigenspace, so j 0 does not satisfy the second condition in Proposition 3.9. 3.12 There exists a pair of 10-dimensional isospectral Riemannian manifolds with boundary, one of which has constant Ricci curvature and the other variable Ricci curvature. There exists a pair of isospectral closed 8-dimensional manifolds, one of which has constant scalar curvature and the other variable scalar curvature.
THEOREM
(i)
(ii)
Proof (i) Choose Q to be an O(6)-invariant domain with boundary in the upper halfspace R6 , and let T be a 3-torus. Let j and j 0 be as in Example 3.11. Then by Proposition 3.9, the manifold (G( j), g˜ 1/2 ) is Einstein while (G( j 0 ), g˜ 1/2 ) is not. Since the local geometry of (T × Q, g( j, 1/2)) is that of (G( j), g˜ 1/2 ), we see that (T × Q, g( j, 1/2)) has constant Ricci curvature while (T × Q, g( j 0 , 1/2)) does not. (ii) It was proven in [GGS] that if j, j 0 : z → so(m) are isospectral linear maps, then for any r > 0 and any lattice L of full rank in z, the closed Riemannian manifolds Nr ( j, L ) and Nr ( j 0 , L ) are isospectral. In particular, choosing j and j 0 as in Example 3.11 and applying Proposition 3.10, we obtain statement (ii) of the theorem. 4. Locally inhomogeneous examples We continue to use the notation of Sections 2.1, 2.2, and 2.4 and Lemma 2.3. We now study the geometry of the manifolds of the form (T × Q, g( j, c)) for which Q is a submanifold of Rm of codimension one. Before considering the manifolds (T × Q, g( j, c)), we first describe the connection and curvature on the ambient space (T × Rm , g( j, c)). We recall that (T × Rm , g( j, c)) is locally homogeneous and locally isometric to (G( j), g˜ c ), so it suffices for this purpose to consider (G( j), g˜ c ). PROPOSITION 4.1 We use the notation of Section 2.2. In particular, the Lie algebra of g( j) is given by
ISOSPECTRAL RIEMANNIAN METRICS
379
a + v + z. For x, x ∗ ∈ v and z, z ∗ ∈ z, we have the following: (i) ∇x x ∗ = (1/2)[x, x ∗ ] + c2 hx, x ∗ iA, (ii) ∇z z ∗ = 2c2 hz, z ∗ iA, (iii) ∇x z = ∇z x = −(1/2) j (z)(x), (iv) ∇x A = −x, (v) ∇z A = −2z, (vi) ∇ A = 0. The proposition is a straightforward computation using the fact that for left-invariant vector fields we have
2 ∇u v, w = [u, v], w + [w, u], v + [w, v], u and recalling that kAk = 1/c. 4.2. A special class of submanifolds For simplicity, we restrict attention in this section to those O(m)-invariant submanifolds Q of Rm of the form Q = (t, x) ∈ Rm : t ∈ [t1 , t2 ] and kxk = r for some interval [t1 , t2 ] and some real number r > 0. In particular, Q has nontrivial boundary. 4.3. The Weingarten map Let Q be given as in Section 4.2. A unit normal vector field n Q along Q in Rm is given by n Q (t, x) = x/r , where on the right-hand side, x denotes the left-invariant vector field on Rm defined by the Lie algebra element x ∈ rm . (See Section 2.1.) A unit normal vector field n along T × Q in (T × Rm , g( j, c)) is given by na = x¯a /r for a = (z, (t, x)) ∈ T × Q, where x¯ is the horizontal lift to T × Rm of the left-invariant ¯ j) as vector field x on Rm . Alternatively, using the map α¯ j to identify T × Rm with G( ¯ j) induced by the left-invariant vector in Section 2.4, then x¯ is the vector field on G( field on G( j) defined by the element x ∈ g( j). For a = (z, (t, x)) ∈ T × Q, the tangent space to T × Q at a is spanned by the values at a of the vector field A¯ and of { y¯ + z¯ : y ∈ v with y ⊥ x, z ∈ z}. (As in the ¯ j) or, equivalently, previous paragraph, for u ∈ g( j), u¯ denotes the vector field on G( on T × Rm , induced by the left-invariant vector field u on G( j).) The Weingarten map B of (T × Q, g( j, c)), defined by B(u) ¯ = ∇u¯ (n) for u¯ a tangent vector to T × Rm , is
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GORDON and SZABO
given at a by 1 t y¯ + [y, x], r 2r 1 B(¯z ) = − j (z)(x), 2r ¯ = 0, B( A) B( y¯ ) =
for y ∈ v with y ⊥ x and for z ∈ z. The expression [y, x] above is the Lie bracket in g( j). The key point here is that the Weingarten map does not depend on the parameter c. This fact can be seen even without the explicit formulas above by the following observations: (i) when x ∈ v and u ∈ g( j) is perpendicular to x, then the invariant vector field ∇u x is independent of c (as can be seen from Proposition 4.1); (ii) the normal vector field n to T × Q takes all its values in v. PROPOSITION 4.4 Given a linear map j : z → so(m) and r > 0, there exists a constant λ( j, r ) > 0 such that the Riemannian manifold (T × Q, g( j, c)), with Q as in Section 4.2, has strictly negative curvature when c > λ( j, r ). Moreover, as c approaches ∞, the maximum of the sectional curvature of (T × Q, g( j, c)) approaches −∞.
Proof ˜ denote the curvature tensor of (T × Rm , g( j, c)), respectively, Let R, respectively, R, (T × Q, g( j, c)). Then for X, Y tangent vectors to T × Q at a point a ∈ T × Q, we have
2 ˜ R(X, Y )Y, X = R(X, Y )Y, X + B(X ), X B(Y ), Y − B(X ), Y . Since the Weingarten map B is independent of c, the proposition follows immediately from Proposition 3.1 and the fact that the curvature of T × Rm is exactly that of (G( j), g˜ c ). 4.5 Suppose that Q is a submanifold of Rm given by Q = {(t, X ) ∈ Rm : t ∈ [t1 , t2 ] and kX k = f (t)}, where f is a smooth function. If f 0 and f 00 are uniformly bounded sufficiently close to zero, then for each j there exists a constant λ( j) > 0 such that (T × Q, g( j, c)) has strictly negative curvature when c > λ( j). Moreover, as c approaches ∞, the maximum of the sectional curvature of (T × Q, g( j, c)) approaches −∞. COROLLARY
ISOSPECTRAL RIEMANNIAN METRICS
381
The corollary follows from Proposition 4.4 by a continuity argument. Remark 4.6 One would not expect to obtain a similar result when f 0 is allowed to vary too greatly. Indeed, in the extreme case when f is zero on the endpoints of its domain [t1 , t2 ] and has infinite slope at these points, then (T × Q, g( j, c)) is a closed manifold that admits a nontrivial isometric action by a torus. By Bochner’s theorem, (T × Q, g( j, c)) cannot have negative Ricci curvature, let alone negative sectional curvature, in this case. 4.7 In the notation of Section 4.2, the manifold (T × Q, g( j, c)) has nonconstant scalar curvature and thus is locally inhomogeneous. PROPOSITION
Proof Letting ρ and Ric denote the scalar curvature and Ricci tensor on the ambient manifold (T × Rm , g( j, c)), and letting ρ˜ denote the scalar curvature on (T × Q, g( j, c)), we obtain ρ˜ = ρ − 2 Ric(n, n) + (trace(B))2 − trace(B 2 ). Since (T × Rm , g( j, c)) is locally homogeneous, ρ is constant. By Section 4.3, we see that as (z, (t, x)) varies over (T × Q, g( j, c)), the normal vector field n depends only on x and thus Ric(n, n) is independent of the coordinate t. On the other hand, Section 4.3 also shows that (trace(B))2 − trace(B 2 ) depends nontrivially on t. Thus ρ(a) ˜ depends nontrivially on t. THEOREM 4.8 In every dimension n ≥ 7, there exist nontrivial continuous isospectral deformations of locally inhomogeneous negatively curved manifolds with boundary. Moreover, given any constant κ > 0, we can choose the isospectral metrics so that their curvature is bounded above by −κ.
Proof The short proof is identical to that of Theorem 3.2 with Q now of the form specified in Section 4.2 and with Proposition 4.4 replacing Proposition 3.1. References [AW1]
R. AZENCOTT and E. N. WILSON, Homogeneous manifolds with negative curvature, I,
Trans. Amer. Math. Soc. 215 (1976), 323 – 362. MR 52:15308 374
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J. Differential Geom., 19 (1984), 241 – 256. MR 85j:58143 356 , The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J. 33 (1986), 253 – 271. MR 87k:58275 356 , Isometry groups of Riemannian solvmanifolds, Trans. Amer. Math. Soc. 307 (1988), 245 – 269. MR 89g:53073 367 , Continuous families of isospectral Riemannian metrics which are not locally isometric, J. Differential Geom. 47 (1997), 504 – 529. MR 99a:58159 356, 359, 366, 367, 368, 375 R. GORNET, A new construction of isospectral Riemannian manifolds with examples, Michigan Math. J. 43 (1996), 159 – 188. MR 97b:58143 356 , Continuous families of Riemannian manifolds, isospectral on functions but not on 1-forms, J. Geom. Anal. 10 (2000), 281 – 298. MR 2001i:58065 356 V. GUILLEMIN and D. KAZHDAN, Some inverse spectral results for negatively curved 2-manifolds, Topology 19 (1980), 301 – 312. MR 81j:58082 356 E. HEINTZE, On homogeneous manifolds of negative curvature, Math. Ann. 211 (1974), 23 – 34. MR 50:5695 374 ´ A. IKEDA, On lens spaces which are isospectral but not isometric, Ann. Sci. Ecole
[K]
A. KAPLAN, On the geometry of groups of Heisenberg type, Bull. London Math. Soc.
[M]
J. MILNOR, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat.
[P]
V. K. PATODI, Curvature and the fundamental solution of the heat operator, J. Indian
[Sch1]
D. SCHUETH, Continuous families of isospectral metrics on simply connected
[GW2] [GW3] [GW4]
[Gt1] [Gt2] [GuK] [Hz]
Norm. Sup. (4) 13 (1980), 303 – 315. MR 83a:58091 356 15 (1983), 35 – 42. MR 84h:53063 375 Acad. Sci. U.S.A. 51 (1964), 542. MR 28:5403 356 Math. Soc. 34 (1970), 269 – 285. MR 58:7744 356, 357
[Sch2] [Su] [Sz] [V]
manifolds, Ann. of Math. (2) 149 (1999), 287 – 308. MR 2000c:58063 356 , Isospectral manifolds with different local geometries, J. Reine Angew. Math. 534 (2001), 41 – 94. CMP 1 831 631 356 T. SUNADA, Riemannian coverings and isospectral manifolds, Ann. of Math. (2) 121 (1985), 169 – 186. MR 86h:58141 356 Z. I. SZABO, Locally non-isometric yet super isospectral spaces, Geom. Funct. Anal. 9 (1999), 185 – 214. MR 2000a:58089 356 ´ M.-F. VIGNERAS , Vari´et´es riemanniennes isospectrales et non isom´etriques, Ann. of Math. (2) 112 (1980), 21 – 32. MR 82b:58102 356
Gordon Department of Mathematics, 6188 Bradley Hall, Dartmouth College, Hanover, New Hampshire 03755, USA; [email protected] Szabo Department of Mathematics and Computer Science, Lehman College, 250 Bedford Park Boulevard West, Bronx, New York 10468, USA; [email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 2,
¨ LOCAL HOLDER REGULARITY FOR SOLUTIONS OF ELLIPTIC SYSTEMS MARIA ALESSANDRA RAGUSA
To the memory of my teacher, Filippo Chiarenza Abstract In this note we prove local L p -regularity for the highest-order derivatives of an elliptic system of arbitrary order in nondivergence form where the coefficients of the principal part are taken in the space of Sarason vanishing mean oscillation (VMO). Lower-order coefficients and the known term belong to suitable Lebesgue spaces. As a consequence, we obtain H¨older regularity results. 1. Introduction In this note the author studies the local regularity in the Lebesgue spaces L p , 1 < p < ∞, for higher-order derivatives of solutions of elliptic system of arbitrary order in nondivergence form with coefficients, which can be discontinuous, belonging to the Sarason class of VMO functions. We generalize the results given by F. Chiarenza, M. Franciosi, and M. Frasca in [5] because we consider the system with lower-order terms. The technique used in this paper rests on the idea, well known as Korn’s trick, used by Chiarenza, M. Frasca, and P. Longo in [6] and [7], and by Chiarenza, Franciosi, and Frasca in [5] and [8], to obtain an explicit representation formula for the highest-order derivatives of local solutions of an elliptic system in nondivergence form in terms of singular integral operators and commutators. Then it is possible to obtain the final estimate by following the line of the proof of [10, Theorem 9.11] and using estimates in Lebesgue spaces of these singular integral operators of nonconvolution type. The obtained L p -estimates are the first available ones for systems with discontinuous coefficients in the full range p ∈ ]1, ∞[ that we know about. In the case of continuous coefficients, we review the results obtained by S. Agmon, A. Douglis, and L. Nirenberg in [1] and [2] and, only for p belonging to a DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 2, Received 11 May 1999. Revision received 4 July 2001. 2000 Mathematics Subject Classification. Primary 35J30, 35J45, 31B10; Secondary 43A15, 34A30.
385
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MARIA ALESSANDRA RAGUSA
neighborhood of 2, by S. Campanato in [4]. 2. Basic assumptions and main results Hereafter we suppose to be an open bounded subset in Rn , n ≥ 3, α a multi-index, and we set ∂ |α| |α| = α1 + · · · + αn , D α = α1 . ∂ x1 · · · ∂ xnαn Let us consider the system Lu ≡
N X X
(α)
ai j (x)D α u j (x) +
N X
j=1 |α|=2s
X
(α)
bi j (x)D α u j (x) = f i (x),
j=1 |α|≤2s−1
i = 1, . . . , N , (2.1) p
where f i ∈ L loc (), i = 1, . . . , N , 1 < p < ∞. We define a local solution of (2.1) as a vector function u = (u 1 , . . . , u N ) with 2s, p u i ∈ Wloc (), ∀i = 1, . . . , N , s ∈ N, satisfying (2.1) a.e. in . We make the following three assumptions, which we refer collectively to as Hypothesis I. (i) We assume that (α)
ai j ∈ VMO(Rn ) ∩ L ∞ (Rn ),
i, j = 1, . . . , N , |α| = 2s,
(α)
M = maxi, j=1,...,N sup |ai j |. |α|=2s
(ii)
Set
(α) α |α|=2s ai j τ ,
P
∀ i, j = 1, . . . , N ; for a.e. x ∈ , the homogeneous
polynomial of degree 2s in τ obtained by replacing D α with τ α = τ1α1 · · · τnαn is X (α) ∃3 > 0 : det ai j (x)τ α ≥ 3|τ |2s N , ∀τ ∈ Rn , a.e. x ∈ . |α|=2s
(iii)
Let us assume that n is odd, as in [9], and that (α)
bi j ∈ L tm (), where tm is defined as follows: n n = m if 1 < p < m , tm = > mn if p = mn , = p if p > mn , m = 2s − |α|
for every |α| ≤ 2s − 1.
¨ LOCAL HOLDER REGULARITY
387
We next state the main results of this note in the following two theorems. 2.1 2s,q Let us consider Hypothesis I true. Also, let 1 < q ≤ p < ∞ and u ∈ Wloc () be a solution of the system THEOREM
Lu ≡
N X X
(α)
ai j (x)D α u j (x) +
j=1 |α|=2s
N X
X
(α)
bi j (x)D α u j (x) = f i (x),
j=1 |α|≤2s−1
i = 1, . . . , N , p
with f i ∈ L loc (), ∀i = 1, . . . , N . Then, given 0 ⊂⊂ 00 ⊂ (0 , 00 open), we have u ∈ W 2s, p (0 ). Moreover, there exists a constant C independent of u i and f i , ∀i = 1, . . . , N , such that N X
kD α u j k L p (0 ) ≤ C
N X
j=1 |α|=2s
ku j k L p (00 ) + k f j k L p (00 ) .
j=1
THEOREM 2.2 2s,q Let us suppose that Hypothesis I is true, that u ∈ Wloc () is a solution of the system p (2.1), and that f i ∈ L loc () for i = 1, . . . , N , 1 < q ≤ p < ∞. Then, given 0 ⊂⊂ 00 ⊂ , where 0 , 00 are open sets, we have
D α u ∈ C 0,β (0 ),
where β = 1 −
n , |α| = 2s − 1, p
and there exists a constant C independent of u i and f i , ∀i = 1, . . . , N , such that N X j=1 |α|=2s−1
α
kD u j kC 0,β (0 ) ≤ C
N X
ku j k L p (00 ) + k f j k L p (00 ) .
j=1
3. The space VMO, singular integrals, and commutators Definition 3.1 Let f be a locally integrable function. We say that f belongs to the John-Nirenberg space BMO of functions with bounded mean oscillation introduced in [13] if Z 1 | f (x) − f B | d x < ∞, k f k∗ ≡ sup B⊂Rn |B| B
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MARIA ALESSANDRA RAGUSA
R where f B stands for the integral average (1/|B|) B f (x) d x of the function f (x) over the set B, and B ranges over the class of balls of Rn . Definition 3.2 If f ∈ BMO and Bρ belongs to the class of balls of radius ρ > 0, we set Z 1 η(r ) = sup | f (x) − f Bρ | d x, Bρ ⊂Rn , ρ≤r |Bρ | Bρ the VMO modulus of f . We say that f belongs to the Sarason class of VMO introduced in [13] if lim η(r ) = 0.
r →0+
(α)
According to the above definition of η(r ), we denote ηi j the VMO modulus of the (α)
function ai j and η(r ) =
N X X
2 1/2 (α) ηi j (r ) .
i, j=1 |α|=2s
Definition 3.3 Let us recall the following classical Sobolev spaces: W k, p () = f : D α f ∈ L p (), 1 < p < ∞, |α| ≤ k ,
k an integer,
with the norm k f kW k, p () = k f k L p () + kD k f k L p () . Definition 3.4 Let k : Rn \ {0} → R. We say that k(x) is a Calder´on-Zygmund kernel (C-Z kernel) if (i) k ∈ C ∞ (Rn \ {0}); (ii) k(x) is homogeneous of degree − n; R n (iii) 6 k(x) d x = 0, where 6 = {x ∈ R : |x| = 1}. THEOREM 3.5 ([8, Theorem 2.3]) Let B be an open subset of Rn , let f ∈ L p (B), 1 < p < ∞, and let a ∈ VMO ∩ L ∞ (Rn ). Let k(x, z) be a C-Z kernel in the z variable for a.a. x ∈ B such that
∂j
max j k(x, z) ∞ = M < +∞, L (B×6) | j|≤2n ∂z
¨ LOCAL HOLDER REGULARITY
389
where 6 denotes the surface of the unit ball in Rn . For any ε > 0, we set Z K ε f (x) = k(x, x − y) f (y) dy, |x−y|>ε, y∈B Z Cε (a, f )(x) = k(x, x − y) a(x) − a(y) f (y) dy. |x−y|>ε, y∈B
Then there exist K f , C(a, f ) ∈ L p (B) such that lim kK ε f − K f k L p (B) = lim kCε (a, f ) − C(a, f )k L p (B) = 0,
ε→0
ε→0
and, moreover, there exists a positive constant c = c(n, p, M, s, N ) such that kK f k L p (B) ≤ ck f k L p (B) ,
kC(a, f )k L p (B) ≤ ckak∗ k f k L p (B) .
THEOREM 3.6 ([8, Theorem 2.4]) Let a ∈ VMO(Rn ). Then, for every > 0, there exists ρ0 > 0 such that if Br is a ball with radius r such that 0 < r < ρ0 , k(x, z) verifies the hypotheses of the previous theorem in Br and f ∈ L p (Br ) for 1 < p < ∞, we have
kC(a, f )k L p (Br ) ≤ ck f k L p (Br ) for some constant c = c( p, n, M, s, N ). 4. Representation formula (α) Let B be a ball, B ⊂ , and let B˜ be the subset of B where ai j are defined and ˜ Let us consider the following linear elliptic satisfy condition (ii) and where x0 ∈ B. differential operator of order 2s N : X (α) L˜0 ≡ det ai j (x0 )D α ,
i, j = 1, . . . , N .
(4.1)
|α|=2s
We also set
t 0(x0 , t) = |t|2s N −n ψ x0 , , |t| where ψ(x0 ) is an opportune analytic function. 0(x0 , t) is the well-known John fundamental solution (see [12]). It follows that |D α 0(x0 , t)| ≤ k(n, |α|, 3, M, s, N )|t|2s N −n−|α| , and if |α| = 2s N , D α 0(x0 , t) is a homogeneous function of degree −n with zero mean value on the sphere |t| = 1.
390
MARIA ALESSANDRA RAGUSA
Let us consider 0 the above fundamental solution; then ∀v ∈ C0∞ (B), Z v(x) = 0(x0 , x − y)L˜0 v(y) dy.
From this formula in [5], integrating by parts and using a technique of Bureau in [3], it is proved that, for any vector field u = (u 1 , . . . , u n ) ∈ C0∞ (B), the derivative D α u i (x), with |α| = 2s, can be written a.e. in B as a linear combination of terms like Z
D γ +α 0(x, x − y)
P.V. B
N X
(α) (α) ak j (x) − ak j (y) D α u j (y) dy,
j=1 |α|=2s
Z
D γ +α 0(x, x − y)
P.V. B
N X X j=1
N X X j=1
(α) ai j (y)D α u j (y) dy,
|α|=2s
(α) ai j (x)D α u j (x) .
|α|=2s
2s, p
We observe that if u ∈ W0 (B), the same representation formula as in the above case u ∈ C0∞ (B) can be obtained by a density argument.
5. Main results LEMMA 5.1 Let us suppose that assumptions (i), (ii), and (iii) are true. Also, let f ∈ L p (Br ), 1 < p < ∞. Then there exist a positive constant c˜ independent of u i , f i , ∀i = 1, . . . , N , and 2s, p ρ0 > 0 such that for any ball Br ⊂⊂ with 0 < r < ρ0 and any u ∈ W0 (Br ) with D α u j ∈ L p (Br ), |α| = 2s, j = 1, . . . , N , we have N X j=1
kD α u j k L p (Br ) ≤ c˜
N X j=1
k f j k L p (Br ) + ku j k L p (Br ) ,
∀α : |α| = 2s.
¨ LOCAL HOLDER REGULARITY
391
Proof Let us consider the operator L˜ (x, D)u ≡ L˜ u =
N X
L˜ (x, D)u j (x) =
j=1
= fi −
j=1
N X j=1
N X X
X
(α) ai j (x)D α u j (x)
|α|=2s
(α) bi j (x)D α u j (x) ,
i = 1, . . . , N .
|α|≤2s−1
Using the representation formula and Theorems 3.5 and 3.6, we obtain N X
kD α u j k L p (Br ) ≤ c˜
j=1 |α|=2s
N X
kL˜ uk L p (Br ) ,
j=1
where c˜ = c(n, ˜ p, r, |α|, M). It follows that N X
kD α u j k L p (Br ) ≤ ck ˜
j=1 |α|=2s
N X
fi −
i=1
≤ c˜
N nX
N X i, j=1
|α|≤2s−1 N X
k f j k L p (Br ) +
j=1
(α) bi j D α u j k L p (Br )
X
o (α) kbi j D α u j k L p (Br ) .
(5.1)
i, j=1 |α|≤2s−1 (α)
To estimate the term containing bi j in the last inequality, let us define (α) B(0) = x ∈ Br : |bi j (x)| ≥ 0 ; then N X
(α)
kbi j D α u j k L p (Br )
i, j=1 |α|≤2s−1
=
N X
Z Br
i, j=1 |α|≤2s−1
≤
N X i, j=1 |α|≤2s−1
(α)
0p
|bi j D α u j | p dy
Z
1/ p
|D α u j | p dy + Br \B(0)
Z
(α)
B(0)
|bi j | p · |D α u j | p dy
1/ p
392
MARIA ALESSANDRA RAGUSA
(and, using the H¨older inequality and then the Sobolev inequality)
≤
N X 0
Z
X
|D α u j | p dy
1/ p
|α|≤2s−1 Br \B(0)
i, j=1
+S
(α) kbi j k L tm (B(0))
X
· kD u j k L p (B(0)) , α
(5.2)
|α|=2s
|α|≤2s−1
where we define kD α uk L p (B) as the usual norm in L p (B) of D α u with |α| = 2s. |α|=2s
Let us denote N X
A≡
α
kD u j k
L p (Br )
,
B≡
j=1 |α|=2s
N X
(α)
kbi j k L p (Br ) ;
i, j=1 |α|≤2s−1
then, using an interpolation inequality (see [10, Theorem 7.27]), we have N X
C≡
(α)
kbi j D α u j k L p (Br ) ≤ 0 A + c()
i, j=1 |α|≤2s−1
+ SA
N X
ku j k L p (Br )
j=1 N X
(α)
kbi j k L p (Br ) .
i, j=1 |α|≤2s−1
By the absolute continuity of the Lebesgue integral, the last sum can be made arbitrarily large for large 0. Namely, for an arbitrary 1 > 0, one can choose 0 such that this sum does not exceed 1 /2S. Then we choose = 1 /20, so that C ≤ 1 A + c(1 )
N X
ku j k L p (Br ) .
j=1
Now (5.1) implies A ≤ c˜
N nX
k f j k L p (Br ) + 1 A + c(1 )
j=1
N X
o ku j k L p (Br ) ,
j=1
and finally, choosing 1 = 1/2c, ˜ we get the desired estimate, with a different constant c. ˜ Proof of Theorem 2.1 We first prove that D α u ∈ L p (0 ) for |α| = 2s.
¨ LOCAL HOLDER REGULARITY
393
Let us recall the representation formula for the derivatives D α u i , |α| = 2s, of any C0∞ (B) vector function u = (u 1 , . . . , u N ), where B is a ball, B ⊂ . As above, it can be written a.e. in B as a linear combination of the terms of Z B
Z
(α) (α) ak j (x) − ak j (y) D α u j (y) dy,
j=1 |α|=2s
D γ +α 0(x, x − y)
P.V.
N X
D γ +α 0(x, x − y)
P.V.
B
N X
N X
L˜ (y, D)u j (y) dy,
j=1
L˜ (x, D)u j (x),
j=1
where |γ | = 2s(N − 1), |α| = 2s, s ∈ N, and B ⊂ is a ball whose radius is to be chosen later. To simplify this representation formula, we define N ≡ (Nα )|α|=2s ≡ P.V.
Z
D γ +α 0(x, x − y) B
N X
L˜ (y, D)u j (y) dy
j=1
+ cγ ,α (x)
N X
L˜ (x, D)u j (x),
j=1
where cγ ,α are bounded functions, and M g(x) = Mγ αk g(x) ≡ P.V.
Z
D γ +α 0(x, x − y) B
N X
akαj (x) − akαj (y) D α g j (y) dy.
j=1 |α|=2s
By Theorem 3.6, it is possible to choose the radius of B so small that X kMγ αk k < 1.
(5.3)
γ αk
We now rewrite D α u i (x), |α| = 2s, as a linear combination of the terms M and N.
By Hypothesis I and the Sobolev lemma, it follows that L˜ u j ∈ L p1 (B), ∀ j = 1, . . . , N , and q < p1 ≤ p. Let us define X F (φ) = F (φσ ) σ =1,...,N n 2s ≡ A(x)M (φσ ) + B(x)N 2s |γ |=2s(N −1)
σ =1,...,N n
394
MARIA ALESSANDRA RAGUSA
for every φσ ∈ L p2 (B) and for every p2 such that q ≤ p2 ≤ p1 , where A(x) and B(x) are bounded functions. Using (5.3), it is easy to prove that F is a contraction in [L p2 (B)]θ , where θ = N n 2s . Then F has a unique fixed point in [L p2 (B)]θ , ∀ p2 such that q ≤ p2 ≤ p1 . Since D α u, |α| = 2s, is a fixed point in [L q (B)]θ and since, moreover, φ is a fixed point in [L p1 (B)]θ ⊆ [L q (B)]θ , the uniqueness implies that the fixed points must coincide. It follows that if p1 = p, we get to the conclusion; otherwise, iterating this procedure a finite number of times, we prove that D α u i ∈ L p (B), ∀i = 1, . . . , N , ∀α : |α| = 2s. The ball B is arbitrary; it then follows that D α u i ∈ L p (0 ), ∀i = 1, . . . , N , ∀α : |α| = 2s. Now we prove the estimate. We introduce some notation, and we follow the line of the proof of [10, Theorem 9.11]. Let σ ∈ (0, 1), let r ≤ dist(0 , ∂), and let {Bσ r } be a finite covering of 0 such S that Br ⊆ 00 . Let η ∈ C02s (Br ) be a cut-off function such that 0 ≤ η ≤ 1, η = 1 in Bσ r , η = 0 for |x| ≥ σ 0r , where σ 0 = (1 + σ )/2; then |Dη| ≤
τ , (1 − σ )r
...,
|D 2s η| ≤
τ (1 − σ )2s r 2s
for some positive constant τ independent of σ and r . Then for every j = 1, . . . , N and |α| = 2s, kD α u j k L p (Bσ r ) = kD α (ηu j )k L p (Bσ r )
(5.4)
because η = 1 on Bσ r . Moreover, because Bσ r ⊆ Br , for every j = 1, . . . , N and |α| = 2s, we have kD α (ηu j )k L p (Bσ r ) ≤ kD α (ηu j )k L p (Br ) . We have, for every h = 1, . . . , m and j = 1, . . . , N , L (ηu) =
N X
(α)
ai j D α (ηu) j +
j=1 |α|=2s
N X
(α)
bi j D α (βu) j
j=1 |α|≤2s−1
N α X n X (α) = c(α) ai j [D τ ηD α−τ u] j j=1 |α|=2s
τ =0
(5.5)
¨ LOCAL HOLDER REGULARITY
395
N X
+
(α)
bi j
α o X [D τ ηD α−τ u] j τ =0
j=1 |α|≤2s−1
(where c(α) is a constant dependent on α) N n X (α) = c(α) ai j [ηD α u] j + j=1 |α|=2s
+
N X
N X
(α)
ai j [u D α η] j +
N X
(α)
bi j [u D α η] j
j=1 |α|≤2s−1 (α)
ai j
α−1 X
[D τ η D α−τ u] j
τ =1
j=1 |α|=2s N X
+
(α)
bi j [ηD α u] j
j=1 |α|≤2s−1
j=1 |α|=2s
+
N X
(α)
bi j
α−1 o X [D τ η D α−τ u] j τ =1
j=1 |α|≤2s−1
N N N α−1 nX X X X (α) = c(α) ηL u j + u jL η + ai j [D τ η D α−τ u] j j=1
+
j=1 N X
(α)
bi j
j=1 |α|≤2s−1
j=1 |α|=2s
τ =1
α−1 o X [D τ ηD α−τ u] j τ =1
= c(α)(I0 + I1 + I2 ). Using Lemma 5.1, we obtain kηu j kW 2s, p (Br ) ≤ C˜ kI0 k L p,λ (Br ) + kI1 k L p,λ (Br ) + kI2 k L p,λ (Br ) k + kηu j k L p,λ (Br ) . (5.6) Moreover, there exist three constants c0 , c1 , and c2 such that kI0 k L p,λ (Br ) ≤ c0 k f j k L p,λ (Br ) , c1 kI1 k L p,λ (Br ) ≤ ku j k L p,λ (Br ) , (1 − σ )2s r 2s and kI2 k L p (Br ) ≤
2s−1 X τ =1
c2 kD τ u j k L p (Br ) . (1 − σ )2s−τ r 2s−τ
(5.7)
396
MARIA ALESSANDRA RAGUSA
From (5.4), (5.5), and (5.6), we have, for every j = 1, . . . , N , ku j kW 2s, p (Bσ r ) ≤ C k f j k L p (Br ) + +
2s−1 X τ =1
1 ku j k L p (Br ) (1 − σ )2s r 2s 1
(1 − σ )2s−τ r
τ p (B ) ; kD u k j L r 2s−τ
then (1 − σ )2s r 2s ku j kW 2s, p (Bσ r ) ≤ C(1 − σ )2s r 2s k f j k L p (Br ) + +
2s−1 X τ =1
1 ku j k L p (Br ) (1 − σ )2s r 2s 1
(1 − σ )2s−τ r
kD u j k L p (Br ) . 2s−τ τ
Let us now consider, for k = 0, 1, . . . , 2s, the weighted seminorm 8k = sup (1 − σ k )r k kD k uk L p (Bσ r ) ; 0<σ <1
following the lines of [10, Theorem 9.11], we obtain, for every j = 1, . . . , N , kD 2s u j k L p (Bσ r ) ≤
c (r 2s k f j k L p (Br ) + ku j k L p (Br ) ). (1 − σ )2s r 2s
Choosing σ = 1/2, the final estimate follows. Proof of Theorem 2.2 Theorem 2.2 follows immediately from Theorem 2.1 and the known properties of Lebesgue spaces for suitable exponent p. Acknowledgments. The author is indebted to the anonymous referee for criticism and very useful suggestions. References [1]
[2]
S. AGMON, A. DOUGLIS, and L. NIRENBERG, Estimates near the boundary for
solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math. 12 (1959), 623 – 727. MR 23:A2610 385 , Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II, Comm. Pure Appl. Math. 17 (1964), 35 – 72. MR 28:5252 385
¨ LOCAL HOLDER REGULARITY
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[3]
F. BUREAU, Divergent integrals and partial differential equations, Comm. Pure Appl.
[4]
S. CAMPANATO, Sistemi parabolici del secondo ordine, non variazionali, a coefficienti
Math. 8 (1955), 143 – 202. MR 16:826a 390
[5]
[6]
[7]
[8]
[9]
[10]
[11] [12] [13]
discontinui, Ann. Univ. Ferrara 23 Sez. VII (N.S.) (1977), 169 – 187. MR 57:13195 386 F. CHIARENZA, M. FRANCIOSI, and M. FRASCA, L p -estimates for linear elliptic systems with discontinuous coefficients, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 5 (1994), 27 – 32. MR 95d:35040 385, 390 F. CHIARENZA, M. FRASCA, and P. LONGO, Interior W 2, p estimates for non-divergence elliptic equations with discontinuous coefficients, Ricerche Mat. 40 (1991), 149 – 168. MR 93k:35051 385 , W 2, p -solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336 (1993), 841 – 853. MR 93f:35232 385 G. DI FAZIO and M. A. RAGUSA, Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal. 112 (1993), 241 – 256. MR 94e:35035 385, 388, 389 A. DOUGLIS and L. NIRENBERG, Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math. 8 (1955), 503 – 538. MR 17:743b 386 D. GILBARG and N. TRUDINGER, Elliptic Partial Differential Equations of Second Order, 2d ed., Grundlehren Math. Wiss. 224, Springer, Berlin, 1983. MR 86C:35035 385, 392, 394, 396 F. JOHN, The fundamental solution of linear elliptic differential equations with analytic coefficients, Comm. Pure Appl. Math. 3 (1950), 273 – 304. MR 13:40h F. JOHN and L. NIRENBERG, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415 – 426. MR 24:A1348 389 D. SARASON, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391 – 405. MR 51:13690 387, 388
Dipartimento di Matematica, Universit`a di Catania, Viale A. Doria, 6, 95125 Catania, Italy; [email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 3,
A POLYNOMIAL BOUND IN FREIMAN’S THEOREM MEI-CHU CHANG
Abstract In this paper the following improvement on Freiman’s theorem on set addition is obtained (see Theorems 1 and 2 in Section 1). Let A ⊂ Z be a finite set such that |A + A| < α|A|. Then A is contained in a proper d-dimensional progression P, where d ≤ [α − 1] and log(|P|/|A|) < Cα 2 (log α)3 . Earlier bounds involved exponential dependence in α in the second estimate. Our argument combines I. Ruzsa’s method, which we improve in several places, as well as Y. Bilu’s proof of Freiman’s theorem. A fundamental result in the theory of set addition is Freiman’s theorem. Let A ⊂ Z be a finite set of integers with small sumset; thus assume |A + A| < α|A|,
(0.1)
A + A = {x + y | x, y ∈ A}
(0.2)
where and | · | denotes the cardinality. The factor α should be thought of as a (possibly large) constant. Then Freiman’s theorem states that A is contained in a d-dimensional progression P, where d ≤ d(α) (0.3) and
|P| ≤ C(α). |A|
(0.4)
(Precise definitions are given in Section 1.) Although this statement is very intuitive, there is no simple proof so far, and it is one of the deep results in additive number theory. G. Freiman’s book [Fr] on the subject is not easy to read, which perhaps explains why in earlier years the result did not get its deserved publicity. More recently, two DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 3, Received 14 February 2001. Revision received 9 July 2001. 2000 Mathematics Subject Classification. Primary 11P70; Secondary 11B13, 11B25.
399
400
MEI-CHU CHANG
detailed proofs have been given. One, due to Bilu [Bi], is close to Freiman’s and very geometric in spirit. The other, due to Ruzsa [Ru2], is less geometric and is based also on results in graph theory, such as Pl¨unnecke’s theorem. More details of Ruzsa’s proof are given later. In (0.3) and (0.4), we denoted by d(α) and C(α) constants that depend on α. In most applications of Freiman’s theorem, it is also important to have some quantitative understanding of this dependence. An optimal result would be to show linear dependence of d(α) in α and exponential dependence of C(α) (trivial examples mentioned in [Ru2] show that this would be optimal). This paper addresses that issue and provides a substantial improvement of what has been obtained so far from either Bilu’s or Ruzsa’s approach. But before getting into details, we mention very briefly some results and problems, subjects of current research, that are intimately related to quantitative versions of Freiman’s theorem. (i) T. Gowers’s work on arithmetic progressions (see [G1], [G2]). A celebrated theorem of E. Szemer´edi [Sz], solving an old conjecture of P. Erd˝os and P. Tur´an, roughly asserts that if S ⊂ Z+ is a set of positive upper density, that is, if lim sup
N →∞
|S ∩ [1, N ]| > 0, N
(0.5)
then S contains arbitrarily long arithmetic progressions a, a + b, a + 2b, . . . , a + jb.
(0.6)
More precisely, there is a function δ(N , j) such that if T ⊂ [1, N ] and |T | > δ(N , j)N ,
(0.7)
then T contains a progression (0.6) of size j. Moreover, for fixed j, δ(N , j) → 0
when N → ∞.
(0.8)
Szemer´edi’s proof was a tour de force in combinatorics, which only few people tried to read, and certainly extracting any quantitative information about the function δ(N , j) from it looks hopeless. Later a more conceptual approach based on ergodic theory was developed by H. Furstenberg and his collaborators (see [Fu], [FKO]). This method applies also in greater generality (see, for instance, [BL] on polynomial versions of Szemer´edi’s theorem), but it has the drawback of providing no quantitative information at all. In recent work Gowers [G1], [G2] established a lower bound δ(N , j) <
1 . (log log N )c( j)
(0.9)
A POLYNOMIAL BOUND IN FREIMAN’S THEOREM
401
Notice that for j = 4 absolutely no estimate was known. (The case j = 3 goes back to K. Roth [Ro].) In fact, even for B. van der Waerden’s theorem on progression in [VdW], published in 1927, bounds expressed by primitively recursive functions were given only a few years ago (see [Sh]). Gowers’s estimate (0.9) is therefore certainly most spectacular. The key ingredient in this approach is a quantitative version of Ruzsa’s proof of Freiman’s theorem. Further progress on this issue is therefore of primary importance to the problematic of progressions in “thin” sets of integers (most notoriously, the set of prime numbers). (ii) The dimension of measurable rings of real numbers. Let S ⊂ R be a measurable set and a ring in the algebraic sense; that is, S + S ⊂ S, S · S ⊂ S. An old conjecture of Erd˝os states that the Hausdorff dimension of S is either 0 or 1. It is known that if 1/2 < dim S ≤ 1, then dim S = 1 (see [Fal]). The problem for 0 ≤ dim S ≤ 1/2 turns out to be much harder and is closely related to the following conjecture of Erd˝os and Szemer´edi [ES]. CONJECTURE
If A is a finite set of integers, then |E 2 (A)| = |A|2−ε
for all ε > 0,
(0.10)
where E 2 (A) = (A + A) ∪ A · A.
(0.11)
In [NT], it is shown that if A1 , A2 ⊂ Z are finite sets and |A1 | = |A2 | = k ≥ 2, then |A1 · A2 | =
|A1 + A2 | ≤ 3k − 4, k 2 . log k
(0.12)
(0.13)
Here one uses the fact that if (0.12) holds, then A1 and A2 are contained in a 1dimensional arithmetic progression. This is a special case of Freiman’s general theorem, where a strong conclusion holds. Related to the general conjecture, the record at this point is (see [El]) |E 2 (A)| > c|A|5/4 ,
(0.14)
obtained from the Szemer´edi-Trotter theorem on line incidences in the plane (see [ST]). (iii) Relation of Freiman’s theorem on set addition to the problem of the dimension of Besicovitch sets in Rd . Recall that a measurable subset A ⊂ Rd , d = 2, is a Besicovitch set if it contains a line segment in every direction. Such sets may be of
402
MEI-CHU CHANG
zero measure, but it is likely that always dim A = d (the maximal dimension). For d = 2, this is a known result, but the question for d > 2 appears to be very hard (for d = 3, the best result so far is Hausdorff-dim A ≥ 5/2, for Minkowski-dim A ≥ 5/2 + (see [KLT])). This is a problem in geometric measure theory with major implications for Fourier analysis in several variables. It has been subject to intensive research during the last decade, a survey of which the reader may find in [W] and [T]. (Relations between this problem and a number of other conjectures on the Erd˝os ring problem are discussed in [KT], [T]; for applications in other subjects such as group theory, coding theory, and integer programming, see [He], [Ru3], [CZ], [Ch].) We now return to the content of the paper. We mostly follow Ruzsa’s method (which provides the best bounds so far) and improve his argument in several places. Basically there are two stages in Ruzsa’s method. First, one generates a large progression P0 ⊂ 2A − 2A by embedding a subset of A in Z N , finding a large progression in this image, and then pushing it back to Z. Next, one enlarges P0 to get a progression P1 ⊃ A. The progression P0 is of dimension d0 ≤ d0 (α), (0.15) and
|A| < C0 (α). |P0 |
(0.16)
Rusza obtains d0 (α) < α 4 and log C0 (α) bounded by some power of α. We improve this here to d0 (α) . α log α, log C0 (α) . α(log α)2
(0.17) (0.170 )
by refining the harmonic analysis part related to the circle method. We do feel, however, that this statement is not optimal, and it does not seem unreasonable to conjecture bounds α ε or even C log α in (0.17). (If true, this last statement would have substantial new applications.) Notice that the construction of the progression P0 inside A is the hard part of the argument. Once P0 is obtained, one considers a maximal set of elements a1 , . . . , as ∈ A such that the sets ai + P0 are naturally disjoint. Then A ⊂ {a1 , . . . , as } + P0 − P0 ,
(0.18)
and we use {a1 , . . . , as } as additional generators for a progression P ⊃ A, whose dimension may be bounded by d(α) ≤ s + dim P0 ≤ C0 (α) + d0 (α).
(0.19)
A POLYNOMIAL BOUND IN FREIMAN’S THEOREM
403
This procedure thus introduces an exponential dependence of d(α) on α in (0.3) because of the C0 (α)-dependence. We present here a more economical procedure, replacing (0.19) by d(α) . α log C0 (α) + d0 (α) . α 2 (log α)2 .
(0.20)
As mentioned earlier, d(α) . α
(0.21)
would be the optimal result here. The progression P obtained is not necessarily proper (see Section 1 for a definition). In [Bi], it is shown how, starting from Ruzsa’s result, one may replace P by a proper progression still satisfying (0.3) and (0.4). Based on a variant of this argument, we obtain Theorem 2, where P ⊃ A is a proper progression of dimension d ≤ [α − 1] and log(|P|/|A|) < Cα 2 (log α)3 . In this paper, Z N always denote Z/N Z. The paper is organized as follows. In Section 1, we give preliminaries and the precise statement of our theorems. We also summarize Ruzsa’s method. In Section 2, we improve Step 4 in Ruzsa’s method. In Section 3, we prove a technical proposition that is used for the improvement of Step 4 in Ruzsa’s method. In Section 4, we prove Theorem 2. 1. Preliminaries and Ruzsa’s method We begin this section by recalling some definitions. For the reader’s convenience, we write here various theorems from [Na] in the form we need. For proofs, please see [Na]. A d-dimensional (generalized) arithmetic progression is a set of the form P = P(q1 , . . . , qd ; `1 , . . . , `d ; a) = {a + x1 q1 + · · · + xd qd | 0 ≤ xi < `i , i = 1, . . . , d}.
(1.1)
The length of P is `(P) =
d Y
`i .
(1.2)
i=1
Clearly, |P| ≤ `(P). (Here |P| is the cardinality of P.) If |P| = `(P), the progression is called proper. Denote A + B = {a + b | a ∈ A, b ∈ B},
(1.3)
(h fold).
(1.4)
hA = A + ··· + A
404
MEI-CHU CHANG
Observe that if P in (1.1) is proper, then |2P| ≤ 2d |P|.
(1.5)
The above makes sense in any abelian group, but we restrict ourselves to Z in this paper. The following result is a structural theorem for a subset of Z with “small” doubling set. FREIMAN ’ S THEOREM Let A ⊂ Z be a finite set, and let
|2A| ≤ α|A|.
(1.6)
Then A is contained in a d-dimensional generalized arithmetic progression P, where d ≤ d(α),
(1.7)
`(P) ≤ C(α)|A|.
(1.8)
Our interest here is in the quantitative aspects. Known bounds (obtained in [Ru1]) for d(α) in (1.7) (resp., for C(α) in (1.8)) are exponential (resp., double exponential) in α. The role of α here is as a possibly large constant. In this paper, the following improvement is obtained. THEOREM 1 Freiman’s theorem holds with d(α) and log C(α) bounded by Cα 2 (log α)2 (C standing for various absolute constants). THEOREM 2 Assume that A ⊂ Z is a finite set satisfying (1.6). Then A ⊂ P, where P is a proper d-dimensional arithmetic progression with
d ≤ [α − 1], |P| log ≤ Cα 2 (log α)3 . |A|
(1.9) (1.10)
Remark 1.1 Compared with Theorem 1, (1.9) is an improvement of (1.7). Moreover, P is proper in Theorem 2. Theorem 2 is deduced from Theorem 1 using an additional argument from [Bi].
A POLYNOMIAL BOUND IN FREIMAN’S THEOREM
405
These statements answer to a satisfactory extent the question raised at the end of [Ru1] (where it is conjectured that one may take d(α), log C(α) . α) and also in [Na]. To prove Theorem 1, we basically follow Ruzsa’s proof in its consecutive steps and present an improvement in two of them. Notice that Ruzsa’s argument, although simpler than Freiman’s, remains fairly nontrivial and combines techniques and results from at least three different fields—graph theory (Pl¨unnecke’s inequalities), geometry of numbers (Minkowski’s second theorem), and harmonic analysis (Bogolyubov’s method). Now, we present some preliminaries. Recall that a Freiman homomorphism of order h (h ≥ 2) is a map φ:A→B
(A, B ⊂ Z)
such that φ(a1 ) + · · · + φ(ah ) = φ(a10 ) + · · · + φ(ah0 )
(1.11)
if a1 , . . . , ah , a10 , . . . , ah0 ∈ A and a1 + · · · + ah = a10 + · · · + ah0 . If φ : A → B is a one-to-one correspondence and satisfies a1 + · · · + ah = a10 + · · · + ah0 if and only if φ(a1 ) + · · · + φ(an ) = φ(a10 ) + · · · + φ(ah0 ),
(1.12)
then φ is called a Freiman isomorphism of order h. We begin with two easy lemmas. Their proofs can be found in [Na, Theorems 8.5 and 8.4]. 1.1 If h = h 0 (k + `) and A, B are Freiman isomorphic of order h, then k A − `A and k B − `B are Freiman isomorphic of order h 0 . LEMMA
LEMMA 1.2 Let P be a d-dimensional arithmetical progression, and let φ : P → Z be a Freiman homomorphism of order h ≥ 2. Then φ(P) is a d-dimensional progression. If P is proper and φ is a Freiman isomorphism, then φ(P) is also proper.
The following is an important inequality due to H. Pl¨unnecke. 1.3 ([Na, Theorem 7.8]) Let A be a finite subset of an abelian group such that PROPOSITION
|2A| = |A + A| ≤ α|A|.
406
MEI-CHU CHANG
Then, for all k, ` > 1, |k A − `A| ≤ α k+` |A|. Now we summarize the main steps in Ruzsa’s proof. Step 1 ([Na, Theorem 8.9]). Fix h ≥ 2, and denote D = h A − h A. Let N be the smallest number such that N > 4h|D|. (1.13) Then there is a subset A1 ⊂ A, |A1 | >
|A| , h
(1.14)
which is Freiman isomorphic of order h to a subset of Z N . Denote by φ : A1 → A01 ⊂ Z N
(1.15)
this h-Freiman isomorphism. From (1.13), (1.14), and Proposition 1.3, we may thus ensure that N < 8h|D| ≤ 8hα 2h |A| < 8h 2 α 2h |A01 |.
(1.16)
Next, one invokes the following fact. Step 2 ([Bo], [Na, Theorem 8.6]). Let R ⊂ Z N , with |R | = λN . Then for some integer d ≤ λ−2 , there exist pairwise distinct elements r1 , . . . , rd ∈ Z N such that 1 B r1 , . . . , rd ; ⊂ 2R − 2R , (1.17) 4 where
gr o n
i B(r1 , . . . , rd , ε) = g ∈ Z N |
< ε, for i = 1, . . . , d N denotes the “Bohr neighborhood.” Also, for x ∈ R, kxk = dist (x, Z).
(1.18)
Remark. The proof of this is a discrete version of the usual circle method (see also [FHR], [Ru2]). Step 3 ([Na, Theorem 8.7]). The Bohr set B(r1 , . . . , rd ; ε) defined in (1.18) contains a (proper) arithmetic progression P ⊂ Z N , dim P = d, and ε d |P| > N . (1.19) d
A POLYNOMIAL BOUND IN FREIMAN’S THEOREM
407
Remark. The main tool involved in the proof is Minkowski’s second theorem on the consecutive minima. Applying Step 2 with R = A01 , λ−1 ≤ 8h 2 α 2h (cf. (1.16)) thus yields a Bohr set B(r1 , . . . , rd ; 1/4) ⊂ 2A01 − 2A01 with d ≤ 64 h 4 α 4h .
(1.20)
Application of Step 3 gives a d-dim progression P 0 ⊂ 2A01 − 2A01 , |P 0 | >
|A| N > . d (4d) h(4d)d
(1.21)
By Lemma 1.1, the map φ in (1.15) induces an (h/4)-Freiman isomorphism ψ : 2A1 − 2A1 → 2A01 − 2A01 ,
(1.22)
and, assuming h/4 ≥ 2, it follows from Lemma 1.2 that P0 = ψ −1 (P 0 ) is a (proper) d-dimensional progression in 2A1 − 2A1 ⊂ 2A − 2A. Moreover, by (1.21), |P0 | >
|A| . h(4d)d
(1.23)
Step 4. This is the final step of the proof. The argument is the same as that in [Chan]. Simply consider a maximal collection {a1 , . . . , as } ⊂ A for which the sets ai + P0 ⊂ Z are mutually disjoint. Hence, for each a ∈ A, we get a + P0 ∩ ai + P0 6 = φ
for some i.
Therefore, a ∈ ai + P0 − P0
for some i = 1, . . . , s;
that is, a ∈ {a1 , . . . , as } + P0 − P0 .
(1.24)
The set in (1.24) is clearly contained in a progression P1 of dimension dim P1 = s + dim P0 = s + d,
(1.25)
and `(P1 ) ≤ 2s 2d `(P0 ) = 2s+d |P0 | ≤ 2s+d |2A − 2A| ≤ 2s+d α 4 |A|.
(1.26)
It remains to bound s. Clearly, from (1.23) and Proposition 1.3, s |A| < s|P0 | = |P0 + {a1 , . . . , as }| ≤ |2A − 2A + A| ≤ α 5 |A|. h(4d)d Hence, s < h(4d)d α 5 .
(1.27)
Observe that (1.20) and (1.27) lead to exponential dependence of s and dim P1 in α.
408
MEI-CHU CHANG
2. Some improvement of Step 4 In this section, we improve Step 4 in Ruzsa’s argument. The improvement is a rather trivial one, but it permits us to replace the exponential α-dependence of d(α) = dim P1 by a powerlike bound d(α) < α C . This bound depends mainly on d = dim P0 for the progression P0 , where P0 is obtained as above from Steps 2 and 3. This section concerns what can be deduced from the following proposition, which is proved in Section 3. 2.1 Let A ⊂ Z be a finite set such that |2A| ≤ α|A|. Then 2A − 2A contains a (proper) progression P with PROPOSITION
d = dim P < C(log α)α
(2.1)
and |P| >
|A| . 8(10d 2 )d
(2.2)
To improve Step 4, we apply Proposition 2.1. This provides an arithmetic progression P ⊂ 2A − 2A
(2.3)
such that, from (2.1) and (2.2), we have d = dim P . α(log α), |A| . |P| > 8(10d 2 )d
(2.4) (2.5)
Assuming that there exists a set S1 ⊂ A, |S1 | = 10α,
(2.6)
such that (P + x) ∩ (P + y) = ∅ we define
for x 6= y in S1 ,
P (1) = P + S1 ⊂ 2A − 2A + A.
(2.7) (2.8)
Then it follows from (2.6) and (2.7) that we have |P (1) | = 10α|P|.
(2.9)
Next, we assume again that there is a subset S2 ⊂ A, |S2 | = 10α,
(2.10)
A POLYNOMIAL BOUND IN FREIMAN’S THEOREM
such that
(P (1) + x) ∩ (P (1) + y) = ∅
409
for x 6 = y ∈ S2 .
(2.11)
Thus we have P (2) = P (1) + S2 ⊂ 2A − 2A + A + A = 2A − 2A + 2A
(2.12)
|P (2) | = (10α)2 |P|.
(2.13)
and
If the process is iterated t times, we obtain P (t) = P + S1 + · · · + St ⊂ 2A − 2A + t A
(2.14)
|P (t) | = (10α)t |P|.
(2.15)
and
It follows from (2.5), (2.15), (2.14), and Proposition 1.3 that we have (10α)t
|A| ≤ |(2 + t)A − 2A| ≤ α 4+t |A|. 8(10d 2 )d
(2.16)
Hence, 10t ≤ 8α 4 (10d 2 )d .
(2.17)
t . log α + d log d . α(log α)2 .
(2.18)
Now, (2.4) gives Therefore, after t steps (note that t is bounded in (2.18)), the set St cannot be defined; that is, there is a set St0 ⊂ A, |St0 | < 10α, such that for each x ∈ A there is a ∈ St0 with (x + P (t−1) ) ∩ (a + P (t−1) ) 6= ∅; hence,
x ∈ a + P (t−1) − P (t−1) ⊂ St0 + P (t−1) − P (t−1) .
(2.19)
It follows, recalling (2.14), that A ⊂ (P − P) + (S1 − S1 ) + · · · + (St−1 − St−1 ) + St0 .
(2.20)
If P = P(q1 , . . . , qd ; `1 , . . . , `d ) (cf. (1.1)), then (2.20) is clearly contained in a translate of the progression [ P¯ = P q1 , . . . , qd , Sr ∪ St0 ; 2`1 , . . . , 2`d , 3, . . . , 3, 2 (2.21) r
of dimension d¯ = dim P¯ = d +
X r
|Sr | + |St0 | < d + 10αt,
(2.22)
410
MEI-CHU CHANG
and ¯ ≤ 2d `(P) · 310αt = 2d 310αt |P| ≤ 2d 310αt α 4 |A|. `( P)
(2.23)
(The last inequality follows from (2.3) and Proposition 1.3.) Together with (2.4) and (2.18), this yields Freiman’s theorem with a d-dimensional progression P¯ satisfying ¯ d¯ . α 2 (log α)2 ,
¯ < C d¯|A|. `( P)
(2.24)
Remark. The progression P¯ need not be proper. 3. Proof of Proposition 2.1 By [Na, Theorem 8.9] (in our Step 1, take h = 8), there is a subset A1 ⊂ A, |A1 | >
|A| , 8
(3.1)
which is 8-isomorphic to a subset R of Z N with N prime and N < 40|8A − 8A| < 40α 16 |A|.
(3.2)
Denote by φ : A1 → R this 8-isomorphism. To prove the proposition, it clearly suffices to produce a d-dimensional progression P in 2R − 2R satisfying (2.1) and (2.2). We begin with some definitions and standard facts. Let f, g : Z N → R be functions. We define the following terms: P (1) fˆ(m) = (1/N ) 0≤k
Let f = χR be the indicator function of the set R ; that is, χR = 1 if x ∈ R and 0 otherwise. Then (e) Supp( f ∗ f 0 ) ⊂ R − R and Supp( f ∗ f ) ⊂ 2R , (f) Supp( f ∗ f 0 ∗ f ∗ f 0 ) ⊂ 2R − 2R , P (g) f ∗ f 0 ∗ f ∗ f 0 (x) = 0≤m
A POLYNOMIAL BOUND IN FREIMAN’S THEOREM
411
P (5) k f k p = ((1/N ) 0≤x
(3.3)
[Rud] If D is dissociated, then
X 1/2 √ X
|an |2 . an e2πi(nx/N ) ≤ C p
p
n∈D
n∈D
LEMMA 3.1 Let R ⊂ Z N with |R | = δ N , and let f = χR be the indicator function of R . Let ρ be a constant. We define 0 = {0 ≤ m < N | fˆ(m)| > ρδ}, and we let 3 be a maximal dissociated subset of 0. Then |3| < ρ −2 log(1/δ).
Proof Let g(x) =
X
an e2πi(nx/N ) ,
(3.4)
n∈3
where an = qP
fˆ(n) m∈3 | f (m)|
Let p0 =
ˆ
.
(3.5)
2
p . p−1
(3.6)
Then Fact (a), (3.4), and (3.5) give P X | fˆ(n)|2 k f k p0 kgk p ≥ |h f, gi| = a¯ n fˆ(n) = qPn∈3 2 ˆ n∈3 m∈3 | f (m)| sX p = | fˆ(n)|2 ≥ ρδ |3|.
(3.7)
n∈3
The last inequality follows from the definition of 0, which contains 3. On the other hand, k f k( p/( p−1)) =
1 X (( p−1)/ p) 1 = δ (( p−1)/ p) N
(3.8)
R
and kgk p ≤ C
√
p.
(3.9)
412
MEI-CHU CHANG
The last inequality follows from Fact (h) (Rudin). Putting these together, we have p √ ρδ |3| ≤ C pδ (( p−1)/ p) .
(3.10)
Now, choosing p = log(1/δ), we have the bound claimed. 3.2 Let R ⊂ Z N with |R | = δ N and f = χ R , the indicator function of R . Let ρ be a constant. We define 0 = {0 ≤ m < N | fˆ(m)| > ρδ}. Denote B = B(0, ε) = {x kmx/N k < ε for every m ∈ 0}, where ε < 1/4. P If ρ 2 δ 3 < ((1 − 2πε)/(2 − 2πε)) 0≤m
Proof First, we note that from trigonometry, for every x ∈ B and for every m ∈ 0, |1 − e2πi(mx/N ) | < 2πε.
(3.11)
To show B ⊂ 2R − 2R , by Fact (f), it suffices to show that B ⊂ Supp( f ∗ f 0 ∗ f ∗ f 0 ). According to Fact (g), it suffices to show that X | fˆ(m)|4 e2πi(mx/N ) 6 = 0
for all x ∈ B.
(3.12)
(3.13)
0≤m
Write X X X | fˆ(m)|4 e2πi(mx/N ) = | fˆ(m)|4 e2πi(mx/N ) + | fˆ(m)|4 e2πi(mx/N ) . (3.14) m
m∈0
m6 ∈0
P to show that m∈0 | fˆ(m)|4 e2πi(mx/N ) is big, while P The idea 4 is2πi(mx/N ) is small. | fˆ(m)| e m6∈0
A POLYNOMIAL BOUND IN FREIMAN’S THEOREM
413
We have X X 4 2πi(mx/N ) 4 ˆ ˆ | f (m)| e | f (m)| − m∈0 m∈0 X X | fˆ(m)|4 e2πi(mx/N ) − | fˆ(m)|4 ≤ m∈0
m∈0
X = | fˆ(m)|4 (e2πi(mx/N ) − 1) m∈0
≤
X
| fˆ(m)|4 |e2πi(mx/N ) − 1|
m∈0
< 2π ε
X
| fˆ(m)|4 .
(3.15)
m∈0
Therefore, X X | fˆ(m)|4 e2πi(mx/N ) > (1 − 2πε) | fˆ(m)|4 m∈0
m∈0
X X = (1 − 2πε) | fˆ(m)|4 − | fˆ(m)|4 .
(3.16)
m6 ∈0
m
On the other hand, X X | fˆ(m)|4 e2πi(mx/N ) ≤ | fˆ(m)|4 .
(3.17)
The definition of 0 and Fact (d) give X X | fˆ(m)|4 ≤ ρ 2 δ 2 | fˆ(m)|2 = ρ 2 δ 3 .
(3.18)
m6 ∈0
m6 ∈0
m6 ∈0
m
Putting (3.16), (3.17), and (3.18) together, we have X X X | fˆ(m)|4 e2πi(mx/N ) ≥ | fˆ(m)|4 e2πi(mx/N ) − | fˆ(m)|4 e2πi(mx/N ) m
m6 ∈0
m∈0
X X X > (1 − 2πε) | fˆ(m)|4 − | fˆ(m)|4 − | fˆ(m)|4 m6 ∈0
m
= (1 − 2πε)
X
≥ (1 − 2πε)
X
m
m
which is positive by our assumption.
| fˆ(m)| − (2 − 2πε) 4
m6∈0
X
| fˆ(m)|4
m6∈0
| fˆ(m)|4 − (2 − 2π ε)ρ 2 δ 3 ,
(3.19)
414
MEI-CHU CHANG
LEMMA 3.3 Let R ⊂ Z N with |R | = δ N be as in inequalities (3.1) and (3.2), and let f = χR be the indicator function of R . Then
X 0≤m
Proof We denote and
δ3 | fˆ(m)|4 > . 8α
(3.20)
f (2) = f ∗ f
(3.21)
S = Supp f (2) .
(3.22)
First, we note that Fact (e), Proposition 1.3, and (3.1) give |S| ≤ |2R | = |2A1 | ≤ |2A| ≤ α|A| < 8α|R | = 8αδ N .
(3.23)
Next, Facts (b) and (d) give X
| fˆ(m)|4 =
0≤m
X
|d f (2) (m)|2 =
0≤m
1 N
X
| f (2) (m)|2 = k f (2) k22 .
(3.24)
0≤m
Now, H¨older’s inequality and Definitions 2 and 4 give δ =kf 2
(2)
k1 ≤ k f
(2)
k2 kχ S k2 = k f
(2)
r k2
|S| . N
(3.25)
Putting (3.23), (3.24), and (3.25) together, we have (3.20). In the following lemma, we use the notation defined in (1.18) and Lemma 3.2 for the Bohr neighborhood. LEMMA 3.4 Let 0 be a subset of Z N , and let 3 ⊂ 0 be a maximal dissociated subset with |3| = d. Then B(3, ε/d) ⊂ B(0, ε).
Proof First, we notice that every m ∈ 0 can be represented as X m= γ j m j , where γ j ∈ {0, 1, −1}. m j ∈3
(3.26)
A POLYNOMIAL BOUND IN FREIMAN’S THEOREM
415
Let x ∈ B(3, ε/d). Then
mx
m x X X ε
j
|γ j | |γ j | ≤ ε;
≤
≤
N N d m j ∈3
(3.27)
m j ∈3
that is, x ∈ B(0, ε). Proof of Proposition 2.1 To apply Lemma 3.2, we choose ρ such that 102 α < ρ −2 < 103 α, and we choose
(3.28)
1 . 10
(3.29)
1 − 2π ε 1 1 < . 100α 2 − 2πε 8α
(3.30)
ε= Now, (3.28) and (3.29) imply ρ2 <
Lemma 3.3 and (3.30) imply that the hypothesis of Lemma 3.2 holds. Inequalities (3.1) and (3.2) give δ>
1 . 320α 16
(3.31)
Lemma 3.1, (3.28), and (3.31) give the bound (2.1) on d. Now use Lemma 3.4 and Step 3. Substitute (3.29) in (1.19), and we have (2.2), the bound on |P|. Remark. Compared with the “usual” argument presented in [Na, Theorem 8.6], the method used above gives a significant improvement of the dimension bound, that is, d . α(log α). It is not unreasonable, however, to conjecture estimates in Proposition 2.1 of the form d < (log α)C (in this respect, cf. comments in [FHR]). If true, one would obtain estimates d(α), log C(α) < α(log α)C
0
in Freiman’s theorem (which would be essentially optimal). 4. Proof of Theorem 2 Starting from Ruzsa’s result (see [Ru1]), [Bi] demonstrates how to pass to a proper progression of dimension less than or equal to [α−1]. Following this and the estimates in [Bi], the resulting estimate on |P| becomes log
|P| < α 3 (log α). |A|
(4.1)
416
MEI-CHU CHANG
In order to preserve the bound (1.10), we proceed a bit more carefully. Here, we adopt terminology and notation from [Bi] and highlight a number of key estimates. For further details, the reader should consult [Bi]. First, we redefine a triple (m, B, ϕ). This means that m ∈ Z+ ,
(4.2)
B ⊂ Rm is a convex symmetric body such that dim(Span B ∩ Zm ) = m,
(4.3)
ϕ : Z → Z is a group homomorphism.
(4.4)
m
An m-dimensional progression P is the image of a parallelepiped in Zm under the obvious ϕ. The fact that ϕ is one-to-one implies that P is proper. To control the dimension of the progression, we use the argument in [Bi, Section 9.3]. However, this argument only implies that A is in the image of a symmetric convex body. We use [Bi, proof of Theorem 1.2, Section 3] for the construction of the parallelepiped. We start with the progression P1 = P(q1 , . . . , qd ; `1 , . . . , `d ) obtained in Theorem 1. We let m 1 = d, and we let ϕ1 : Zm 1 → Z be the homomorQd phism defined by ϕ(ei ) = qi . Let B1 be the box i=1 [−`i + 1, `i − 1]. Thus ϕ1 (B1 ∩ Zm 1 ) ⊃ A, Volm 1 (B1 ) ≤ 2 `(P); hence, m 1 and log
(4.5)
d
(4.6)
Vol (B1 ) < Cα 2 (log α)2 . |A|
(4.7)
First, we use the construction in [Bi, proof of Proposition 9.3 in Section 9.2] by letting T = 2, and we obtain a triple (m 2 , B2 , ϕ2 ) satisfying m2 ≤ m1, ϕ2 (B2 ∩ Z ) ⊃ A. m2
(4.8) (4.9)
The restriction ϕ2 T B
m 2 ∩Z 2
is one-to-one,
(4.10)
and Volm 2 (B2 ) ≤ (2m 1 T )m 1 −m 2 Volm 1 (B1 ).
(4.11)
Hence, from (4.7), log
Vol (B2 ) Vol (B1 ) ≤ Cd log α + log < Cα 2 (log α)3 . |A| |A|
(4.12)
A POLYNOMIAL BOUND IN FREIMAN’S THEOREM
417
Next, follow [Bi, Section 9.3], and replace (m 2 , B2 , ϕ2 ) by (m 0 , B 0 , ϕ 0 ) satisfying, in particular, m 0 ≤ [α − 1],
(4.13)
m0
ϕ 0 (B 0 ∩ Z ) ⊃ A, Volm 0 (B 0 ) ≤ m 2 !
(4.14) m m 2 2
2
Volm 2 (B2 ).
(4.15)
Hence, from (4.12),
Vol B 0 ≤ Cα 2 (log α)3 . (4.16) |A| At this stage, what we gain is the estimate (4.13) on the dimension. Next, we need to replace B2 by a parallelepiped. We first apply [Bi, proof of Proposition 9.3], again with (m 1 , B1 , ϕ1 ) replaced by (m 0 , B 0 , ϕ 0 ), and we take log
T = 2α([α]!)2 .
(4.17)
We get a triple (m 00 , B 00 , ϕ 00 ) such that m 00 ≤ m 0 ≤ [α − 1], m 00
ϕ (B ∩ Z ) ⊃ A, the restriction ϕ 00 T B 00 ∩Zm 00 is one-to-one, 00
00
m 0 −m 00
Volm 00 (B 00 ) ≤ (2m 0 T )
Volm 0 (B 0 ).
(4.18) (4.19) (4.20) (4.21)
Hence, from (4.16) and (4.17), log
Vol B 00 ≤ Cα 2 log α + Cα 2 (log α)3 < Cα 2 (log α)3 . |A|
(4.22)
To replace the body B by a parallelepiped, we use the argument in [Bi, Section 3]. This finally yields a proper m 00 -dim progression A ⊂ P satisfying 3 m 00 00 |P| ≤ (m 00 !) 21−m (m 00 !)2 Vol B 00 ; (4.23) 2 thus, log
|P| ≤ Cα 2 (log α)3 . |A|
(4.24)
This proves Theorem 2. Acknowledgments. The author would like to thank J. Bourgain for helpful discussions, particularly for explaining Ruzsa’s method and various mathematics (in fact, most of the introduction) related to Freiman’s theorem. The author would also like to thank T. Gowers for pointing out some errors in an earlier version of the paper.
418
MEI-CHU CHANG
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and sets of Furstenburg type, New York J. Math. 7 (2001), 149 – 187. CMP 1 856 956 402 J. LOPEZ and K. ROSS, Sidon Sets, Lecture Notes in Pure and Appl. Math. 13, Dekker, New York, 1975. MR 55:13173 M. B. NATHANSON, Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Springer, New York, 1996. MR 98f:11011 403, 405, 406, 410, 415 M. NATHANSON and G. TENENBAUM, “Inverse theorems and the number of sums and products” in Structure Theory of Set Addition, Ast´erisque 258, Soc. Math. France, Montrouge, 1999, 195 – 204. MR 200h:11110 401 K. ROTH, On certain sets of integers, J. London Math. Soc 28 (1953), 104 – 109. MR 14:536g 401 W. RUDIN, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203 – 227. MR 22:6972 411 I. RUZSA, Arithmetic progressions in sumsets, Acta Arith. 60 (1991), 191 – 202. MR 92k:11009 404, 405, 415 , Generalized arithmetic progressions and sumsets, Acta Math. Hungar. 65 (1994), 379 – 388. MR 95k:11011 400, 406 , “An analog of Freiman’s theorem in groups” in Structure Theory of Set Addition, Ast´erisque 258, Soc. Math. France, Montrouge, 1999, 323 – 326. MR 2000h:11111 402 S. SHELAH, Primitive recursive bounds for van der Waerden numbers, J. Amer. Math. Soc. 1 (1988), 683 – 697. MR 89a:05017 401 ´ , On sets of integers containing no k elements in arithmetic progression, E. SZEMEREDI Acta Arith. 27 (1975), 199 – 245. MR 51:5547 400 ´ and W. TROTTER, Extremal problems in discrete geometry, E. SZEMEREDI Combinatorica 3 (1983), 381 – 392. MR 85j:52014 401 T. TAO, From rotating needles to stability of waves: Emerging connections between combinatorics, analysis, and PDE, Notices Amer. Math. Soc. 48 (2001), 294 – 303. MR 2002b:42021 402 B. L. VAN DER WAERDEN, Beweis einer Baudetsche Vermutung, Nieuw Arch. Wisk. 15 (1927), 212 – 216. 401 T. WOLFF, “Recent work connected with the Kakeya problem” in Prospects in Mathematics (Princeton, 1996), Amer. Math. Soc., Providence, 1999, 129 – 162. MR 2000d:42010 402
Department of Mathematics, University of California at Riverside, Riverside, California 92521-0135, USA; [email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 3,
´ SUR NOMBRE DE POINTS DE HAUTEUR BORNEE LES SURFACES DE DEL PEZZO DE DEGRE´ 5 ´ ` REGIS DE LA BRETECHE
R´esum´e Nous e´ tablissons la conjecture de Manin dans le cas particulier des surfaces V de del Pezzo d´eploy´ees de degr´e 5 sur Q. Autrement dit, nous montrons que, pour un ouvert U ⊂ V , on a NU (Q) (B) := card P ∈ U (Q) : h(P) ≤ B ∼ C B(log B)4 (B → +∞). La constante C est conforme a` l’expression conjectur´ee par E. Peyre. Abstract We state Manin’s conjecture in the particular case of the split del Pezzo’s surfaces of degree 5 over Q. We show that, for an open set U ⊂ V , NU (Q) (B) := card P ∈ U (Q) : h(P) ≤ B ∼ C B(log B)4 (B → +∞). The constant C is the one conjectured by E. Peyre. Sommaire 1. Introduction . . . . . . . . . . . . . . . . . . 1.1. Pr´esentation . . . . . . . . . . . . . . . ´ 1.2. Enonc´ e du r´esultat . . . . . . . . . . . . 1.3. Conformit´e avec la conjecture de Peyre . 1.4. Les m´ethodes utilis´ees . . . . . . . . . . 2. Utilisation des sym´etries et r´eduction du domaine 2.1. Calcul de pgcd . . . . . . . . . . . . . . 2.2. R´eduction a` des variables positives . . . . ´ 2.3. Equations de Pl¨ucker . . . . . . . . . . . ´ 2.4. Etude du domaine en x . . . . . . . . . . ´ 2.5. Etude du domaine en d . . . . . . . . . .
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DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 3, Received 21 January 2000. Revision received 24 July 2001. 2000 Mathematics Subject Classification. Primary 14G05; Secondary 11D72, 11G35, 11D09.
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Estimation de N1 (B) . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Inversion de M¨obius . . . . . . . . . . . . . . . . . . . . . . . 3.2. D´emonstration de la Proposition 2 . . . . . . . . . . . . . . . . 4. Estimation de C 0 pour les δ ∈ 1(T ) : d´emonstration de la Proposition 3 5. Majoration de la contribution des δ 6 ∈ 1(T ) : d´emonstration de la Proposition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Pr´eparation des variables et r´eduction du√probl`eme . . . . . . . . 5.2. Introduction de la condition δ1 δ2 δ3 δ4 ≥ T . . . . . . . . . . . 5.3. Cl´e de la d´emonstration . . . . . . . . . . . . . . . . . . . . . 5.4. Fin de la d´emonstration : le Lemme 13 . . . . . . . . . . . . . . Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.
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442 442 444 451 457 457 458 459 461 462
1. Introduction 1.1. Pr´esentation Nous nous int´eressons a` l’estimation asymptotique du nombre de points de hauteur born´ee par B sur les vari´et´es de Fano. Yu. Manin a conjectur´e avec certains de ses collaborateurs (voir [1], [10], [12], [13]) que, pour V une vari´et´e de Fano sur k un corps de nombres qu’on munit d’une hauteur h relative au fibr´e anticanonique, il existe un ouvert U de V , tel que NU (B) := card P ∈ U : h(P) ≤ B ∼ C B(log B)r −1 , (1.1) o`u r est le rang du groupe de Picard de V et C une constante non nulle. La restriction a` un ouvert U permet d’´eviter les sous-vari´et´es accumulatrices. Par exemple, dans le cas de surfaces cubiques, on exclut les points des droites contenues dans V . Dans [4], V. Batyrev et Y. Tschinkel e´ tablissent que la puissance du logarithme n’est pas celle pronostiqu´ee pour toutes les vari´et´es de Fano. Cependant, la conjecture de Manin a e´ t´e e´ tablie dans une large classe d’exemples. Batyrev et Tschinkel (voir [2], [3], [5]) montrent (1.1) pour les vari´et´es toriques en utilisant de l’analyse harmonique fine. Leur m´ethode d´ependant de l’action de groupe sur les vari´et´es toriques peut difficilement s’appliquer dans certains cas. Par ailleurs, la m´ethode du cercle permet de traiter le cas des intersections compl`etes lisses sur Q de grande dimension (voir [6], [10], ou plus r´ecemment [18]). La n´ecessit´e d’avoir un nombre important de variables empˆeche d’utiliser efficacement la m´ethode du cercle dans tous les cas. Dans [15], Peyre a donn´e une expression conjecturale de la constante C dans la conjecture de Manin. La valeur de cette constante est reconnue comme tr`es int´eressante car elle s’exprime en fonction d’invariants g´eom´etriques. Dans [19] et
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[16], cette expression est pr´ecis´ee avec l’ajout d’un facteur multiplicatif. Ce terme multiplicatif suppl´ementaire qui est l’ordre du groupe de Brauer est apparu pour la premi`ere fois dans [2] (voir [2, page 593]). Un des cas qui r´esistent aux sp´ecialistes est le cas des surfaces de del Pezzo de degr´e 5. Les deux m´ethodes pr´ecit´ees ne s’appliquent pas. On pourra pour plus de renseignements se reporter au volume e´ dit´e tr`es r´ecemment par Emmanuel Peyre, Nombre et r´epartition de points de hauteur born´ee (voir [16], [19]). Appelons V la surface de del Pezzo d´eploy´ee de degr´e 5 obtenue en e´ clatant quatre points rationnels en position g´en´erale sur P2Q et U le compl´ementaire des courbes de premi`ere esp`ece dans V . Ici, en position g´en´erale signifie qu’il n’y en a pas trois align´es ni quatre coplanaires. Manin et Tschinkel ont montr´e dans [14] la majoration NU (Q) (B) = O(B(log B)6 ). D’autre part, P. Salberger a annonc´e en 1993 qu’il avait d´emontr´e (conf´erence a` Berne en juin 1993, voir [20]) la majoration NU (Q) (B) = O(B(log B)4 ) pour B ≥ 2. Cependant, la preuve n’a pas encore e´ t´e publi´ee. Le groupe de Picard Pic(V ) de V est isomorphe a` Z5 (voir [16, exemples 3.3.4 et 4.2.4]). D’apr`es la conjecture de Manin, la puissance attendue du log est donc bien 4 = rang(Pic(V )) − 1. Dans cet article, nous e´ tablissons la conjecture de Manin dans le cas particulier des surfaces de del Pezzo d´eploy´ees de degr´e 5 sur Q. Nous nous efforcerons de toujours nous placer a` un niveau e´ l´ementaire. L’´etude d’un cas particulier est propice a` ce type de parti pris et permet d’´eclairer la th´eorie g´en´erale. ´ 1.2. Enonc´ e du r´esultat Pour e´ noncer notre r´esultat de mani`ere e´ l´ementaire, nous introduisons les notations suivantes. Soient Pσ et Q σ les polynˆomes de R[X 1 , X 2 , X 3 ] d´efinis par Pσ (X 1 , X 2 , X 3 ) := X σ (1) X σ (2) (X σ (1) − X σ (3) )
(σ ∈ S3 ),
Q σ (X 1 , X 2 , X 3 ) := X σ (2) (X σ (1) − X σ (2) )(X σ (1) − X σ (3) )
(σ ∈ S3 ).
Les Pσ engendrent un Z-module libre dans Z[X 1 , X 2 , X 3 ] de rang 6. Ils permettent un plongement de V dans P5 . Les polynˆomes Q σ satisfont Q (2,3) = P(2,3) − P(1,3,2) + P(1,3) , Q (1,2) = P(1,2) − PId + P(2,3) , Q (1,2,3) = P(1,2,3) − P(1,3) + P(1,3,2) ,
Q (1,3) = P(1,3) − P(1,2,3) + P(1,2) , Q Id = PId − P(1,2) + P(1,2,3) , Q (1,3,2) = P(1,3,2) − P(2,3) + PId ,
o`u (i, j) d´esigne la transposition de S3 qui e´ change i et j, (1, 2, 3) et (1, 3, 2) sont les cycles d’ordre de 3 de S3 . Soit U l’ouvert du plan projectif P2 d´efini par U := x : Pσ (x) 6 = 0 (∀σ ∈ S3 ) .
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Cet ouvert est naturellement isomorphe au compl´ementaire des courbes de premi`ere esp`ece sur la surface de del Pezzo V . Pour x ∈ U (Q) un point Q-rationnel de U , repr´esent´e par le triplet x ∈ Z3 , la hauteur h(x) que nous consid´erons est donn´ee par h(x) :=
max{|Pσ (x)|, |Q σ (x)|, σ ∈ S3 } . pgcd{Pσ (x), Q σ (x), σ ∈ S3 }
(1.2)
Cette hauteur correspond au syst`eme lin´eaire anticanonique sur V qui correspond au plongement de V dans l’espace projectif P5 donn´e par les formes homog`enes de degr´e 3, dans le plan P2 qui s’annulent aux quatre points donn´es (voir [15, paragraphe 1.3] ou [16, paragraphe 2.2]). Elle est li´ee a` la norme N de R6 d´efinie par le maximum des |ti | (1 ≤ i ≤ 6), (1.3) |t1 − t6 + t2 |, |t2 − t5 + t3 |, |t3 − t4 + t1 |, |t − t + t |, |t − t + t |, |t − t + t | 4
3
5
5
2
6
6
1
4
par la formule suivante : h(x) :=
N (P(2,3) (x), P(1,3) (x), P(1,2) (x), PId (x), P(1,2,3) (x), P(1,3,2) (x)) ∗ . pgcd{Pσ (x), σ ∈ S3 }
Cette hauteur est invariante par de plus nombreuses transformations que celle plus souvent utilis´ee, max{|Pσ (x)|, σ ∈ S3 } (x ∈ Z3 ), pgcd{Pσ (x), σ ∈ S3 } qui est li´ee a` la norme maxi=1,...,6 {|ti |} de R6 . Ce choix de cette m´etrique ad´elique est crucial pour la d´emonstration. Il s’agit donc d’estimer asymptotiquement la quantit´e N (B) = NU (Q) (B) =
o`u l’ensemble U (B) est d´efini par ( U (B) := x ∈ Z : 3
1 card U (B) , 2
(x1 , x2 , x3 ) = 1, h(x) ≤ B Pσ (x) 6 = 0
(∀σ ∈ S3 )
) .
Notre r´esultat est le suivant. a pgcd{Pσ (x), Q σ (x), σ ∈ S3 } = pgcd{Pσ (x), σ ∈ S3 } puisque les polyˆomes Q σ sont des e´ l´ements du Z-module engendr´e par les Pσ . ∗ On
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´ OR E` ME TH E
1 Pour B ≥ 3, on a l’estimation N (B) = B(log B)4
avec C0 =
n π2 o 1 C0 + O 3 × 4! log log B
Y 1 5 5 1 1− 1+ + 2 . p p p p
(1.4)
(1.5)
Nous n’avons pas cherch´e a` obtenir le meilleur terme d’erreur possible. En majorant les fonctions diviseurs intervenant dans les d´emonstrations en moyenne et non terme a` terme, il est probable que l’on obtienne un terme d’erreur en B(log B)3 (log log B). ` la lumi`ere de tous les autres exemples connus, le terme attendu est en B(log B)3 . A 1.3. Conformit´e avec la conjecture de Peyre Pour comparer ce r´esultat avec le terme conjectur´e par Peyre [15], on remarque que le terme donn´e par la place en l’infini est∗ 3 Vol x ∈ R3 : max{|Pσ (x)|, |Q σ (x)|, σ ∈ S3 } ≤ 1 Z2 Z dx2 dx3 = = 2π 2 . 2 max{|P (1, x , x )|, |Q (1, x , x )|, σ ∈ S } σ σ 2 3 2 3 3 R
C∞ =
Comme la constante C0 d´efinie en (1.5) correspond aux places finies, la constante de Tamagawa (voir [15, d´efinition 2.3]) correspondant a` h est donc Y 1 5 5 1 τh (V ) = C0 C∞ = 2π 2 1− 1+ + 2 . p p p p La constante de Peyre est d´efinie par CPeyre (V ) = α(V )τh (V ) (voir [15, D´efinition 2.5]), o`u∗ ( ) 2t + 2t − t − t ≤ 1, 1 i j k ` α(V ) := Vol t ∈ [0, +∞[4 : 3 ∀(i, j, k, `) tel que {i, j, k, `} = {1, 2, 3, 4} = ∗ On
1 . 6 × 4!
a ici effectu´e le changement de variables u 1 = x13 , u 2 = x2 /x1 , u 3 = x3 /x1 , puis int´egr´e en u 1 . calculer le volume, on remarque que
∗ Pour
24 × Vol{t ∈ [0, +∞[4 : t4 ≤ t3 ≤ t2 ≤ t1 , 2t1 + 2t2 − t3 − t4 ≤ 1} 3 Z 1/2 Z (1+t4 )/3 Z (1+t3 +t4 )/4 Z (1−2t2 +t3 +t4 )/2 1 . =8 dt1 dt2 dt3 dt4 = 6 × 4! 0 t4 t3 t2
α(V ) =
Pour la justification du facteur 1/3, se reporter a` [15, lemme 7.1.6].
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Nous trouvons bien la constante conjectur´ee par Peyre : CPeyre (V ) =
π2 C0 . 3 × 4!
1.4. Les m´ethodes utilis´ees Les m´ethodes de th´eorie analytique des nombres utilis´ees sont usuelles. Notre d´emonstration s’appuie sur la compr´ehension de toutes les sym´etries du domaine sur lequel nous comptons. ` la section 2.2, nous ramenons le probl`eme du comptage des e´ l´ements x de A U (B) a` celui des x = (x1 , x2 , x3 ) satisfaisant de plus a` x1 > x2 > x3 > 0. Ensuite, a` la section 2.3, on pourra se restreindre au cas o`u d = (d1 , d2 , d3 , d4 ) d´efini par d1 = (x2 , x3 ),
d2 = (x1 , x3 ),
d3 = (x1 , x2 ),
d4 = (x1 − x2 , x1 − x3 )
est dans le domaine ) ( d ∈ N4∗ : di d j dk ≤ (Bd1 d2 d3 d4 )1/3 /B A/ log log B , D (B) := ∀(i, j, k, `) tel que {i, j, k, `} = {1, 2, 3, 4} ( ) d ∈ N4∗ : di2 d 2j dk2 d`−1 ≤ B 1−3A/ log log B , , = ∀(i, j, k, `) tel que {i, j, k, `} = {1, 2, 3, 4} o`u A est une constante suffisamment grande.∗ Dans cette e´ tape, nous serons amen´e a` compter des points sur un sousensemble du torseur universel au-dessus de V . Cette notion a e´ t´e introduite par J.-L. Colliot-Th´el`ene et J.-J. Sansuc en liaison avec l’´etude du principe de Hasse et de l’approximation faible (voir [7], [8]). Nous renvoyons le lecteur a` [16] et [19] pour l’utilisation des torseurs universels pour des probl`emes de comptage. Dans le cas des vari´et´es toriques, le torseur universel est un ouvert d’un espace affine. En utilisant cette propri´et´e, Salberger se ram`ene a` un comptage de points sur un r´eseau [19]. Ici, le torseur universel au-dessus de V est un ouvert du cˆone de Pl¨ucker C d´efini par les e´ quations sur Q (voir [16, exemple 4.2.4]) z 1,2 z 3,4 − z 1,3 z 2,4 + z 1,4 z 2,3 = 0, z 1,2 z 3,5 − z 1,3 z 2,5 + z 1,5 z 2,3 = 0, z 1,2 z 4,5 − z 1,4 z 2,5 + z 1,5 z 2,4 = 0, z 1,3 z 4,5 − z 1,4 z 3,5 + z 1,5 z 3,4 = 0, z z − z z + z z = 0. 2,3 4,5
2,4 3,5
2,5 3,4
pr´esence d’un facteur 1/B 3A/ log log B est uniquement technique et peut eˆ tre omise dans un premier temps. ∗ La
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Ce passage aux torseurs universels, implicite dans [15] et [17], d´ej`a utilis´e par Salberger pour la majoration en B(log B)4 , a e´ t´e d´evelopp´e de mani`ere g´en´erale par Salberger [19] et Peyre [16] pour e´ tendre la conjecture sur la constante de (1.1). L’exemple le plus simple de passage a` un comptage sur les torseurs universels est le cas de l’espace projectif P1 . L’ouvert A2 r {(0, 0)} au-dessus de P1 est un Gm torseur. Nous nous int´eressons aux points Q-rationnels que nous repr´esentons par les points a` coordonn´ees enti`eres homog`enes. La quantit´e a` estimer est donc 1 NU (Q) (B) = card (x1 , x2 ) ∈ (Z∗ )2 : (x1 , x2 ) = 1, max{x12 , x22 } ≤ B . 2 La condition de primalit´e (x1 , x2 ) = 1 permet d’affirmer que (x1 , x2 ) 6= (0, 0) dans toute sp´ecialisation. Consid´erons maintenant le cas de la surface V obtenue en e´ clatant P2 en quatre points P0 = (1, 0, 0),
P1 = (0, 1, 0),
P2 = (0, 0, 1),
P3 = (1, 1, 1).
Soit Gr(2, 5) la grassmanienne des sous-espaces de dimension 2 dans Q5 . D’apr`es A. Skorobogatov [22], l’ensemble Gr(2, 5)ss des points semi-stables de Gr(2, 5) v´erifie les conditions (z i, j , z i,k ) 6= (0, 0) et (z i,k , z j,k ) 6= (0, 0) (1 ≤ i < j < k ≤ 5). Soit C le cˆone de Pl¨ucker au-dessus de Gr(2, 5) qui est d´efini par le syst`eme d’´equations (4). On d´efinit C ss l’image r´eciproque de Gr(2, 5)ss dans C , l’ensemble des points semi-stables du cˆone de Pl¨ucker au-dessus de Gr(2, 5)ss . Les points de cet ouvert satisfont (z i, j , z i,k ) 6= (0, 0) et (z i,k , z j,k ) 6 = (0, 0) (1 ≤ i < j < k ≤ 5). L’ouvert C ss est un G5m -torseur au-dessus de V . L’ouvert U de V est le compl´ementaire des courbes de premi`ere esp`ece dans V . L’ouvert CUss image r´eciproque de U dans C ss est l’ensemble sur lequel on compte les points Q-rationnels. On repr´esente ces points par les points a` coordonn´ees enti`eres z = (z i, j )1≤i< j≤5 satisfaisant aux conditions z i, j 6= 0 (1 ≤ i < j ≤ 5); (z i, j , z i,k ) = 1,
(z i,k , z j,k ) = 1
(1 ≤ i < j < k ≤ 5).
On a ainsi le diagramme suivant : /
CUSS
C SS
Gr(2, 5) SS U
/
V
/
C Gr(2, 5)
/
/
P5
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La hauteur h est d´efinie par image r´eciproque par le plongement de V dans P5 a` partir du faisceau anticanonique et la m´etrique ad´elique li´ee a` (1.3) sur P5 . Dans une derni`ere e´ tape, un proc´ed´e bien connu dans son principe mais, ici, techniquement relativement d´elicat permettra de conclure. Ce proc´ed´e a des points communs avec celui choisi par Schanuel pour les espaces projectifs [21], par Peyre [15] pour certains e´ clatements de Pn , par Salberger [19] pour les vari´et´es toriques rationnelles, et par Fouvry [9] et Heath-Brown et Moroz [11] pour une certaine surface cubique. Tous les exemples du paragraphe pr´ec´edent sont des vari´et´es toriques. En sch´ematisant, on peut dire que cette m´ethode pour les surfaces de del Pezzo de degr´e 5 conduit encore a` estimer des cardinaux d’ensemble d’uplets avec une condition sur chaque coordonn´ee. Cependant, ces conditions de congruences ne se r´eduisent pas toujours a` des conditions de divisibilit´e des coordonn´ees. Par exemple, dans [19], les points compt´es par le cardinal Ad (B) introduit en [19, 11.6.b] sont les points d’un certain volume d´efinis par les conditions de divisibilit´e di | qi . Si l’on avait des conditions qi ≡ ai (mod di ) o`u les ai ∈ [0, q[ ne seraient pas ` notre connaissance, tous nuls, la majoration du lemme 11.7 ne serait plus valable. A c’est cette difficult´e-l`a qui devait eˆ tre d´epass´ee. 2. Utilisation des sym´etries et r´eduction du domaine Dans toute la suite, les indices i, j, k dans une formule o`u ` n’interviendra pas d´ecriront de mani`ere implicite les indices 1, 2, 3 de sorte que {i, j, k} = {1, 2, 3} et les indices i, j, k, ` d´ecriront de mani`ere implicite les indices 1, 2, 3, et 4 de sorte que {i, j, k, `} = {1, 2, 3, 4}. 2.1. Calcul de pgcd On exprime de mani`ere simple le pgcd intervenant dans la d´efinition (1.2) de h(x). LEMME 1 Pour tout x ∈ Z3∗ tel que (x1 , x2 , x3 ) = 1 et xi 6 = x j , on a pgcd Pσ (x), Q σ (x), σ ∈ S3 = (x1 , x2 )(x2 , x3 )(x3 , x1 )(x1 − x2 , x1 − x3 ).
D´emonstration Consid´erons x ∈ Z3∗ et d ∈ N4∗ v´erifiant (xi , x j ) = dk , (di , d j ) = 1, xi 6 = x j , ∀(i, j, k) tel que {i, j, k} = {1, 2, 3}, d4 = (x1 − x2 , x1 − x3 , x2 − x3 ) = (x1 − x2 , x2 − x3 ) = (x2 − x3 , x1 − x3 ) = (x1 − x2 , x1 − x3 ).
(2.1)
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Nous introduisons les notations x1 , d2 d3 x2 − x3 , z1 = d1 d4 y1 =
x2 , d1 d3 x1 − x3 z2 = , d2 d4 y2 =
x3 , d1 d2 x1 − x2 z3 = . d3 d4
y3 =
(2.2)
Il est facile de remarquer que d4 , y1 , y2 , y3 (resp. y3 , d3 , z 2 , z 1 , resp. z 1 , y2 , d2 , z 3 , resp. z 3 , z 2 , y1 , d1 ) sont premiers deux a` deux. Posant n = d1 d2 d3 d4 , on a pour tout (i, j, k) tel que {i, j, k} = {1, 2, 3}, ( |xi x j (xi − xk )| = n |yi y j z j d j dk |, (2.3) |x j (xi − x j )(xi − xk )| = n |y j z j z k dk d4 |. On en d´eduit que n | pgcd Pσ (x), Q σ (x), σ ∈ S3 . Supposons maintenant qu’il existe un p premier qui divise 1 pgcd Pσ (x), Q σ (x), σ ∈ S3 . n ´ Etudions par exemple le cas p | d4 . Il en d´ecoule que p | yi y j z j d j dk . Les di et yi e´ tant premiers a` d4 , on a p | z j pour j = 1, 2, 3. Cela contredit la coprimalit´e des z j . 2.2. R´eduction a` des variables positives Nous notons (xi , x j ) = dk , (di , d j ) = 1, 7 F := (x, d) ∈ N∗ : ∀(i, j, k) tel que {i, j, k} = {1, 2, 3} (x1 − x2 , x1 − x3 ) = d4 , x1 > x2 > x3 > 0
. (2.4)
Chacune des 24 composantes connexes de {x ∈ R∗ 3 : x1 6= x2 , x1 6= x3 , ` l’int´erieur x2 6= x3 } dans le domaine U (B) fournit la mˆeme contribution a` N (B). A de chacune d’entre elles, le maximum intervenant dans la d´efinition de la hauteur est domin´e par un seul polynˆome. Ainsi, le lemme suivant affirme que l’on peut se ramener au cas x1 > x2 > x3 > 0. LEMME
2
On a N (B) = 12 card (x, d) ∈ F : x1 x2 (x1 − x3 ) ≤ Bd1 d2 d3 d4 .
(2.5)
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D´emonstration Posant N
+
(B) := card U (B) ∩ N3 ,
N
−
(B) := card U (B) ∩ N2 × (−N) ,
on a N (B) = N
+
(B) + 3N − (B),
(2.6)
puisque les xi jouent le mˆeme rˆole et que les polynˆomes Pσ et Q σ sont homog`enes. Par le mˆeme argument, on obtient les formules N + (B) = 6 card U (B) ∩ {x ∈ N3 : x1 > x2 > x3 } , N − (B) = 2 card U (B) ∩ {x ∈ N2 × (−N) : x1 > x2 } . Faisant le changement de variables x10 = x1 − x3 , x20 = x2 − x3 , x30 = −x3 , on a |Pσ (x0 )|, |Q σ (x0 )|, (σ ∈ S3 ) = |Pσ (x)|, |Q σ (x)|, (σ ∈ S3 ) , puisque Q (x0 ) = −P(2,3) (x), Q (1,3) (x0 ) = P(1,2) (x), Q (1,2) (x0 ) = −Q Id (x), (2,3) Q (x0 ) = −Q 0 0 Id (1,2) (x), Q (1,2,3) (x ) = P(1,2,3) (x), Q (1,3,2) (x ) = PId (x), P(2,3) (x0 ) = −Q (2,3) (x), P(1,3) (x0 ) = −P(1,3,2) (x), P(1,2) (x0 ) = Q (1,3) (x), PId (x0 ) = Q (1,3,2) (x), P(1,2,3) (x0 ) = Q (1,2,3) (x), P(1,3,2) (x0 ) = −P(1,3) (x). On en d´eduit que N
−
(B) = 2 card U (B) ∩ {x ∈ N3 : x1 > x2 > x3 } .
Lorsque x1 > x2 > x3 > 0, on a max |Pσ (x)|, |Q σ (x)|, σ ∈ S3 = max |Pσ (x)|, σ ∈ S3 = x1 x2 (x1 − x3 ), puisqu’alors on a x1 x3 (x2 − x3 ) < x1 x2 (x2 − x3 ) < x1 x2 (x1 − x3 ), x2 x3 (x1 − x2 ) < x1 x3 (x1 − x2 ) < x1 x2 (x1 − x3 ), x x (x − x ) < x x (x − x ). 2 3 1 3 1 2 1 3 En utilisant le Lemme 1 et en remarquant que la condition (x1 , x2 , x3 ) = 1 e´ quivaut a` (di , d j ) = 1 (i 6= j), on en d´eduit N (B) = 12 card (x, d) ∈ F : x1 x2 (x1 − x3 ) ≤ Bd1 d2 d3 d4 , ce qui fournit le r´esultat.
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´ 2.3. Equations de Pl¨ucker Le torseur universel au-dessus de V est d´ecrit comme un ouvert du cˆone C d´efini en (4) avec les conditions suppl´ementaires z i, j 6 = 0 pour 1 ≤ i < j ≤ 5 (voir [16, exemple 4.2.4] et [22]). D’une part, deux e´ quations de (4) peuvent eˆ tre facilement d´eduites des trois autres. Montrons par exemple que les trois premi`eres impliquent la quatri`eme e´ quation du syst`eme (4). Pour cela, on note qi le membre de gauche de la i-`eme ligne de (4). On v´erifie facilement la formule z 1,5 q1 − z 1,3 q2 − z 1,4 q3 = z 1,2 q4 . Ceci permet de d´eduire que q4 = 0 lorsque q1 = q2 = q3 = 0, et z 1,2 6= 0. D’autre part, on obtient chaque e´ quation a` partir de la premi`ere en d´ecalant tous les indices d’une mˆeme quantit´e pourvu que l’on ait identifi´e z i, j a` z j,i , z i+5, j , et z i, j+5 . LEMME
3
On a N (B) = 12 card z ∈ C ∩ N10 ∗ :
z 1,3 z 1,4 z 2,4 z 2,5 z 3,5 ≤ B, (z i, j , z i,k ) = 1, (z i,k , z j,k ) = 1, pour tout 1 ≤ i < j < k ≤ 5
. (2.7)
D´emonstration Il existe une bijection φ entre l’ensemble F d´efini en (2.4) et l’ensemble des dix-uplets (z i, j )1≤i< j≤5 appartenant au cˆone C ∩ N10 e ∗ satisfaisant aux conditions de primalit´ (z i, j , z i,k ) = 1 et (z i,k , z j,k ) = 1 pour tout 1 ≤ i < j < k ≤ 5. Pour e´ tablir cela, on utilise les notations (2.1), (2.2) et on pose φ(x, d) = z = (z 1,2 , z 1,3 , z 1,4 , z 1,5 , z 2,3 , z 2,4 , z 2,5 , z 3,4 , z 3,5 , z 4,5 ) = (d4 , d3 , d2 , d1 , y3 , y2 , y1 , z 1 , z 2 , z 3 ). La fonction r´eciproque est d´efinie par φ −1 (z) = (x, d) = (z 1,4 z 1,3 z 2,5 , z 1,5 z 1,3 z 2,4 , z 1,4 z 1,5 z 2,3 , z 1,5 , z 1,4 , z 1,3 , z 1,2 ). D’apr`es la formule (2.2) et la remarque qui la suit, les conditions de coprimalit´e sont bien v´erifi´ees. On peut donc se ramener a` un probl`eme de compte sur le cˆone C a` coordonn´ees positives. La formule x1 x2 (x1 − x3 )/d1 d2 d3 d4 = d3 d2 y2 z 2 y1 et la bijection φ fournissent donc bien le r´esultat.
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Soit F 0 le sous-ensemble des sept-uplets (x, d) de F d´efini par les in´egalit´es x1 x2 x3 ≥ n 34 , (x2 − x3 )(x1 − x3 )x3 ≥ n 3 , 3 (x2 − x3 )(x1 − x2 )x2 ≥ n 32 , (x1 − x3 )(x1 − x2 )x1 ≥ n 31 ,
(2.8)
o`u l’on a pos´e n = d1 d2 d3 d4 et n i = n/di = d j dk d` lorsque {i, j, k, `} = {1, 2, 3, 4}. De mˆeme, on d´efinit F 0 le sous-ensemble de F d´efini par les in´egalit´es strictes dans (2.8). L’´etape suivante est cruciale pour la suite de la d´emonstration. Elle permet de se ramener a` un probl`eme de compte sur F 0 . LEMME
4
Posant N 0 (B) = card (x, d) ∈ F 0 : x1 x2 (x1 − x3 ) ≤ Bd1 d2 d3 d4 , N 0 (B) = card (x, d) ∈ F 0 : x1 x2 (x1 − x3 ) ≤ Bd1 d2 d3 d4 ,
(2.9)
60N 0 (B) ≤ N (B) ≤ 60N 0 (B).
(2.10)
on a
D´emonstration L’id´ee de la d´emonstration est d’utiliser les sym´etries pour r´eduire le probl`eme a` compter dans le domaine F 0 . Le domaine F 0 est a` des erreurs n´egligeables pr`es le cinqui`eme de F. Nous d´efinissons θ la bijection de {z i, j }i< j dans {z i, j }i< j d´efinie par θ(z i, j ) = z i+1, j+1 avec la convention z i, j = z j,i = z i+5, j = z i, j+5 . Par abus de notation, θ(z) d´esigne le dix-uplet ayant pour coordonn´ees l’image par θ des coordonn´ees de z. Il est tr`es important a` remarquer que l’ensemble intervenant dans la formule (2.7) est invariant par θ. En effet, on a θ(C ) = C et θ(z 1,3 ) θ(z 1,4 ) θ(z 2,4 ) θ (z 2,5 ) θ(z 3,5 ) = z 1,3 z 1,4 z 2,4 z 2,5 z 3,5 .
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Il est clair que l’on a Y θ(z 1,5 ) θ(z 1,4 ) θ(z 1,3 ) θ(z 1,2 ) = z 2, j , j6 = 2 Y 2 (z ) θ 2 (z ) θ 2 (z ) θ 2 (z ) = θ z 3, j , 1,5 1,4 1,3 1,2 j6 =3 Y θ 3 (z 1,5 ) θ 3 (z 1,4 ) θ 3 (z 1,3 ) θ 3 (z 1,2 ) = z 4, j , j6 =4 Y 4 (z ) θ 4 (z ) θ 4 (z ) θ 4 (z ) = θ z 5, j . 1,5 1,4 1,3 1,2 j6 =5
Quitte a` appliquer θ , on peut donc supposer que z 1,5 z 1,4 z 1,3 z 1,2 = min
i=1,...,5
nY
o z i, j .
(2.11)
j6 =i
Le dix-uplet z satisfait aux conditions (2.11) si, et seulement si, φ −1 (z) = (x, d) appartient au sous-ensemble F 0 de F. Des Lemmes 2 et 3, nous d´eduisons que (z i, j , z i,k ) = 1, z 1,3 z 1,4 z 2,4 z 2,5 z 3,5 ≤ B Y N (B) ≤ 60 card z ∈ C ∩ N10 : ∗ z 1,5 z 1,4 z 1,3 z 1,2 ≤ z i, j (∀i 6= 1) j6 =i ≤ 60 card (x, d) ∈ F 0 : x1 x2 (x1 − x3 ) ≤ Bd1 d2 d3 d4 = 60N 0 (B). (2.12) L’in´egalit´e est l`a pour parer le cas o`u le minimum d´efini au membre de droite de (2.11) est atteint sur plusieurs indices. De mˆeme, on a (z i, j , z i,k ) = 1, z 1,3 z 1,4 z 2,4 z 2,5 z 3,5 ≤ B Y N (B) ≥ 60 card z ∈ C ∩ N10 ∗ : z 1,5 z 1,4 z 1,3 z 1,2 < z i, j (∀i 6= 1) j6=i ≥ 60 card (x, d) ∈ F 0 : x1 x2 (x1 − x3 ) ≤ Bd1 d2 d3 d4 = 60N 0 (B). (2.13)
Dans la suite, nous n’´etudierons que le cardinal N 0 (B), e´ tant entendu que les estimations e´ nonc´ees pour celui-l`a sont encore valables pour N 0 (B). ´ 2.4. Etude du domaine en x Notons ( E(H ) := x ∈ ]0, +∞[3 :
x1 > x2 > x3 x1 x2 (x1 − x3 ) ≤ H 3
) (2.14)
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de sorte que N 0 (B) = card (x, d) ∈ F 0 : x ∈ E (Bd1 d2 d3 d4 )1/3 . L’ensemble E est un ensemble homoth´etique puisque o n 1 E(H ) = x ∈ ]0, +∞[3 : x ∈ E(1) . H Nous introduisons la fonction h 1 sur [41/3 , +∞[ d´efinie par q x12 − x14 − 4x1 2 q h 1 (x1 ) = = . 2x1 x 2 + x 4 − 4x 1
1
1
Notons, pour la suite, que l’on a h 1 (x1 ) ≤ min
nx
2o . 2 x12 1
,
(2.15)
Nous d´efinissons les sous-ensembles E i (H ) de E(H ) par n o 1 E i (H ) = x ∈ ]0, +∞[3 : x ∈ E i (1) H et √ E 1 (1) := x ∈ ]0, +∞[3 : x2 < 1, x2 < x1 ≤ 1/ x2 , x3 < x2 , 1 < x1 < 41/3 3 2 E 2 (1) := x ∈ ]0, +∞[ : 1/x1 < x2 < x1 , x1 − 1/x1 x2 ≤ x3 < x2 x1 ≥ 41/3 3 2 E 3 (1) := x ∈ ]0, +∞[ : 1/x1 < x2 < h 1 (x1 ) , (2.16) x1 − 1/x1 x2 ≤ x3 < x2 x1 ≥ 41/3 3 E 4 (1) := x ∈ ]0, +∞[ : x1 − h 1 (x1 ) < x2 < x1 . x1 − 1/x1 x2 ≤ x3 < x2 Il est facile de constater le lemme suivant. 5 Les ensembles E i (H ) forment une partition de E(H ). LEMME
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2 1.8 2
1.6 1.8
1.4 1.6 1.2
1.4 1
1.2 1
0.8
y
z
0.4
x
0.8
2
0.6
0.6 1
0.4
0.2
0.2
0
Figure 1
D´emonstration Il suffit de consid´erer le cas H = 1. On a 1 E(1) = x ∈ ]0, +∞[3 :
x2 < x1 . 1 ≤ x3 < x2 x1 − x1 x2
Lorsque x2 ≤ 1/x12 , on a x1 − 1/x1 x2 ≤ 0 et x2 < 1. Lorsque x1 > x2 > 1/x12 et x1 ∈ ]1, 41/3 [, on a toujours x1 − 1/x1 x2 < x2 . Lorsque x1 > x2 > 1/x12 et x1 ≥ 41/3 , on a x1 − 1/x1 x2 < x2 si, et seulement si, x2 ∈ ]1/x12 , h 1 (x1 )[ et x2 ∈ ] x1 − h 1 (x1 ), x1 [. On en d´eduit le r´esultat annonc´e. Enfin pour illustrer notre d´emarche, nous avons repr´esent´e la surface d´efinie par x ∈ ]0, +∞[3 : max{|Pσ (x)|, σ ∈ S3 } = 1 (voir la Figure 1). Il s’agit de compter les points a` coordonn´ees enti`eres qui satisfont certaines conditions arithm´etiques et qui se trouvent sur les droites joignant l’origine et un point
436
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P = (x1 , x2 , x3 ) tel que l’homoth´etique de ce point P 0 = (x1 /H, x2 /H, x3 /H ) soit sur cette surface. La “chemin´ee de ce volcan” a une direction donn´ee par le vecteur (1, 1, 1). ´ 2.5. Etude du domaine en d Constatons pour commencer que les d compt´es dans la premi`ere ligne de (2.9) sont des e´ l´ements de D1 (B) := d ∈ N4∗ : di d j ≤ B (1 ≤ i < j ≤ 4) . (2.17) Pour montrer cela, on remarque que pour tout (i, j, k) tel que {i, j, k} = {1, 2, 3}, on a di d j | xk , d4 di | (x j − xk ), xk ≤ x1 , |x j − xk | ≤ x1 . Donc, pour tout 1 ≤ i < j ≤ 4, on a di d j ≤ x1 . Par ailleurs, n | x2 (x1 − x3 ). Pour tout 1 ≤ i < j ≤ 4, on a donc ndi d j ≤ x1 x2 (x1 − x3 ) ≤ Bn, ce qui fournit les in´egalit´es recherch´ees. Le Lemme 4 implique le r´esultat suivant. PROPOSITION 1 Soient A > 0 et T = B A/ log log B . Il existe A0 > 0 tel que, pour tout A ≥ A0 , on ait pour B ≥ 3, (log B)4 N 0 (B) = N1 (B) + O A B log log B avec ( ) x ∈ E((Bd1 d2 d3 d4 )1/3 ) N1 (B) := card (x, d) ∈ F : d ∈ D (B)
et D (B) := d ∈ N4 : n i ≤ (Bd1 d2 d3 d4 )1/3 /T (1 ≤ i ≤ 4) . D´emonstration D’apr`es la d´efinition (2.9), nous avons N 0 (B) = card (x, d) ∈ F 0 : x ∈ E (Bd1 d2 d3 d4 )1/3 .
La premi`ere e´ tape consiste a` rajouter la condition d ∈ D (B) qui est e´ quivalente a` n i ≤ H/T pour tout i o`u H = (Bd1 d2 d3 d4 )1/3 .
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LEMME 6 Lorsque (x, d) est un e´ l´ement de F 0 , le sous-ensemble de F d´efini en (2.8), on a n i < x1 pour 1 ≤ i ≤ 4 et les implications
x ∈ E 1 (H ) =⇒ (n 2 ≤ H, n 3 ≤ H, n 4 ≤ H ), x ∈ E 2 (H ) =⇒ (n 1 ≤ 31/3 H, n 2 ≤ H, n 3 ≤ H, n 4 ≤ 41/3 H ), x ∈ E 3 (H ) =⇒ (n 2 ≤ H, n 3 ≤ H, n 4 ≤ H ),
(2.18)
x ∈ E 4 (H ) =⇒ (n 1 ≤ H, n 2 ≤ H, n 3 ≤ H ). Ainsi, ce lemme permet de construire une partition de E(H ) pour laquelle dans chacun de ses e´ l´ements, au moins trois des n i sont born´es en fonction de H . D´emonstration On remarque tout d’abord que tous les membres de gauche du syst`eme (2.8) sont strictement inf´erieurs a` x13 . Cela fournit n i < x1 lorsque i ∈ {1, 2, 3, 4}. Les in´egalit´es concernant n 2 et n 3 se d´eduisent directement de (2.8) puisque n 33 ≤ (x2 − x3 )(x1 − x3 )x3 ≤ (x1 − x3 )x1 x2 ≤ H 3 , n 32
≤ (x2 − x3 )(x1 − x2 )x2 ≤ x2 (x1 − x3 )x1 ≤ H , 3
x ∈ E(H ) .
Lorsque x ∈ E 2 (H ) ∪ E 4 (H ), on obtient n 31 ≤ (x1 − x3 )(x1 − x2 )x1 ≤ (x1 − x3 )x1 x2
x1 − x2 x1 − x2 ≤ H3 . x2 x2
Les in´egalit´es x1 ≤ 41/3 H et x2 ≥ 4−2/3 H valables dans E 2 (H ) et l’in´egalit´e x1 ≤ 2x2 valable dans E 4 (H ) permettent d’´etablir les majorations de n 1 recherch´ees dans E 2 (H ) ∪ E 4 (H ). L’in´egalit´e n 34 ≤ x1 x2 x3 de (2.8) implique dans E 1 (H ), 3/2
n 34 ≤ H 3/2 x2
≤ H 3.
Elle implique dans E 2 (H ) la majoration n 34 ≤ x13 ≤ 4H 3 . Lorsque x ∈ E 3 (H ), nous obtenons de (2.8) et de l’in´egalit´e h(x1 /H ) ≤ 2H 2 /x12 issue de (2.15) n 34 ≤ x1 x2 x3 ≤ x1 x22 ≤ x1 H 2 h 1 (x1 /H )2 ≤ 4H 6 /x13 ≤ H 3 . Cela ach`eve la d´emonstration du Lemme 6.
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Nous majorons la quantit´e Q(H, d) := card x :
x1 > max{n 1 , n 4 }, x ∈ E(H ) di d j | xk , xi ≡ x j (mod d4 ) ∀(i, j, k) tel que {i, j, k} = {1, 2, 3}
,
en vue d’´ecarter les d ∈ / D (B). Le lemme suivant implique la Proposition 1. 7 Soit d ∈ N4∗ tel que (di , d j ) = 1. Lorsque n 3 ≤ H , n 2 ≤ H , on a LEMME
H3 . n2 Soit T ≥ 2. Lorsque n 3 ≤ H , n 2 ≤ H , max{n 1 , n 4 } > H T , on a
(2.19)
Q(H, d)
Q(H, d)
1 H3 . T n2
(2.20)
Il est a` noter que la premi`ere majoration du lemme permet de retrouver la majoration de Salberger en O(B(log B)4 ). D´emonstration Nous notons lorsque 1 ≤ m ≤ 4, Q m (H, d) := card x :
x1 > max{n 1 , n 4 }, x ∈ E m (H ) di d j | xk , xi ≡ x j (mod d4 ) ∀(i, j, k) tel que {i, j, k} = {1, 2, 3}
et Q 1,2,3 (H, d) = Q 1 (H, d) + Q 2 (H, d) + Q 3 (H, d). Les seuls r´esultats utilis´es sont les estimations e´ l´ementaires suivantes : ( card{x ≤ X : x ≡ a (mod q)} ≤ 2X/q (X ≥ q), card{q < x ≤ X : x ≡ a (mod q)} ≤ X/q. Nous commenc¸ons par la majoration de Q 1,2,3 (H, d). Puisque h 1 (x1 ) ≤ 2/x12 , on a √ E 1 (H ) ∪ E 2 (H ) ∪ E 3 (H ) ⊂ x : x2 ≤ 2H, x1 ≤ 2H 3/2 / x2 , x3 ≤ 2H . Lorsque x2 est fix´e, les conditions d2 d3 | x1 et x1 ≡ x2 (mod d4 ) sont e´ quivalentes a` x1 ≡ a1 (mod n 1 ), o`u a1 d´epend de la classe de congruence de x2 modulo d4 . De mˆeme pour x3 , on a une condition de la forme x3 ≡ a3 (mod n 3 ). Il en d´ecoule que X X X 1. (2.21) Q 1,2,3 (H, d) ≤ x2 ≤2H n 1 <x1 ≤2H 3/2 /√x2 x3 ≤2H d1 d3 |x2 x ≡a (mod n ) x3 ≡a3 (mod n 3 ) 1 1 1
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L’in´egalit´e n 3 ≤ H fournit Q 1,2,3 (H, d) ≤ 4
X
X
x2 ≤2H n 1 <x1 ≤2H 3/2 /√x2 d1 d3 |x2 x ≡a (mod n ) 1 1 1
H . n3
(2.22)
La condition n 1 < x1 permet d’affirmer que Q 1,2,3 (H, d) ≤ 4
X 2H 3/2 H H 5/2 X 1 ≤8 √ √ . n 1 x2 n 3 n1n3 x2
x2 ≤2H d1 d3 |x2
(2.23)
x2 ≤2H d1 d3 |x2
On utilise le param`etrage x2 = kd1 d3 et la majoration suivante issue d’une int´egration par parties∗ X x2 ≤2H d1 d3 |x2
1 1 √ =√ x2 d1 d3
X k≤2H/d1 d3
1 √ k
1 [2H/d1 d3 ] =√ + √ 2H/d1 d3 d1 d3
2H/d1 d3
Z 1
√ [t] 2H . dt ≤ 2 3/2 d1 d3 2t
(2.24)
La formule n 1 n 3 d1 d3 = n 2 permet d’obtenir Q 1,2,3 (H, d) H 3 /n 2 lorsque n 3 ≤ H . Ici, l’in´egalit´e n 2 ≤ H n’a pas e´ t´e utilis´ee. Regardons maintenant le cas n 1 > H T . D’apr`es le Lemme 6, toutes les contributions sont nulles sauf peut-ˆetre celle de E 1 et de E 3 . Au vu de (2.15), on a dans E 1 (H ) ∪ E 3 (H ) l’in´egalit´e x2 ≤ 2H 3 /x12 . On d´eduit de x1 > n 1 la condition suppl´ementaire x2 ≤ 2H 3 /n 21 . De (2.5), il vient Q 1,2,3 (H, d) ≤ 4
X
X
√ x2 ≤2H 3 /n 21 n 1 <x1 ≤2H 3/2 / x2 x1 ≡a1 (mod n 1 ) d1 d3 |x2
≤8
H 5/2 n1n3
X x2 ≤2H 3 /n 21 d1 d3 |x2
H n3
√ √ H3 H4 1 ≤ 16 2 2 . √ ≤ 16 2 2 x2 Tn n 1 n 3 d1 d3
(2.25)
Dans le cas n 4 > H T , on a Q 1,2,3 (H, d) = 0 puisque E 1 (H ) = E 2 (H ) = E 3 (H ) = ∅ au vu du Lemme 6. L’inclusion ( ) t2 = x1 − x3 , t3 = x1 − x2 E 4 (H ) ⊂ x : (2.26) √ √ t2 ≤ 2H, x1 ≤ H 3/2 2/ t3 , t3 ≤ 2H ∗ Ici,
[t] d´esigne la partie enti`ere de t.
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fournit Q 4 (H, d) ≤
X
X
X
1,
(2.27)
√ t3 ≤2H n 4 <x1 ≤H 3/2 2/√t3 t2 ≤2H d4 d3 |t3 0 t2 ≡a20 (mod n 3 ) x1 ≡a1 (mod n 4 )
o`u a10 et a20 d´ependent de la classe de congruence de t3 modulo d1 . Les d´etails de la preuve de la majoration Q 4 que nous n’´ecrivons pas sont donc semblables au cas trait´e en (2.24). √ √ Avec la condition suppl´ementaire n 4 > H T , les in´egalit´es x1 ≤ H 3/2 2/ t3 et x1 > n 4 implique t3 ≤ 2H 3 /n 24 . De la mˆeme mani`ere qu’en (2.25) et (2.27), on a X
X
t3 ≤2H 3 /n 24 d4 d3 |t3
√ n 4 <x1 ≤H 3/2 2/ t3 x1 ≡a10 (mod n 4 )
Q 4 (H, d) ≤
≤4
X √
X
X
√ √ t3 ≤2H 3 /n 24 n 4 <x1 ≤H 3/2 2/ t3 0 d4 d3 |t3 x1 ≡a1 (mod n 4 )
√ H 5/2 ≤4 2 n4n3
X t3 ≤2H 3 /n 24 d4 d3 |t3
1
t2 ≤2H t2 ≡a20 (mod n 3 )
H n3
(2.28)
1 H4 H3 ≤ 16 2 . √ ≤ 16 2 t3 Tn n 4 n 3 d4 d3
En rassemblant les majorations de Q 1,2,3 et de Q 4 , on obtient les in´egalit´es recherch´ees. Nous pouvons maintenant comparer N 0 (B) a` ( N10 (B) := card (x, d) ∈ F 0 :
x ∈ E (Bd1 d2 d3 d4 )1/3 d ∈ D (B)
) .
Pour tout e´ l´ement (x, d) de F 0 , on a x1 > max{n 1 , n 4 }. De la remarque pr´eliminaire de la section 2.5, il vient X N 0 (B) = N10 (B) + O Q(H, d) . (2.29) d∈D1 (B)rD (B)
D’apr`es le Lemme 6, on a forc´ement i = 1 ou i = 4 dans l’in´egalit´e n i > H T et n 2 ≤ H , n 3 ≤ H . L’in´egalit´e (2.20) du Lemme 7 s’applique donc. Il en d´ecoule X d∈D1 (B)rD (B) ∃i : n i >H T
Q(H, d)
B(log B)4 . T
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D’autre part, d’apr`es le Lemme 6, on a n 2 ≤ H , n 3 ≤ H , ce qui permet d’utiliser l’in´egalit´e (2.19) du Lemme 7. On obtient X Q(H, d) B(log B)3 log T. d∈N4∗ , d∈D1 (B) ∃i : H/T
En reportant ces majorations dans (2.29), on en d´eduit que (log B)4 N 0 (B) = N10 (B) + O A B . log log B
(2.30)
La deuxi`eme e´ tape de la preuve de la Proposition 1 consiste a` montrer que les (x, d) ∈ F r F 0 ont une contribution n´egligeable dans N1 (B). Tout d’abord, puisque F 0 ⊂ F, on a N10 (B) ≤ N1 (B).
Ensuite, on consid`ere l’ensemble F 00 := (x, d) ∈ F : m(x) ≤ H/T , o`u l’on a pos´e m(x) := min x1 , x2 , x3 , x1 − x2 , x2 − x3 , x1 − x3 . Lorsque d ∈ D (B) et min{x1 , x2 , x3 , x1 − x2 , x2 − x3 , x1 − x3 } > H/T , les quatre in´egalit´es (2.8) d´efinissant F 0 sont v´erifi´ees. Par cons´equent, on a (x, d) ∈ F : d ∈ D (B) r (x, d) ∈ F 0 : d ∈ D (B) ⊂ (x, d) ∈ F 00 : d ∈ D (B) . Il est facile de majorer la contribution des (x, d) ∈ F 00 . Notons tout d’abord que lorsque x1 > x2 > x3 > 0, on a m(x) = min{x3 , x1 − x2 , x2 − x3 }. Nous adaptons la d´emonstration du Lemme 7 aux nouvelles conditions. Au vu de (2.5) et (2.26), il suffit de majorer les contributions de la r´eunion d’ensemble E 1 (H ) ∪ E 2 (H ) ∪ E 3 (H ) lorsque l’on a l’une des deux conditions suppl´ementaires min{x3 , x2 − x3 } ≤ H/T , x1 − x2 ≤ H/T (dont la r´eunion contient bien la condition m(x) ≤ H/T ) √ x2 ≤ 2H, x1 ≤ 2H 3/2 / x2 , x3 ≤ 2H 0 , Q 1 (H, d) := card x : x1 > max{n 1 , n 4 }, min{x3 , x2 − x3 } ≤ H/T di d j | xk , xi ≡ x j (mod d4 ), ({i, j, k} = {1, 2, 3}) 3/2 √ x ≤ 2H, x ≤ 2H / x , x ≤ 2H 2 1 2 3 Q 02 (H, d) := card x : x1 > max{n 1 , n 4 }, x1 − x2 ≤ H/T , di d j | xk , xi ≡ x j (mod d4 ), ({i, j, k} = {1, 2, 3})
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et celle de l’ensemble E 4 (H ) lorsque l’on a l’une des deux conditions suppl´ementaires min{x2 − x3 , x1 − x2 } ≤ H/T , x3 ≤ H/T , Q 001 (H, d) := √ t3 = x2 − x3 ≤ 2H, x1 ≤ 2H 3/2 / t3 , t2 = x1 − x3 ≤ 2H card x : x1 > max{n 1 , n 4 }, min{t3 , t2 − t3 } ≤ H/T di d j | xk , xi ≡ x j (mod d4 ), ({i, j, k} = {1, 2, 3})
Q 002 (H, d) := √ t3 = x2 − x3 ≤ 2H, x1 ≤ 2H 3/2 / t3 , t2 = x1 − x3 ≤ 2H card x : x1 > max{n 1 , n 4 }, x1 − t2 ≤ H/T di d j | xk , xi ≡ x j (mod d4 ), ({i, j, k} = {1, 2, 3})
,
.
On se contente de donner les d´etails des majorations de Q 01 (H, d) et Q 02 (H, d). Plagiant (2.21) et (2.22), on obtient lorsque d ∈ D (B), X X X X 1+ 1 Q 01 (H, d) ≤ x2 ≤2H n 1 <x1 ≤2H 3/2 /√x2 d1 d3 |x2 x ≡a (mod n ) 1 1 1
X
X
x2 ≤2H n 1 <x1 ≤2H 3/2 /√x2 d1 d3 |x2 x ≡a (mod n ) 1 1 1
x3 ≤H/T x3 ≡a3 (mod n 3 )
x2 −H/T ≤x3 ≤x2 x3 ≡a3 (mod n 3 )
H H3 T n3 T n2
(2.31)
et Q 02 (H, d) ≤
X
X
X
1
x2 ≤2H x2 ≤x1 ≤x2 +H/T x3 ≤2H d1 d3 |x2 x1 ≡a1 (mod n 1 ) x3 ≡a3 (mod n 3 )
X x2 ≤2H d1 d3 |x2
H2 H3 . T n3n2 T n2
(2.32)
Ici, la restriction a` d ∈ D (B) c’est-`a-dire a` n i ≤ H/T pour tout 1 ≤ i ≤ 4 permet de conclure tr`es facilement. On en d´eduit tout d’abord N10 (B) = N1 (B) + O B(log B)4 /T , puis la relation de la Proposition 1 grˆace a` (2.30).
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3. Estimation de N1 (B) Pour e´ tablir le Th´eor`eme 1, il suffit de montrer le r´esultat suivant. PROPOSITION 2 On a lorsque B ≥ 3, l’estimation
n N1 (B) = B(log B)4 Vol E(1)
log log B o C0 +O . 10 × 4! log B
(3.1)
On constate que la Proposition 2 coupl´ee avec la Proposition 1 et le Lemme 4 implique le Th´eor`eme 1. En effet, un calcul de volume fournit Vol(E(1)) = π 2 /18. Le changement de variables u 1 = x13 , u 2 = x2 /x1 , u 3 = x3 /x1 fournit 1 Vol E(1) = 3
1 Z u2
Z 0
0
du 3 du 2 1 = u 2 (1 − u 3 ) 3
1
Z
− log(1 − u 2 )
0
du 2 π2 = . u2 18
(3.2)
On en d´eduit le Th´eor`eme 1 grˆace a` la Proposition 1 et le Lemme 4. 3.1. Inversion de M¨obius On a N1 (B) =
X
C (Bd1 d2 d3 d4 )1/3 , d ,
(3.3)
d∈D ∗ (B)
o`u l’on a pos´e D ∗ (B) := d ∈ D (B) : (di , d j ) = 1
(3.4)
et C(H, d) := card x ∈ N3 ∩ E(H ) :
(x2 , x3 ) = d1 , (x1 , x3 ) = d2 , . (x1 , x2 ) = d3 , (x1 − x2 , x1 − x3 ) = d4
(3.5)
Une inversion de M¨obius fournit C(H, d) =
4 XY δ∈N4∗
i=1
µ(δi )C 0 (H, d, δ)
(3.6)
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avec∗ C 0 (H, d, δ) := card x ∈ N3 ∩ E(H ) :
[δi di , δ j d j ] | xk ,
∀(i, j, k) tel que {i, j, k} = {1, 2, 3} . (3.7) x2 ≡ x1 (mod δ4 d4 ) x3 ≡ x1 (mod δ4 d4 )
Nous utiliserons syst´ematiquement dans la suite pour tout (i, j, k, `) tel que {i, j, k, `} = {1, 2, 3, 4} les notations n := d1 d2 d3 d4 , D` := d` δ` ,
H := (Bd1 d2 d3 d4 )1/3 ,
n 0 := δ1 δ2 δ3 δ4 ,
D := (D1 , D2 , D3 , D4 ) = (δ1 d1 , δ2 d2 , δ3 d3 , δ4 d4 ),
Mi, j := [Di , D j , (Dk , D` )],
(3.8)
Ni := [D j , Dk , D` ].
Une analyse des conditions de congruence fournit M2,3 | x1 , M1,3 | x2 M1,2 | x3 C 0 (H, d, δ) := card x ∈ N3 ∩ E(H ) : x2 ≡ x1 (mod D4 ) x3 ≡ x1 (mod D4 )
.
(3.9)
La condition [D3 , D2 , (D1 , D4 )] | x1 , par exemple, d´ecoule de l’implication D4 | x1 − x2 , D1 | x2 =⇒ (D1 , D4 ) | x1 . Les conditions (3.9) sont des congruences pour x1 modulo M2,3 , pour x2 modulo ´ N2 (x1 e´ tant fix´e), pour x3 modulo N3 (x1 e´ tant fix´e). Etant donn´e que E(H ) est un 0 domaine homoth´etique, le terme principal attendu de C (H, d, δ est donc H3 f (D) Vol(E(H )) = Vol E(1) = Vol E(1) H 3 2 2 2 2 , M2,3 N2 N3 M2,3 N2 N3 D1 D2 D3 D4 o`u nous d´esignons par f la fonction d´efinie par f (D) =
(i 6 = j).†
la suite, le ppcm de deux entiers m 1 et m 2 sera not´e [m 1 , m 2 ]. remarquera que le produit Mi, j Ni N j ne d´epend pas du choix de (i, j).
∗ Dans † On
D12 D22 D32 D42 Mi, j Ni N j
(3.10)
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On calcule les plus petits communs multiples d´efinis en (3.8) en fonction des plus grands communs diviseurs. De [Di , D j ](Dk , D` ) ([Di , D j ], (Dk , D` )) Di D j (Dk , D` ) (Di , D j , Dk , D` ) = , (3.11) (D j , Di )(D j , Dk , D` )(Di , Dk , D` ) [D j , Dk ]D` D j Dk D` (D j , Dk , D` ) Ni = [D j , Dk , D` ] = = , ([D j , Dk ], D` ) (D j , Dk )(D j , D` )(D` , Dk )
Mi, j = [Di , D j ], (Dk , D` ) =
on d´eduit que f (D) =
(D1 , D3 )(D1 , D2 )(D3 , D2 )(D1 , D4 )(D2 , D4 )(D3 , D4 ) . (D1 , D2 , D3 , D4 )
3.2. D´emonstration de la Proposition 2 Notons √ 1(T ) := δ ∈ N4∗ : n 0 = δ1 δ2 δ3 δ4 ≤ T . Nous d´emontrons que la Proposition 2 est impliqu´ee par les r´esultats suivants que nous d´emontrerons aux sections suivantes. Nous traitons s´epar´ement le cas δ ∈ 1(T ) et le cas δ ∈ / 1(T ). PROPOSITION 3 Soit T la quantit´e d´efinie dans l’´enonc´e de la Proposition 1. Lorsque d ∈ D (B) et δ ∈ 1(T ), on a pour B ≥ 3, 1 f (D) C 0 (H, d, δ) = H 3 Vol E(1) + O . (3.12) √ (D1 D2 D3 D4 )2 T
Soit S(B, T ) :=
X
X
C 0 (H, d, δ),
/ ) d∈D ∗ (B) δ ∈1(T
o`u le cardinal C 0 est d´efini en (3.7). Nous montrerons a` la section 5 le r´esultat suivant. 4 On a pour B ≥ 3 et T ≥ 1, PROPOSITION
S(B, T ) B
B O(1/ log log B) , T 1/16
o`u S(B, T ) est la quantit´e d´efinie en (3.13).
(3.13)
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En reportant les estimations des Propositions 3 et 4 dans les formules (3.3) et (3.6), nous obtenons 4 X Y
X
N1 (B) =
µ(δi )C 0 (Bd1 d2 d3 d4 )1/3 , d, δ + O S(B, T )
d∈D ∗ (B) δ∈1(T ) i=1 4 X Y f (D) B µ(δi ) 2 2 2 2 d1 d2 d3 d4 δ1 δ2 δ3 δ4 δ∈1(T ) i=1 d∈D ∗ (B)
X = Vol E(1) B +O √ T
X d∈D ∗ (B)
(3.14)
4 X Y 1 f (D) µ(δi )2 2 2 2 2 + S(B, T ) . d1 d2 d3 d4 δ1 δ2 δ3 δ4 δ∈1(T ) i=1
On aura besoin du lemme suivant. LEMME
8
Posant X
R(t, σ ) :=
d∈N4∗ (di ,d j )=1
4 XY 1 f (D) µ(δi )2 , 1+σ (d1 d2 d3 d4 ) (δ1 δ2 δ3 δ4 )2−t 4 δ∈N∗ i=1
on a uniform´ement lorsque σ ∈]0, 1] et t ∈ [0, 1/4], R(t, σ ) σ −4 .
(3.15)
D´emonstration La quantit´e s(d) :=
4 XY
µ(δi )2
δ∈N4∗ i=1
f (D) (δ1 δ2 δ3 δ4 )2−t
est la somme d’une fonction multiplicative de quatre variables δ1 , δ2 , δ3 , et δ4 , disons h. Elle s’´ecrit donc sous la forme d’un produit eul´erien. Il vient s(d) =
1 X 1 X 1 X 1 Y X p
h( p ν1 , p ν2 , p ν3 , p ν4 ) .
ν1 =0 ν2 =0 ν3 =0 ν4 =0
Il reste a` calculer les valeurs de h intervenant dans ce produit. Si p - n, on a 1 si ν1 + ν2 + ν3 + ν4 = 0, −(2−t) si ν1 + ν2 + ν3 + ν4 = 1, p h( p ν1 , p ν2 , p ν3 , p ν4 ) = p −(3−2t) ≤ p −(2−t) si ν1 + ν2 + ν3 + ν4 = 2, p −(3−3t) ≤ p −(2−t) si ν1 + ν2 + ν3 + ν4 = 3, p −(3−4t) ≤ p −(2−t) si ν + ν + ν + ν = 4. 1
2
3
4
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´ Etant donn´e que la valeur de h est invariante par permutation, nous ne calculons les autres valeurs h( p ν1 , p ν2 , p ν3 , p ν4 ) que pour p | d1 . Si p | d1 et ν1 = 1, on a p −(2−t) ≤ p −(1−3t) si ν2 + ν3 + ν4 = 0, p −(3−2t) ≤ p −(1−3t) si ν + ν + ν = 1, 2 3 4 h( p ν1 , p ν2 , p ν3 , p ν4 ) = −(3−3t) −(1−3t) p ≤p si ν2 + ν3 + ν4 = 2, −(3−4t) −(1−3t) p ≤p si ν2 + ν3 + ν4 = 3. Si p | d1 et ν1 = 0, on a 1 p −(1−t) ≤ p −(1−3t) h( p ν1 , p ν2 , p ν3 , p ν4 ) = p −(1−2t) ≤ p −(1−3t) −(1−3t) p
si ν2 + ν3 + ν4 = 0, si ν2 + ν3 + ν4 = 1, si ν2 + ν3 + ν4 = 2, si ν2 + ν3 + ν4 = 3.
De ces calculs, il d´ecoule Y Y 15 15 15 Y s(d) ≤ 1 + 2−t 1 + 1−3t 1 + 1/4 . p p p p|n
p-n
p|n
En reportant cette majoration dans la somme d´efinissant R(t, σ ), on obtient X ∞ 1 Y 15 4 R(t, σ ) 1 + 1/4 ζ (1 + σ )4 σ −4 . 1+σ p d p|d d =1 1 1
1
D’une part, en prenant t = 0 et σ = 1/ log B dans le Lemme 8, on obtient X d∈D ∗ (B)
4 XY 1 f (D) µ(δi )2 2 2 2 2 R(0, 1/ log B) (log B)4 . (3.16) d1 d2 d3 d4 δ1 δ2 δ3 δ4 4 δ∈N∗ i=1
D’autre part, en prenant t = 1/4 et σ = 1/ log B, il vient X d∈D ∗ (B)
4 1 X Y 1 f (D) 1 µ(δi )2 2 2 2 2 1/8 R , 1/ log B d1 d2 d3 d4 4 T δ1 δ2 δ3 δ4 δ ∈1(T / ) i=1
(log B)4 . T 1/8
(3.17)
De (3.14), nous en d´eduisons N1 (B) = Vol E(1) B
X d∈D ∗ (B)
B 1+O(1/ log log B) g1 (d) +O , d1 d2 d3 d4 T 1/16
(3.18)
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o`u l’on a pos´e g1 (d) :=
4 XY δ∈N4∗
µ(δi )
i=1
f (D) δ12 δ22 δ32 δ42
.
Lorsque (di , d j ) = 1 (1 ≤ i < j ≤ 4), un simple calcul fournit Y 4 3 Y 1 1 1 g1 (d) = 1− 2 + 3 1− − 2 + 3 , p p p p p
(3.19)
p|n
p-n
o`u l’on rappelle la notation n = d1 d2 d3 d4 . On d´efinit sur l’ensemble des fonctions arithm´etiques de N4∗ une op´eration convol´ee de la mani`ere suivante : h 3 (d) = (h 1 ∗ h 2 )(d) d d d d X X X X 1 2 3 4 h 1 (m 1 , m 2 , m 3 , m 4 )h 2 = , , , . m1 m2 m3 m4 m 1 |d1 m 2 |d2 m 3 |d3 m 4 |d4
On a la relation H3 (s) = H1 (s)H2 (s) o`u Hi (s) sont les s´eries de Dirichlet associ´ees X h i (m) Hi (s) := s1 s2 s3 s4 . m m 1 2 m3 m4 4 m∈N∗
Nous notons χ la fonction caract´eristique des quatre-uplets d tels que (di , d j ) = 1. Nous d´esignons par g2 la fonction arithm´etique d´efinie par (1 ∗ g2 )(d) = g1 (d)χ(d),
(3.20)
o`u 1 est la fonction constante un sur N4∗ . Cette relation s’´ecrit en terme de s´eries de Dirichlet X g1 (d)χ(d) X 1 g2 (m) G 2 (s) := . (3.21) s s1 s2 s3 s4 = ζ (s1 )ζ (s2 )ζ (s3 )ζ (s4 ) m1 m2 m3 m4 d s1 d s2 d 3 d s4 4 4 1 2 3 4 d∈N∗
m∈N∗
L’introduction de la fonction auxiliaire g2 est une m´ethode classique en th´eorie analytique des nombres. Ainsi, en dimension 1, lorsqu’on a a` estimer asymptotiquement P une somme de la forme d≤B g(d)/d o`u g est une fonction multiplicative telle que les valeurs g( p) sont proches de 1, on introduit la fonction g ∗µ qui a des valeurs en p P proches de z´ero et on compare la somme e´ tudi´ee a` d≤B 1/d plus facile a` estimer. Pour cela, nous aurons besoin du lemme calculatoire suivant. LEMME
9
On a G 2 (1, 1, 1, 1) = C0 ,
X m∈N4∗
log(m 1 m 2 m 3 m 4 )
|g2 (m)| < ∞. m1 m2 m3 m4
(3.22)
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D´emonstration La formule (3.19) montre que g1 ne d´epend que de n = d1 d2 d3 d4 . De plus, lorsque les di sont premiers deux a` deux, on a pour tout p | n un i = i p tel que v p (di ) = v p (n), o`u v p d´esigne la valuation en p. On d´eduit alors de (3.19) que G(s) :=
X g1 (d)χ(d) (d1 d2 d3 d4 )s 4
d∈N∗
=
∞ Y X 4 3 Y 1 1 4ω(n) 1 1− − 2 + 3 1− 2 + 3 p ns p p p p p|n
n=1 p-n
Y n 1 4 1 3 1 2 1 o 1− = 1+ − 2 + s 1− 1+ , p p p −1 p p p p o`u ω(n) d´esigne le nombre de facteurs premiers distincts de n. De (3.21), il vient G 2 (1, 1, 1, 1) =
Y n 1 5 1 3 4 4 o 1− = C0 . 1+ − 2 + + 2 p p p p p p
Pour e´ tablir la convergence de la s´erie du Lemme 9, remarquons qu’il suffit d’´etablir que la s´erie X |g2 (m)| G 3 (t) := (m 1 m 2 m 3 m 4 )1+t 4 m∈N∗
est born´ee uniform´ement lorsque |t| ≤ 1/3. En effet, la formule de Cauchy donnant la d´eriv´ee de la fonction analytique G 3 en fonction de la moyenne de G 3 permet de conclure. Pour faciliter le calcul, on a introduit les notations C1 :=
Y 4 3 1− 2 + 3 , p p p
αp =
1 − 1/ p 2 . 1 + 1/ p − 3/ p 2
Un calcul du d´eveloppement eul´erien de G 2 (s) fournit 4
G 2 (s) =
Y X C1 1 1 + αp ζ (s1 )ζ (s2 )ζ (s3 )ζ (s4 ) p p si − 1 i=1
= C1
Yn p
1 + (α p − 1)
4 X
p −si +
i=1
+ p −s1 −s2 −s3 −s4 + α p
X
p −si −s j −
i< j 4 X i=1
p −si
X
p −si −s j −sk
i< j
Y j6 =i
o .
(1 − p −s j ) − 1
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Pour tout p, on a α p ≤ 1. Il vient X |g2 (m)| s
m∈N4∗
m s11 m s22 m 33 m s44 = C1
Yn
1 + (1 − α p )
p
4 X
p −si +
X
p −si −s j +
i< j
i=1
+ p −s1 −s2 −s3 −s4 + α p
4 X
p −si
X
p −si −s j −sk
i< j
Y
o (1 + p −s j ) − 1 .
j6 =i
i=1
On en d´eduit que, lorsque |t| ≤ 1/3, on a Y G 3 (t) ≤ 1 + 4(1 − α p ) p −2/3 + 6 p −4/3 + 4 p −2 + p −8/3 + 28α p p −4/3 1, p
o`u l’on a utilis´e que (1 − α p ) 1/ p. Nous sommes maintenant en mesure d’estimer la somme du membre de droite de (3.18). La relation de convolution (3.20) fournit X X X g1 (d) g2 (m) 1 = . d d d d m m m m d d 1 2 3 4 1 2 3 4 1 2 d3 d4 ∗ 4 4 d∈D (B)
m∈N∗
d∈N∗ (m 1 d1 ,m 2 ,d2 ,m 3 d3 ,m 4 d4 )∈D (B)
(3.23) Notant D (B1 , B2 , B3 , B4 ) := d ∈ N4∗ : (d1 d2 d3 d4 )2 /di3 ≤ Bi /T 3 (∀i) , on a X d∈N4∗ (m 1 d1 ,m 2 d2 ,m 3 d3 ,m 4 d4 )∈D (B)
1 = d1 d2 d3 d4
X d∈D (B1 ,B2 ,B3 ,B4 )
1 d1 d2 d3 d4
avec Bi := Bm i3 /(m 1 m 2 m 3 m 4 )2 . En reportant la formule X X 1 1 = d1 d2 d3 d4 d1 d2 d3 d4 d∈D (B1 ,B2 ,B3 ,B4 )
d∈D (B)
+ O (log B)3 max{| log(B/Bi )| + 1} dans (3.23), on en d´eduit que X X X g1 (d) g2 (m) 1 = + O (log B)3 d1 d2 d3 d4 m1m2m3m4 d1 d2 d3 d4 ∗ 4 d∈D (B)
m∈N∗
= C0
X d∈D (B)
d∈D (B)
1 + O (log B)3 , d1 d2 d3 d4
(3.24)
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o`u l’on a utilis´e (3.22). Enfin, une m´ethode classique de comparaison de somme et d’int´egrale (voir par exemple [11, pages 393 et 394]) fournit Z Z Z Z X 1 du 1 du 2 du 3 du 4 = + O (log B)3 , d1 d2 d3 d4 u1u2u3u4 d∈D (B)
o`u le domaine d’int´egration en u est u ∈ [1, +∞[4 : (u 1 u 2 u 3 u 4 )2 /u i3 ≤ B/T 3 (∀i) . Un changement de variables u i = (B/T 3 )ti implique alors Z Z Z Z 4 du 1 du 2 du 3 du 4 = log(B/T 3 ) Vol(P ), u1u2u3u4 o`u l’on a pos´e P := t ∈ [0, ∞[4 : 2t1 + 2t2 + 2t3 + 2t4 − 3ti ≤ 1 (∀i) . De (3.18) et (3.24), on en d´eduit que n N1 (B) = Vol E(1) Vol(P ) C0 B(log B)4 1 + O
o 1 , log log B
o`u l’on utilis´e la relation log T = A log B/ log log B = O(log B/ log log B). Il reste a` montrer la relation 1 . (3.25) Vol(P ) = 10 × 4! En r`egle g´en´erale, ce type de calcul peut se r´ev´eler assez d´elicat. Ici, de simples manipulations aboutissent. Grˆace a` la sym´etrie des variables, on obtient Vol(P ) = 24 Vol t ∈ [0, +∞[4 : t4 ≤ t3 ≤ t2 ≤ t1 , 2t1 + 2t2 + 2t3 − t4 ≤ 1 Z 1/5 Z (1+t4 )/6 Z (1−2t3 +t4 )/4 Z (1−2t2 −2t3 +t4 )/2 = 24 dt1 dt2 dt3 dt4 0
t4
t3
t2
1 = . 10 × 4! On obtient donc bien (3.25). Pour prouver le Th´eor`eme 1, il reste donc a` montrer les Propositions 3 et 4. 4. Estimation de C 0 pour les δ ∈ 1(T ) : d´emonstration de la Proposition 3 On dira que E est un ensemble de type I s’il existe des r´eels λ1 , λ2 , et des fonctions g1 , g2 , f 1 , f 2 tels que 0 ≤ λ1 < x2 < λ2 3 E := x ∈ R+ : 0 ≤ g1 (x2 ) < x1 < g2 (x2 ) , (4.1) 0 ≤ f 1 (x1 , x2 ) < x3 < f 2 (x1 , x2 )
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o`u les trois conditions suivantes sont satisfaites : (i) la fonction f 3 := f 2 − f 1 est une fonction monotone de la variable x1 et born´ee; (ii) la fonction g3 d´efinie par Z g2 (x2 ) g3 (x2 ) = f 3 (t, x2 ) dt g1 (x2 )
est une fonction monotone et born´ee; √ il existe K > 0 tel que g2 (x2 ) ≤ K / x2 lorsque x2 ∈ ]λ1 , λ2 ]. De plus, par extension,∗ on dira que E est aussi de type I lorsque son image par l’application (x1 , x2 , x3 ) → (x1 , t3 = x1 − x2 , x3 ) ou par permutation des indices est de type I ou lorsqu’on remplace dans la d´efinition (4.1) de E certaines in´egalit´es strictes par des in´egalit´es larges. On a le r´esultat suivant. (iii)
LEMME 10 Tous les ensembles E i (1) sont des r´eunions finies d’ensembles de type I.
D´emonstration L’´ecriture √ E 1 (1) = x ∈ ]0, +∞[3 : x2 < 1, x2 < x1 ≤ 1/ x2 , x3 < x2 ,
E 2 (1) =
E 3 (1) =
E 4 (1) =
x ∈ ]0, +∞[3 :
1/x12 < x2 < x1
x1 − 1/x1 x2 ≤ x3 < x2
x2 ≤ 2−1/3
∗ Cette
1 < x1 < 41/3
x ∈ ]0, +∞[3 :
1 √ < x1 < x2
x2 +
q
,
x22 + 4/x2 2
x1 − 1/x1 x2 ≤ x3 < x2
x ∈ ]0, +∞[3 :
t3 = x1 − x2 , t3 ≤ 2−1/3 q t3 + t32 + 4/t3 41/3 ≤ x1 < 2 1 x1 − ≤ x 3 < x 1 − t3 x1 (x1 − t3 )
,
extension est n´ecessaire pour affirmer que E 4 est une r´eunion finie d’ensemble de type I.
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permet de v´erifier la condition (iii) et le fait que f 3 et g3 sont des fonctions born´ees. Les fonctions f 3 associ´ees aux ensembles E 1 (1), E 3 (1), et E 4 (1) sont respectivement e´ gales a` f 3 (x1 , x2 ) = x2 ,
f 3 (x1 , x2 ) = x2 −x1 +
1 , x1 x2
f 3 (x1 , t3 ) =
1 −t3 . (x1 − t3 )x1
Elles sont toutes monotones par rapport a` leur premi`ere variable dans les domaines consid´er´es. Les fonctions g3 associ´ees sont respectivement e´ gales a` √ g3 (x2 ) = x2 − x22 , q 3/2 q x + x23 + 4 √ 1 2 1 2 g3 (x2 ) = x2 + x22 + 4/x2 − 4 x2 + log , 4 x2 2 1 28/3 q g3 (t3 ) = log 2 t3 t (41/3 − t ) t + t 2 + 4/t 3
3
3
3
3
q t3 − t3 + t32 + 4/t3 − 25/3 . 2 La fonction g3 relative a` E 1 (1) est croissante sur ]0, 2−4/3 ] et d´ecroissante sur ]2−4/3 , 1], ce qui permet d’affirmer que E 1 (1) est une r´eunion de deux sousensembles de type I. Le fait que les fonctions g3 associ´ees a` E 3 (1) et E 4 (1) soient la restriction sur ]0, 2−1/3 [ de fonctions analytiques sur ]0, 2−1/3 ] implique que leur d´eriv´ee change un nombre fini de fois de signe dans les intervalles consid´er´es sauf peut-ˆetre au voisinage de z´ero. Il reste pour affirmer que les ensembles E 3 (1) et E 4 (1) sont des r´eunions finies de sous-ensembles de type I a` e´ tudier le voisinage de z´ero. √ Pour ce faire, nous posons z 2 = x2 et nous consid´erons q r 3 + z6 + 4 z 2 1 1 1 4 2 2 z2 + 1 + z 26 − 4z 2 + 2 log ge3 (z 2 ) = 4 z2 4 2 z2 de sorte que pour z 2 ∈ R+ on ait g3 (x2 ) = g3 (z 22 ) = ge3 (z 2 ) et g30 (z 22 ) = ge3 0 (z 2 )/2z 2 . La fonction he3 d´efinie par 1 he3 (z 2 ) = ge3 (z 2 ) − 2z 2 est analytique en z´ero. La d´eriv´ee de ge3 s’´ecrit sous la forme 0
ge3 0 (z 2 ) =
2z 22 he3 (z 2 ) − 1 2z 22
et admet un nombre fini de z´ero au voisinage de z´ero. On en d´eduit le r´esultat recherch´e concernant E 3 (1).
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Pour le cas de l’ensemble E 4 (1), nous e´ tudions la fonction q 1 4 28/3 1 6 + 4 − 25/3 . q − z + z z ge3 (z 3 ) = 2 log 3 3 3 2 2 z3 (41/3 − z 32 ) z 33 + z 36 + 4 On a pour z 3 ∈ R+ et t3 = z 32 les relations g3 (t3 ) = g3 (z 32 ) = ge3 (z 3 ) et g30 (z 22 ) = ge3 0 (z 2 )/2z 2 . Cette fonction ge3 est analytique au voisinage de z´ero et donc sa d´eriv´ee admet un nombre fini de z´ero au voisinage de z´ero. On en d´eduit le r´esultat recherch´e concernant E 4 (1). Nous d´etaillons le d´ecoupage explicite de l’ensemble E 2 (1) en r´eunion finie de sous-ensembles de type I. Soient β l’unique racine dans ]1, 41/3 [ de l’´equation 9/2
4 = x1
+ 3x13 log x1
et β 0 l’unique racine dans ]1, 41/3 [ de l’´equation 3/2
3 log x1 = 4 + x1
− 2x13 .
On introduit la partition de E 2 (1) suivante : 1 < x1 ≤ β √ E 2,1 (1) = x ∈ ]0, +∞[3 : 1/x12 < x2 ≤ 1/ x1 x1 − 1/x1 x2 ≤ x3 < β < x1 < 41/3 √ E 2,2 (1) = x ∈ ]0, +∞[3 : 1/x12 < x2 ≤ 1/ x1 x1 − 1/x1 x2 ≤ x3 < 1 < x1 ≤ β 0 √ E 2,3 (1) = x ∈ ]0, +∞[3 : 1/ x1 < x2 < x1 x1 − 1/x1 x2 ≤ x3 < β 0 < x1 < 41/3 √ E 2,4 (1) = x ∈ ]0, +∞[3 : 1/ x1 < x2 < x1 x1 − 1/x1 x2 ≤ x3 <
x2
x2
,
x2
,
x2
,
.
Pour ces quatre domaines, la fonction f 3 associ´ee est d´efinie par f 3 (x2 , x1 ) = x2 − x1 +
1 . x1 x2
√ Elle est d´ecroissante par rapport a` la variable x2 lorsque 1/x12 < x2 ≤ 1/ x1 et √ croissante par rapport a` la variable x2 lorsque 1/ x1 < x2 < x1 . La fonction g3
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associ´ee aux ensembles E 2,1 (1) et E 2,2 (1) est d´efinie par g3 (x1 ) =
√ 1 3 log x1 3 − 4 − x1 + 2x1 2x1 2x1
de d´eriv´ee
9/2
g30 (x1 )
=
4 − 3x13 log x1 − x1 2x15
.
La fonction g3 est donc croissante sur ]1, β] et d´ecroissante [β, 41/3 [. La fonction g3 associ´ee aux ensembles E 2,3 (1) et E 2,4 (1) est d´efinie par 3/2
g3 (x1 ) := de d´eriv´ee g30 (x1 ) =
3 log x1 − (x1 2x1
− 1)2
1 3/2 (−3 log x1 + 4 + x1 − 2x13 ). 2x12
La fonction g3 est croissante sur ]1, β 0 [ et d´ecroissante sur ]β 0 , 41/3 [. L’ensemble E 2 (1) est donc bien une r´eunion finie de quatre ensembles de type I. On va utiliser a` de nombreuses occasions le r´esultat classique suivant. LEMME 11 Soit h une fonction r´eelle monotone sur l’intervalle [b1 , b2 ] avec b1 , b2 ∈ R. Soient a et q des entiers positifs. Alors on a Z X 1 b2 h(t) dt + O |h(b1 )| + |h(b2 )| . h(x) = q b1 b1 <x
Le r´esultat est le mˆeme lorsque que l’on change certaines in´egalit´es strictes sur l’indice de sommation x en in´egalit´es larges. Soit E un sous-ensemble de R3 . Estimons C E0 (H, d, δ) := card x ∈ N3 ∩ E(H ) :
[δ2 d2 , δ3 , d3 ] | x1 , [δ1 d1 , δ3 , d3 ] | x2 [δ1 d1 , δ2 d2 | x3 . (4.2) x2 ≡ x1 (mod δ4 d4 ), x3 ≡ x1 (mod δ4 d4 ),
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LEMME 12 Lorsque E est un ensemble de type I, lorsque d ∈ D (B) et δ ∈ 1(T ), B ≥ 3, T ≥ 1, et H = (Bd1 d2 d3 d4 )1/3 , on a l’estimation 1 f (D) Vol(E) + O . (4.3) C E0 (H, d, δ) = H 3 √ (D1 D2 D3 D4 )2 T
Ce r´esultat est aussi valable pour E(1) puisqu’il est une partition finie d’ensembles de type I. Il permet donc d’´etablir la Proposition 3. D´emonstration √ On a l’in´egalit´e Ni ≤ n i n 0 ≤ H/ T pour tout i. La d´efinition de C E0 (H, d, δ) fournit X X X 1, C E0 (H, d, δ) = M1,3 |x2 g1 (x2 /H )<x1 /H
o`u a2 et a3 d´ependent de la congruence modulo D4 de x2 . Une premi`ere application du Lemme 11 avec h(x) = 1 fournit C E0 (H, d, δ) X =
X
H x1 x2 f3 , + O(1) N3 H H
H H x1 x2 f3 , +O √ . N3 H H T N3
M1,3 |x2 g1 (x2 /H )<x1 /H
X
=
X
M1,3 |x2 g1 (x2 /H )<x1 /H
√ Pour majorer le terme r´esiduel, disons R1 , on utilise la majoration g2 (x2 ) ≤ K / x2 . Il vient R1 √
√
H T N3 H T N3
X
X
√ M1,3 |x2 0<x1 ≤K H 3/2 / x2 λ1 <x2 /H <λ2 x1 ≡a1 (mod N1 )
1≤ √
H T N3
X K H 3/2 √ +1 N1 x 2
M1,3 |x2 x2 <λ2 H
H2 H H3 + √ , N1 M1,3 M1,3 T N1 M1,3 N3
o`u l’on a utilis´e une majoration semblable a` (2.24). Une deuxi`eme application du Lemme 11 avec h(x) = f 3 (x/H, x2 /H ) permet
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d’en d´eduire que C E0 (H, d, δ)
H = N3
=
X
M1,3 |x2 λ1 <x2 /H <λ2
H2 N1 N3
H H x2 H3 g3 +O √ +O √ N1 H T N1 T M1,3 N1 N3
X
g3
x
M1,3 |x2 λ1 <x2 /H <λ2
2
H
+O √
H3
T M1,3 N1 N3
.
Une derni`ere application du Lemme 11 avec h(x) = g3 (x/H ) fournit enfin C E0 (H, d, δ) =
H3 H3 Vol(E) + O √ , N1 N3 M1,3 T M1,3 N1 N3
Rλ o`u on a utilis´e que λ12 g3 (t) dt = Vol(E). La d´efinition (3.10) permet de clore la d´emonstration du Lemme 12 et de la Proposition 3.
5. Majoration de la contribution des δ 6 ∈ 1(T ) : d´emonstration de la Proposition 4 5.1. Pr´eparation des variables et r´eduction du probl`eme Au vu de (3.9), on peut e´ crire C (H, d, δ) = card x ∈ N3 ∩ E(H ) : 0
Mi, j | xk Mi,4 | x j − xk
.
(5.1)
Posant yi =
xi , M j,k
z1 =
x2 − x3 , M1,4
z2 =
x1 − x3 , M2,4
z3 =
x1 − x2 , M3,4
on en d´eduit que C 0 compte le nombre de six-uplets (y1 , y2 , y3 , z 1 , z 2 , z 3 ) a` coordonn´ees dans N∗ tels que y1 y2 z 2 M2,3 M1,3 M2,4 ≤ Bn
(5.2)
M1,3 y2 − M1,2 y3 = M1,4 z 1 , M2,3 y1 − M1,2 y3 = M2,4 z 2 , M y − M y = M z . 2,3 1 1,3 2 3,4 3
(5.3)
et tels que
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On d´efinit les entiers Di0 par les relations Di = Di0
(Di , D j )(Di , Dk )(Di , D` )(D1 , D2 , D3 , D4 ) (Di , D j , Dk )(Di , Dk , D` )(Di , D j , D` )
de sorte que les Di0 sont premiers deux a` deux. La formule (3.11) fournit alors Mr,s =
Dr0 Ds0
Q (D1 , D2 , D3 , D4 )3 i< j (Di , D j ) Q . 2 i< j
Il en d´ecoule que C 0 compte le nombre de six-uplets (y1 , y2 , y3 , z 1 , z 2 , z 3 ) a` coordonn´ees dans N∗ tels que Q 6 n i< j
(5.5)
Ce syst`eme est semblable au syst`eme (4) de Pl¨ucker puisque, rappelons-le, ce syst`eme est d´efini a` partir de trois des cinq e´ quations. Notre probl`eme est sensiblement diff´erent puisque les conditions de coprimalit´e sont diff´erentes. √ 5.2. Introduction de la condition δ1 δ2 δ3 δ4 ≥ T Faisons tout d’abord quelques remarques pr´eliminaires. De simples manipulations sur les pgcd fournissent Q Y i< j (Di , D j ) 0 [D1 , D2 , D3 , D4 ] = Di Q (D1 , D2 , D3 , D4 )3 (5.6) 2 (D , D , D ) i j k i< j
et [D1 , D2 , D3 , D4 ]
Y
(Di , D j )1/8 ≥ n × n 01/8 .
(5.7)
i< j
En effet, l’in´egalit´e (5.7) est de caract`ere multiplicatif. Il suffit donc de la d´emontrer pour des puissances de nombres premiers. Nous e´ crivons Di = p νi en supposant de plus que ν1 ≥ ν2 ≥ ν2 ≥ ν4 . On a Y ν2 + 2ν3 + 3ν4 v p [D1 , D2 , D3 , D4 ] (Di , D j )1/8 = ν1 + . 8 i< j
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Si ( p, n) = 1, on a bien ν +ν +ν +ν Y 1 2 3 4 v p [D1 , D2 , D3 , D4 ] (Di , D j )1/8 ≥ . 8 i< j
Puisque d est un e´ l´ement de D ∗ (B), l’ensemble d´efini en (3.4), on a (di , d j ) = 1 et 0 donc il existe au plus un indice i tel que νi0 ≥ 1. Si p νi k di , νi = νi0 + νi00 et νi0 ≥ 1, on a Y ν 00 + ν j + νk + ν` v p [D1 , D2 , D3 , D4 ] (Di , D j )1/8 ≥ νi0 + i , 8 i< j
ce qui permet d’achever la d´emonstration de (5.7). Q4 ´ Etant donn´e que pour tout d ∈ D , on a n i3 ≤ Bn, on a n 9 = i=1 n i3 ≤ B 4 n 4 et 4/5 9/5 par cons´equent n ≤ B . De (5.2), on en d´eduit que Di ≤ Bn ≤ B . Les di et les δi sont des diviseurs de [D1 , D2 , D3 , D4 ] qui divise lui-mˆeme Y Y Di0 (Di , D j ) ≤ ( max Di )10 ≤ B 18 . i
i=1,...,4
i< j
Lorsque les ai, j := (Di , D j ) et les Di0 sont fix´es, le nombre de (d, δ) est major´e par 8 max τ (m) B O(1/ log log B) .
m≤B 18
(5.8)
De plus, d’apr`es (5.4), (5.6), et (5.7), on a√pour les points somm´es dans C 0 (H, d, δ) avec d ∈ D ∗ (B) et d ∈ / 1(T ) (donc n 0 > T ) la suite d’in´egalit´es Q 4 Bn i< j
Q c(a) := Y
(5.10)
i< j
Il convient de noter que la fonction c est une fonction multiplicative des six variables a1,2 , a1,3 , a1,4 , a2,3 , a2,4 , a3,4 et a` valeurs dans [0, 1].
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5.3. Cl´e de la d´emonstration Le r´esultat suivant implique la Proposition 4. LEMME
13
Soit ( K (B) := card
(D10 , D20 , D30 , D40 , y1 , y2 , y3 , z 1 , z 2 , z 3 ) ∈ N10 ` (5.5) ∗ satisfait a y1 y2 z 2 D20 D30 ≤ B
) .
(5.11) On a pour B ≥ 1, K (B) = B 1+O(1/ log log(3B)) . D´emonstration D’apr`es (5.8) et (5.9), on obtient S(B, T ) ≤
X a∈N6∗
K
B 8 c(a) max τ (m) . 1/16 18 T m≤B
(5.12)
On raisonne en fonction de la valeur de Bc(a)/T 1/16 pour montrer que pour tout a, on a B B K c(a) B O(1/ log log(3B)) 1/16 c(a). (5.13) 1/16 T T Lorsque Bc(a)/T 1/16 ≥ B 1/(log log(3B)) , on a log log(3Bc(a)/T 1/16 ) log log(3B) et, donc, d’apr`es le Lemme 13, K
B B c(a) B O(1/ log log(3B)) 1/16 c(a). 1/16 T T
Lorsque 1 ≤ Bc(a)/T 1/16 ≤ B 1/(log log(3B)) , on a K
B B c(a) K (B 1/(log log(3B)) ) B O(1/ log log(3B)) 1/16 c(a). 1/16 T T
Lorsque Bc(a)/T 1/16 < 1, on a K
B B c(a) = 0 1/16 c(a), T 1/16 T
ce qui e´ tablit (5.13). En reportant cette majoration dans (5.12), on obtient S(B, T )
B 1+O(1/ log log(3B)) X B 1+O(1/ log log(3B)) c(a) , T 1/16 T 1/16 6 a∈N∗
ce qui d´emontre la Proposition 4.
(5.14)
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Il reste a` e´ tablir la convergence de la s´erie a` six variables de terme g´en´eral c(a). On introduit les variables b0 := pgcd(ai, j ), i< j
bk,`
pgcd(ai, j , ai,k , a j,k , ai,` , a j,` ) := b0
(1 ≤ k < ` ≤ 4),
puis (ai, j , ai,k , a j,k ) (1 ≤ ` ≤ 4), b0 b`,k b`,i b`, j ai, j Q := (1 ≤ i < j ≤ 4). b0 b` bk {u,v}6={`,k} bu,v
b` := bi,0 j
Les entiers bk,` sont premiers entre eux deux a` deux et les b` sont premiers entre eux trois a` trois. Il est clair que la derni`ere formule fournit un param´etrage des ai, j en fonction des 17 entiers not´es par la lettre b. En ne tenant pas compte des conditions de coprimalit´e entre ces entiers, on obtient X a
c(a) ≤
1
X
Y X
5/4 k<` bk,` ∈N b0 ∈N b0
1
4 X Y
11/8 bk,` `=1 b` ∈N
1
1
Y X
17/8 b` i< j b0 i, j ∈N
15/8 b0 i, j
1,
le r´esultat recherch´e. 5.4. Fin de la d´emonstration : le Lemme 13 En renommant les variables, on a K (B) = card z ∈ C ∩ N10 ∗ : z 1,3 z 1,4 z 2,4 z 2,5 z 3,5 ≤ B . La seule diff´erence avec le membre de droite de (2.7) d´efini au Lemme 3 est la condition (z i, j , z i,k ) = 1 qui a e´ t´e omise. Quitte a` faire op´erer sur les indices la permutation (1, 4)(2, 3) qui laisse le monˆome z 1,3 z 1,4 z 2,4 z 2,5 z 3,5 invariant et a` multiplier le cardinal par deux, on peut supposer z 2,4 ≥ z 1,2 . Le dix-uplet z = (z 1,2 , z 1,3 , z 1,4 , z 1,5 , z 2,3 , z 2,4 , z 2,5 , z 3,4 , z 3,5 , z 4,5 ) est enti`erement d´efini par huit-uplet (z 1,2 , z 1,3 , z 1,4 , z 1,5 , z 2,3 , z 2,4 , z 2,5 , z 3,5 ), puisque, d’apr`es la premi`ere et la cinqui`eme e´ quations de (4), on a z 3,4 =
z 1,3 z 2,4 − z 1,4 z 2,3 , z 1,2
z 4,5 =
z 3,5 z 2,4 − z 2,5 z 3,4 . z 2,3
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La premi`ere e´ quation fournit z 1,3 z 2,4 ≡ z 1,4 z 2,3
(mod z 1,2 ).
Si cette congruence a des solutions en z 2,4 , ces solutions sont sup´erieures ou e´ gales a` z 1,2 et appartiennent a` une progression arithm´etique de raison z 1,2 /(z 1,3 , z 1,2 ). Ainsi, les entiers t d´efinis par t := [z 2,4 (z 1,3 , z 1,2 )/z 1,2 ] sont tous distincts et sup´erieurs ou e´ gaux a` (z 1,3 , z 1,2 ) ≥ 1. On remarque que z 1,2 z 3,5 z 1,4 t z 1,3 z 2,5 ≤ z 2,4 z 3,5 z 1,4 z 1,3 z 2,5 (z 13 , z 1,2 ) ≤ B(z 13 , z 1,2 ). Posant b1 := z 1,2 z 3,5 , b2 := z 1,3 z 2,5 , cela implique z 1,4 t b1 b2 ≤ B(z 1,3 , z 1,2 ) ≤ B(b1 , b2 ). En utilisant la deuxi`eme relation de (4), on d´eduit l’in´egalit´e X X X K (B) ≤ τ (b1 )τ (b2 ) z 1,4 t≤B(b1 ,b2 )/b1 b2
b1 b2 ≤B
B(log B)
X
τ (b1 )τ (b2 )τ (b2 − b1 )
b1 b2 ≤B
1
(z 1,5 ,z 2,3 ) z 1,5 z 2,3 =b2 −b1
(b1 , b2 ) b1 b2
3 X (b1 , b2 ) B(log B) max{τ (m)} B 1+O(1/ log log(3B)) . m≤B b1 b2 b1 b2 ≤B
Cette m´ethode a d´ej`a e´ t´e utilis´ee par Manin et Tschinkel dans [14] pour montrer que N (B) = O(B(log B)6 ). ´ Remerciements. Je tiens a` exprimer ma plus profonde gratitude a` Etienne Fouvry pour les conseils et les encouragements qu’il n’a pas cess´e de me prodiguer lors de l’´elaboration de ce travail. Je remercie aussi Per Salberger pour ses avis et la communication du r´esum´e [20], Emmanuel Peyre pour les r´eponses par courrier e´ lectronique qui ont e´ clair´e ce travail, et Alain Genestier pour ses remarques pertinentes et ses patientes explications. Je remercie l’arbitre anonyme pour sa relecture attentive et ses conseils avis´es. Bibliographie [1]
V. V. BATYREV et YU. I. MANIN, Sur le nombre des points rationnels de hauteur
born´e des vari´et´es alg´ebriques, Math. Ann. 286 (1990), 27 – 43. MR 91g:11069 421
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compactifications of anisotropic tori, Internat. Math. Res. Notices 1995, 591 – 635. MR 97a:14021 422 , Height zeta functions of toric varieties, J. Math. Sci. 82 (1996), 3220 – 3239. MR 98b:11067 422 , Rational points on some Fano cubic bundles, C. R. Acad. Sci. Paris S´er. I Math. 323 (1996), 41 – 46. MR 97j:14023 422 , Manin’s conjecture for toric varieties, J. Algebraic Geom. 7 (1998), 15 – 53. MR 2000c:11107 422 B. J. BIRCH, Forms in many variables, Proc. Roy. Soc. Ser. A. 265A (1961/1962), 245 – 263. MR 27:132 422 ´ ENE ` J.-L. COLLIOT-THEL et J.-J. SANSUC, “La descente sur les vari´et´es rationnelles” dans Journ´ees de g´eom´etrie alg´ebrique d’Angers (Angers, 1979), Sijthoff et Noordhoff, Alphen aan den Rijn, Netherlands, 1980, 223 – 237. MR 82d:14016 426 , La descente sur les vari´et´es rationnelles, II, Duke Math. J. 54 (1987), 375 – 492. MR 89f:11082 426 E. FOUVRY, “Sur la hauteur des points d’une certaine surface cubique singuli`ere” dans Nombre et r´epartition de points de hauteur born´ee (Paris, 1996), Ast´erisque 251, Soc. Math. France, Montrouge, 1998, 31 – 49. MR 2000b:11075 427 J. FRANKE, YU. I. MANIN , et Y. TSCHINKEL, Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989), 421 – 435. MR 89m:11060 421, 422 D. R. HEATH-BROWN et B. Z. MOROZ, The density of rational points on the cubic surface X 03 = X 1 X 2 X 3 , Math. Proc. Cambridge Philos. Soc. 125 (1999), 385 – 395. MR 2000f:11080 427, 450 YU. I. MANIN, Notes on the arithmetic of Fano threefolds, Compositio Math. 85 (1993), 37 – 55. MR 94c:11053 421 , “Problems on rational points and rational curves on algebraic varieties” dans Surveys in Differential Geometry (Cambridge, Mass., 1993), Vol. II, Internat. Press, Cambridge, Mass., 1995, 214 – 245. MR 97a:14022 421 YU. I. MANIN et YU. TSCHINKEL, Points of bounded height on del Pezzo surfaces, Compositio Math. 85 (1993), 315 – 332. MR 94k:11070 422, 461 E. PEYRE, Hauteurs et mesures de Tamagawa sur les vari´et´es de Fano, Duke Math. J. 79 (1995), 101 – 218. MR 96h:11062 422, 423, 424, 425, 426, 427 , “Terme principal de la fonction zˆeta des hauteurs et torseurs universels” dans Nombre et r´epartition de points de hauteur born´ee (Paris, 1996), Ast´erisque 251, Soc. Math. France, Montrouge, 1998, 259 – 298. MR 2000f:11081 422, 423, 426, 430 M. ROBBIANI, Rational points of bounded height on del Pezzo surfaces of degree six, Comment. Math. Helv. 70 (1995), 403 – 422. MR 96g:11069 426 , On the number of rational points of bounded height on smooth bilinear hypersurfaces in biprojective space, J. London Math. Soc. (2) 63 (2001), 33 – 51. CMP 1 801 715 422 P. SALBERGER, “Tamagawa measures on universal torsors and points of bounded
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height on Fano varieties” dans Nombre et r´epartition de points de hauteur born´ee (Paris, 1996), Ast´erisque 251, Soc. Math. France, Montrouge, 1998, 91 – 258. MR 2000d:11091 422, 426, 427
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, Counting rational points on del Pezzo surfaces of degree 5, summary of talks in Bern (June 1993) and Paris (January 1994), 2 pages, communiqu´e a` l’auteur en novembre 1999 lors du colloque sur les points rationnels a` Luminy (France) en octobre 1999. 422, 462 S. H. SCHANUEL, Heights in number fields, Bull. Soc. Math. France 107 (1979), 433 – 449. MR 81c:12025 427 A. N. SKOROBOGATOV, On a theorem of Enriques-Swinnerton-Dyer, Ann. Fac. Sci. Toulouse Math. (6) 2 (1993), 429 – 440. MR 95b:14018 426, 430 G. TENENBAUM, Introduction a` la th´eorie analytique et probabiliste des nombres, 2`eme e´ d., Cours Spec. 1, Soc. Math. France, Paris, 1995. MR 97e:11005a
Laboratoire d’Arithm´etique et de G´eom´etrie Alg´ebrique d’Orsay, D´epartement de Math´ematiques, Universit´e Paris XI, Bˆatiment 425, 91405 Orsay CEDEX, France; [email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 3,
GEOMETRIC BRANCHED COVERS BETWEEN GENERALIZED MANIFOLDS JUHA HEINONEN and SEPPO RICKMAN
Abstract We develop a theory of geometrically controlled branched covering maps between metric spaces that are generalized cohomology manifolds. Our notion extends that of maps of bounded length distortion, or BLD-maps, from Euclidean spaces. We give a construction that generalizes an extension theorem for branched covers by I. Berstein and A. Edmonds. We apply the theory and the construction to show that certain reasonable metric spaces that were shown by S. Semmes not to admit bi-Lipschitz parametrizations by a Euclidean space nevertheless admit BLD-maps into Euclidean space of same dimension. 0. Introduction It is a difficult problem to determine when a given metric space is locally bi-Lipschitz equivalent to an open subset of Euclidean space. Recall that a map f : X → Y is L-Lipschitz if | f (a) − f (b)| ≤ L|a − b| for each pair of points a, b ∈ X , and for some L ≥ 1 independent of the points. A homeomorphism f is L-bi-Lipschitz if both f and f −1 are L-Lipschitz. (Generally, in this paper, we use the distance notation |x − y| in every metric space.) In 1979 L. Siebenmann and D. Sullivan [SS] noted the curious fact that there are, for each n ≥ 5, compact pure n-dimensional polyhedra that are topological manifolds but do not admit local bi-Lipschitz parametrizations. The double suspension of a homology sphere with nontrivial π1 serves as an example. After some interesting positive results due to T. Toro [T1], [T2], Semmes exhibited a family of geometrically nice metrics on S 3 for which no local bi-Lipschitz parametrizations exist (see [S4], [S3]). At this point, it is not clear if a simple geometric characterization can be found for the metric spaces that admit local bi-Lipschitz parametrizations. (Dimension n = 2 could be special here (cf. [S1], [DS2], [HK2]; see, however, Rem. 0.6.) DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 3, Received 17 February 2000. Revision received 29 May 2001. 2000 Mathematics Subject Classification. Primary 57M12; Secondary 30C65. Heinonen’s work supported by National Science Foundation grant number DMS 9970427. 465
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The situation is interestingly different if one allows for parametrizations by maps with branching. In this work, we consider maps that are natural multivalent analogs of bi-Lipschitz maps, called BLD-maps. We show, in particular, that none of the metrics d exhibited by Semmes on S 3 carries an obstruction for the existence of “branched Lipschitz,” or “BLD-coordinates.” In fact, global BLD-maps f : (S 3 , d) → S 3 exist. The same is true for the double suspension examples, but this is easier, for the polyhedra in question can be triangulated so that an Alexander-type PL-branched cover is possible. J. Heinonen and Sullivan [HSu] have recently outlined an approach to the parametrization problem which is based on differential (Whitney) forms. In particular, in [HSu] a characterization of a “branched Euclidean metric gauge” is given. Among other things, the results of the present paper demonstrate the sharpness of the characterization in [HSu] for dimensions n ≥ 3. In the first part of this work, we develop a basic theory of geometrically controlled branched covers between generalized metric manifolds. A generalized n-manifold is a finite-dimensional locally compact Hausdorff space whose local cohomology groups over the integers in dimensions n − 1 and higher are similar to those of an n-manifold (for a precise definition, see Def. 1.6). We call a continuous map between topological spaces a branched cover if it is open and if the preimage of each point consists of isolated points. The latter property is also expressed by saying that the map is discrete. Thus, “branched cover” in this paper means “discrete and open.” The algebraic condition in the definition of a generalized manifold allows for a local degree theory for continuous maps between oriented generalized manifolds. In particular, it makes sense to speak about sense-preserving maps (see Section 2). The main class of maps we study in this paper is the class of BLD-maps. Definition 0.1 A sense-preserving branched cover f : X → Y between two oriented generalized metric n-manifolds is said to be an L-BLD, or an L-bounded length distortion map, L ≥ 1, if 1 length α ≤ length f ◦ α ≤ L length α (0.2) L for all (nonconstant) paths α in X . Note that requirement (0.2) has interesting content only if the spaces X and Y are rich in rectifiable curves. We emphasize that “length” here means “length of a path”; thus, for example, every (sense-preserving) Riemannian covering map is a BLD-map. BLD-maps in Euclidean spaces were first studied by O. Martio and J. V¨ais¨al¨a [MV¨a]. They gave the following analytic (quantitatively equivalent) characterization
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of BLD-maps in R n : a continuous map f : G → R n , where G ⊂ R n is open and n ≥ 2, is BLD if and only if f is locally uniformly Lipschitz and the Jacobian determinant det D f is positive and uniformly bounded away from zero almost everywhere in G. Lipschitz maps are almost everywhere differentiable, so the assumption on the Jacobian makes sense. This description of a BLD-map in Euclidean space does not include the assumption that the map is discrete and open, nor that it is sense-preserving. In fact, maps satisfying the above analytic condition form a (strict) subclass of quasiregular maps, or maps of bounded distortion, introduced by Yu. Reshetnyak in the 1960s. It follows from a deep theorem of Reshetnyak [Re1] that each (locally nonconstant) quasi-regular map is discrete, open, and sense-preserving. BLD-maps in R n can be characterized as those quasi-regular maps that are both Lipschitz and have the Jacobian determinant bounded away from zero uniformly (in the almost everywhere sense). The difference between the two conditions is typified by the following two examples: for an integer p ≥ 2, the complex polynomial z 7 → z p is holomorphic and hence quasi-regular in R 2 but not BLD; the “winding map” (r, θ, z) 7→ (r, pθ, z) in cylindrical coordinates in R n , n ≥ 2, is BLD. These remarks about BLD and quasi-regular maps hold on smooth Riemannian manifolds. In this work, we do not attempt to extend the theory of general quasiregular maps from the smooth setting; it seems reasonable first to study the BLD case, where the tools of Lipschitz analysis are available. We emphasize that BLD-maps can already exhibit quite complicated (local) behavior. Indeed, one can show that (at least locally) each quasi-regular map f : R n → R n factors f = ϕ ◦ g, where g is a quasisymmetric homeomorphism from R n to a metric space X f , and ϕ : X f → R n is a BLD-map; moreover, the metric space X f , the “Riemann surface of f ,” is of type A124, as defined in Section 5. On the other hand, it is true that the global behavior of BLD-maps is more limited than that of general quasi-regular maps (cf. Th. 6.8). We refer to [Re2], [Ri2], and [MV¨a] for the basic theory and further examples of quasi-regular and BLD-maps. In the second part of this work, we prove the following general existence theorem for branched covers in dimension n = 3; this theorem is crucial in showing the existence of branched coordinates for metric spaces as discussed above. 0.3 Let W be a connected, compact, oriented P L 3-manifold with boundary ∂ W consisting of p ≥ 2 components M0 , . . . , M p−1 with the induced orientation. We assume that W is a polyhedron in some R m and is equipped with the metric induced from R m . There exists an integer d0 ≥ 3 such that the following holds. Let N be an oriented polyhedral 3-sphere in R 4 , and let B0 , . . . , B p−1 be disjoint closed polyhedral 3-balls in N ; the boundaries ∂ B0 , . . . , ∂ B p−1 are given the orientation induced from THEOREM
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S p−1 W 0 = N \ j=0 int B j . For each j = 0, 1, . . . , p − 1, let ϕ j : M j → ∂ B j be a sense-preserving P L-branched cover of degree d = 3i ≥ d0 . Then there exists a sense-preserving P L-branched cover ϕ : W → W 0 of degree d that extends the ϕ j ’s. Theorem 0.3 was proved by Berstein and Edmonds [BE, Th. 6.2] for p = 2 and d0 = 3, and without the restriction d = 3i. The restriction d = 3i on the degree is needed for our proof, and we do not know if it is really necessary. Similarly, in our construction the degree lower bound d0 depends on W . It could well be that one can choose d0 = d0 ( p, m), but we do not know this. In our proof we use the techniques of [Ri1] to reduce the general case to the case considered by Berstein and Edmonds. This reduction is presented here in a self-contained manner; no reference to [Ri1] is needed. A special case of Theorem 0.3 was proved in [HR], where we showed the existence of a BLD-map f : S 3 → S 3 whose branch set contains a wild Cantor set. The general construction of the present paper simplifies that of [HR], except the use of the Berstein-Edmonds theorem. (In [HR] we did not use [BE] but relied instead on S. Rickman’s paper [Ri1] in a more substantial way.) By the aid of Theorem 0.3 we can construct BLD-maps from some interesting compact generalized 3-manifolds onto S 3 . Semmes [S3] showed that the classical (compact) decomposition spaces arising from the Whitehead continuum, Bing’s dogbone space and Bing’s double, for example, all admit metrics that are smooth Riemannian outside a totally disconnected closed singular set and that are indistinguishable from the standard metric on S 3 by means of classical analysis. We show that all these spaces, although bi-Lipschitz inequivalent to the standard or PL 3-sphere, can be mapped onto standard S 3 by a BLD-map. As alluded to above, the claims made by Semmes, as well as our constructions, work for each space arising in a self-similar manner from an “excellent package,” as defined in [S3]. All this is made more precise in Section 8. At present, the method of constructing interesting branched covers in the spirit of [Ri1] and [HR] is limited to dimension n = 3. When n = 2, every generalized manifold is actually a manifold, and the existence and nature of branched covers is well understood. In higher dimensions, every closed orientable n-pseudomanifold can be mapped onto S n by a BLD-branched cover. For smooth manifolds this was proved by J. Alexander [A] in 1920, and his method works for orientable pseudomanifolds, too. Recall that a closed n-pseudomanifold (cf. [ST, Sec. 24]) is a pure n-dimensional finite complex where every (n −1)-simplex is a face of precisely two n-simplexes and where every two n-simplexes can be joined by an arc that only meets n- or (n − 1)simplexes. An n-pseudomanifold X is orientable if Hn (X ) = Z. A particular example
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of a closed n-pseudomanifold is a finite complex X that satisfies H∗ (X, X \ {x}) = H∗ (R n , R n \ {0})
(0.4)
for all x ∈ X . Complexes X that satisfy (0.4) are generalized n-manifolds as defined in this paper, and they can be characterized by the property that the link of each vertex has the homology of the (n −1)-sphere. The homology groups above, and everywhere in this paper, are integral. Finally, let us compare the BLD-maps as defined above in Definition 0.1 with another related class of maps. Remember that condition (0.2) becomes uninteresting if the spaces X and Y contain no rectifiable paths. In [D], G. David introduced “regular” maps between “regular” subsets of Euclidean space. Later David and Semmes [DS1], [DS2], [DS3], [S4] took up a more general definition, valid in arbitrary metric spaces. Definition 0.5 A map f : X → Y between metric spaces is called regular with data (L , M, N ) if f is L-Lipschitz and if for each closed ball B in Y the preimage f −1 (B) can be covered by M closed balls in X , each of diameter at most N times the diameter of B. Regular maps as defined in Definition 0.5 are natural finite-to-one analogs of biLipschitz maps. They serve as a natural generalization of BLD-maps to spaces where one cannot define sense-preserving maps, or where no rectifiable curves are to be found. For most of our stronger results, the assumption that we have a sensepreserving map is necessary; the proofs often rely on the path-lifting property of sense-preserving branched covers between oriented generalized manifolds as explained in Section 3.3. On the other hand, many results could be true in more generality. There is room for further research here. In Part I of this paper, we study the relationship between regular and BLD-maps on generalized manifolds. It turns out that under quite general circumstances, the sense-preserving regular maps are precisely the BLD-maps of finite maximal multiplicity (see Th. 4.5.) We give examples that illustrate the relationship between various classes of maps. In Part I we also introduce a particular kind of generalized metric manifolds, called spaces of “type A,” and we show that BLD-maps between such spaces behave in many respects like BLD-maps between Riemannian manifolds. In particular, for maps from spaces of type A into R n , an analytic description akin to the one used by Martio and V¨ais¨al¨a in R n is available. We point out some further analytic and metric properties of BLD-maps between spaces of type A, including a result on value distribution (see Th. 6.8). Part II contains a proof of the generalized Berstein-Edmonds theorem, Theorem 0.3, together with some examples and applications. In particular, we carefully study the geometric decomposition spaces of Semmes, and we show how they admit
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BLD-branched coordinates in R 3 . We also point out how one obtains bounded quasiregular maps in the open 3-ball in R 3 which fail to have radial limits in a large set on the boundary. Moreover, we construct Lipschitz maps S 3 → S 3 with nonzero degree having as fibers some strange continua, for example, the Whitehead continuum, but with dilatation function in a local Lebesgue space L p for p arbitrarily close to 2. These examples are quite interesting in view of the fact that each Lipschitz map S 3 → S 3 is a branched cover if the integrability exponent for the dilatation exceeds 2 (see Sec. 9.4 for the details). Finally, we discuss how BLD-maps preserve Poincar´e inequalities. Remark 0.6 (Added in February 2001) After this paper was submitted, Laakso [L] constructed a metric d on the 2-sphere S 2 such that the space (S 2 , d) is of type A124 as in Definition 5.1 (for n = 2) but does not admit local BLD-maps into R 2 . On the other hand, Laakso’s space is not of type A3. The authors do not know of any example of a metric space that is a topological 2manifold and of type A as in Definition 5.1 (for n = 2) but is not locally bi-Lipschitz equivalent to an open subset of R 2 . We also found a paper by U. Hirsch [Hi], where the author proves a result that is related to our Theorem 0.3. The result, [Hi, Th. 3.1], asserts that every orientable, compact PL 3-manifold with p boundary components admits a 3-fold PL-branched covering of the 3-sphere with p balls removed. This result neither implies nor is implied by Theorem 0.3, but it could have been used, for example, to obtain our Theorem 8.17 (see Rem. 8.19(b)).
Part I: BLD and regular maps 1. Generalized manifolds All spaces in this paper are assumed to be locally compact, Hausdorff, connected, and locally connected. Hc∗ (X ) denotes the Alexander-Spanier cohomology groups of a space X with compact supports and coefficients in Z. Definition 1.1 A space X is called an n-dimensional, n ≥ 2, cohomology manifold (over Z), or a cohomology n-manifold, if (a) the cohomological dimension dimZ X is at most n, and (b) the local cohomology groups of X are equivalent to Z in degree n and to zero in degree n − 1.
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p
Condition (a) means that Hc (U ) = 0 for all open U ⊂ X and p ≥ n + 1 (in general, dimZ X is the least integer n for which this is true; see [Br, II, Th. 16.14]). Condition (b) means that for each point x ∈ X , and for each open neighborhood U of x, there is another open neighborhood V of x contained in U such that ( Z if p = n, p Hc (V ) = 0 if p = n − 1, and the standard homomorphism Hcn (W ) → Hcn (V )
(1.2)
is a surjection whenever W is an open neighborhood of x contained in V . Remarks 1.3 (a) Our definition for a cohomology n-manifold is weaker than what typically appears in the literature, for we require that the local groups are similar to that of an n-manifold in degrees n − 1 and higher only. Cohomology manifolds as defined in [Br, V, Def. 16.7] and [Bo, I, Def. 3.3], for example, satisfy our requirements. (b) The cohomological dimension dimZ X (of a separable metric X ) equals the topological dimension of X provided the latter is finite. This is proved in [HW, VIII, Th. 3, p. 151] for X compact; in general, dimZ X is the supremum of dimensions dimZ Y , where Y ranges over compact subspaces of X (see [Br, II, Prop. 16.7]). (c) An n-dimensional cohomology manifold X need not be finite-dimensional, but dimZ X = n always (see [Br, p. 115 and p. 122], [Bo, I, Th. 2.3]). (d) A (connected) open subset of a cohomology manifold is a cohomology manifold of the same dimension. If V ⊂ X is open, connected, and satisfies Hcn (V ) = Z, then the standard homomorphism in (1.2) is an isomorphism whenever W is connected (see [Br, V, Th. 16.16] or [Bo, I, Th. 4.3(2)]). Examples 1.4 (a) Every topological n-manifold is a cohomology manifold. Conversely, every (separable, metric) cohomology 2-manifold is an actual 2-manifold (see [Br, V, Th. 16.32] or [W, IX, Th. 5.5, p. 277]). For each n ≥ 3 there are compact subsets of Euclidean space that are n-cohomology manifolds but not manifolds (cf. Exam. 1.4(d)). (b) A locally finite simplicial complex is a cohomology n-manifold if the link of every vertex has the homology of an (n − 1)-sphere. (c) Some authors call an ENR space X a homology n-manifold if, for each x ∈ X , H∗ (X, X \ {x}) ' H∗ (R n , R n \ {0}),
(1.5)
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where H∗ denotes the singular homology with coefficients in Z. Every homology nmanifold according to this definition is a finite-dimensional cohomology n-manifold. This follows from [Br, V, Th. 16.8]. On the other hand, the latter need not be locally contractible (see the example in [W, p. 245]). Recall that an ENR space can be characterized as a locally compact, finite-dimensional, and locally contractible metric space. (d) If G is a cell-like decomposition of an n-manifold M, then the decomposition space M/G is a homology n-manifold as defined in (c) provided it is finite dimensional. In particular, M/G is a cohomology n-manifold (see [Da, p. 191] and also Section 8). Next, we define a generalized manifold in a way that suits our purposes in this paper. Definition 1.6 A space X is called a generalized n-manifold, n ≥ 2, if it is a finite-dimensional cohomology n-manifold. It follows from Remark 1.3(b) that a generalized n-manifold has topological dimension n. Thus, a space X is a generalized n-manifold if and only if it is n-dimensional and conditions (a) and (b) in Definition 1.1 hold. The geometric conditions imposed on most spaces in this paper automatically imply that the dimension is finite. Because of the assumption on finite-dimensionality, a generalized manifold is tacitly assumed to be separable. If not otherwise stated, we assume that a generalized manifold is always given a metric. 2. Orientation and local degree If a generalized n-manifold X satisfies Hcn (X ) ' Z, then X is said to be orientable and a choice of a generator g X in Hcn (X ) is called an orientation; X together with g X is an oriented generalized n-manifold. If X is oriented, we can simultaneously choose an orientation gU for all connected open subsets U of X via the isomorphisms Hcn (U ) → Hcn (X ) (see Rem.1.3(d)). Next let X and Y be oriented generalized n-manifolds, and let f : X → Y be continuous. For each relatively compact domain D in X (a domain is an open connected set) and for each component V of Y \ f (∂ D), the map f | f −1 (V ) ∩ D : f −1 (V ) ∩ D → V is proper. Hence we have a sequence of maps Hcn (V ) → Hcn f −1 (V ) ∩ D → Hcn (D),
(2.1)
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where the first map is induced by f and the second map is the standard homomorphism as in (1.2). The composition of these two maps sends the generator gV to an integer multiple of the generator g D ; this integer, denoted by µ(y, D, f ), is called the local degree of f at a point y ∈ V with respect to D. The local degree is an integer-valued locally constant function y 7 → µ(y, D, f ) defined in Y \ f (∂ D). If V ∩ f (D) = ∅, then µ(y, D, f ) = 0 for y ∈ V . Definition 2.2 A continuous map f : X → Y between two oriented generalized n-manifolds is said to be sense-preserving if µ(y, D, f ) > 0 whenever D ⊂ X is a relatively compact domain and y ∈ f (D) \ f (∂ D). 2.3. Basic properties of the local degree The following properties of the local degree are easily ascertained. (a) If f, g : X → Y are homotopic through proper maps h t , 0 ≤ t ≤ 1, such that y ∈ Y \ h t (∂ D) for all t, then µ(y, D, f ) = µ(y, D, g). (b) If y ∈ Y \ f (∂ D) and if f −1 (y) ⊂ D1 ∪ · · · ∪ D p , where Di are all disjoint domains and contained in D such that y ∈ Y \ f (∂ Di ), then µ(y, D, f ) =
p X
µ(y, Di , f ).
i=1
(c) If f : D → f (D) is a homeomorphism, then µ(y, D, f ) = ±1 for each y ∈ f (D). In particular, if f is a local homeomorphism, there is for each x ∈ X a connected neighborhood D such that µ( f (x), D, f ) = ±1. More generally, if f is discrete and open and x ∈ X , then there is a relatively compact neighborhood D of x such that { f −1 ( f (x))} ∩ D = {x}; the number µ( f (x), D, f ) = i(x, f ) is independent of D and called the local index of f at x. (Recall that X is locally connected by assumption.) (d) If f is open, discrete, and sense-preserving, then for each x ∈ X there is a connected neighborhood D as in (c) above such that f (∂ D) = ∂( f (D)); D is called a normal neighborhood of x, and i(x, f ) = max card{ f −1 (y) ∩ D}. y∈ f (D)
(2.4)
If D is any domain such that f (∂ D) = ∂( f (D)), then D is called a normal domain.
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3. Sense-preserving branched covers Let f : X → Y be a continuous map between generalized n-manifolds. Recall from the introduction that f is said to be a branched cover if f is a discrete and open map. 3.1. Branch set The branch set B f of f is the closed set of points in X where f does not define ˇ a local homeomorphism. A theorem of A. Cernavski˘ ı [C1], [C2] and V¨ais¨al¨a [V2] implies that for a branched cover f : X → Y the branch set B f has topological dimension at most n − 2. (This result is often stated for maps between manifolds, but V¨ais¨al¨a’s proof, in particular, extends to the present situation; see [V2, Rem. 7, p. 9]). It follows that X \ B f is connected if f : X → Y is a branched cover by the proofs in [Bo, I, Cor. 4.7] and [Br, V, Th. 16.20]. Remark 3.2 If f : X → Y is a branched cover between two oriented generalized nmanifolds, then f is either sense-preserving or sense-reversing. The latter means that µ(y, D, f ) < 0 for all y ∈ f (D) \ f (∂ D) (for a proof, see [V1, Sec. 5.2]). 3.3. Path-lifting We next explain a path-lifting property of branched covers. Let f : X → Y be a sense-preserving branched cover between oriented generalized manifolds, and let β : [a, b) → Y be a path. A maximal f -lifting of β starting at a point x ∈ f −1 (β(a)) is a path α : [a, c) → X, a < c ≤ b, such that (i) α(a) = x, that (ii) f ◦ α = β|[a, c), and that (iii) if c < c0 ≤ b, then there does not exist a path α 0 : [a, c0 ) → X such that α = α 0 |[a, c) and f ◦ α 0 = β|[a, c0 ). If x1 , . . . , xk are k distinct preimages of β(a), then there exists a maximal sequence α1 , . . . , αm of f -liftings of β starting at the points x1 , . . . , xk . This means that m=
k X
i(xi , f ),
(3.4)
i=1
each α j : [a, c j ) → X is a maximal f -lifting of β, card j : α j (a) = xi = i(xi , f ),
(3.5)
card j : α j (s) = x ≤ i(x, f ) for all x ∈ X and all s.
(3.7)
(3.6)
and The existence of a maximal sequence of f -liftings in the present generality can be proved as in [Ri2, II, Sec. 3]; the original idea of that proof is due to E. Poletski˘ı [P].
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4. BLD and regular maps In this section, we study the elementary properties of BLD and regular maps. Recall Definitions 0.1 and 0.5. Notation 4.1 Throughout this paper, we denote by B(x, r ) an open ball of radius r centered at x in a metric space X , and by B(x, r ) the closure of B(x, r ). By a closed ball (of radius r centered at x) we mean the set Bc (x, r ) = {y ∈ X : |x − y| ≤ r }. Note that B(x, r ) ⊂ B(x, r ) ⊂ Bc (x, r ) and that all inclusions can be strict. We often let C denote a generic positive constant whose value may vary from line to line. 4.2 A regular map f : X → Y between arbitrary metric spaces with data (L , M, N ) is at most M-to-one. LEMMA
Proof Suppose that x1 , . . . , x M+1 are distinct points in X with image y. Let 0<ε<
1 min |xi − x j |. 2N i6= j
Then diam Bc (y, ε) = d ≤ 2ε, and by assumption the points x1 , . . . , x M+1 belong to M balls in X each of diameter at most N d ≤ 2N ε < min |xi − x j |. i6 = j
This is a contradiction, and the lemma follows. Remarks 4.3 (a) By Lemma 4.2, regular maps are always discrete. They need not be branched covers according to our definition; the folding R 2 → R 2 , (x, y) 7→ (x, |y|), is a regular map but not open. (b) An open regular map between oriented generalized manifolds is either sensepreserving or sense-reversing by Remark 3.2. In fact, a regular map between oriented generalized manifolds is sense-preserving or sense-reversing if and only if it is open (see the proof in [Ri2, VI, Lem. 5.6]). (c) A BLD-map f : X → Y need not be regular. For example, a (sensepreserving) Riemannian covering map is BLD and has infinite multiplicity in general. More generally, if f : X → Y is locally uniformly bi-Lipschitz, then f is BLD if it
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is sense-preserving. We show in Theorem 4.5 that under certain conditions on X and Y , BLD-maps that have finite multiplicity are regular maps. 4.4. Quasi-convex spaces A metric space is said to be c-quasi-convex, c ≥ 1, if every pair of points x, y in it can be joined by a curve whose length does not exceed c|x − y|. THEOREM 4.5 Let X and Y be quasi-convex oriented generalized n-manifolds, let X be complete, and let f : X → Y be sense-preserving. Then f is regular if and only if it is a BLDmap of finite maximal multiplicity, quantitatively.
Remark 4.6 None of the conditions (a) X is quasi-convex, (b) Y is quasi-convex, or (c) X is complete can be left out in Theorem 4.5. We show this by the following simple examples. First take X to be the open strip {(x1 , x2 ) ∈ R 2 : x1 > 1, |x2 | < 1/x12 }, and let f 1 : X → R 2 = Y be a bounded injective map such that f 1 is locally 2-biLipschitz. Then (a) and (b) hold, but not (c). The map f 1 is BLD but not regular. Next 2 consider the infinite cylinder X = {(x1 , x2 , x3 ) ∈ R 3 : x22 + x32 = 1/(1 + x12 ) }, and let f 2 : X → f 2 (X ) = Y ⊂ R 3 be a bounded, injective locally 2-bi-Lipschitz map. Then (a) and (c) hold, but not (b). The map f 2 is BLD but not regular. Finally, to get a case where (b) and (c) are fulfilled but not (a), one takes f 3 = fˆ2−1 , where fˆ2 : X → fˆ2 (X ) ⊂ R 3 is an unbounded version of the above f 2 ; the image fˆ2 (X ) wiggles more and more the further out it goes. Then f 3 is BLD but not regular because it is not Lipschitz. On the other hand, if we know a priori that f is Lipschitz in Theorem 4.5, then the assumption that X is quasi-convex is not needed. This is clear from the proof. A regular map always satisfies (0.2). LEMMA 4.7 A regular map f : X → Y between arbitrary metric spaces satisfies
1 length α ≤ length f ◦ α ≤ C length α C
(4.8)
for all paths α in X , where C ≥ 1 depends only on the data of f . Naturally, condition (4.8) is interesting only if X and Y have nontrivial rectifiable paths. The proof of Theorem 4.5 shows that neither completeness nor quasiconvexity is needed in the “only if” direction.
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Proof Let f : X → Y be a regular map between arbitrary metric spaces with data (L , M, N ). Because f is L-Lipschitz, we always have length f ◦ α ≤ L length α
(4.9)
for any path α in X , whether of finite or infinite length. Next we show that the left inequality in (4.8) holds for a given (nonconstant) path α : [0, 1] → X with C = C(L , M, N ) ≥ 1. To this end, let 0 = t0 ≤ · · · ≤ tk = 1 be a partition, where without loss of generality |ti − ti+1 | = ε for all i ∈ {0, . . . , k − 1}. Let αi be the subpath α|[ti , ti+1 ] and |αi | its image. From Lemma 4.10 it follows that diam f (|αi |) ≥
1 1 diam |αi | ≥ |α(ti ) − α(ti+1 )|. 2M N 2M N
Therefore we have for some t 0 < t 00 in [ti , ti+1 ], |α(ti ) − α(ti+1 )| ≤ 2M N | f ◦ α(t 0 ) − f ◦ α(t 00 )| ≤ 2M N | f ◦ α(ti ) − f ◦ α(t 0 )| + | f ◦ α(t 0 ) − f ◦ α(t 00 )| + | f ◦ α(t 00 ) − f ◦ α(ti+1 )| , whence k−1 X
|α(ti ) − α(ti+1 )| ≤ 2M N length f ◦ α.
i=0
Because the partition was arbitrary, this and (4.9) imply (4.8) with C max{2M N , L}.
=
LEMMA 4.10 ([DS3, Lem. 12.4]) Let f : X → Y be a regular map between arbitrary metric spaces with data (L , M, N ), and let γ ⊂ X be a connected set. Then
diam γ ≤ 2M N diam f (γ ). Proof We can assume that diam f (γ ) = d > 0. Then f (γ ) belongs to the closed ball Bc (y, d) for some y ∈ f (γ ). We have diam Bc (y, d) ≤ 2d, and by regularity we can cover γ by M closed balls of diameter at most 2N d. This proves the lemma.
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Proof of Theorem 4.5. We have already seen that a regular map f : X → Y satisfies (4.8) for arbitrary metric spaces X and Y . If, in addition, the spaces are oriented generalized n-manifolds and if f is sense-preserving, then f is open by Remark 4.3(b), and hence it is a BLD-map. The proof shows that if f has data (L , M, N ), then we can take max{2M N , L} for the BLD-constant. Next assume that f : X → Y is an L-BLD map onto Y with maximal multiplicity p = max card f −1 (y) < ∞ y∈Y
and that Y is cY -quasi-convex. Let Bc (y0 , R) be a closed ball in Y . We may assume that there exists z 0 ∈ Bc (y0 , R) such that ∅ 6= {x1 , . . . , xq } = f −1 (z 0 ), We claim that
1 ≤ q ≤ p.
q
f
−1
[ Bc (y0 , R) ⊂ Bc (xi , 2LcY R).
(4.11)
i=1
Because we may assume that diam Bc (y0 , R) ≥ R/2, it follows from (4.11) and (4.12) that f is regular with data (Lc X , p, 8LcY ) if X is c X -quasi-convex. To prove (4.11), let x ∈ X be such that | f (x) − y0 | ≤ R. Let γ : [0, 1] → Y be a path with γ (0) = f (x), γ (1) = z 0 such that length γ ≤ cY | f (x) − z 0 | ≤ cY 2R. By the path-lifting property of sense-preserving branched covers, explained in Section 3.3, there is a maximal f -lifting α of γ starting at x. Because f is a BLD-map and because γ has finite length, α stays bounded; moreover, by completeness of X, α ends at a point u ∈ X with f (u) = z 0 . Thus u = xi for some i = 1, . . . , q, which implies |x − xi | ≤ length α ≤ L length γ ≤ 2LcY R. Thus (4.11) follows. To show that f is Lipschitz, fix a, b ∈ X and pick a curve γ from a to b such that length γ ≤ c X |a − b|. Then | f (a) − f (b)| ≤ length f ◦ γ ≤ L length γ ≤ Lc X |a − b|,
(4.12)
which shows that f is Lc X -Lipschitz if it is L-BLD and X is c X -quasi-convex. This completes the proof of Theorem 4.5.
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In the next proposition, we establish a useful geometric description of BLD-maps; it is essentially a corollary to Theorem 4.5. PROPOSITION 4.13 Let X and Y be quasi-convex oriented generalized n-manifolds, let X be complete, and let f : X → Y be a surjective BLD-map of finite multiplicity. Then there is a constant C ≥ 1 with the following property: for every ball B = B(x, r ) in X , we have B f (x), r/C ⊂ f (B) ⊂ B f (x), Cr , (4.14)
and for every ball B = B(y, r ) in Y we have [ B(x, r/C) ⊂ f −1 (B) ⊂ x∈ f −1 (y)
[
B(x, Cr ).
(4.15)
x∈ f −1 (y)
The constant C above depends only on the data associated with X , Y , and f including the multiplicity. Proof The second inclusion in (4.14) is clear because f is Lipschitz by Theorem 4.5. To prove the first inclusion, let ` denote the distance from f (x) to ∂ f (B), where B = B(x, r ) ⊂ X . Note that ` > 0 because f is open and that there is nothing to prove if ∂ f (B) = ∅. Let β be a path from f (x) to ∂ f (B) of length no more than a fixed constant times `; such a path β exists because Y is quasi-convex. Then any lift α of β, starting from x, must hit the boundary of B by the completeness of X (cf. Sec. 3.3). Thus r ≤ length α ≤ C length β ≤ C `, as desired. This proves (4.14). Now the first inclusion in (4.15) is an easy consequence of the fact that f is Lipschitz (see Th. 4.5), while the second inclusion follows from (4.11). (Note that we can take z 0 = y in the proof of (4.11).) The proposition is thereby proved. Remarks 4.16 (a) The quasiconvexity of X is not needed in Proposition 4.13 if f is assumed to be, in addition, Lipschitz (cf. the comment at the end of Rem. 4.6). (b) It is clear that Proposition 4.13 has a local version, valid for BLD-maps of arbitrary multiplicity and for source spaces X that need not be complete. (Recall our standing assumption that all spaces are locally compact.) (c) It follows from (4.14) in Proposition 4.13 that BLD-maps between quasiconvex, complete, oriented generalized n-manifolds are Lipschitz quotient maps in the sense of S. Bates, W. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman
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[BJLPS]. Indeed, the assumption on finite multiplicity is not needed for (4.14), as the proof shows. The definition for a BLD-map f : X → Y consists of four conditions: the map has to be open, discrete, and sense-preserving, and it has to satisfy (0.2). Of these, only the requirement “sense-preserving” needs X and Y to be generalized manifolds. We next discuss what happens if these conditions are relaxed or replaced with other similar conditions. 4.17. Light maps A continuous map between topological spaces is light if the preimage of every point is totally disconnected, that is, if all its components are points. 4.18 A sense-preserving and light map between generalized manifolds is discrete and open. PROPOSITION
In the case of manifolds, Proposition 4.18 is due to C. Titus and G. Young [TY]. The proof extends easily (see [Ri2, VI, Lem. 5.6]). 4.19 A sense-preserving map between generalized manifolds is a BLD-map if and only if it is light and (0.2) holds. COROLLARY
Remarks 4.20 (a) Light maps between manifolds need not be discrete, even if they satisfy (0.2); thus the assumption “sense-preserving” in Corollary 4.19 is necessary. (b) There is a light map of the 2-sphere onto itself that satisfies (0.2) and maps a Cantor set to a point. To construct such a map, consider the standard Cantor ternary set C on [0, 1] and define a map f : [0, 1] → [0, 1] by f (x) = 6 dist(x, C). By projecting f to the unit circle, we obtain an onto map S 1 → S 1 that takes a Cantor set to a point. To get a desired map on S 2 , we suspend. (c) We do not know whether a continuous map between path-connected metric spaces that satisfies (0.2) is always light. Naturally, no fiber of such a map can contain a nonconstant path. (d) V¨ais¨al¨a [V4] (for 1 ≤ n ≤ 3) and P. Church [Ch] (all n ≥ 1) have shown that every countable map between n-manifolds is a local homeomorphism in a dense set. (A map is countable if it has countable fibers.) In particular, if f : X → Y is a regular map between metric quasi-convex manifolds, then there is an open dense set f ⊂ X such that f defines a locally bi-Lipschitz map, either sense-preserving
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or sense-reversing, in each component of f . Recall from Section 3.1 that f is connected if f is, in addition, open. We study the metric size of B f = X \ f in Section 6. 5. Spaces of type A In this section, we isolate a class of generalized manifolds whose Lipschitz analysis is similar to that on Riemannian manifolds (cf. [HSu]). Definition 5.1 Let X be a metric space, and let n ≥ 2 be an integer. Consider the following properties that X may or may not possess: (A1) X is n-rectifiable and has locally finite Hn -measure; (A2) X is locally Ahlfors n-regular; (A3) X is locally bi-Lipschitz embeddable in Euclidean space; (A4) X is locally linearly contractible. We call X a space of type A1, A2, . . . , A12, . . . , if it satisfies axioms (A1), (A2), . . . , (A1) and (A2), . . . . If X satisfies all four axioms (A1) – (A4), then we simply call X of type A. Note that in this terminology, the dimension n should be understood as fixed. The first condition in (A1) means, by definition, that X is a countable union of Lipschitz images of subsets of R n plus a set of Hausdorff n-measure zero. (Observe the differing terminology in some of the literature: for subsets X of Euclidean space, the above condition is called countable (Hn , n)-rectifiability in [F, Sec. 3.2.14].) Condition (A2) means, by definition, that for each compact set K ⊂ X there are constants C K ≥ 1 and r K > 0 such that the Hausdorff n-measure Hn in X satisfies 1 n r ≤ Hn (Br ) ≤ C K r n CK
(5.2)
for each ball Br of radius 0 < r ≤ r K centered at a point in K . The meaning of (A3) is clear. The dimension of the receiving Euclidean space is allowed to depend on the local embedding. The last condition (A4) means, by definition, that for each compact set K ⊂ X there are constants C K ≥ 1 and r K > 0 such that for each 0 < r ≤ r K and for each x ∈ K the ball B(x, r ) is contractible in B(x, C K r ). (Note that our requirement is weaker than the one bearing a similar term in [S2].) Obviously, every Riemannian manifold is a space of type A. It is also easy to see that every Lipschitz n-manifold admits metrics that make the manifold into a space of type A. In particular, by Sullivan’s theorem [Su], every topological n-manifold,
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n 6= 4, admits type A metrics. On the other hand, spaces of type A need not be manifolds (see [S3] and Section 8). One can show that spaces of type A allow for differential calculus akin to that in Euclidean spaces. Mainly this follows from a work of Semmes [S2] which shows that (locally) a space of type A admits a Poincar´e-type inequality (as defined in Sec. 9.6). The validity of a Poincar´e inequality is crucial for the proof of one of our main results, Theorem 6.18. Properties (A1) and (A2) guarantee that the usual Lipschitz calculus on rectifiable sets is at our disposal (cf. [F]). We note that conditions (A1) – (A4) are not optimal for the purposes of such calculi. For instance, (A1) and (A3) rule out socalled Carnot geometries (see also Remark 5.12). We refer to [Che], [HaK], [HK3], [S2], [S5], and [Sh] for further information about calculus on metric measure spaces. 5.3. Change-of-variables formula Let X n and Y n be two spaces of type (A1), and let f : X → Y be a Lipschitz map. Under these assumptions, we have the following proposition. 5.4 There is a locally integrable nonnegative function J ( f, x) on X such that Z Z u ◦ f (x)J ( f, x) d x = u(y)N ( f, y) dy PROPOSITION
X
(5.5)
Y
for each nonnegative measurable function u on Y . Throughout this paper, we use the notation N ( f, y) = card{ f −1 (y) : y ∈ Y }, N ( f, A, y) = card{ f −1 (y) ∩ A : y ∈ Y },
A ⊂ X,
(5.6)
if f : X → Y is a map. Moreover, if the underlying Borel measure in a space is understood, we write d x, dy, . . . inside integrals; for instance, in (5.5) we have d x = d Hn (x) and dy = d Hn (y). Proposition 5.4 is essentially a result of B. Kirchheim [K], who proved that if A ⊂ R n is (Lebesgue) measurable and if g is a Lipschitz map of A into an n-rectifiable metric space Z (in particular, Z can be a space of type A1 as defined in Def. 5.1), then there is a nonnegative locally integrable function J (g, ·) on A such that Z Z v ◦ g(a)J (g, a) da = v(z)N (g, A, z) dz (5.7) A
Z
for each nonnegative measurable function v on Z (see [K, Cor. 8(ii)]). The “Jacobian” J (g, ·) is explicitly described in [K], where the notation J (M D(g, ·)) is used. If
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Z is a subset of some Euclidean space, J (g, ·) is the approximate Jacobian of g described in [F, Cor. 3.2.20]; in this case, formula (5.7) is proved in [F, Cor. 3.2.20] with the assumption Hn (Z ) < ∞. We deduce (5.5) from these known formulas. (In fact, we found Prop. 5.4 also in a recent preprint of L. Ambrosio and Kirchheim [AK].) To this end, write X as a countable disjoint union of subsets A01 , A02 , . . . plus a set E of n-measure zero, ∞ [ X= A0k ∪ E, k=1
such that = λk (Ak ) for some Borel set Ak ⊂ R n and for some bi-Lipschitz map 0 λk : Ak → Ak for each k. From Lemma 4 and from [K, proof of Th. 7], together with assumption (A1), such sets A0k and maps λk can be found (see also [F, Lem. 3.2.18]). Fix a measurable function u : Y → [0, ∞). Define A0k
J ( f, x) =
J ( f ◦ λk , λ−1 k (x)) J (λk , λ−1 k (x))
for x ∈ A0k . Because λk is bi-Lipschitz, the function J (λk , ·) is bounded away from 0 and ∞ almost everywhere on Ak . Then, by Kirchheim’s formula (5.7), we have Z u ◦ f ◦ λk (a)J ( f ◦ λk , a) da Ak Z Z = u(y)N ( f ◦ λk , y) dy = u(y)N ( f, A0k , y) dy. (5.8) Y
Y
On the other hand, we can apply (5.7) again and obtain that the left-hand side of (5.8) is equal to Z
u ◦ f ◦ λk (a) Ak
J ( f ◦ λk , λ−1 k ◦ λk (a)) J (λk , λ−1 k ◦ λk (a))
J (λk , a) da
Z = A0k
u ◦ f (x)J ( f, x) d x.
Thus (5.5) follows. 5.9. Metric orientation Assume that X is an oriented generalized n-manifold (with metric), as defined in Definition 1.6, and assume that X is of type A123. Let U be an open connected neighborhood of a point in X that can be embedded bi-Lipschitzly in some R N . Thus, without much loss of generality, we assume that U ⊂ R N and that U has finite Hausdorff n-measure. Because of properties (A1) and (A2), the set U has a tangent
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n-plane Tx U at a.e. point x ∈ U . The collection of these planes is called a measurable tangent bundle of U , and it is denoted by T U . Note, in particular, that (A2) guarantees that the tangent cone is an n-plane at a.e. point in X (see [F, Sec. 3.2.16, Th. 3.2.19]). Each n-plane Tx U , whenever it exists, is an n-dimensional subspace of R N , and a measurable choice of orientations ξ = (ξx ) on each Tx U is called an orientation of the tangent bundle T U . Such orientations always exist (see [F, Lem. 3.2.25]). Because X is an oriented generalized n-manifold, there is another orientation on U , provided U is connected; this is a generator gU in the group Hcn (U ) = Z determined by the fixed orientation on X (see Sec. 2). Now fix a point x ∈ U such that Tx U exists. Then the projection πx : R N → Tx U + x satisfies x 6 ∈ πx (∂ D) whenever D is a sufficiently small open connected neighborhood of x in U . Thus, if V is the x-component of (Tx U + x) \ πx (∂ D), we have πx∗
Hcn (Tx U ) ← Hcn (V ) −→ Hcn πx−1 (V ) ∩ D
→ Hcn (D) → Hcn (U ),
(5.10)
where the three unnamed arrows represent a canonical isomorphism. If the orientations ξx and gU correspond to each other under the map in (5.10), we say that Tx U and U are coherently oriented at x by ξx and gU ; if a measurable coherent orientation ξ = (ξx ) is chosen at a.e. point, we say that T U is metrically oriented by ξ and gU . We need the following fact. PROPOSITION 5.11 ([HSu, Exam. 3.10]) Every generalized n-manifold of type A can be (locally) metrically oriented.
We remark that condition (A4) is important in the proof of Proposition 5.11 (cf. [HSu, (3.11)]). Remark 5.12 In the ensuing sections, we could often replace assumption (A4) by the weaker requirement that X admit a Poincar´e inequality as in (9.7). More precisely, the assumption on linear local contractibility (A4) is only used to guarantee that such an inequality holds (via Semmes’s work in [S2]) and that spaces of type A can be locally metrically oriented (as in Proposition 5.11). Thus, if only the validity of a Poincar´e inequality is assumed, one needs, in addition, to stipulate the metric orientability of X . We have chosen to work with conditions (A1) – (A4) for their relative simplicity. There is room left for research as to what conditions exactly are needed for the results in Section 6 (cf. the discussion in [HSu, Sec. 5]).
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6. Analytic and geometric properties of BLD-maps In [MV¨a], Martio and V¨ais¨al¨a established interesting analytic and geometric properties of BLD-maps in Euclidean space. In this section, we prove results along these lines for BLD-maps between generalized manifolds. Many properties of BLD-maps in [MV¨a] were deduced from the more general theory of quasi-regular maps. In our case, no such theory is currently available. All spaces in this section are assumed to be oriented generalized manifolds with metric, unless otherwise stated. Our first proposition follows from Theorem 4.5 because (this is easy to see) regular maps satisfy (6.2), quantitatively (cf. Lem. 4.10). PROPOSITION 6.1 Let f be a BLD-map of finite multiplicity of a complete quasi-convex space X into a quasi-convex space Y . Then
1 H p (A) ≤ H p f (A) ≤ C H p (A) C
(6.2)
for every Borel set A ⊂ X and for every number p > 0, where H p denotes the Hausdorff p-measure in respective spaces and C ≥ 1 depends only on p and on the data associated with f, X , and Y . In particular, BLD-maps between (locally) quasiconvex spaces preserve the Hausdorff dimension of every set. The next proposition follows from Propositions 6.1 and 4.13. Note that Proposition 6.3 admits a local version by Remark 4.16(b). 6.3 Assume that f , X , and Y are as in Proposition 4.13. Then, if either of the spaces X or Y is Ahlfors n-regular, so is the other one, quantitatively. PROPOSITION
Next we prove a theorem concerning the size of the branch set of a BLD-map. The fact that both B f and f (B f ) have measure zero is of great importance in the theory of quasi-regular maps in Euclidean space. We do not know whether Theorem 6.4 remains valid for BLD-maps between more general spaces. THEOREM 6.4 Let f : X n → Y n be a BLD-map between locally quasi-convex spaces of type A. Then Hn (B f ) = Hn ( f (B f )) = 0.
Proof By the proof of Theorem 4.5, f is locally Lipschitz. Thus it suffices to show that
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Hn (B f ) = 0. Because the issue is local, and the conclusion is bi-Lipschitz invariant, we may assume that X and Y are metrically oriented subsets of some Euclidean space by the discussion in Section 5.9 and by Proposition 5.11. In this framework, f is approximately differentiable at Hn almost everywhere in X (see [F, Sec. 3.2.16, Th. 3.2.19] and Sec. 6.5). The theorem therefore follows from Proposition 6.6.
6.5. Approximate differentiability Recall that a map f of an n-rectifiable subset X of R N into R M is approximately differentiable at x0 ∈ X if x0 has a neighborhood in X of finite Hn -measure and if there exists y0 ∈ R M and a linear map T : R N → R M such that lim Hn ({x ∈ X : |x − x0 | < r and | f (x) − y0 − T (x − x0 )| ≥ ε|x − x0 |})r −n = 0
r →0
for each ε > 0. If a tangent n-plane Tx0 X exists at x0 , the restriction of T to Tx0 X is the approximate differential of f at x0 , and it is denoted by ap D f (x0 ). (The approximate differential ap D f (x0 ) is unique, if it exists, at every point x0 such that X has positive lower n-density at x0 ; because almost every point is such a point, we can assume that approximate differentials are unique.) Next assume that X and Y are two metrically oriented generalized n-manifolds of type A lying in some Euclidean space. Denote the respective orientations by (ξ, g X ) and (η, gY ). Let O TX be the set of all points in X such that Tx X exists, that Tx X and X are coherently oriented by ξx and g X , and that X has Hausdorff n-measure density 1 at x (see [F, Sec. 2.10.19, Th. 3.2.22]). The set O TX has full measure in X . We define O TY similarly. If f : X → Y is as in Theorem 6.4, it follows from Proposition 6.1 that the sets −1 f (O TY )∩O TX and f (O TX )∩O TY have full measure in X and f (X ), respectively. 6.6 Let f : X → Y be as in Theorem 6.4, with the added assumption that X and Y are metrically oriented subsets of some Euclidean space. Then f cannot be approximately differentiable at any point in B f ∩ f −1 (O TY ) ∩ O TX . PROPOSITION
Proof Let x0 ∈ f −1 (O TY ) ∩ O TX be a point where f is approximately differentiable. Because f is a regular map near x0 by Theorem 4.5, it is easy to see that the approximate differential ap D f (x0 ) cannot be degenerate; that is, it has to map the tangent plane Tx0 X onto the tangent plane Ty0 Y , where f (x0 ) = y0 (cf. [F, Th. 3.2.22]). Moreover, if we fix a sufficiently small neighborhood D of x0 in X , then it is easy to see that the maps ap D f (x0 ) ◦ πx0 and π y0 ◦ f are homotopic as maps D → Ty0 Y + y0 , where π∗ denotes the projection map as in Section 5.9, and apD f (x0 ) is the affine
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map between the planes Tx0 X + x0 and Ty0 Y + y0 determined by the approximate differential ap D f (x0 ). By using this and the definition for coherent orientation, it is clear that f must have local index i(x0 , f ) = 1 at x0 (see Sec. 2.3). In particular, f is a local homeomorphism at x0 , and the proposition follows. Remarks 6.7 (a) The assumption that f is sense-preserving in Proposition 6.6 is crucial. To give an example of a regular map f : R n → R n which is differentiable at a point in the branch set, consider the following sequence of n-balls B j = B( j −1 e1 , ( j!)−1 ), where e1 = (1, 0, . . . , 0) and j = 5, 6, . . . , and define a map f that is the identity outside the union of these balls and makes a little “flip” inside each B j . The definition of f can easily be made so that f is both regular and not locally injective near the origin; clearly, f is differentiable at the origin. (b) Theorem 6.4 is sharp in the following sense: for each n ≥ 3 and ε > 0 there is a BLD-map f : R n → R n such that the Hausdorff dimension of B f is at least n − ε. On the other hand, in R n , the branch set of an L-BLD-map has Hausdorff dimension at most c = c(n, L) < n. We do not know in what generality this latter statement is true (see [MV¨a, Sec. 4.27] for the above assertions). In the next theorem, we give a growth estimate for the multiplicity function of a BLDmap. A similar result in R n was established in [MV¨a]. Recall the notation (5.6). THEOREM 6.8 Let f : X n → Y n be an L-BLD-map between oriented generalized manifolds of type A. Assume, in addition, that X is c X -quasi-convex and that Y is cY -quasi-convex. If x0 ∈ X, r > 0, and if λ > 1 such that the ball B(x0 , λr ) has compact closure in X , then Hn (B(x0 , λr )) N f, B(x0 , r ), y ≤ (Lc X )n Hn B(y, (λ − 1)r/LcY )
for each y ∈ Y . The following corollary is noteworthy. COROLLARY 6.9 Let f : X → Y be as in Theorem 6.8. Assume, in addition, that closed balls in X are compact and that the following mass bounds are satisfied: Hn B(x, r ) ≤ V X r n , (6.10) n Hn B(y, r ) ≥ VY r , (6.11)
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for all x ∈ X, y ∈ Y , and for all positive radii r < 2 diam X . Then the maximal multiplicity of f is finite, and it is at most 2n L 2n (c X cY )n V X VY−1 . Proof Apply Theorem 6.8 to x0 ∈ X, λ = 2, to r = (3/2) diam X in the case where diam X < ∞ and r arbitrary for diam X = ∞. Proof of Theorem 6.8. We follow the argument in [MV¨a, p. 436]. Pick y ∈ Y , and let {x1 , . . . , xk } = f −1 (y) ∩ B(x0 , r ). The preimage f −1 (y) of y in B(x0 , r ) is finite because B(x0 , λr ) is assumed to be precompact; we may also assume that f −1 (y) is nonempty. Fix a radius s≤
(λ − 1)r , LcY
(6.12)
and fix a point e ∈ B(y, s) \ f (B f ). Because f is an open map and because Hn ( f (B f )) = 0 by Theorem 6.4, the set B(y, s) \ f (B f ) is indeed nonempty. Join y to e by a path γ with (λ − 1)r length γ ≤ cY s ≤ , L and consider a maximal sequence α1 , . . . , αm of f -liftings of γ starting at the points x1 , . . . , xk (see Sec. 3.3). Because length αi ≤ L length f ◦ αi ≤ L length γ ≤ (λ − 1)r, all the lifts stay in the ball B(x0 , λr ). Therefore, because B(x0 , λr ) is precompact in X , all the lifts αi are defined on the full interval [0, l(γ )] and f ◦ αi = γ , i = 1, . . . , m. (We assume here that γ is parametrized by the arc length.) It follows from property (3.7) of f -liftings that the point e gets hit from B(x0 , λr ) at least k X m= i(xi , f ) ≥ k i=1
times because e 6 ∈ f (B f ). Thus Z Z J ( f, x) d x = N f, B(x0 , λr ), w dw B(x0 ,λr ) Y Z ≥ N f, B(x0 , λr ), w dw B(y,s)\ f (B f )
≥ k Hn B(y, s) .
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On the other hand, because X is c X -quasi-convex, f is Lc X -Lipschitz, and it is easy to see that J ( f, x) ≤ (Lc X )n . By combining these two estimates, we have k ≤ (Lc X )n
Hn (B(x0 , λr )) Hn (B(y, s))
for each s as in (6.12). The theorem follows. 6.13. Analytic description of BLD-maps Let f : X n → Y n be an L-BLD-map between complete quasi-convex (oriented) generalized n-manifolds of type A. Then f is Lipschitz and in view of Propositions 5.4 and 6.1, the “Jacobian” J ( f, z) of f satisfies J ( f, z) ≥ c > 0
(6.14)
for a.e. z in X and for some positive constant c depending only on the data associated with f and the spaces X and Y . The Jacobian in (6.14) was defined in Section 5.3, but it can alternatively be described by Hn ( f (B(z, r )) r →0 Hn (B(z, r ))
J ( f, z) = lim
(6.15)
for a.e. z ∈ X , in view of (5.5) and Theorem 6.4. To be more precise about the constant c in (6.14), we first observe that f is a local homeomorphism almost everywhere by Theorem 6.4. Thus, by Proposition 6.1 and by (6.15), c depends only on the BLD constant L and on the parameters associated with X and Y ; the latter consist of the quasi-convexity constants and the Ahlfors regularity constants, including the dimension n. There is one caveat here: the Ahlfors regularity is stipulated only locally for spaces of type A, so that, in principle, c may vary together with the location of z, but it remains bounded on each compact set. In many applications the Ahlfors regularity holds uniformly over the whole space. Finally, because of Theorem 6.4, the local index has no bearing on the constant c in (6.14), as the inequality is required to hold only almost everywhere. Using local embeddings in some Euclidean space and metric orientations as in Section 5.9, the sign of the determinant det ap Dg is well defined almost everywhere for each Lipschitz map g : X → Y such that J (g, z) > 0 for a.e. z ∈ X (cf. Props. 5.4 and 6.6). In particular, for each such g we can define a signed Jacobian Jˆ(g, z) by Jˆ(g, z) = sgn det ap Dg(z) J (g, z) (6.16) for a.e. z. In the following, the notation Jˆ(g, z) > 0 a.e. tacitly means that J (g, z) > 0 a.e.
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PROPOSITION 6.17 Let g : X n → Y n be a Lipschitz map between two generalized n-manifolds of type A such that Jˆ(g, z) > 0 a.e. Then g is sense-preserving.
Proof Let D be a precompact domain in X , and let y ∈ g(D) \ g(∂ D). Fix x ∈ D such that g(x) = y, and fix a domain W ⊂ D such that x ∈ W and that g(W ) ∩ g(∂ D) = ∅. By condition (A1), and by the change-of-variables formula (5.5), we have Z Z Jˆ(g, z) dz ≤ C Hn (D) < ∞. N (g, D, b) db = g(D)
D
On the other hand, by the hypothesis, Jˆ(g, z) > 0 a.e., and so we have Z Z N (g, W, b) db = Jˆ(g, z) dz > 0. g(W )
W
It follows that there exists a set E ⊂ W of measure zero such that for some b ∈ g(W ) \ g(E) we have (1) #{g −1 (b) ∩ D} < ∞, and (2) Jˆ(g, a) > 0 for each a ∈ g −1 (b) ∩ D. By changing the set E if needed, we also have approximate tangent planes existing at every point in D \ E. This statement implicitly involves a use of local embeddings in Euclidean space. We remark that at this point we have used assumptions (A1) and (A3) in our proof, while assumptions (A2) and (A4) have implicitly been used via Proposition 5.11 (see, however, Remark 5.12). Let {a1 , . . . , ak } = g −1 (b) ∩ D, where b is as in (1) and (2). It follows from (1) and (2), and from an easy homotopy argument using projections to the tangent spaces, that there exist domains D j in D, with pairwise disjoint closures, containing points a j , j = 1, . . . , k, such that b ∈ / g(∂ D j ) and that µ(b, D j , g) = 1 for each j. In conclusion, by the properties of local index as described in Section 2, we find that µ(y, D, g) = µ(b, D, g) =
k X
µ(b, D j , g) = k > 0,
j=1
as required. The proposition follows. Martio and V¨ais¨al¨a [MV¨a] proved that condition (6.14) characterizes BLD-maps among sense-preserving, locally uniformly Lipschitz maps between Euclidean domains. The most difficult point in proving this characterization is Reshetnyak’s theorem in [Re1] which implies that (locally) Lipschitz maps in R n , with Jacobian determinant positive and bounded away from zero, are discrete and open. A generalization
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of this result is given in [HSu]. Thus we are able to extend the characterization from [MV¨a] to the following general setting. THEOREM 6.18 Let f : X n → R n be a Lipschitz map from a quasi-convex oriented generalized nmanifold X of type A into R n such that the signed Jacobian (see (6.16)) is a.e. positive and that (6.14) holds. Then f |W is a BLD-map for each precompact domain W in X , quantitatively. In particular, f is a sense-preserving, discrete, and open map.
In addition to the result from [HSu], the proof of Theorem 6.18 requires some metric space analysis from [HK3] and [S2]. First we need two lemmas and some notation. For a map f : X → R n the linear dilatation H (x, f ) and the inverse linear dilatation H ∗ (x, f ) are defined as L(x, r ) , r →0 l(x, r ) L ∗ (x, s) H ∗ (x, f ) = lim sup ∗ , s→0 l (x, s) H (x, f ) = lim sup
where L(x, r ) = sup | f (y) − f (x)|, |y−x|=r
l(x, r ) =
inf
|y−x|=r
| f (y) − f (x)|,
L ∗ (x, s) = sup |x − z|, z∈∂Us
l (x, s) = inf |x − z|, ∗
z∈∂Us
where Us = U (x, s) is the x-component of f −1 (B( f (x), s)). LEMMA 6.19 Let X be as in Theorem 6.18 with the additional assumption that X is geodesic. Let f : X → R n be a sense-preserving, discrete, and open M-Lipschitz map such that (6.14) holds a.e., and let x ∈ X . Then there are constants Hx , Hx∗ < ∞, depending only on the data of X , on c, M, and the local index i(x, f ), such that the linear dilatation H (x, f ) and the inverse linear dilatation H ∗ (x, f ) satisfy
H (x, f ) < Hx ,
(6.20)
Hx∗ .
(6.21)
H (x, f ) < ∗
Proof Write y = f (x). There exists σ = σx > 0 such that (see Sec. 2.3(d))
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(1) Us is a normal neighborhood of x for 0 < s ≤ σ and diam Us → 0 as s → 0, (2) Us = Uσ ∩ f −1 (B(y, s)) for 0 < s ≤ σ , (3) ∂Us = Uσ ∩ f −1 (∂ B(y, s)) for 0 < s < σ , (4) l ∗ (x, L(x, r )) = L ∗ (x, l(x, r )) = r for 0 < r < l ∗ (x, σ ). The proof of (1) – (4) given in [Ri2, II, Lem. 4.1] for Euclidean space applies as such to X . In particular, because X is geodesic, open balls are connected, which fact is sometimes used in [Ri2, II, Lem. 4.1]. To prove (6.20), we modify the proof of [Ri2, II, Lem. 4.2] to the present situation. Let 0 < r < l ∗ (x, σ ), and suppose l = l(x, r ) < L = L(x, r ). Then by (4), F = U l is a continuum with x ∈ F and diam F ≥ r , and the set E = U σ \ U L is a continuum with dist(E, F) < r . The connectedness of E follows easily by lifting paths lying in the ring B(y, σ ) \ B(y, L). For small r we also have diam E ≥ r . We want to apply a local version of [HK3, Th. 3.6] to the condenser (E 0 , F; Uσ ), where E 0 = E ∩ Uσ . First define v : B(y, σ ) → [0, 1] by setting |z − y| L −1 log , v(z) = log l l
l ≤ |z − y| ≤ L ,
and by extending v as 0 in B(y, l) and as 1 in B(y, σ ) \ B(y, L). Then u = v ◦ f |Uσ satisfies u|E 0 = 1 and u|F = 0; hence u is admissible for the condenser (E 0 , F; Uσ ) (for the terminology, see [HK3, Sec. 2]). Set | f (ζ ) − f (ξ )| . |ζ − ξ | |ζ −ξ |→0
L f (ξ ) = lim sup
We claim that the Borel function ρ = (|∇v| ◦ f )L f is an upper gradient of u in Uσ , that is, that Z |u(ξ ) − u(ζ )| ≤ ρ ds (6.22) γξ ζ
whenever γξ ζ is a rectifiable curve in Uσ from ξ to ζ . Since f is Lipschitz, f ◦ γξ ζ is rectifiable for each such γξ ζ , and, moreover, f is absolutely continuous on γξ ζ (see [V3, Def. 5.2]). Clearly, Z |u(ξ ) − u(ζ )| ≤ |∇v| ds f ◦γξ ζ
for each γξ ζ , and by [V3, Th. 5.3] we get Z Z |∇v| ds ≤ f ◦γξ ζ
Thus (6.22) follows.
γξ ζ
ρ ds.
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It follows from the assumptions that X satisfies locally the condition for a Loewner space as defined in [HK3, Sec. 3]. To be more precise here, the assumptions (A1) – (A4) guarantee the validity of a Poincar´e inequality (locally) on X by [S2, Th. B.10], and then [HK3, Th. 5.7] implies the Loewner condition. (Semmes in [S2] considers only manifolds, but his proofs are valid under the weaker hypotheses of a generalized manifold; see [S2, Rem. A. 35, p. 272].) See Section 9.6 for further discussion on Poincar´e inequalities. Without going deeper into the Loewner property here, it suffices to say that it implies, in the present context, that the n-capacity of the condenser (E 0 , F; Uσ ) satisfies 0 < κ ≤ capn (E 0 , F; Uσ ), where κ depends only on the data of X in Uσ . Clearly, L f (ξ ) ≤ M for all ξ ∈ X . By the definition (see [HK3, Sec. 2.11]) of the n-capacity, by (6.14), and by Proposition 5.4, we obtain Z Z κ≤ ρn = |∇v|n f (ξ ) L nf (ξ ) dξ Uσ Uσ Z −1 n ≤c M |∇v|n f (ξ ) J ( f, ξ ) dξ Uσ Z L 1−n −1 n = c M i(x, f ) , |∇v|n (η)dη = c−1 M n i(x, f )ωn−1 log l B(y,σ )
which gives (6.20) with 1/(n−1) Hx = exp 2κ −1 c−1 M n i(x, f )ωn−1 , where ωn−1 is the (n − 1)-measure of the unit sphere in R n . For the proof of (6.21), we similarly use a modification of the proof of [Ri2, II, Th. 4.4]. In particular, we obtain the bound 2i(x, f )c−1 M n
Hx∗ = Hx
.
We leave the details to the reader and conclude the proof of the lemma. LEMMA 6.23 Let X be as in Theorem 6.18. Let f : X → R n be a sense-preserving, discrete, and open M-Lipschitz map such that (6.14) holds a.e., and let x ∈ X . There exists a constant Q < ∞, depending only on the data of X and on c, such that
L ∗ (x, s) ≤ Q Hx∗ s,
0 < s < σ = σx .
(6.24)
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Proof We may assume that X is geodesic because X is quasi-convex. Indeed, L ∗ (x, f ), H ∗ (x, f ), c, M, and the data given in the definition of type A change only up to a bounded factor when a bi-Lipschitz change is made on the metric of X . A (locally) geodesic bi-Lipschitz equivalent metric on X is obtained as the infimum of the lengths of the paths joining a given pair of points in X . (Recall that X is assumed to be locally compact.) Let σ = σx be as in the proof of Lemma 6.19. We may assume f (x) = 0. We apply the idea from [MV¨a, Lem. 2.12]. For a Borel set E ⊂ S(s) = ∂ B(0, s), let ϕ(E) = Hn f −1 (C E ) ∩ U s , where C E is the cone {tv : 0 ≤ t ≤ 1, v ∈ E} and where Us = U (x, s) the x component of f −1 (B( f (x), s)) . Fix a point v on S(s) such that the derivative ϕ 0 (v) exists and is finite; because the set function ϕ is completely additive and finite, almost every point on S(s) is such a point. Write t βv (t) = v, s
0 ≤ t ≤ s,
and let αv : [0, s] → U s be any lifting of βv starting at x. The path αv defines a function gv : [0, s] → [0, ∞) by gv (t) = |αv (t) − x|. Slight modification of a part of the proof of [Ri2, II, Lem. 5.3], starting on p. 41 there, gives that gv is absolutely continuous. Here we need (6.21) and local n-regularity of X . In [Ri2, II, Lem. 5.3] we also need the fact that (z, τ ) 7 → l ∗ (z, τ ) is locally upper semicontinuous and (z, τ ) 7→ L ∗ (z, τ ) locally lower semicontinuous. The proofs of these statements can be copied for our setting from the Euclidean case as presented in [Ri2, II, Lem. 4.7]. Because f is Lipschitz and satisfies (6.14) a.e., the preceding discussion implies that for almost every v ∈ S(s) the following two conditions hold for almost every t ∈ [0, s]: (a) J (αv (t)) ≥ c, (b) gv is absolutely continuous and has a finite derivative gv0 (t). Fix v and 0 < t < s satisfying (a) and (b) for a lifting αv as above. Set u = αv (t). By (6.15) and (6.19), there exists ρu > 0 such that Hn ( f (B(u, ρ))) c ≤ , 2 Hn (B(u, ρ)) L ∗ u, L(u, ρ) < Hu∗l ∗ u, L(u, ρ) ,
GEOMETRIC BRANCHED COVERS
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for 0 < ρ ≤ ρu . Moreover, we can choose ρu so small that B(u, ρu ) ⊂ Uσ and that l ∗ (u, L(u, ρ)) = ρ for 0 < ρ ≤ ρu (see (4) in the proof of Lem. 6.19). Fix 0 < ρ ≤ ρu , and write L = L(u, ρ). Then, by the local n-regularity of X and by choosing ρu still smaller if necessary, there exists a constant cx such that c Hn ( f (B(u, ρ))) Ln Ln ≤ ≤ cx n = cx ∗ 2 Hn (B(u, ρ)) ρ l (u, L)n n L ≤ cx Hu∗n ∗ , L (u, L)n
(6.25)
where Hu∗ is the bound given in Lemma 6.19. It follows that gv0 (t) ≤
2c 1/n x
c
Hu∗ .
(6.26)
By (2.4), we have i(u, f ) ≤ i(x, f ), and hence, by the proof of Lemma 6.19, Hu∗ has an upper bound C0 Hx∗ , where C0 depends only on the data on X in a neighborhood of x, and on c and M. From this and the absolute continuity of gv , we conclude that gv (s) ≤ C0
2c 1/n x
c
Hx∗ s.
(6.27)
By choosing suitable lifts αv , we find a sequence αvi (s) tending to a point z ∈ ∂Us with |z − x| = L ∗ (x, s) and such that (6.27) holds with v = vi . This completes the proof with 2c 1/n x Q = C0 , (6.28) c and the lemma follows. Proof of Theorem 6.18. It follows from Proposition 6.17 that f is sense-preserving, and from [HSu] that f is a discrete and open map. It thus remains to show that f |W satisfies (0.2) for each precompact W and for some L ≥ 1 depending on the data. Since f is M-Lipschitz for some M, the second inequality of (0.2) holds with constant M for all paths α in X . To prove the first inequality of (0.2), we use the idea from [MV¨a, Lem. 2.15]. To this end, let α be a path in W and set β = f ◦ α. We may assume that l(β) = length β < ∞, and we let β : [0, l(β)] → R n be a parametrization by arc length. Since W is compact, the numbers cW = sup cx , x∈W ∗ HW
= sup Hx∗ x∈W
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are finite, where cx is the constant in (6.25) and where Hx∗ is the bound in Lemma 6.19. For a given partition P of [0, l(β)], there is a refinement P 0 = {t0 , . . . , tk } of P and numbers r0 , . . . , rk > 0 such that Ui = U (α(ti ), ri ) is a normal neighborhood of α(ti ), α[ti−1 , ti ] ⊂ Ui−1 ∪ Ui , and ri < σα(ti ) , where σα(ti ) is given in Lemma 6.23. For each i = 1, . . . , k, choose si ∈ [ti−1 , ti ] with α(si ) ∈ Ui−1 ∩ Ui . Then Lemma 6.23 implies |α(si ) − α(ti−1 )| ≤ L 0 (si − ti−1 ), |α(ti ) − α(si )| ≤ L 0 (ti − si ), where L 0 = CW
where C W
2c 1/n W
c corresponds to C0 in (6.28). Hence
k X i=1
|α(ti ) − α(ti−1 )| ≤ L 0
k X
∗ HW ,
(6.29)
(ti − si ) + (si − ti−1 ) = L 0l(β),
i=1
and so l(α) ≤ L 0l(β). (Note: the condition β[ti−1 , ti ] ⊂ B(β(ti−1 ), ri−1 ) ∪ B(β(ti ), ri ) in the proof of [MV¨a, Lem. 2.15] does not guarantee that α(si ) ∈ Ui−1 ∩ Ui ; hence the presentation in [MV¨a] is not quite right and should be corrected along the lines given above.) We have obtained that f |W is L-BLD with L = max(M, L 0 ). In L 0 the factor ∗ depends also on max{i(x, f ) : x ∈ W }. To get rid of the dependence on the HW local index, we deduce from Theorem 6.4 that in the proof of Lemma 6.23 we can, in addition to conditions (a) and (b), require that H1 ( f (B f ) ∩ |βv |) = 0. This gives by (6.26) a bound for gv0 (t) almost everywhere on [0, s] which is independent of i(u, f ). Consequently, (6.24) can be replaced by L ∗ (x, s) ≤ Px s, where Px depends only on the data of X in a neighborhood of x, on M, and on c. ∗ in (6.29) can be replaced by a constant independent of the local It follows that HW index. This completes the proof of Theorem 6.18. 6.30. BLD-maps into R n In [HSu], an oriented generalized n-manifold X was termed locally BLD-Euclidean if every point in X has a neighborhood U together with a Lipschitz BLD-map f : U → R n . We close this section by pointing out the following property of locally BLD-Euclidean spaces (for a further property of locally BLD-Euclidean spaces, see Sec. 9.6).
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PROPOSITION 6.31 A locally BLD-Euclidean space is of type A12 and is a Lipschitz manifold outside a closed singular set of topological dimension at most n − 2.
Proof If X n is a locally BLD-Euclidean space, then X is locally Ahlfors n-regular by Proposition 6.3 and Remarks 4.16(a) and (b). To prove that X is rectifiable, we may assume without loss of generality that there exists a Lipschitz BLD-map f : X → R n of finite multiplicity. It is easy to see that f is locally bi-Lipschitz outside the branch set B f , by the path-lifting property. Hence it suffices to show that B f is n-rectifiable. To this end, fix a normal neighborhood U of a point x0 ∈ B f , and let C0 = {x ∈ U : i(x, f ) = i(x0 , f )} (see Sec. 2.3(c) and (d)). Then f |C0 is injective, and it is easy to see that f −1 | f (C0 ) : f (C0 ) → C0 is locally Lipschitz, by the path-lifting property. Thus C0 is nrectifiable. Because the function x 7 → i(x, f ) is upper semicontinuous, the set U \ C0 is open. Hence, the proof is easily completed by exhausting U \ C0 by a countable number of compact sets, and proceeding by downward induction with respect to the local index. It follows that X is of type A12. The second assertion follows because X \ B f is connected and because X is easily seen to be a Lipschitz manifold outside the branch set B f . This completes the proof of Proposition 6.31. Remarks 6.32 (a) We do not know whether locally BLD-Euclidean spaces are of type A3, that is, whether they are (locally) bi-Lipschitz embeddable in some finite-dimensional Euclidean space. (b) We do not know if a locally BLD-Euclidean space X of type A3 is locally metrically orientable. It is easy to see that such an X is locally metrically orientable if for some (all) Lipschitz BLD-maps f : X → R n we have that the branch set B f has zero Hausdorff n-measure. We conjecture that this is the case. (c) Locally BLD-Euclidean spaces need not be locally contractible, a fortiori not of type A4. In particular, we cannot use Proposition 5.11 to solve the problem posed in Remark 6.32(b). An example to this effect can be given by using the space in [W, p. 245] and the generalized Berstein-Edmonds Theorem 0.3. We next sketch such an example. Consider in R 3 a standard self-similar construction of a Cantor set, where one begins with a PL 3-ball B0 containing two similar but disjoint PL 3-balls B1 and B2 . By mapping B0 onto B1 and B2 by similarities T1 and T2 , respectively, we get balls
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Bi j = Ti (B j ) inside Bi for i, j = 1, 2. By continuing in the same manner, we get balls Bi1 ...i p , where i k = 1, 2. Next let H1 be a triangulated homology 3-sphere with nontrivial fundamental group. Remove a PL 3-ball D1 from H1 . We assume that the triangulation is so chosen that H1 \ D1 can be glued along the boundary to the ball B2 , so that the resulting connected sum M1 = B0 #H1 is a triangulated 3-manifold (with one boundary component ∂ B0 ). A scaled version H2 \ D2 of H1 \ D1 can similarly be glued to M1 along the boundary of B12 . We obtain a manifold M2 , which is B0 with two homology spheres “sticking out.” By continuing in this manner, we obtain a sequence of PL 3-manifolds M1 , M2 , M3 . . . with the obvious limiting space X . This space is a generalized 3-manifold and not locally contractible; it is described in [W, p. 245]. We can use the original Berstein-Edmonds theorem (see [BE, Th. 6.2]) to map the manifold with boundary M10 , consisting of M1 minus the interior of B1 , onto B0 minus the interior of B1 by a finite-to-one PL-branched cover. By continuing, using the self-similarity of the construction, we get manifolds Mi0 and maps f i from Mi0 onto B0 minus B1...1 (i iterates). The obvious limiting map f : X → B0 is the required BLD-branched cover.
Part II: Extension of BLD-maps with applications 7. Proof of Theorem 0.3 This section is devoted to the proof of the generalized Berstein-Edmonds Theorem 0.3. Notation 7.1 We use [Hu] as a standard reference for PL theory. We consider complexes that are polyhedra in some Euclidean space. If A is a simplex, A˙ denotes its boundary as a simplicial complex, A˚ its interior, and Aˆ its barycenter. The join of A with another simplex B is denoted by A ∗ B. The latter notation is used also for joinable simplicial complexes (see [Hu, p. 6]). We write I k = [−1, 1]k for the closed k-cube. The interior of a closed manifold M is denoted by int M, and cl denotes closure. If v0 , . . . , vk are points in some Euclidean space, we denote the (unoriented) k-simplex with vertices v0 , . . . , vk by hv0 , . . . , vk i. 7.2. Separating complex The idea of the proof of Theorem 0.3 is based, on the one hand, on the techniques from [HR, Sec. 5], which originate in [Ri1, Sec. 7.2, Sec. 8.3], and on the other hand, on [BE, Th. 6.2] in Section 7.7. Apart from [BE], our presentation is self-contained; in
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particular, no reference to [Ri1] or [HR] is necessary (see, however, the discussion in [Ri1, Sec. 1], or in [Ri3], for the motivation for some of the constructions to follow). In view of the Berstein-Edmonds theorem, we may assume p ≥ 3. At the end, we reduce the construction to the case where p = 2, and then we invoke the BersteinEdmonds extension. We start out by defining boundary collars in W to M1 , . . . , M p−1 as follows. We choose for j ≥ 1 a PL embedding h j : M j × [0, 1] → W such that h j (x, 0) = x and such that the images W j = h j (M j × [0, 1]) are disjoint and do not meet M0 (see [Hu, p. 36]). Set p−1 [ ˜ W = cl W \ Wj , j=1
M 1j
= h j (M j × {1}),
j = 1, . . . , p − 1.
Next fix a closed PL 3-ball B in int W˜ and a PL homeomorphism 8 : B → B 0 , where B 0 is the polyhedron in R 3 of the form B0 = Y × I 1, and Y ⊂ R 2 = R 2 × {0} ⊂ R 3 is a regular (2 p − 2)-gon with zero as center and diameter 2. Let the faces of Y be E 1 , . . . , E 2 p−2 in this order. We connect the (combinatorial) face F2 j = 8−1 (E 2 j × I 1 ) (see [Hu, p. 30] for terminology) of B to M 1j by a PL cylinder A j for j = 1, . . . , p−1, with a PL homeomorphism ψ j : [0, 1]× I 2 → A j such that ψ j ({0} × I 2 ) ⊂ M 1j , ψ j ({1}) × I 2 ) ⊂ F2 j , ψ j ((0, 1) × I 2 ) ⊂ int(W˜ \ B), and A j ’s are all disjoint. The cylinder A j can be constructed by taking some regular neighborhood of a polygonal arc joining M 1 and F2 j in int(W˜ \ B). j
Let C 0j be the join of E j × I 1 and J = (0, 0) × I 1 , and set C j = 8−1 (C 0j ), Z = W˜ \ int
j = 1, . . . , 2 p − 2,
p−1 [
(A j ∪ C2 j ) .
j=1
The set Z satisfies the condition for a PL 3-manifold at points in Z \ 8−1 (J ), and Z has two boundary components, M0 and |K |, where K is a simplicial 2-complex and a subcomplex of a triangulation Q of W (see Fig. 1). The complex K has the following properties: (a) |K | satisfies the condition of a PL 2-manifold at points of |K | \ 8−1 (J ); (b) |K | separates M0 , M1 , . . . , M p−1 from each other in W ; (c) W \ |K | consists of p components, V0 , . . . , V p−1 , such that M j ⊂ Vj,
j = 0, . . . , p − 1;
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Figure 1
(d)
there exists a PL homeomorphism g j : M j × [0, 1] → V j ,
g j (x, 0) = x, j = 1, . . . , p − 1.
We may assume that the triangulation Q of W is so chosen that K satisfies the following additional properties: (1) if v is a vertex of K and v 6∈ 8−1 (J ), then v belongs to an even number of 2-simplexes of K ; (2) if v is a vertex of K in 8−1 (J ), then v belongs to an even number of 2simplexes of K in each 8−1 (X ), where X is any of the four 2-cells that are faces of C20 j and meet J, j = 1, . . . , p − 1. Conditions (1) and (2) are satisfied as soon as K is a barycentric subdivision of some triangulation of |K |. (In [HR, Sec. 5.4] many of the complications arise from certain symmetry requirements.) If P is a simplicial complex, we let P i be the set of i-simplexes in P. The subcomplex of K whose space is |K | ∩ V j is denoted by K ∩ V j or K j . It follows from (1) and (2) that the number #K 2j of 2-simplexes of K j is even for j = 0, . . . , p − 1. Here we use the fact that |K j | is homeomorphic to an orientable closed 2-surface for
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j ≥ 1, and, on the other hand, #K 02
=
p−1 X
#K 2j .
j=1
For each j = 1, . . . , p − 1, the complex K j is a map complex in the terminology of [HR, Sec. 4.4], which means that each vertex of K j belongs to an even number of 2-simplexes. If A ∈ K 2 , then there is a unique j = 1, . . . , p − 1 such that A ⊂ ∂ V0 ∩ ∂ V j , and we say that A has classes 0, j. Conditions (1) and (2) divide the vertices of K into three classes α, β, γ , which are uniquely determined if we fix the vertices a and b of a side so that a ∈ α and b ∈ β. Next we define positive and negative simplexes in K 2 as follows (cf. [HR, Sec. 5.2]). Let A ∈ K 2 have classes 0, j. There is exactly one 3-simplex C in Q ∩ V j having A as a face. Let v be the vertex of C not in A, and let ω be the orientationpreserving affine map of C into R 3 such that ω(a) = 0, ω(c) = e1 , ω(b) = e2 . Here a, b, c are the vertices of A in classes α, β, γ , respectively, and ei is the standard ith basis vector in R 3 . If ω(v)3 > 0, then A is positive; otherwise it is negative. It follows from the construction that in each K j , j ≥ 1, the number of positive and negative 2-simplexes is the same. Consequently, this is true for the whole K . Fix a 1-simplex s of K in 8−1 (J ). Let j = 1, . . . , p − 1. In |K j | we first define a tree T j0 with the properties (a) the vertices of T 0 are the barycenters Aˆ of the simplexes A ∈ K 2 , and j
(b)
j
ˆ l] ˆ ∪ [l, ˆ B], ˆ where A and B are in K 2 with a each side of T j0 is of the form [ A, j ˆ l] ˆ is the line segment with common side l, which is not in 8−1 (J ). (Here [ A, ˆ endpoints Aˆ and l.)
The existence of the trees T j0 follows easily since |K j | is a surface and 8−1 (J ) does ˆ sˆ ], not separate. To T 0 we add the barycenter sˆ of the fixed 1-simplex s and a side [ A, j
where A is one of the two 2-simplexes in K j having s as a side. The resulting tree is denoted by T j . From the trees T j we build trees L k , k = 1, . . . , p − 1, by setting Lk =
p−1 [
Tj .
j=1 j6 =k
We give L k a partial ordering by letting the vertex sˆ be the last element. Hence for each side of L k with endpoints Aˆ and (A0 )∧ we know which one of Aˆ and (A0 )∧ precedes the other.
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7.3. Sheets Following the principles in [HR, Sec. 5.3], we now replace each 2-simplex A in K by a certain number of nearby PL disks, called sheets, the number depending on whether A is positive or negative. The teatment differs from [HR, Sec. 5.3] because this time we are dealing with an embedding into R m and not just into R 3 . Fix j ∈ {1, . . . , p − 1}, and let A ∈ K j . There is exactly one 3-simplex C in Q ∩ V j having A as a face. We ˆ C] ˆ joining the barycenters of choose a point q A in the interior of the line segment [ A, A and C. Bounds for the distance from q A to Aˆ are given later. If A is positive, we replace A by 2 p − j PL 2-disks A = A0 , . . . , A2 p− j−1 defined by Aµ = A˙ ∗ aµ , where A˙ is the boundary of A, ∗ is the join operation, and aµ = Aˆ +
µ ˆ (q A − A). 2p − j − 1
If A is negative, we replace it by p + j PL disks A0 , . . . , A p+ j−1 defined by Aµ = A˙ ∗ bµ , where
µ ˆ (q A − A). p+ j −1 We say that the Aµ ’s are inherited from A. With these definitions the sheets all lie in C. If A0 ∈ K 2j is another 2-simplex, then the sheets that are inherited from A0 can meet the sheets Aµ along their boundaries, but this happens only when A and A0 share a side. Notice that A and A0 can have a common 3-simplex C as above. We next perform a gluing process among the sheets that are inherited from simplexes A, A0 ∈ K 2j with a common side. This process includes a way of opening certain connections among different layers between the sheets. After the gluing, the sheets have been deformed near their boundaries; we continue to call them sheets inherited from A. After the gluings have been performed for all such pairs, and for all j = 1, . . . , p − 1, |K | is replaced by a considerably more complicated branched surface, a 2-complex that we call R. The gluing process is done so that the complement 0 W \ |R| consists of p components V00 , . . . , V p−1 such that V j0 is PL homeomorphic to V j (see Sec. 7.6), that bµ = Aˆ +
M j ⊂ V j0 ,
j = 0, . . . , p − 1,
and that for each A ∈ K and each j = 0, . . . , p − 1, there are (deformed) sheets that 0 are inherited from A and are contained in V j . A more detailed description of the gluing process is as follows. (The precise technique is given in Sec. 7.4.) We use the partially ordered tree L k given in Section 7.2 as
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Figure 2
a guideline to get for every A ∈ K 2j , j ≥ 1, j 6 = k, a connection within Vk0 from Mk to some of the sheets inherited from A. The principles of getting these connections (in the case where one has the space embedded in R 3 ) are described in [HR, Sec. 5.3] 0 and [Ri1, Sec. 7.2, Sec. 8.3]. The sets V00 , . . . , V p−1 appear in a cyclic order when ˆ To illustrate the order of the sets V 0 near one follows a line segment such as [q A , A]. j
the chosen side s of K in 8−1 (J ), we consider the case where p = 4. In Figure 2 we have drawn schematically the sheets inherited from the six simplexes A1 , . . . , A6 in K 2 with s as a side and from six other simplexes B 1 , . . . , B 6 in K 2 such that Ai and B i belong to a common K j , j ≥ 1, and share a side. The center of the picture represents the side s, and the appearance of the sets V j0 is given by numbers. In addition, the curves with arrows show the way toward M1 of connections from the layers between the sheets associated with V10 . For sheets that are inherited from 2-simplexes A in some K j , j > 1, the connections are determined by the tree L 1 . The partial
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Figure 3
ordering of L 1 determines the direction of the arrows in Figure 2. When there are two layers between the sheets inherited from the same simplex and these two layers have the same index, the tree L k (in Fig. 2, look at L 1 ) does not tell the rule for the connections between these layers. In fact, there is a choice to be made. The principle is that no loops should appear, and this principle could be presented through a new tree with layers associated with one Vk0 as vertices. To avoid still extra notation, we do not introduce such a tree. Connections for the layers of sheets inherited from simplexes A ∈ K 1 , and associated with V10 , are simpler. In Figure 2, there are four curves with arrows to indicate these kinds of connections. Figure 3 represents (for p = 4) a generic case of sheets inherited from some A, A0 ∈ K 12 with a common side together with the connections in each V j0 toward M j . Note the different type of choices of connections of layers with index 2 and 3. The gluing technique is also illustrated in [HR, Fig. 5]. Note, however, that the picture there represents the gluing in the so-called cave refinement process that is not needed now; here we are dealing with the final gluing in the terminology of [Ri1, Sec. 7]. 0 It follows from the construction that each V j ∩ R has the same even number 2l of sheets and that l = 3q, where q is half of the number of 2-simplexes in K 2 . To see this, let j = 0, . . . , p − 1. If A, A0 is a pair of adjacent 2-simplexes in some 0 K k , then V j ∩ R always contains exactly six sheets inherited from A and A0 . Hence 0
3#K 2 = #(R ∩ V j )2 , where we have extended the notation of the set of 2-simplexes to sheets. Recall that #K 2 is even.
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7.4. PL gluings Since we are working inside R m , we need some extra care in performing the gluings. Moreover, we give a precise description of how to do the gluing in a PL manner. (In this respect our presentation differs from that of [Ri1] and [HR].) So first let A and A0 be in K 2j , j ≥ 1, with a common 1-simplex l, which is not in 8−1 (J ). Let v be one of the vertices of l. Let H = star(v, Q ∩ V j ) be the star of v in Q ∩ V j as a 3-complex. (Recall that Q is a fixed triangulation of our 3-manifold W .) Then there is a PL homeomorphism ψ : |H | → I 2 × [0, 1] such that ψ(v) = 0 and ψ(|H | ∩ |K j |) = I 2 × {0}. The sheets inherited from A and A0 lie in 3-simplexes C and C 0 , which both belong to H . (Here C and C 0 can coincide.) Therefore, we can first perform the gluings of the images of the sheets under the map ψ in I 2 × [0, 1] near ψ(l). The problem now is that the images of the sheets under ψ can, in general, not be projected to a 2-plane so that the projection is one-to-one. Let H1 be a subdivision of H such that ψ is simplicial with respect to H1 , let l1 , . . . , lr be the 1-simplexes of H1 ∩ l in this order, let vi−1 and vi be the vertices of li , and let Bi ⊂ A and Bi0 ⊂ A0 be the two 2-simplexes in H1 ∩ |K j | with li as a common side. As before, there are uniquely determined 3-simplexes Ci and Ci0 in H1 with Bi and Bi0 as a face, where Ci and Ci0 can coincide. The images under ψ are denoted by li∗ = ψ(li ), and so on. In li∗ we choose points u i∗ and wi∗ such that ∗ ∗ |vi−1 − u i∗ | = |wi∗ − vi∗ | = σ/3, where σ = |vi−1 − vi∗ |. For some τ ∈ (0, σ/6) ∗ ∗ the intersection Dτ ∩ Aµ is contained in Ci for every sheet Aµ inherited from A, where Dτ is the τ -neighborhood of the line segment [u i∗ , wi∗ ] in R 3 , provided we have chosen q A near enough Aˆ (see the beginning of Sec. 7.3). In such neighborhoods Dτ we have good control of the shape of the images of the sheets under ψ since each ψ|Ci is affine. We take τ so small that the corresponding property holds for the sheets inherited from A0 , too. As l is not in 8−1 (J ), we can separate three different cases of connections to be performed by gluings (see Fig. 3): (a) connection from the layer between sheets Aµ−1 and Aµ to the layer between sheets A0ν−1 and A0ν ; (b) connection from the layer between sheets Aµ−1 and Aµ to the layer between sheets Aν−1 and Aν ; (c) connection from the layer between sheets Aµ−1 and Aµ to some Vk , k = 0, . . . , p − 1. The meaning of these three cases becomes clearer below. Case (a). We look at the image under ψ and perform the gluing in a neighborhood Dτ of some [u i∗ , wi∗ ]. Let T1 be the plane in R 3 containing li∗ and orthogonal to I 2 × {0}. We choose points d, e ∈ [u i∗ , wi∗ ] with δ = |d − e| < τ/4. Let T2 be the plane {ξ ∈ R 3 : |ξ − d| = |ξ − e|}. Let z 1 ∈ T1 ∩ T2 , ζ ∈ A∗µ ∩ T2 , ζ 0 ∈ A0∗ ν ∩ T2
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be points such that their distance to li∗ is δ, and that the third coordinate z 31 of z 1 is ˆ is small enough so that the parts of the images of the positive. We assume that |q A − A| sheets inherited from A and contained in Dτ both are projected injectively to I 2 × {0} by x 7→ (x1 , x2 , 0) and lie on “one side” of T1 , and similarly for A0 in the opposite side of T1 . Further, let z 0 be the midpoint of [d, e], and let z 2 be a point in T1 ∩ T2 with |z 2 − z 0 | = 2δ and positive third coordinate. Let λ be the simplicial map of the 3-simplex G = hd, e, z 2 , ζ i which maps affinely hd, z 0 , z 2 , ζ i onto hd, z 1 , z 2 , ζ i and he, z 0 , z 2 , ζ i onto he, z 1 , z 2 , ζ i, preserving the order of vertices. Let λ0 be the corresponding map with ζ replaced by ζ 0 . Now the connection for Case (a) is obtained as follows. The images 0∗ A∗0 , . . . , A∗µ−1 ,A0∗ 0 , . . . , Aν−1 remain unchanged. The rest of the sheet images inherited from A, say, A∗µ , . . . , Ar∗ , we deform to Aˇ ∗µ , . . . , Aˇ r∗ by using λ and setting Aˇ ∗h = A∗h ∩ ((I 2 × [0, 1]) \ G) ∪ λ(G ∩ A∗h ),
h ≥ µ.
Similarly, we deform the rest of the image sheets inherited from A0 by using λ0 . As a ˇ 0∗ result, A∗µ−1 ∪ Aˇ ∗µ ∪ A0∗ ν−1 ∪ Aν bounds a PL 3-ball. Case (b). We may assume µ < ν − 1. Let d, e, and z 0 be as in Case (a). Let ζ ∈ A∗µ ∩ T2 and ζ 2 ∈ A∗ν−1 ∩ T2 be points at distance δ from li∗ , and let ζ 1 be the midpoint of [ζ, z 0 ]. We map the 3-simplex G 1 = hd, e, ζ 2 , ζ i simplicially by a map λ1 that takes affinely hd, z 0 , ζ 2 , ζ i onto hd, ζ 1 , ζ 2 , ζ i and he, z 0 , ζ 2 , ζ i onto he, ζ 1 , ζ 2 , ζ i, preserving the order of vertices. The images A∗0 , . . . , A∗µ−1 and A∗ν , . . . , Ar∗ remain unchanged. The images A∗µ , . . . , A∗ we deform to Aˇ ∗µ , . . . , Aˇ ∗ by setting ν−1
Aˇ ∗h = A∗h ∩ ((I 2 × [0, 1]) \ G 1 ) ∪ λ1 (G 1 ∩ A∗h ),
ν−1
µ ≤ h ≤ ν − 1,
which clearly gives the desired connection. Again, A∗µ−1 ∪ Aˇ ∗µ ∪ Aˇ ∗ν−1 ∪ A∗ν bounds a PL 3-ball. Case (c). Suppose first that k 6 = 0. Then we use exactly the rule from Case (b) with ζ 2 ∈ Ar∗ ∩ T2 , where r is the last index. If k = 0, we use the rule from Case (b) with ζ ∈ A∗0 ∩ T2 , that is, µ is then the first index. Various connections in some neighborhood of l ∗ can be performed independently by choosing the line segments corresponding to [d, e] above small and far enough from each other. Moreover, connections that are performed near different sides l do not interact, provided the line segments [d, e] are small and stay away from the endpoints of the sides l. It remains to consider connections at 8−1 (J ). The only 1-simplex of K that has to be considered is s, which we fixed in Section 7.2 in connection with the definition
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of the trees L k . This time we make use of the PL map 8 : B → B 0 . For each pair k, j ∈ {1, . . . , p − 1}, k 6= j, we have to build a connection from Vk to the layer ˆ is a side in T j (see Sec. 7.2). between some Aµ−1 and Aµ , where A ∈ K 2j and [ˆs , A] We use a technique similar to the one in Case (a). Let us use ∗ to denote images under 8. As in Case (a), we choose points d, e ∈ s ∗ ⊂ J , and we assume that δ = |d − e| is small enough. Let T1 be the plane in R 3 through J such that C20 j is the reflection of 0 with respect to T , and let T = {ξ ∈ R 3 : |ξ − d| = |ξ − e|}. The line containing C2k 1 2 J divides T1 into two half-planes; let T1+ be the one of these such that the angle near [d, e] between T1+ and A∗µ−1 is smaller than the angle between T1+ and A+ µ. + ∗ 1 Next choose z ∈ T1 ∩ T2 and ζ ∈ Aµ−1 ∩ T2 with distance δ to J , and choose z 2 ∈ T1+ ∩ T2 with distance 2δ to J . Let z 0 be the midpoint of [d, e]. There are two elements in K k2 with s as a side. Let A0 be the one with the smaller angle between A0∗ and T1+ near J . Suppose A00 , . . . , Ar0 are the sheets inherited from A0 . Then let ζ 0 ∈ Ar0∗ ∩ T2 be the point with distance δ to J . Now we are in a position to define maps λ and λ0 exactly as in Case (a). More precisely, except for the image sheets 0∗ 0 A∗0 , . . . , A∗µ−1 and A0∗ 0 , . . . , Ar , which are deformed by using λ and λ , all other image sheets containing s ∗ remain unchanged. This then gives the required connection. There are no other types of connections at sides in 8−1 (J ) (see Fig. 2). 7.5. PL homeomorphism of Vk0 onto Vk Recall the definition of Vk0 from Section 7.3, which is now completed through the gluing process in Section 7.4. The space |R| of the 2-complex R is the union of all the new sheets. In the following, we use the word connection to mean any of the opening processes defined in Section 7.4. Fix k = 1, . . . , p−1. We start by collapsing inductively that “part” of Vk0 , where the boundary of Vk0 is inherited from 2-simplexes in some K j , j ≥ 1, j 6= k. After the gluing process, the original sheets have been deformed. According to our convention, we continue to call the deformed objects sheets. In particular, the new sheets inherited from some A are now denoted by A˜ µ . Note that we used the notation Aˇ µ in Section 7.4, Cases (a) and (b), but Aˇ µ was not the final deformed sheet because other connections are performed, too. Now let A ∈ K 2j , j ≥ 1, j 6 = k, be such that its barycenter Aˆ is an end of the tree L k (see Sec. 7.2). Suppose first that Vk0 appears only in one “layer” between A˜ µ−1 and A˜ µ , that is, that A˜ µ−1 ∪ A˜ µ ⊂ ∂ Vk0 only for one index µ. There is a unique A0 ∈ K 2j such that L k has a side connecting Aˆ and (A0 )∧ . The connection for V 0 to a layer between A˜ 0 and A˜ 0ν is performed by Case (a) in Section 7.4. k
ν−1
Recall that in Section 7.4 we denoted by A∗ , and so on, the image of A, and so on, under maps ψ; now let # denote the image under ψ −1 (see the beginning of Sec. 7.4). Using the notation in Case (a), we first observe that the open set bounded by A˜ ∗µ−1 ∪ A˜ ∗µ ∪ hd, e, z 1 i is the interior of a PL ball. Hence Vk0 collapses to U =
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Vk0 \# ∪ hd, e, z 1 i# , and there is a PL homeomorphism 3 : Vk0 → U ; more precisely, there is a subdivision Q 1 of the original triangulation Q so that 3 is simplicial with respect to Q 1 . We then continue similarly by collapsing U to U1 = U \ hd, e, z 1 , ζ 0 i# and obtain a PL homeomorphism 31 : U → U1 . Combining these two operations, we have reduced the number of sides of the tree L k by one, since U1 is precisely a “Vk0 ” if the side of L k connecting Aˆ and (A0 )∧ is not there. Suppose then that Vk0 appears in two “layers,” one between A˜ µ−1 and A˜ µ , and the other between A˜ ν−1 and A˜ ν . (There cannot be more such layers; see Fig. 2.) Assume µ < ν − 1. Suppose first that from both these layers there are connections to exactly one layer in Vk0 between A˜ 0h−1 and A˜ 0h . Collapsing both layers corresponding to A separately as above, we have again reduced the tree L k by one side. The other choice is that the connection from the layer between A˜ ν−1 and A˜ ν is performed to the layer between A˜ µ−1 and A˜ µ with the rule in Case (b) in Section 7.4. Then one of the layers, say, the one between A˜ ν−1 and A˜ ν , must have a connection to exactly one layer in Vk0 between A˜ 0h−1 and A˜ 0h . Using the notation in Case (b), now let be the open PL ball bounded by A˜ ∗µ−1 ∪ A˜ ∗µ ∪ hd, e, ζ 1 i. We first collapse Vk0 to U = Vk0 \ # ∪ hd, e, ζ 1 i# and then collapse U to U \ hd, e, ζ 1 , ζ 2 i# . After this we continue as in the case of only one layer, which then results again in the reduction of the number of sides of L k by one. The induction step is similar to the process described above except for the last ˆ sˆ ]. This is considered after steps, namely, the removing of the sides of the form [ A, the next paragraph. Next let A ∈ K k2 , and suppose that Vk0 appears in the layer between the sheets A˜ µ−1 and A˜ µ . There is always exactly one such layer (see Fig. 2). To “remove” that layer, we use collapsing as above with Case (b). This works because gluing is done according to Case (c), and this can be considered as a degenerate case of Case (b). ˆ sˆ ] in L k . If V 0 appears Finally, we turn to the last steps of removing the sides [ A, k ˜ ˜ only in one layer between some sheets Aµ−1 and Aµ , we use collapsing in a manner similar to Case (a) (see also the end of Sec. 7.4). If Vk0 appears also in another layer, say, between A˜ ν−1 and A˜ ν , there is a connection to the layer between A˜ µ−1 and A˜ µ . We first use collapsing associated with the connection as in Case (b), and then we continue as in the case of only one layer. With all the collapsings understood, we obtain a PL homeomorphism ϑ1 : Vk0 → G k , where ∂G k consists of Mk and another polyhedron Nk that is PL homeomorphic to |K k |. In fact, Nk is a result of a number of small deformations of the simplexes of K k . First each A ∈ K k is mapped onto the sheet Ar (r maximum index), then pullbacks by ψ of simple maps like λ, λ0 in Case (a), λ1 in Case (b), and so on, are applied. Hence, we easily get a PL homeomorphism ϑ2 : G k → V k by a small adjustment of Nk . The required PL homeomorphism is then ηk = ϑ2 ◦ ϑ1 : Vk0 → Vk .
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Finally, let k = 0. This case is easier since we only have connections of Case (c) in Section 7.4. The corresponding collapsings were considered above. As a result we get a PL homeomorphism η0 : V00 → V0 with the help of an auxiliary adjustment corresponding to ϑ2 above. 7.6. Increasing #K 2 In order to catch all degrees stated in Theorem 0.3, we need a method to increase #K 2 . The construction of K in Section 7.2 gives an even number 2q = #K 2 of 2-simplexes in K . We perform two types of subdivisions of the simplicial complex Q giving an increase in #K 2 . Recall from Section 7.2 that |Q| = W and K is a subcomplex of Q. To perform the first subdivision, pick any 2-simplex A in K with vertices a, b, c, and then add three vertices a 0 , b0 , c0 as in Figure 4. This gives a new complex that we call K 0 . To extend this subdivision to a subdivision Q 0 of Q, we simply use the join operation to d and e, where A ∗ d and A ∗ e are the simplexes in Q having A as a face. The complex K 0 is still a map complex (every vertex belongs to an even number of 2-simplexes), and it clearly also satisfies conditions (1) and (2) in Section 7.2. The number #K 2 is increased by six.
Figure 4
In the second subdivision we consider the case of a vertex v of K , which is not in 8−1 (J ) (see Sec. 7.2), such that v belongs to exactly four 2-simplexes of K . If l = hv, wi and l1 = hv, W1 i are opposite 1-simplexes in K with common vertex v, we add two new vertices lˆ and lˆ1 to K and get the subdivision in Figure 5. Again we extend this to Q by first using the join operation to adjacent 3-simplexes as in the first subdivision, and then also to those 3-simplexes in Q which are of the form l ∗ l ∗ and l1 ∗l1∗ , where l ∗ and l1∗ are some 1-simplexes in Hv = star(v, Q). For example, if l ∗l ∗ ˆ ∗ l ∗ and hl, ˆ wi ∗ l ∗ . is such a 3-simplex in Hv , it is replaced by two 3-simplexes, hv, li
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Figure 5
With this definition |Q| \ int |Hv | remains untouched. This second subdivision has the effect of increasing #K 2 by four. We can always start with the first type of subdivision and then continue with the second type because after the first type of subdivision the vertices a 0 , b0 , c0 each belong to exactly four 2-simplexes. Consequently, using these types alternately, we can have a new complex K 0 with (1/2)#K 02 any prescribed integer i ≥ (1/2)#K 2 + 5 = i 0 . (Note that if Q is obtained as a barycentric subdivision, then one can start with the second type.) By change of notation, let K be the complex with #K 2 = 2i. 7.7. Definition of the map ϕ Let i 0 be the integer from Section 7.6, and assume i ≥ i 0 . We start by defining ϕ on |R| (see the beginning of Sec. 7.5). We take S 3 embedded in R 4 as the unit sphere. In the equator S 2 of S 3 , we consider the unit sphere S 1 = (R 2 × {0}) ∩ S 3 . Let ω be a rotation of S 3 with dihedral angle 2π/ p and fixed point set S 1 . Let S + be one of the two components of S 2 \ S 1 , and set Yk0 = ωk S¯ + , Y0 =
p−1 [
k = 0, . . . , p − 1, (ω0 = id),
Yk0 .
k=0
We may assume that S 3 is given a C 1 -triangulation g : N → S 3 in the sense of [M, p. 81] such that the sets Yk0 are subcomplexes of S 3 . Write Yk = g −1 (Yk0 ), Y = g −1 (Y 0 ), and Y p = Y0 . The set N \Y consists of p components. Through a preliminary
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PL homeomorphism of N onto itself, we may assume that the ball Bk lies in the component bounded by Yk ∪ Yk+1 , k = 0, . . . , p − 1. On g −1 (S 1 ) we fix three points a 0 , b0 , c0 in an order to be fixed below. Each sheet 0 0 0 in |R| ∩ V k is also a sheet of (i) |R| ∩ V k−1 or (ii) |R| ∩ V k+1 . In case (i) we map a 0 given sheet in |R| ∩ V k by a PL homeomorphism onto Yk , and in case (ii) onto Yk+1 , such that vertices of classes α, β, γ are taken to a 0 , b0 , c0 , respectively. This defines ϕ on |R|. Recall from the end of Section 7.5 the definition of the PL homeomorphism ηk : 0 Vk → Vk , k = 0, . . . , p − 1. The definition of ϕ on |R| induces via ηk , and its boundary correspondence, a map ϕ˜k of |K k | onto the PL 2-sphere Yk ∪ Yk+1 . Let k ≥ 1. The set |K k | is one of the two boundary components of the PL 3-manifold V k , the other boundary component being Mk (see Sec. 7.2(d)). The map ϕ˜k is a branched cover of degree d = 3i. By the Berstein-Edmonds theorem ([BE, Th. 6.2]) cited in the introduction, we can extend ϕk and ϕ˜k to a PL sense-preserving branched cover ϕˆk : V k → X k of degree d, where X k is the closed PL ring bounded by Yk ∪ Yk+1 and ∂ Bk , provided we have chosen a 0 , b0 , c0 in the right order. The same order works 0 for all k ≥ 1. Using the map ηk again, we get the required extension of ϕ||R| to V k ; 0 0 that is, we have defined ϕ|V k : V k → X k for k ≥ 1. It remains to consider the case where k = 0. The difference now is that |K 0 | (= |K |) is not a manifold. This is not a serious problem, however, because the only trouble is in a neighborhood of 8−1 (J ) (see Sec. 7.2). Using the PL homeomorphism 8 : B → B 0 , we can collapse V0 to U0 = V0 \ B by a PL homeomorphism ζ0 : V0 → U0 . The set U 0 is a PL 3-manifold with two boundary components. One boundary component is M0 ; we call the other one P0 . The maps ζ0 ◦η0 : V00 → U0 and ϕ||R| induce by boundary correspondence a branched cover ϕ˜0 : P0 → Y0 ∪ Y1 with degree d. The orientations are compatible so that we can again apply [BE, Th. 6.2] to extend ϕ˜0 and ϕ0 to a sense-peserving PL-branched cover ϕˆ0 : U 0 → X 0 . Fi0 0 nally, with the help of ζ0 ◦ η0 we get the required extension ϕ|V 0 : V 0 → X 0 in the remaining part. This completes the construction of the required extension ϕ:W →N\
p−1 [
int Bk ,
k=0
which is a sense-preserving PL-branched cover. The proof of Theorem 0.3 is complete. 8. Geometric decomposition spaces and BLD-maps In [S3], Semmes defined a class of decomposition spaces of R 3 and provided them with metrics with a special self-similarity property. In fact, his spaces are all subsets of R 4 with the induced metric, and they are constructed by the aid of similarities from
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what Semmes calls an excellent package (see Sec. 8.1). Here we consider spaces that arise from an excellent package with an additional contractibility condition (see Sec. 8.1(h)), which then leads to a cell-like decomposition. As a result, we are then in the framework of generalized 3-manifolds by [Da, Cor. 1A, p. 191]. Moreover, it follows from Semmes’s work in [S3] that such spaces are of type A as defined in Definition 5.1. In this section, we show by means of Theorem 0.3 that the compact generalized 3-manifolds constructed from the excellent packages with a contractibility condition all admit BLD-maps onto S 3 (see Th. 8.17). The special example arising from the Bing doubling decomposition space leads to a space X that is homeomorphic to S 3 but is not quasi-symmetrically equivalent to S 3 [S3, p. 244]. This particular example shows a sharp contrast between finite-to-one and injective cases. 8.1. Excellent packages We use the book [Da] as a general reference on decomposition spaces. We are concerned with special decomposition spaces of R 3 arising through a defining sequence (Ci ) in the terminology of [Da, p. 61]. This means that each Ci ⊂ R 3 is a compact (not necessarily connected) 3-manifold with boundary and Ci+1 ⊂ int Ci . The sequence (Ci ) defines a decomposition space R 3 /G, where the decomposition G is the T T set of components of i Ci and the singletons of R 3 \ i Ci (see [Da, p. 8]). Following Semmes [S3, p. 200], an initial package consists of bounded smooth domains D, D1 , . . . , Dk ⊂ R 3 = R 3 × {0} ⊂ R 4 , with D j ⊂ D and D i ∩ D j = ∅ for i 6= j, together with sense-preserving diffeomorphisms ϕ j : U → U j , where U (resp., U j ) is a neighborhood of D (resp., D j ) and where ϕ j (D) = D j . An initial package determines a defining sequence (Ci ) by setting C0 = D, C1 = S S j ϕ j C 1 , and so on. An excellent package [S3, p. 202] consists j D j , C2 = of an initial package as above, bounded smooth domains , ω1 , . . . , ωk , 1 , . . . , k of R 4 , sense-preserving similarities ψ j : R 4 → R 4 , j = 1, . . . , k, with dilation factor ρ ∈ (0, 1/10), and a diffeomorphism θ : R 4 → R 4 such that (a) ω j ⊂ , j ⊂ , ωi ∩ ω j = ∅, i ∩ j = ∅ for i 6= j; (b) ∩ R 3 = D, ω j ∩ R 3 = D j , and ∂, ∂ω j intersect R 3 transversally; (c) ψ j (R 3 ) = R 3 , ψ j () = j ; (d) θ is the identity on a neighborhood of R 4 \ ; 3 (e) θ (ω j ) = j and θ = ψ j ◦ ϕ −1 j on a neighborhood of D j in R . Following [S3, p. 203], we set S0 = {∅} and Sl = (a1 , . . . , al ) : a j ∈ {1, . . . , k} , l ≥ 1. For α = (a1 , . . . , al ) ∈ Sl , l ≥ 1, we define recursively a similarity ψα : R 4 → R 4 and a domain α ⊂ R 4 by setting ψα = ψα 0 ◦ ψal , α = ψα 0 (al ), where α 0 =
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(a1 , . . . , al−1 ) if l > 1 and α 0 = ∅ if l = 1. Here we understand that ψ∅ is the identity map and ∅ = . To build the decomposition space with a self-similarity property, Semmes starts out with the compact 3-manifold k [ 6=θ D\ Dj ,
(8.2)
j=1
smoothly embedded in R 4 (see [S3, p. 205]). For j ≥ 1, set M = (R \ D) ∪ j
3
j−1 [ [
[ 6α ∪ α ∩ R 3 , α∈S j
l=0 α∈Sl
where 6α = ψα (6), 6∅ = 6. The set M j is a smoothly embedded 3-manifold in R 4 which converges in the Hausdorff topology to M = (R 3 \ D) ∪
∞ [ [
6α ∪ F,
(8.3)
l=0 α∈Sl
where F=
∞ [ \
α .
l=1 α∈Sl
The set F is a Cantor set in R 3 if k ≥ 2, and a point if k = 1, and M \ F is a smoothly embedded 3-manifold in R 4 . By construction, M satisfies the self-similarity condition α ∩ M = ψα ( ∩ M) for all α.
(8.4)
We provide M with the metric induced from R 4 . The decomposition space R 3 /G determined by the defining sequence (Ci ) of the given initial package is homeomorphic to M by [S3, Lem. 3.21]. In order to get into the framework of BLD-maps with good analytic properties, we assume that the excellent package satisfies the following additional conditions: (f) 4kρ < 1, where ρ is the dilation factor of ψ j ; (g) the sets k [ R3 \ D and E = D\ Dj j=1
(h)
are connected; D j is contractible in D.
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Condition (f) is a stronger condition than kρ 3 < 1, which appears in [S3, Lem. 3.45, p. 212]. Condition (g) is [S3, (3.69), p. 218]. In the specific examples considered in [S3], (h) is satisfied. Note that (g) and (h) are conditions on the initial package only. Condition (h) is used in Proposition 8.8 and the strengthened condition (f) in Theorem 8.17. For future use we introduce the notation E α = ϕα (E), where ϕα is defined analogously to ψα from the maps ϕ1 , . . . , ϕk . 8.5. Properties of M We fix an excellent package and assume that the conditions (f) – (h) are also satisfied. PROPOSITION 8.6 Let M be the space defined in (8.3). Then (1) M is of type A123 (see Def. 5.1), and (2) M is quasi-convex.
Proof The condition (A1) is clear with the observation that H3 (F) = 0 for the set F in (8.3). The space M lies in R 4 with the induced metric, so (A3) is true even globally. [S3, Lem. 3.45, p. 212] shows that M is (A2) with uniform bounds; here condition (f) is used. By [S3, Prop. 3.70, p. 218], M is quasi-convex; here condition (g) is used. The proposition follows. 8.7. Cell-like sets Recall that a compact set C in an n-manifold Y is cell-like if for every neighborhood U of C there is another neighborhood V of C such that V is contractible in U . A decomposition G of Y is cell-like if G is an upper semicontinuous decomposition (see [Da, p. 13] for the definition) and if each g ∈ G is a cell-like set. The following proposition is an easy consequence of condition (h) (see also [Da, Prop. 4, p. 248]). PROPOSITION 8.8 The decomposition G of R 3 defined by the given initial package is cell-like.
If G is a cell-like decomposition of some n-manifold Y , then (1.5) holds for every x ∈ X = Y/G by [Da, Prop. 1, p. 191]. If dim(Y/G) < ∞, then Y/G is ENR by [Da, Cor. 12B, p. 129]. If, moreover, n = 3, then dim Y/G ≤ 3 by [Da, Th. 12, p. 141]. From these facts, together with Proposition 8.8 and Example 1.4(c), we obtain the following proposition.
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PROPOSITION 8.9 The space M defined in (8.3) is a locally contractible generalized 3-manifold.
Remarks 8.10 (a) The space M satisfies, in fact, the stronger condition described in Example 1.4(c) (for n = 3). (b) From M ⊂ R 4 we immediately get the weaker estimate dim R 3 /G ≤ 4 for the dimension, which would suffice to prove Proposition 8.9. An initial package fits also into the setting of [Da, Prop. 3, p. 248] from which dim R 3 /G ≤ 3 could alternatively be concluded. PROPOSITION 8.11 The space M is of type A4 (see Def. 5.1).
Proof The idea is to use the local contractibility from Proposition 8.9 together with the self-similarity expressed in (8.4). Since 6, as defined in (8.2), has a smooth open neighborhood in M, we can fix a smaller such neighborhood 6 0 of 6 where the condition of local linear contractibility holds; that is, there exist C > 1 and r0 > 0 such that for each x ∈ 6 0 and 0 < r ≤ r0 , B 4 (x, r ) ∩ M is contractible in B 4 (x, Cr ) ∩ M.
(8.12)
The complement M \ is just R 3 \ , so we may assume that (8.12) holds also for x ∈ M \ and 0 ≤ r ≤ r0 . Let R0 be the distance of ∂ to the set K0 =
k [
j ∩ M.
j=1
Then for each x ∈ K 0 there exists r x > 0 such that B 4 (x, 2r x ) ∩ M contracts in B 4 (x, R0 ) ∩ M. We cover K 0 by sets B 4 (xi , r xi ) ∩ M, i = 1, . . . , q. Let r00 = min {r x1 , . . . , r xq , r0 }. Then (8.12) is true for some C > 1 if (i) x ∈ (M \ ) ∪ 6 0 and 0 < r ≤ r00 , or (ii) x ∈ K 0 and ρr00 ≤ r ≤ r00 . Recall that ρ is the dilation factor. In order to get (8.12) for smaller r in K 0 , we use the similarities ψα . To illustrate this, let us look at the first step. Condition (8.12) is clearly true for x ∈ ψ j (6 0 ), j = 1, . . . , k, and 0 < r < ρr00 by (i). Condition (ii) implies that (8.12) is true for x ∈ ψ j (K 0 ), j = 1, . . . , k, and ρ 2r00 ≤ r ≤ ρr00 . Since K0 ⊂
k [ j=1
ψ j (6 0 ) ∪
k [ j=1
ψ j (K 0 ),
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we have in the first step extended the range of r ’s to ρ 2r00 ≤ r ≤ ρr00 (with the same constant C). By continuing similarly, we conclude the proof of the proposition. 8.13. Summary We have now proved that M is a generalized 3-manifold of type A, and, in addition, quasi-convex. Moreover, M satisfies a weak (1,1)-Poincar´e inequality as defined in [HK3, Sec. 5] (see Sec. 9.6). This latter property follows from [S3, Prop. 10.8]. Remark 8.14 In [S3] Semmes also considers a space M˜ obtained from M as follows. First we fix a sequence of disjoint 4-balls B 4 (z j , 2r j ), j = 1, 2, . . . , converging to the origin with z j ∈ R 3 . Choose a ball B = B 4 (0, R) containing . Fix similarities A j : R 4 → R 4 such that A j (R 3 ) = R 3 , A j (B) = B j = B(z j , r j ). Then define ∞ ∞ [ [ Bj ∪ A j (B ∩ M j ). M˜ = R 3 \ j=1
j=1
The set M˜ is homeomorphic to R 3 by construction. Semmes uses examples of spaces M˜ to study various questions of parametrization by R 3 (see [S3, p. 232 – 240]). We now build a compact space X by using M. Let B 4 (0, R) be a ball containing . 4 We take the part B (0, 2R) ∩ M of M and attach to it the hemisphere 3 S+ (0, 2R) = {x ∈ R 4 : |x| = 2R, x4 ≥ 0};
that is,
4
3 X = S+ (0, 2R) ∪ B (0, 2R) ∩ M
(8.15)
with the metric induced from R 4 . The space X has all the properties listed in Section 8.13. PROPOSITION 8.16 The space X is orientable.
Proof As in the proof of Proposition 8.11, we make use of the self-similarity given in (8.4). Choose a point x ∈ F. (Recall F from (8.3); F is either a Cantor set or, if k = 1, a single point.) Since X is a generalized 3-manifold (in the stronger sense of Exam. 1.4(c); see Rem. 8.10 (a)), there is a small open neighborhood V of x such that Hc3 (V ) = Z, q
Hc (V ) = 0,
q 6 = 3.
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Since the standard homomorphism in (1.2) is an isomorphism if W ⊂ V is connected, we may assume that V = α ∩ M for some α. Let U = ∩ M and A = X \ U . By (8.4), Hc3 (U ) = Z, q
Hc (U ) = 0,
q 6 = 3.
3
3 (0, 2R) ∪ (B (0, 2R) \ D), A is homeomorphic to a compact subset Since A = S+ 0 3 A of S with S 3 \ A0 a smooth domain. By Alexander duality (see [Do, p. 301]),
Hc2 (A0 ) ' H˜ 0 (S 3 \ A0 ) = 0, Hc3 (A0 ) ' H˜ −1 (S 3 \ A0 ) = 0, where H˜ i denotes reduced singular homology. From the exact sequence δ
τ
i∗
· · · → Hc2 (A) −→ Hc3 (U ) −→ Hc3 (X ) −→ Hc3 (A) → · · · , we obtain Z = Hc3 (U ) ' Hc3 (X ), as required. We fix the orientation on X such that the orientation of the subset B 3 (0, 2R) \ D of X agrees with the standard orientation of R 3 . THEOREM 8.17 Let X be the space defined in (8.15). Then there exists an integer d0 ≥ 3 with the following property: for each integer d = 3i ≥ d0 there exists a BLD-map f : X → S 3 of degree d.
Proof Let 6 be the compact oriented smooth 3-manifold defined in (8.2). By (g), the domain D has one boundary component. Hence 6 has k + 1 boundary components, all diffeomorphic to each other. By [M, Th. 10.6], there exists a C 1 -triangulation h : W → 6 (in the sense of [M, p. 81]) of a polyhedron W ⊂ R 4 with the following property. If M0 = h −1 (∂ D), M j = h −1 (∂(R 3 ∩ j )), j = 1, . . . , k, then each boundary map ψ 0j = h −1 ◦ ψ j ◦ h : M0 → M j , j ≥ 1, is simplicial with respect to the simplicial complexes M0 and M j . Here we recall the notations ψ j and j from Section 8.1. Let τ : N → S 3 be a C 1 -triangulation of S 3 , where N is the space |P| of a simplicial 3-complex P in R 4 such that P contains one (closed) regular 3-simplex A0 . We give W and N orientations induced by h and τ . Next fix sense-preserving similarities η j of A0 into itself with dilation factor ρ (given in the definition of X ) such that the simplexes B j = η j A0 satisfy dist(B j , Bi ) ≥ ρ diam A0 , j 6 = i, and
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dist(∂ A0 , B j ) ≥ ρ diam A0 , j = 1, . . . , k. This is possible because of the condition (f) in Section 8.1. Write B0 = N \ int A0 . We want to apply Theorem 0.3 to W and the PL balls B0 , . . . , Bk in N . To this end, let d0 be the bound given in Theorem 0.3, and let ϕ0 : M0 → ∂ B0 = ∂ A0 be a sense-preserving PL-branched cover of degree d = 3i ≥ d0 (with the orientations given in Theorem 0.3). On each M j , j ≥ 1, define a sense-preserving PL-branched cover ϕ j : M j → ∂ B j of the same degree d by ϕ j = η j ◦ ϕ0 ◦ h −1 ◦ ψ −1 j ◦ h|M j . Theorem 0.3 gives an extension of the ϕ j ’s to a sense-preserving PL-branched cover S ϕ : W → N \ kj=0 intB j of degree d. We transfer ϕ to 6 via h and get a map ξ0 = ϕ ◦ h −1 : 6 → N \
k [
intB j .
j=0
For α ∈ Sl we defined ψα in Section 8.1. From the η j ’s we define similarly ηα . Then extend ξ0 to ( ∩ M) \ F as a map ξ by setting ξ |6α = ηα ◦ ξ0 ◦ ψα−1 .
(8.18)
The map ξ is BLD because ξ0 is BLD and because ψα and ηα are similarities with the same dilation factor ρ |α| . The map ξ extends by construction continuously to F. Let ξ1 be this extension. In fact, it follows from (8.18) that ξ1 |F is bi-Lipschitz; accordingly, ξ1 is BLD. 3 The next step is to extend ξ1 to C = B (0, 2R) \ D = B 4 (0, 2R) ∩ M \ ( ∩ M). Here we only need to use the theorem of Berstein and Edmonds [BE, Th. 6.2], because C has just two boundary components. Fix a closed PL 3-ball E in N \ A0 , fix a C 1 -triangulation of C, and, with appropriate orientations, fix a PL-branched cover σ : S 2 (0, 2R) → ∂ E of degree d. Then [BE, Th. 6.2] applied to C and the boundary maps σ and ξ |∂ D gives a PL-branched cover of degree d from C onto N \int(E ∪ A0 ). 3 (0, 2R); call it ξ . It remains to extend Hence we obtain an extension of ξ1 to X \int S+ 2 3 ξ2 to S+ (0, 2R). This is done simply by coning the boundary map σ with respect to 3 (0, 2R) and int E. We have obtained a sense-preserving suitable vertex points in int S+ 0 PL-branched cover f : X → N of degree d. Composing with the triangulation τ gives the required map f = τ ◦ f 0 : X → S 3 , and the theorem follows. Remark 8.19 (a) The proof of Theorem 8.17 shows that the space M (as defined in (8.3)) admits a finite-to-one BLD-map onto R 3 . (b) After this paper was submitted, we learned about a result of Hirsch [Hi, Th. 3.1], which can be used to show that, in the situation of Theorem 8.17, there exists a degree 3 BLD-map from X to S 3 .
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9. Examples and applications In this section, we display some examples and applications of the results in previous sections. 9.1. Decomposition spaces Specific choices of excellent packages as in Section 8.1 lead to some familiar decomposition spaces of geometric topology. Three such spaces from the point of view of Section 8 were discussed by Semmes in [S3]: the Whitehead continuum space, Bing’s dogbone space, and Bing’s double. Only the last of the three is a manifold. Semmes shows in [S3, p. 244] that the space X in (8.15), associated with Bing’s double, is not bi-Lipschitz or even quasi-conformally homeomorphic to S 3 , not even locally near the points in the Cantor set F. By Propositions 8.6, 8.11, and 8.16, X is an oriented generalized 3-manifold of type A, and by Theorem 8.17 there exists a BLD-map f : X → S 3 . We refer to [S3] for a more in-depth discussion of these examples and the question of parametrization. There is one particular consequence of the preceding discussion that merits listing here: there exists a branched cover F : S 3 → S 3 such that for no homeomorphism h : S 3 → S 3 is F ◦ h : S 3 → S 3 quasi-regular. To see this, consider the space X in (8.15) associated with Bing’s double. By precomposing the BLD-map f : X → S 3 by a sense-preserving homeomorphism ϕ : S 3 → X , we obtain a branched cover F = f ◦ϕ : S 3 → S 3 . If g = F ◦h = f ◦ϕ ◦h is quasi-regular (see Sec. 9.2) for some homeomorphism h : S 3 → S 3 , one infers from the basic properties of quasi-regular maps (as in [Ri2, II.4]), and from the properties of BLD-maps (as in Prop. 4.13 in this paper), that the map g = ϕ ◦ h : S 3 → X is quasi-conformal in the sense of the infinitesimal metric definition. The results in [HK3, Sec. 4], together with the result in [S3, p. 244], then give a contradiction, for they imply that the space X is not quasi-conformally equivalent to the standard sphere. We still do not know if there are homeomorphisms h i : S 3 → S 3 , i = 1, 2, such that h 1 ◦ F ◦ h 2 is quasi-regular if F : S 3 → S 3 is as above (cf. [HSe, Question 29]). Heinonen and Sullivan discuss in [HSu] the problem of finding (local) biLipschitz coordinates for generalized manifolds. Examples like those above demonstrate the sharpness of the conditions given in [HSu]. Namely, by pulling back the standard coframe on S 3 , we obtain differential 1-forms ρ1 , ρ2 , ρ3 on X with bounded measurable coefficients that are closed (in the sense of distributions) and satisfy ρ1 ∧ ρ2 ∧ ρ3 ≥ c > 0 almost everywhere in X . In [HSu] a collection of forms as above is called a CartanWhitney presentation of the metric gauge determined by X . The existence of such a presentation is thus guaranteed by Proposition 8.16, and Semmes’s result about the nonexistence of local bi-Lipschitz parametrizations implies that the residue,
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Res(ρ, p), of no Cartan-Whitney presentation ρ = (ρ1 , ρ2 , ρ3 ) can be 1 at a point p ∈ F (for the terminology here, see [HSu]). Similar remarks apply to the examples based on the Whitehead continuum, Bing’s dogbone space, and other similar constructions, for example, those mentioned in Remark 8.14. The beautiful book of R. Daverman [Da] offers a thorough discussion of various decomposition spaces (see, in particular, [Da, Sec. II.9]). We leave it to the reader to visualize the particular examples of BLD-maps that arise in this manner. 9.2. Nonexistence of radial limits Recall that a continuous map f from a domain in R n into R n is quasi-regular if f is 1,n in the local Sobolev space Wloc and if the (formal) differential d f (x) of f satisfies n |d f (x)| ≤ K det d f (x) for a.e. x, for some finite K ≥ 1 (see [Re2], [Ri2] for the basic theory of these maps). BLD-maps are particular examples of quasi-regular maps by Section 6.13. The boundary behavior of BLD-maps is often easy to describe by using only definition (0.2) (cf. [MV¨a]). In contrast, the boundary behavior of bounded quasi-regular maps of the unit ball B n , say, is not well understood. For instance, it is not known whether a quasi-regular map f : B n → B n has a single radial limit if n ≥ 3. (Dimension n = 2 is special here because quasi-regular maps then factor as quasi-conformal maps composed with holomorphic functions by Bojarski’s measurable Riemann mapping theorem. In particular, radial limits exist in a set of positive Hausdorff dimension.) Martio and U. Srebro [MS] have recently shown, by an explicit construction, that there exist bounded quasi-regular maps of B n with no radial limits in an exceptional set of any prescribed Hausdorff dimension less than n − 1 on the boundary ∂ B n . The exceptional sets provided by Martio and Srebro are all totally disconnected. Next we indicate how the generalized Berstein-Edmonds Theorem 0.3 can be used to give a large family of similar examples. In our examples, the exceptional sets can be complicated topologically. The examples of Martio and Srebro correspond, in a sense, to the case where p = 0 in the proof of Theorem 9.3. One should note, however, that the maps constructed in [MS] are locally injective; our examples are necessarily branched. 9.3 Let 0 be a geometrically finite torsion free Kleinian group without parabolic elements acting on the 2-sphere S 2 with the limit set 30 not the whole sphere. Then there exists a bounded quasi-regular map f : B 3 → B 3 such that f has no radial limit at points in 30 . THEOREM
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Proof The group 0 acts in the hyperbolic 3-space, in its ball model B 3 , with the quotient M 3 = B 3 / 0 a hyperbolic 3-manifold. The action of 0 on S 2 \ 30 uniformizes finitely many Riemann surfaces 60 , . . . , 6 p , p ≥ 0, and we can compactify M by adding these surfaces to it. The resulting manifold W is a compact 3-manifold with boundary, and the assumption of geometric finiteness guarantees that the interior of W , in some fixed PL-metric, is quasi-conformally equivalent to the hyperbolic metric in M. Thus the map f = F ◦ π, where π : B 3 → M is a locally isometric covering projection and F : W → R 3 is a bounded BLD-map provided by Theorem 0.3, is a bounded quasi-regular map. (Note that the case of one boundary component, p = 0, can be reduced to Theorem 0.3 as well; we leave the details to the reader.) Clearly, the limit f (r w), r → 1, cannot exist for points w ∈ 30 . The theorem follows. It is well known that the limit set 30 , as in Theorem 9.3, must have Hausdorff dimension less than 2, but it can be arbitrarily close to 2 (see, e.g., [BJ] or [Ca] for further discussion and references). 9.4. Lipschitz maps with strange fibers In nonlinear elasticity, one is interested in finding topological properties in a mapping ˇ [MTY], for example). Reshetnyak from its analytic data only (see [Ba], [HK1], [IS], proved in 1966 (see [Re1]) that quasi-regular maps, as defined in Section 9.2, are branched covers or constants. One way to express Reshetnyak’s theorem is to consider the dilatation function |d f (x)|n K f (x) = det d f (x) 1,n for a map f ∈ Wloc , where it is understood that K (x, f ) = 1 at points x where the formal differential does not exist, or where det d f (x) = 0. Reshetnyak’s theorem states that if K f is in L ∞ , then f is a discrete and open map or constant (locally). ˇ ak [IS] ˇ proved that in dimension n = 2, it suffices to assume T. Iwaniec and V. Sver´ 1 K f ∈ L loc for the same conclusion. They also conjectured that K f ∈ L n−1 loc suffices p in all dimensions. It is now known that K f ∈ L loc for some p > n − 1 suffices (see [MVi], [ViM]). Earlier, J. Ball [Ba, Exam. 1, p. 317] had exhibited examples of nonconstant p maps f : R n → R n that take a line segment to a point and that have K f ∈ L loc for all p < n − 1. In the next theorem, we point out how the geometric decomposition space theory, as in Section 8, and the generalized Berstein-Edmonds Theorem 0.3 can be
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used to give examples of Lipschitz maps f : S 3 → S 3 , of nonzero degree, with more exotic fibers than line segments, and with K f still in some local Lebesgue space. THEOREM 9.5 For each excellent package as defined in Section 8.1 and for each 0 < p < 2, there exist an equivalent excellent package with defining sequence (Ci ) and a sensepreserving Lipschitz map f : S 3 → S 3 such that (a) f |S 3 \ ∩i Ci is a finite-to-one branched cover; T (b) each component of i Ci is a fiber of f ; p (c) K f ∈ L loc .
We call two excellent packages equivalent if there is a diffeomorphism of R 4 that conjugates the pertinent maps in the packages. In (c), we use Lebesgue 3-measure and define K f in Euclidean terms in R 3 . Note that f has nonzero degree. The map f to be constructed depends on the particular choice of exponent p. Proof Fix a defining sequence (Ci ) and 0 < p < 2; we have the freedom to choose (Ci ) up to a diffeomorphism. Here and below we use the notation from Section 8. Each domain D, D1 , . . . , Dk in the package is a handle-body in R 3 with smooth boundary. We assume that D is a smooth regular neighborhood of its core, where by a core we mean a 1-polygon inside D. We assume that D is essentially a 1neighborhood of γ , up to a small but fixed error. We also assume that the length `0 of the core is so large that we can choose D to satisfy |D| ≤ 4π`0 , where | · | denotes Lebesgue 3-measure. Now fix ε > 0 small depending only on p and the package, to be defined later. The cores γi of Di lie inside D subject to some topological constraints; in any case, we can choose sense-preserving diffeomorphisms ϕi : D → Di , ϕi (γ ) = γi , such that det dϕi ≤ ηε 2 and that Lip ϕi−1 ≤ ε−1 , where η > 1 is a fixed constant and Lip stands for the Lipschitz constant. Thus the Di ’s are essentially ε-neighborhoods of the γi ’s. The dilation factor ρ of the similarities ψ1 , . . . , ψk is at our disposal, and we choose ρ = ε. It follows that Lip(ψα ◦ θ ◦ ϕα−1 ) ≤ C ρ n ε−n = C
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and that det dψα det dϕα−1 ≥ (ρ 3 /ηε2 )n = (ε/η)n for |α| = n. Here and below, C denotes any positive constant independent of α. Now define f = F ◦ ψα ◦ θ ◦ ϕα−1 on E α , where F is a BLD-map from the decomposition space M(= R 3 /G) to R 3 . The existence of F is guaranteed by the study in Section 8 (see Rems. 8.19). We claim that f has the desired properties. To this end, first observe that f is globally Lipschitz. Second we note that det d f ≥ C(ε/η)n on E α if |α| = n, and then we compute Z XZ (det d f )− p d x = D
(det d f )− p d x Eα
α
≤C
X
≤C
X
|E α |ε− p|α| η p|α|
α
k n ηn ε2n− pn η pn < ∞,
n
provided we choose ε small enough so that kε2− p η p+1 < 1. Finally, outside the domain D the integrability requirement is clearly satisfied. This completes the proof of the theorem. 9.6. BLD-maps and Poincar´e inequalities Spaces that admit a Poincar´e inequality have turned out to be a proper environment for abstract first-order calculi. This fact was implicitly used in the proof of our Theorem 6.18. The validity of such an inequality is a bi-Lipschitz invariant property of a space. We next demonstrate that, to some extent, this invariance remains true when BLDmaps are used. (For calculus in spaces with a Poincar´e inequality, see [Che], [HaK], [HK3], [S5], [Sh], for example.) Recall (see [HK3]) that a metric measure space (X, µ) admits a (weak) (1, 1)Poincar´e inequality if there exist constants C ≥ 1 and τ ≥ 1 such that Z Z |u − u B | dµ ≤ C diam B ρ dµ (9.7) τB
B
for all balls B in X , for all bounded measurable functions u in the enlarged ball τ B, and for all Borel functions ρ : X → [0, ∞] with the property that Z |u(a) − u(b)| ≤ ρ ds γab
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whenever γab is a rectifiable curve joining two points a and b in τ B. Thus, ρ is an upper gradient of u in τ B (cf. the proof of Lem. 6.19). In (9.7), u B denotes the mean value of u in B. THEOREM 9.8 Let X be a complete, quasi-convex oriented generalized n-manifold, and assume that there exists a BLD-map of finite multiplicity either from X onto R n or from R n onto X . Then X admits a weak (1, 1)-Poincar´e inequality with constants depending only on the data associated with X and f .
The harder case in Theorem 9.8 is to establish the Poincar´e inequality on X if X admits a BLD-map onto R n . To this end, we rely on the work of Semmes [S2], in addition to the results from Part I of this paper. It is particularly crucial here to have the path-lifting property of BLD-maps, as explained in Section 3.3. The case where R n is the receiving space is also more interesting in light of the existing examples (cf. Section 8). Proof Assume first that there exists a surjective BLD-map f : X → R n of finite multiplicity. Then X is Ahlfors n-regular by Proposition 6.3. Under these conditions, the validity of a weak (1, 1)-Poincar´e inequality essentially follows from the work [S2], but a few explanatory remarks are due. Semmes shows in [S2] that a weak (1, 1)-Poincar´e inequality holds in every complete Ahlfors n-regular metric space X , provided there exists a constant C > 1 so that the following assertion is true: given x, y ∈ X , there exists a Lipschitz map F : X × (0, |x − y|) → S n with Lipschitz constant satisfying Lip F|X × (ε, |x − y| − ε) ≤ C ε−1 ,
0 < ε < |x − y|/2,
such that the maps Ft = F(·, t) : X → S n , 0 < t < |x − y|, all have nonzero degree and satisfy Ft |X \ B(x, Ct) ≡ σ, 0 < t < |x − y|/2, and Ft |X \ B(y, C(|x − y| − t)) ≡ σ,
|x − y|/2 < t < |x − y|,
for some fixed point σ ∈ S n (see [S2, Sec. 12 and App. B]). To show the existence of a map F as above, we use Proposition 4.13, and postcompose our BLD-map f : X → R n with an appropriate Lipschitz map R n → S n . Indeed, if B(x, r ) is a ball in X , then there is a ball B 0 = B( f (x), r 0 ) in R n ,
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with r 0 comparable to r , such that B 0 ⊂ f (B) by Proposition 4.13; moreover, the x-component U of f −1 (B 0 ) lies in B(x, r1 ), and it contains B(x, r2 ), with both r1 and r2 comparable to r . Choose a homotopically nontrivial C/r 0 -Lipschitz map h B 0 : R n → S n such that h B 0 |R n \ B 0 ≡ σ for some fixed point σ ∈ S n . Then define a map FB by FB = h B 0 ◦ f in U , and FB ≡ σ in X \ U . This map FB : X → S n is homotopically nontrivial, and it takes the value σ outside U ; in particular, F|X \ B(x, r1 ) ≡ σ . Because the radii r 0 , r1 , and r2 are all comparable to r , it is easy to see how, given a pair of points x and y in X , the above construction of the map FB provides a continuous family of Lipschitz maps Ft , as required by Semmes’s criterion. We leave the details to the interested reader. This proves the first part of the theorem. Now assume that there exists a surjective BLD-map f : R n → X of finite multiplicity. The validity of the Poincar´e inequality in X is a rather straightforward application of the change-of-variables formula (5.5) (see also Sec. 6.13) and Prop. 4.13. We leave the details to the reader. This completes the proof of the theorem. Remarks 9.9 (a) The proof of Theorem 9.8 shows that R n can often be replaced by a more general space in order to get similar results. On the other hand, the other hypotheses such as completeness, quasiconvexity, and finite multiplicity are generally needed (cf. the examples in Rem. 4.6). (b) If we assume in Theorem 9.8, in addition, that f : X → R n is Lipschitz, then quasiconvexity of X need not be assumed; it follows as a consequence. Indeed, Proposition 4.13 remains valid as pointed out in Remark 4.16(a). Next one shows without difficulty that X is pathwise connected and (as in the proof of Theorem 9.8) that X admits a Poincar´e inequality. Quasiconvexity of X now follows, as demonstrated in [HaK]. Acknowledgments. We thank Frank Raymond and Dennis Sullivan for valuable discussions. We are most grateful to the referees of this paper for their extraordinarily careful reading of the entire manuscript and for their many useful comments. References [A]
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Heinonen Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109, USA; [email protected] Rickman Department of Mathematics, University of Helsinki, P.O. Box 4 (Yliopistonkatu 5), FIN-00014 Helsinki, Finland; [email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 3,
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE SHU-YEN PAN
Abstract In this paper, we prove that the depths of irreducible admissible representations are preserved by the local theta correspondence for any type I reductive dual pairs over a nonarchimedean local field. 1. Introduction Let F be a nonarchimedean local field, and let W be a finite-dimensional nondegenerate symplectic space over F. Let (G, G 0 ) be a (type I) reductive dual pair in the symplectic group Sp(W ). It is known that G, G 0 and G · G 0 are naturally embedded ^ e as subgroups of Sp(W ). Let Sp( W ) be the metaplectic cover of Sp(W ), and let G 0 0 e ^ (resp., G ) be the inverse image of G (resp., G ) in Sp(W ). By restricting the Weil e0 and pulling back to G e0 via the ^ e·G e×G representation of Sp( W ) to the subgroup G 0 0 e e e e homomorphism G × G → G · G , we obtain a correspondence called local theta correspondence or Howe duality between some irreducible admissible representations e0 . It has been proved by e and some irreducible admissible representations of G of G R. Howe (see [MVW]) and J.-L. Waldspurger [W] that this correspondence is one-toone when the residual characteristic of F is odd. Determining the explicit local theta correspondence is an important but extremely difficult problem. Only a few special examples are known so far. Instead of determining the explicit correspondence, a more accessible task is to investigate what kind of properties of the representations are preserved by the correspondence. Several results in this direction are known. For example, Howe [H] shows that spherical representations are preserved by the correspondence for unramified reductive dual pairs, and A.-M. Aubert [A] shows that irreducible admissible representations with nontrivial vectors fixed by an Iwahori subgroup are also preserved for unramified reductive dual pairs. In this paper, we study the (unrefined) minimal K -types of the irreducible admissible representations paired by the theta correspondence. In particular, we prove DUKE MATHEMATICAL JOURNAL c 2002 Vol. 113, No. 3, Received 28 March 2000. Revision received 24 July 2001. 2000 Mathematics Subject Classification. Primary 22E50; Secondary 1F27, 20C33.
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that the depths of irreducible admissible representations are preserved by the local theta correspondence. The concept of minimal K -types of irreducible admissible representations of a padic reductive group was introduced in [MP1] and [MP2] by A. Moy and G. Prasad. Associated to each point x in the Bruhat-Tits building of a reductive group G over F, an increasing sequence {ti }i≥0 of nonnegative real numbers with t0 = 0, a (decreasing) filtration G x,ti of the parahoric subgroup G x , and a filtration g∗x,−ti of the dual space g∗ of the Lie algebra g of G are defined. It is known that G x,t j is a normal subgroup of G x,ti for j ≥ i. The quotient group G x,ti /G x,ti+1 is a finite reductive group for i = 0 and is a finite abelian group for i > 0. It is also known that there is a natural isomorphism between the Pontrjagin dual (G x,ti /G x,ti+1 )∧ and g∗x,−ti /g∗x,−ti−1 for ti > 0. An (unrefined) minimal K -type for G is a pair (G x,ti , ζ ), where ζ is an irreducible representation of G x,ti /G x,ti+1 such that either ζ is cuspidal when ti = 0, or ζ , realized as a residue class in g∗x,−ti /g∗x,−ti−1 , contains no nilpotent element when ti > 0 (cf. Section 3.2). We say that an irreducible admissible representation (π, V ) of G contains a minimal K -type (G x,ti , ζ ) if the restriction of π to G x,ti contains ζ . Moy and Prasad prove that every irreducible admissible representation of a reductive group G contains a minimal K -type (G x,ti , ζ ) with x satisfying a certain optimal property, called an optimal point, and any two minimal K -types (G x,ti , ζ ), (G x 0 ,t 00 , ζ 0 ) coni tained in an irreducible admissible representation must be associated in some sense, in particular, ti = ti00 . Therefore, they define the common number ti to be the depth of that representation. It is known that the depth of any irreducible admissible representation is always a nonnegative rational number. With little modification the concept of minimal K -types can be extended to metaplectic covers of p-adic classical groups. In particular, the depth of an irreducible ade is defined and is again a nonnegative rational number. missible representation of G The following is the major result (see Theorem 6.6) of this work. THEOREM
Let (G, G 0 ) be a reductive dual pair. Let (π, V ) (resp., (π 0 , V 0 )) be an irreducible e0 ). Suppose that π and π 0 are paired in the e (resp., G admissible representation of G theta correspondence. Then the depth of π is equal to the depth of π 0 . e → G and A reductive dual pair (G, G 0 ) is said to be split if both extensions G 0 0 0 e G → G split. Suppose that (G, G ) is a split reductive dual pair. When we fix e0 , an irreducible admissible representation of G e and G 0 → G e (resp., splittings G → G 0 e G ) can be regarded as an irreducible admissible representation of G (resp., G 0 ) via the splittings. Therefore, we have a one-to-one correspondence between some irreducible admissible representations of G and some irreducible admissible representations of
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G 0 . It is clear that we need to specify nice splittings of the metaplectic covers to have a nice description of this correspondence. A result about the splittings is in [P1]. Suppose that (G, G 0 ) is a reductive dual pair of unitary groups or of symplectic and e0 be the splittings obtained in e and β 0 : G 0 → G orthogonal groups. Let β : G → G [P1]. It is not difficult to see that the depth of an irreducible admissible representation e is equal to the depth of the irreducible admissible representation π ◦ β of G. π of G Hence, we have the following corollary (see Theorem 12.2). COROLLARY Let (G, G 0 ) be
a split reductive dual pair of unitary groups or symplectic-orthogonal e0 be the splittings obtained in [P1]. Let π e and β 0 : G 0 → G groups. Let β : G → G e0 ) such that π e (resp., G (resp., π 0 ) be an irreducible admissible representation of G 0 e is equal and π correspond in the local theta correspondence. Then the depth of π ◦ β 0 0 e. to the depth of π ◦ β The idea of the proof of our main result is in fact very simple. From Waldspurger’s result, [W, cor. III.2], we know that the vectors of a (generalized) lattice model of the e modulo the action of Weil representation fixed by certain compact subgroups of G 0 e the other group G are functions of support bounded by some lattice in W . And we know that functions with support in some lattice in W must fixed by some compact e0 . Therefore, the depth of one representation, π, does give us an upper subgroup of G bound for the depth of the other representation, π 0 . However, two difficulties must be conquered to ensure that the depths are really equal. The first difficulty is to refine Moy and Prasad’s result on minimal K -types, at least for classical groups. In particular, we need the result that every irreducible admissible representation of a p-adic classical group has a minimal K -type (G x,ti , ζ ) such that x is a generalized barycenter in a fixed chamber of the building. This part, which is in Sections 3 and 4, mostly relies on or is directly from J.-K. Yu’s work in [Y2] and [Y1]. The second difficulty is to sharpen Waldspurger’s theorem. This is the major content of Sections 5 – 10. The occurrences of nongenuine barycenters in minimal K -types do make the situation more complicated. However, Waldspurger has done the first and most difficult step in [W]. And we just follow his strategy. The content of this paper is as follows. In Section 2, we establish the basic setting of this work and define the notation that is used throughout this paper. In Section 3, we recall Moy and Prasad’s and Yu’s results on minimal K -types and give the modification for metaplectic groups. In Section 4, we have another description of minimal K -types for p-adic classical groups in terms of regular small admissible lattice chains. This is the form that is needed for proving our main theorem. In Section 5, we recall the generalized lattice model of the Weil representation and Waldspurger’s
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result in [W]. In Section 6, we state the main propositions (Propositions 6.3 and 6.5), which can be regarded as a generalization of Waldspurger’s result. Our main theorem (Theorem 6.6) follows this proposition. In Sections 7 and 8, we prove several lemmas that are needed for the proof of Proposition 6.3 in Section 10. Results on Cayley transforms and generalized lattice models of the Weil representation are introduced in these two sections. In Section 9, we prove Proposition 6.2. The proofs of Proposition 6.3 and Corollary 6.4 are in Section 10. In Section 11, we prove Proposition 6.5. The material of this section is also in [P2]. We put it here for completeness. In the last section, we give the result on depth preservation for split reductive dual pairs by using the admissible splitting defined in [P1]. 2. Local theta correspondence In this section, we introduce the notation and the basic setting of this paper. The material is well known, and the references are [K1], [MVW], [Pr], and [W]. 2.1. Notation Let F be a nonarchimedean local field, let O F be the ring of integers of F, let p F be the prime ideal, let $ F be a uniformizer of O F , let f := O F /p F be the residue field, and let τ F be the identity automorphism of F. Let 5O F denote the quotient map from O F to f. We always assume that the characteristic of f is odd. Let ψ be a fixed nontrivial (additive) character of F. Let F 0 be a quadratic extension of F, let O F 0 be the ring of integers of F 0 , let $ F 0 be a uniformizer of O F 0 , let f0 be the residue field of F 0 , and let τ F 0 be the nontrivial automorphism of F 0 over F. We make the choice such that $ F 0 is equal to $ F if F 0 is unramified, and τ F 0 ($ F 0 ) is equal to −$ F 0 if F 0 is ramified. Let D 0 be a central quaternion algebra over F, let $ D 0 be a uniformizer, and let τ D 0 be the canonical involution of D 0 over F. We make the choice such that τ D 0 ($ D 0 ) is equal to −$ D 0 . Let (D, $, τ ) be one of the triples (F, $ F , τ F ), (F 0 , $ F 0 , τ F 0 ), or (D 0 , $ D 0 , τ D 0 ). Let O be the ring of integers of D, let p be the prime ideal, and let d be the residue field of D. Let 5O denote the quotient map from O to d. Let V be a finite-dimensional right vector space over D, and let be 1 or −1. A map h, i : V × V → D is called an -hermitian form if it satisfies the following conditions: hx + y, zi = hx, zi + hy, zi, hxa, ybi = τ (a)hx, yib,
hx, y + zi = hx, yi + hx, zi, hx, yi = τ hy, xi
(2.1.1)
for any x, y ∈ V and a, b ∈ D. The form is nondegenerate if hx, yi = 0 for all y ∈ V implies x = 0. The pair (V , h, i) is called a (nondegenerate) -hermitian space when h, i is a nondegenerate -hermitian form on V . In particular, V is called a symplectic
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space if D = F and = −1. A vector v ∈ V is called isotropic if hv, vi = 0. The space V is called anisotropic if it contains no nontrivial isotropic vectors. If a is a real number, let bac denote the largest integer not greater than a, and let dae denote the smallest integer not less than a. 2.2. Heisenberg groups and the Weil representations Let W , hh, ii be a symplectic space over F. Define the Heisenberg group H(W ) associated to W , hh, ii to be the group with underlying set W × F and multiplication defined by 1 (w1 , t1 ) · (w2 , t2 ) := w1 + w2 , t1 + t2 + hhw1 , w2 ii 2
(2.2.1)
for w1 , w2 ∈ W and t1 , t2 ∈ F. It is known that the center of H(W ) is {0} × F, which is identified with F, and it is also known that there is only one (up to equivalence) irreducible representation ρψ of H(W ) with central character ψ by the Stone–von Neumann theorem. Let (ρψ , S ) be an irreducible representation of H(W ) with central character ψ. The symplectic group Sp(W ) acts on H(W ) by the formula g · (w, t) := (g · w, t) ^ for g ∈ Sp(W ) and (w, t) ∈ H(W ). Define the metaplectic cover Sp( W ) of Sp(W ) to be the topological subgroup of Sp(W ) × Aut(S ) consisting of the pairs (g, Mg ) satisfying Mg ◦ ρψ (h) = ρψ (g · h) ◦ Mg (2.2.2) ^ for g ∈ Sp(W ), Mg ∈ Aut(S ), and any h ∈ H(W ). It is clear that (g, Mg ) ∈ Sp( W) × ^ implies (g, z Mg ) ∈ Sp( W ) for any z ∈ C . We have a short exact sequence of group homomorphisms β α ^ 1 −→ C× −→ Sp( W ) −→ Sp(W ) −→ 1
(2.2.3)
^ given by α : z 7 → (1, z1) and β : (g, Mg ) 7 → g. The metaplectic group Sp( W ) comes equipped with a representation ωψ on the space S given by ωψ (g, Mg ) := Mg .
(2.2.4)
This representation (ωψ , S ) is called the Weil representation or the oscillator repre^ sentation of Sp( W ). 2.3. Reductive dual pairs and local theta correspondence Let (G, G 0 ) be a (type I irreducible) reductive dual pair in Sp(W ); that is, there exist a division algebra D with involution τ , a right -hermitian D-space (V , h, i), and a right 0 -hermitian D-space (V 0 , h, i0 ) such that 0 = −1, and W = V ⊗ D V 0 in such a
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way that G = U (V ), G 0 = U (V 0 ), the groups of isometries, and the skew-symmetric bilinear form hh, ii := Trd D/F h, i ⊗ τ ◦ h, i0 , where Trd D/F denotes the reduced trace from D to F, and , 0 are 1 or −1. From the definition of the form hh, ii, we know that there are embeddings ιV 0 : U (V ) → Sp(W )
and
ιV : U (V 0 ) → Sp(W ).
^ ^ ^ Let U (V ) be the inverse image of ιV 0 (U (V )) in Sp( W ). The group U (V ) is called 0 ^ the metaplectic cover of U (V ). Let U (V ) be defined similarly. One can check that 0 ^ ^ \ U (V ) and U (V ) commute with each other. Let U (V ) be the two-fold cover of U (V ) ^ \ in U (V ). We know that U (V ) is a totally disconnected group. A representation (π, V ) ^ of U (V ) is called admissible if π|C× (z) is multiplication by z and π|U[ is an ad(V ) missible representation of a totally disconnected group. ^ It is known that (ωψ , S ) is an admissible representation of Sp( W ). Then (ωψ , 0 ^ ^ ^ S ) can be regarded as a representation of U (V )× U (V ) via the restriction to U (V )· 0 0 0 ^ ^ ^ ^ ^ U (V ) and the homomorphism U (V ) × U (V ) → U (V ) · U (V ). Let (π, V ) (resp., ^ (π 0 , V 0 )) be an irreducible admissible representation of the metaplectic group U (V ) 0 ^ (resp., U (V )). The representation (π, V ) is said to correspond to the representation ^ ^ (π 0 , V 0 ) if there is a nontrivial (U (V ) × U (V 0 ))-map 5 : S −→ V ⊗C V 0 .
(2.3.1)
This establishes a correspondence, called the local theta correspondence or Howe ^ duality, between some irreducible admissible representations of U (V ) and some ir0 ^ reducible admissible representations of U (V ). It has been proved by Howe (see [MVW, chap. 5]) and Waldspurger [W] that the local theta correspondence is oneto-one when the residue characteristic of F is odd. 2.4. Induction principle A two-dimensional -hermitian space is called a hyperbolic plane if it admits a onedimensional subspace of isotropic vectors. Let V00 be an anisotropic 0 -hermitian space over D, and let Vk0 be an 0 -hermitian space isomorphic to the direct sum of V00 and k copies of hyperbolic planes. The Witt index of Vk0 is k. There is an embed0 ding Vk0 ⊂ Vk+1 given by v 7→ (v, 0, 0) for v ∈ Vk0 . The chain V00 ⊂ V10 ⊂ V20 ⊂ · · · 0 of -hermitian spaces is called a Witt tower or a Witt series. If n > k, then U (Vn0 ) has a parabolic subgroup Pn,k whose Levi component is isomorphic to U (Vk0 ) × T , where ^ en,k (resp., T e) be the inverse image of Pn,k (resp., T ) in U T is a split torus. Let P (V 0 ). n
If
π0
^ is an irreducible admissible representation of U (Vk0 ) and ξ is a character of T ,
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en,k . We have then π 0 ⊗ ξ can be lifted as an irreducible admissible representation of P the following result, which is called the induction principle, from [K1]. PROPOSITION
^ Let (π, V ) be an irreducible admissible representation of U (V ). Suppose that π occurs in the theta correspondence for the dual pair (U (V ), U (Vk0 )) and is paired with ^ an irreducible admissible representation (π 0 , V 0 ) of U (V 0 ). Then the representation k
π occurs in the theta correspondence for the dual pair (U (V ), U (Vn0 )) for any n > k and is paired with an irreducible component of the parabolic induced representation ^ U (Vn0 )
Ind Pe
n,k
(π 0 ⊗ ξ ), where ξ is an unramified character of T .
2.5. Stable range A reductive dual pair (U (V ), U (V 0 )) is in the stable range if the dimension of one of the spaces V , V 0 is less than or equal to the Witt index of the other space. The following proposition is well known. PROPOSITION
Let (U (V ), U (V 0 )) be a reductive dual pair. Suppose that the dimension of V is less than or equal to the Witt index of V 0 . Then every irreducible admissible representation ^ of U (V ) occurs in the theta correspondence for (U (V ), U (V 0 )). 3. Unrefined minimal K -types for p-adic classical groups, I In the first two sections, we review the important result of minimal K -types from [MP1] and [MP2]. The definition of generalized barycenters in Section 3.4 is from [Y2]. 3.1. Moy-Prasad filtrations Let G be (the F-rational points of) a connected classical group over F, let g be the Lie algebra of G, and let g∗ be the dual space of g. Let x be a point in the Bruhat-Tits building B (G) of G. Let {G x,t }t∈R+ ∪{0} , {gx,t }t∈R , and {g∗x,t }t∈R be the decreasing S filtrations associated to x defined in [MP1] and [MP2]. Define G x,t + := s>t G x,s , S S gx,t + := s>t gx,s , and g∗x,t + := s>t g∗x,s . We know that filtrations {G x,t }t∈R+ ∪{0} , {gx,t }t∈R , and {g∗x,t }t∈R depend on the discrete valuation of D. Here we normalize the discrete valuation such that the value group of D × is Z. Therefore, we have gx,t $ = gx,t+1 and $ g∗x,t = g∗x,t+1 . This normalization is different from the normalization used in [MP1]. However, gx,t here is just gx,lt in [MP1], where l := e0 /e and e (resp., e0 ) is the ramification index of D (resp., the splitting field of G) over F. Let G x be the subgroup of G of elements that fix x. Let {ti | i ≥ 0} be the sequence (depending on x) such that t0 := 0, G x,ti 6= G x,t + , and G x,t + = G x,ti+1 . It is i
i
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known that G x,t j is a normal subgroup of G x,ti for j ≥ i and that the quotient group G x,ti /G x,ti+1 is a finite classical group when i = 0 and a finite abelian group when i > 0. There is a nondegenerate f-bilinear (G x /G x,0+ )-invariant pairing g∗x,−ti /g∗x,−ti−1 × gx,ti /gx,ti+1 −→ f
(3.1.1)
by ( X¯ , Y¯ ) 7→ X (Y ) (mod p F ) for ti > 0, where X¯ (resp., Y¯ ) is the residue class of X ∈ g∗x,−ti (resp., Y ∈ gx,ti ) in g∗x,−ti /g∗x,−ti−1 (resp., gx,ti /gx,ti+1 ). Therefore, we have a natural isomorphism between the Pontrjagin dual (G x,ti /G x,ti+1 )∧ of G x,ti / G x,ti+1 and the quotient g∗x,−ti /g∗x,−ti−1 for ti > 0. 3.2. Unrefined minimal K -types An element X of g∗x,−ti is called a nilpotent element for the coadjoint action of G x on g∗x,−ti if there is a one-parameter subgroup λ : F × → G such that limt→0 Ad(λ(t)) · X = 0, where Ad denotes the coadjoint action of G on g∗ . An irreducible representation ζ of G x,ti /G x,t + is called a nondegenerate representation if i (i) ζ is a cuspidal representation of G x,ti /G x,t + when ti = 0; i (ii) ζ , viewed as a residue class in g∗x,−ti /g∗x,(−t )+ , contains no nilpotent elements i when ti > 0. An (unrefined) minimal K -type of an irreducible admissible representation (π, V ) of G is a pair (G x,t , ζ ), where ζ is a nondegenerate representation of G x,t /G x,t + , and V G x,t + , the subspace of vectors fixed by G x,t + , is nontrivial and, as a representation of G x,t /G x,t + , contains ζ . The following is the fundamental result of Moy and Prasad (see [MP1, Th. 5.2], [MP2, Th. 3.5]) on minimal K -types. PROPOSITION
Let (π, V ) be an irreducible admissible representation of group G. Then there is a unique nonnegative rational number ρ such that ρ is the smallest number t such that there exists an optimal point y with V G y,t + nontrivial. Moreover, if ρ is positive and y is a point such that V G y,ρ + is nontrivial, then the space V G y,ρ + decomposes into nondegenerate representations of G y,ρ /G y,ρ + . Conversely, if t is positive and V G x,t + contains a nondegenerate representation of G x,t /G x,t + for some x, then t is equal to ρ. This unique rational number ρ is called the depth of the representation (π, V ). 3.3. Nonconnected classical groups The definition of filtration subgroups G x,t can be extended to a nonconnected classical group as follows. Let G be a classical group, and let G ◦ be the connected component of G. We define the building of G to be the building of G ◦ . We have G x,t = G ◦x,t
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for any x and positive t. The groups G and G ◦ have the same Lie algebra g. Hence, they have the same filtrations of the Lie algebras and of the dual spaces of the Lie algebras. Therefore, we still have the natural isomorphism (G x,ti /G x,ti+1 )∧ ' g∗x,−ti /g∗x,−ti−1
(3.3.1)
for any classical group G and positive ti . The details can be found in [Y1]. It is not difficult to see that Proposition 3.2 is also true for a nonconnected classical group. The proof of the existence of minimal K -types of positive depths is analogous to the proof of Proposition 3.2. For the existence of minimal K -types of depth zero, the proof depends only on Harish-Chandra’s theory of Eisenstein series for reductive groups over a finite field, which is also valid for nonconnected finite classical groups. 3.4. Generalized barycenters Let G := U (V ) be a classical group. Let δ be the smallest positive rational number such that φ + tδ is an affine root for any affine root φ and any integer t. Let 1af := {α0 , . . . , αn } be the system of simple affine roots of G. There are unique integers ci Pn such that i=0 ci αi = δ. Let be the subgroup of the extended Weyl group of G defined in [IM]. It is known that acts on the local Dynkin diagram of G. A simple affine root αi is called terminal if there is at most one other simple affine root joined to αi . A simple affine root αi is called exceptional if it is terminal and ci = 2. A pair of simple affine roots {αi , α j } is called an exceptional pair if αi 6 = α j and if αi and α j form an orbit under the action of on the local Dynkin diagram. Let νi be the vertex corresponding to αi in a fixed Weyl chamber C0 . Let 4 be a nonempty subset of 1af . Then 4 determines a subsimplex C4 of C0 . Assign weights u i to αi ∈ 4 in the following way: if {αi , α j } is an exceptional pair such that both αi and α j are in 4, then (u i , u j ) := (1, 1), (1, 2), or (2, 1); if αi is an exceptional simple affine root, then u i := 1 or 2; if αi is not exceptional and there is no α j such that {αi , α j } is an exceptional pair contained in 4, then u i := 2. The weighted barycenter .X X x4 := u i νi ui (3.4.1) αi ∈4
αi ∈4
of the subsimplex determined by 4 with the weights {u i } is called a generalized barycenter of C4 . If u i = ci for all αi ∈ 4, then the corresponding generalized barycenter is called the (genuine) barycenter. Example Let D be an unramified quadratic extension of F, and let V be a three-dimensional hermitian space over D such that the Witt index of V is 1. Assume that we have the decomposition V = v1 D⊕v2 D⊕v3 D such that v1 D is a totally isotropic subspace of
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V with the dual space v3 D and hv1 , v3 i = hv2 , v2 i = 1. Let G := U (V ) = U2,1 (F) be the group of isometries. We know that G has two simple affine roots α0 , α1 with the relation α0 + 2α1 = 1; that is, c0 = 1, c1 = 2, and δ = 1. An apartment of the building of G is one-dimensional, and a closed chamber is a (closed) segment. Let ν1 , ν2 be the vertices of the Weyl chamber such that νi is related to αi . There are three nonempty subsets of the set of simple affine roots, namely, 41 := {α0 }, 42 := {α1 }, and 43 := {α0 , α1 }. Now α1 is exceptional and α0 is not. Therefore, both 41 and 42 have unique barycenters x41 and x42 , respectively, but 43 has two generalized barycenters, namely, the genuine barycenter x43 corresponding to the weights u 0 = 1, 0 corresponding to the weights u 1 = 2, and the nongenuine generalized barycenter x4 3 u 0 = 1, u 1 = 1. We can identify the Weyl chamber with the closed segment [0, 1] such that ν0 = 0, ν1 = 1. Then it is clear that x41 = 0, x42 = 1, x43 = 2/3, and 0 = 1/2. x4 3
3.5. A refinement of Proposition 3.2 The following result, which can be viewed as a refinement of Proposition 3.2 for classical groups, is from [Y2]. PROPOSITION
The point y in Proposition 3.2 can be chosen from the set of generalized barycenters in a fixed Weyl chamber. 3.6. Minimal K -types for metaplectic covers Let IG be a fixed Iwahori subgroup of G. We know that IG = G x0 , where x0 is the barycenter of a Weyl chamber C0 in the Bruhat-Tits building of G. Since IG is compact, it is known that the extension e IG → IG splits, where e IG is the inverse image e e e of IG in G. Fix a splitting β IG : IG → IG . It is known that G x,r ⊆ IG for x ∈ C0 and eIG (G x,0+ ); that is, we regard r > 0. Therefore, we can and do identify G x,0+ with β e e G x,0+ as a subgroup of G via the splitting β IG . Those subgroups G x,r for x ∈ C0 are e And it is not difficult to enough to define minimal K -types for metaplectic groups G. see that Proposition 3.5 still holds for metaplectic groups when we define the building e to be the building of G. (It was suggested by Jiu-Kang Yu that we need to fix a of G splitting of an Iwahori subgroup to define unrefined minimal K -types for metaplectic groups.) 3.7. Depth and parabolic induction In [MP2], Moy and Prasad prove that the depth is preserved by parabolic induction. More precisely, the depth of an irreducible subquotient of IndGP π is equal to the depth of π, where π is (the lifting to P of) an irreducible admissible representation of the
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Levi component M of a parabolic subgroup P of G. Clearly, Moy and Prasad’s result can be extended to a metaplectic cover of G. More precisely, we know that I M := IG ∩ M is an Iwahori subgroup of M, where IG is a fixed Iwahori subgroup of G as in Section 3.6. Let the depth of an irreducible e be defined with respect to I M , where M e is the inverse admissible representation of M e e Then the depth of an irreducible component of IndG e image of M in G. eπ, where P P e is the inverse image of P in G, is equal to the depth of π , where π is an irreducible e admissible representation of M. 4. Unrefined minimal K -types for p-adic classical groups, II Section 4.1 is from [W]. Some terminologies such as regular small admissible lattice chain and numerical invariant were suggested by Jiu-Kang Yu. Also, the definition (4.8.1) is due to him. 4.1. Good lattices in a hermitian space, I Let D be defined as in Section 2.1. Let V , h, i be a finite-dimensional right hermitian space over D. Let L be a lattice in V , that is, a free right O -module whose rank is equal to the dimension of V . Two lattices L 1 , L 2 are said to be similar if L 1 = L 2 $ k for some k. Two lattices L 1 , L 2 are said to be equivalent if there is an element g ∈ U (V ) such that g · L 1 = L 2 . A decomposition V = X ⊕ V ◦ ⊕ Y of vector spaces is called L-admissible if X, Y are totally isotropic, in duality, and orthogonal to V ◦ , and if L = (L ∩ X ) ⊕ (L ∩ V ◦ ) ⊕ (L ∩ Y ). Fix an integer κ for V (more specific information about κ is given in (5.2.1)). Define L ∗ := v ∈ V hv, l i ∈ pκ for all l ∈ L . (4.1.1) It is clear that L ∗ is also a lattice in V . The lattice L ∗ is called the dual lattice of L (with respect to the integer κ). The lattice L is self-dual if L ∗ is equal to L. A lattice L is called a good lattice if it satisfies the condition L ∗$ ⊆ L ⊆ L ∗.
(4.1.2)
Let L be a good lattice in V . Then l∗ := L ∗/L and l := L/L ∗ $ are vector spaces over d. Let 5 L ∗ : L ∗ → l∗ , 5 L : L → l be the quotient maps. Define sesquilinear forms h, il∗ , h, il on l∗ and l, respectively, by
5 L ∗ (w), 5 L ∗ (w0 ) l∗ := 5O hw, w0 i$ 1−κ ,
5 L (v), 5 L (v 0 ) l := 5O hv, v 0 i$ −κ (4.1.3)
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for w, w0 ∈ L ∗ , v, v 0 ∈ L, and 5O is the quotient map O → d. Note that the forms h, il∗ and h, il are nondegenerate and depend on the choice of a prime element $ . A good lattice is said to be maximal (resp., minimal) if it is a maximal element (resp., a minimal element) in the set of all good lattices with the partial order defined by inclusion. It is easy to see that a good lattice L is maximal (resp., minimal) if and only if the space (l∗ , h, il∗ ) (resp., (l, h, il )) is anisotropic. Clearly, any lattice L satisfying the condition L ⊆ L ∗ is contained in a maximal good lattice. That is, if L ⊆ L ∗ , then there is a maximal good lattice 0 in V such that L ⊆ 0 ⊆ 0 ∗ ⊆ L ∗ . The following lemma is from [W]. LEMMA
Let 0 be a maximal good lattice in V . (i) There is a 0-admissible decomposition V = X ⊕ V ◦ ⊕ Y such that V ◦ is anisotropic; that is, 0 = (0 ∩ X ) ⊕ (0 ∩ V ◦ ) ⊕ (0 ∩ Y ) and 0 ∗ = (0 ∩ X ) ⊕ (0 ∩ V ◦ )∗ ⊕ (0 ∩ Y ), where (0 ∩ V ◦ )∗ is the dual lattice of 0 ∩ V ◦ in V ◦ . (ii) Suppose that 0 0 is another maximal good lattice in V . Then there is an element g ∈ U (V ) such that g · 0 = 0 0 . (iii) Suppose that { z i | i ∈ I } for some index set I is a finite subset of 0 and that l is a positive integer such that the set {50 (z i ) | i ∈ I } is linearly independent in 0/ 0 ∗ $ and hz i , z j i ≡ 0 (mod pl+κ ). Then there exists a subset {vi | i ∈ I } of 0 and a 0-admissible decomposition V = X 1 ⊕ V ◦ ⊕ Y1 such that the set {vi | i ∈ I } is a basis of X 1 and vi − z i ∈ 0$ l for every i ∈ I . Proof Part (i) is [W, cor. I.9], (ii) is [W, lem. I.10], and (iii) is [W, cor. I.8]. 4.2. Good lattices in a hermitian space, II We know that V can be written as an orthogonal direct sum V ◦ ⊕ V 1 , where V ◦ is anisotropic and V 1 is a direct sum of hyperbolic planes. There is only one good lattice A◦ in V ◦ . Let V 1 = X ⊕ Y be a complete polarization, and let m be the Witt index of V . Suppose that x1 , . . . , xm is a basis of X and y1 , . . . , ym is a basis of Y such that hxi , x j i = hyi , y j i = 0 and hxi , y j i = δi j $ κ . Then it is clear that Bi := x1 p + · · · + xi p + xi+1 O + · · · + xm O + ym O + · · · + y1 O
(4.2.1)
is a good lattice in V1 for each 0 ≤ i ≤ m. LEMMA
The lattice Ni := A◦ + Bi is a good lattice in V , and every good lattice in V is equivalent to Ni for some i = 0, . . . , m.
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Proof It is obvious that Ni is good lattice because Ni∗ = A◦∗ + x1 O + · · · + xm O + ym O + · · · + yi+1 O + yi p−1 + · · · + y1 p−1 . Let L be any good lattice in V . We know from Lemma 4.1(i) that V has an Ladmissible decomposition X 0 ⊕ V 0◦ ⊕ Y 0 such that V 0◦ is anisotropic, X 0 is totally isotropic, and Y 0 is dual to X 0 . Now L ∩ Y 0 is a lattice in Y 0 because the decomposition is L-admissible. Therefore, there exists a basis {y10 , . . . , ym0 } of Y 0 such that L ∩ Y 0 = y10 O + · · · + ym0 O . Let {x10 , . . . , xm0 } be the basis of X 0 dual to {y10 , . . . , ym0 } (i.e., hxi0 , x 0j i = hyi0 , y 0j i = 0 and hxi0 , y 0j i = δi j $ κ ). Then after reordering the index, we may conclude that 0 0 O O + · · · + xm L ∩ X 0 = x10 p + · · · + xi0 p + xi+1
for some i. Now let g be the element in U (V ) such that g · x 0j = x j , g · y 0j = y j , and g|V 0 ◦ is the identity. Therefore, it is clear that g · L = Ni for some i. 4.3. Admissible decompositions Let L be a lattice in V . A basis {vi }i∈I of V for some finite index set I is called L L-admissible if we have L = i∈I (L ∩ vi D). Let L n ⊆ L n−1 ⊆ · · · ⊆ L 1 ⊆ L 0 = L n $ −1 be lattices in V . Then from [W, lem. I.14] we know that there exists a basis of V which is L i -admissible for each i = 1, . . . , n. Therefore, we have the following lemma. LEMMA
Let L n ⊆ L n−1 ⊆ · · · ⊆ L 1 ⊆ L 0 be a set of good lattices in V . Then there exist a decomposition V = X ⊕ V ◦ ⊕ Y , a finite set {xi , v j , yi }i∈I, j∈J , and a disjoint union `n+1 I = i=0 Ii satisfying the following conditions: (i) the space V ◦ is anisotropic, and X, Y are dual to each other; (ii) {xi , yi }i∈I is a self-dual basis of X ⊕ Y (i.e., hxi , x j i = hyi , y j i = 0 and hxi , y j i = δi j $ κ ); S S (iii) the set l≤k {xi $ }i∈Il l>k {xi }i∈Il ∪ {yi }i∈I ∪ {v j } j∈J is an O -basis of L k . 4.4. Lattice chains A nonempty collection of lattices L := {L i }i∈Z in V is called a lattice chain in V if it satisfies the following conditions: (i) L is totally ordered by inclusion; that is, L i ⊆ L i+1 for each i; (ii) each lattice L i is similar to a good lattice or the dual lattice of a good lattice; (iii) there exists a number n such that L i+n = L i $ for all i.
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The number n is called the period of a lattice chain L . A lattice chain L is said to be regular if L i 6= L j whenever i 6 = j. Two lattice chains L := {L i }i∈Z , L 0 := {L i0 }i∈Z are said to be equivalent if there exist an element g ∈ U (V ) and a number k such that 0 g · L i = L i+k for all i. 4.5. Small admissible lattice chains A lattice chain L is called a small admissible lattice chain with numerical invariant (n, n 0 ), where n is a positive integer and n 0 = 0 or 1, if it satisfies the following conditions: (i) the period of L is n; (ii) L i∗ = L −i−n 0 for all i when n is even and n 0 = 1, L i∗ = L −i−n 0 for all i 6≡ 0 or n/2 (mod n) when n is even and n 0 = 0, L i∗ = L −i−n 0 for all i 6≡ (n − 1)/2 (mod n) when n is odd and n 0 = 1, L i∗ = L −i−n 0 for all i 6≡ 0 (mod n) when n is odd and n 0 = 0; (iii) L ∗b(n−1−n 0 )/2c $ ⊆ L b(n−1−n 0 )/2c ⊆ · · · ⊆ L 1 ⊆ L 0 ⊆ L ∗0 and L ∗−1 ⊆ L −1 ⊆ L −2 ⊆ · · · ⊆ L −b(n+n 0 )/2c ⊆ L ∗−b(n+n 0 )/2c $ −1 ⊆ L −b(n+n 0 )/2c−1 . A lattice chain L is called a big admissible lattice chain with numerical invariant (n, n 0 ) if it satisfies (i), (ii), and (iii0 ) L b(n−1−n 0 )/2c+2 ⊆ L ∗b(n−1−n 0 )/2c+1 $ ⊆ L b(n−1−n 0 )/2c+1 ⊆ · · · ⊆ L 2 ⊆ L 1 ⊆ L ∗1 and L ∗0 ⊆ L 0 ⊆ L −1 ⊆ · · · ⊆ L −b(n+n 0 )/2c+1 ⊆ L ∗−b(n+n 0 )/2c+1 $ −1 . A lattice chain is called admissible if it is small or big admissible. Let L := {L i }i∈Z be a lattice chain. Define ]
]
L ] := {L i | L i := L ∗−i−n 0 , i ∈ Z}. ]
]
]
(4.5.1)
]
Clearly, L i ⊂ L i−1 , L i+n = L i $ for any i, and L ] is a lattice chain. Moreover, ]
]
we have (L i )] = L i . From the definition it is easy to check that if L i 6= L i , then ] L i = L i∗ $ k for some integer k. It is not difficult to see that L is small (resp., big) admissible if and only if L ] is big (resp., small) admissible. It is clear from (4.5.1) that ] hL i , L j i ⊆ pκ+b(i+ j+n 0 )/nc (4.5.2) ]
for all i, j. A lattice chain L is called self-dual if L i = L i for all i. LEMMA
Suppose that L := {L i }i∈Z is a small admissible lattice chain of period n. Then ] L i ⊆ L i ⊆ L i−1 for any i.
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Proof First, suppose that (n, n 0 ) = (1, 0). From (iii) we know that L 0 ⊆ L ∗0 ⊆ L −1 . Now ] ] we know that L 0 = L ∗0 . Therefore, we have L 0 ⊆ L 0 ⊆ L −1 . Next, suppose that (n, n 0 ) = (1, 1). From (iii) we know that L −1 ⊆ L ∗−1 $ −1 ⊆ ]
]
L −2 . Now we know that L −1 = L ∗0 = L ∗−1 $ −1 . Therefore, we have L −1 ⊆ L −1 ⊆ L −2 . Now suppose that (n, n 0 ) = (2, 0). From (iii) we know that L 0 ⊆ L ∗0 ⊆ L −1 ⊆ ] ] ∗ L −1 $ −1 ⊆ L −2 . Now we know that L 0 = L ∗0 , L −1 = L ∗1 = L ∗−1 $ −1 . Hence, we ]
]
have L 0 ⊂ L 0 ⊂ L −1 ⊂ L −1 ⊂ L −2 . Finally, suppose that (n, n 0 ) 6 = (1, 0), (1, 1), or (2, 0). Then we know from the ] ] ] definition that L i = L i or L i−1 = L i−1 for any i. Because L i ⊆ L i−1 and L i ⊆ ]
L i−1 , the lemma is clearly true. 4.6. An example Keep the notation as in Example 3.4, and assume that κ = 0. Then we have the following two (equivalence classes of) good lattices in V : B0 := v1 O + v2 O + v3 O , B1 := v1 p + v2 O + v3 O . Let B2 := v1 p + v2 p + v3 O . We have B0∗ = B0 , B1∗ = B2 $ −1 , and B2 ⊂ B1 ⊂ B0 . We know that B0 is a maximal good lattice while B1 is minimal. Then we have five (equivalence classes of) regular small admissible lattice chains: L1 : · · · ⊂ B0 $ 3 ⊂ B0 $ 2 ⊂ B0 $ ⊂ B0 ⊂ B0 $ −1 ⊂ B0 $ −2 ⊂ · · · , L2 : · · · ⊂ B2 $ ⊂ B1 $ ⊂ B2 ⊂ B1 ⊂ B2 $ −1 ⊂ B1 $ −1 ⊂ · · · , L20 : · · · ⊂ B2 $ 3 ⊂ B2 $ 2 ⊂ B2 $ ⊂ B2 ⊂ B2 $ −1 ⊂ B2 $ −2 ⊂ · · · , L3 : · · · ⊂ B0 $ ⊂ B2 ⊂ B1 ⊂ B0 ⊂ B2 $ −1 ⊂ B1 $ −1 ⊂ · · · , L30 : · · · ⊂ B2 $ ⊂ B0 $ ⊂ B2 ⊂ B0 ⊂ B2 $ −1 ⊂ B0 $ −1 ⊂ · · · .
We know that L1 , L2 , L3 are regular self-dual lattice chains with numerical invariant (1, 0), (2, 1), (3, 0), respectively. The lattice chains L20 , L30 are not self-dual, and we have 0]
L2 : · · · ⊂ B1 $ 3 ⊂ B1 $ 2 ⊂ B1 $ ⊂ B1 ⊂ B1 $ −1 ⊂ B1 $ −2 ⊂ · · · , 0]
L3 : · · · ⊂ B1 $ ⊂ B0 $ ⊂ B1 ⊂ B0 ⊂ B1 $ −1 ⊂ B0 $ −1 ⊂ · · · .
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The numerical invariants of L20 , L30 are (1, 1), (2, 0), respectively. Let L := {L i }i∈Z be one of the above lattice chains. The index of L is normalized such that 0 B0 when L is L1 , L3 , or L3 , L 0 = B1 when L is L2 , B2 when L is L20 . 4.7. Bruhat-Tits buildings Let G := U (V ) be a classical group, and let B be the Bruhat-Tits building of G. In [BT], F. Bruhat and J. Tits give a realization of B as the set of all admissible norms α : V → R ∪ {+∞} which are maximally subordinate to the -hermitian form h, i. Based on their result, Yu in [Y1] gives another realization of the building as the set of all admissible filtrations of lattices in V . Under the realization by Yu, the admissible filtrations of lattices corresponding to a generalized barycenter can be rewritten as a regular small admissible lattice chain. We give an example to illustrate this relation. More detail about this can be found in [Y1]. Example Keep the notation as in Sections 3.4 and 4.6. The correspondence of the generalized barycenters and the regular small admissible lattice chains described in Section 4.6 0 ↔ L 0 . Note that the two lattice chains L , L 0 is x4i ↔ Li , x42 ↔ L20 , and x4 2 3 2 3 correspond to the same vertex x42 . 4.8. Open compact subgroups Let L := {L i }i∈Z be a regular small admissible lattice chain in V of period n. For any nonnegative integer d, we define G L := g ∈ G g · L i = L i for all i , ] ] G L ,(d/n)+ := g ∈ G (g − 1) · L i ⊆ L i+d+1 , (g − 1) · L i ⊆ L i+d for all i , ] ] G L ,(d+1)/n := g ∈ G (g − 1) · L i ⊆ L i+d+1 , (g − 1) · L i ⊆ L i+d+1 for all i . (4.8.1) It is easy to check that G L ,d/n = {g ∈ G | (g − 1) · L i ⊆ L i+d for all i} and G L ,(d/n)+ = G L ,(d+1)/n when L is self-dual. It is known that G L ,(d+1)/n and G L ,(d/n)+ are open compact subgroups of G. For any d ∈ Z, we define ] ] gL ,(d+1)/n := X ∈ g X · L i ⊆ L i+d+1 , X · L i ⊆ L i+d+1 for all i , ] ] gL ,(d/n)+ := X ∈ g X · L i ⊆ L i+d+1 , X · L i ⊆ L i+d for all i . (4.8.2) The following proposition is from [Y2].
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PROPOSITION
Let L be a regular small admissible lattice chain of period n. Let x be a the corresponding point in the building of G. Then G L ,(i+1)/n = G x,(i+1)/n and G L ,(i/n)+ = G x,(i/n)+ for any i ≥ 0. If L is a regular self-dual lattice chain with numerical invariant (1, 0), (1, 1), or (2, 1), then L is determined by its unique good lattice L = L 0 . In this case, we denote the group G L ,(0/n)+ (resp., G L ,n/n , G L ) by G L ,0+ (resp., G L ,1 , G L ); that is, we have G L := g ∈ G g · L = L , G L ,0+ := g ∈ G (g − 1) · L ∗ ⊆ L , (g − 1) · L ⊆ L ∗ $ , G L ,1 := g ∈ G (g − 1) · L ⊆ L$ . (4.8.3) 4.9. Minimal K -types of an irreducible admissible representation As in Section 3.6, a splitting IG → e IG of a fixed Iwahori subgroup of G has been e via fixed. Then G L ,d/n , G L ,(d/n)+ can be regarded as open compact subgroups of G the splitting IG → e IG when G L ,d/n and G L ,(d/n)+ are contained in IG . Suppose that d is a positive integer. Then from Propositions 3.2 and 4.8, an irreducible admissible e of positive depth has an unrefined minimal K -type of the representation (π, V ) of G form (G L ,d/n , ζ ) for some regular small admissible lattice chain L with numerical invariant (n, n 0 ). In particular, (π, V ) with a nonzero vector fixed by G L ,(d/n)+ must have depth less than or equal to d/n. Now we consider irreducible admissible representations of depth zero. It is known that if x is a point in the building of G, then there exists a vertex ν of a chamber such that G ν,0+ ⊆ G x,0+ . Therefore, an irreducible admissible representation of G is of depth zero if and only if it has nontrivial vectors fixed by G ν,0+ for some vertex ν. Moreover, a vertex ν in the building corresponds to a regular self-dual lattice chain generated by a single good lattice L in V under the correspondence. Therefore, we e is of depth zero if and know that an irreducible admissible representation (π, V ) of G only if V has a nontrivial vector fixed by G L ,0+ for some good lattice L such that G L ,0+ is contained in a fixed Iwahori subgroup IG of G. 5. A result of Waldspurger The material in this section is mostly from [W]. 5.1. Generalized lattice model of the Weil representation In this section we define a special realization of the Weil representation with respect to a good lattice in W .
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Let pλFF be the conductor of a character ψ of F, and let A be a good lattice in W with respect to the number λ F (i.e., the number κ in (4.1.1) is λ F here). Let a∗ denote the quotient A∗/A, and let 5 A∗ be the projection from A∗ to a∗ . We know that a∗ is a vector space over f. Define hh, iia∗ : a∗ × a∗ → f by hh5 A∗ (a1 ), 5 A∗ (a2 )iia∗ := 5O F $ F1−λ F hha1 , a2 ii , (5.1.1) where a1 , a2 ∈ A∗ and 5O F is the quotient map O F → f. It is easy to check that hh, iia∗ is a nondegenerate skew-symmetric f-bilinear form. We remark here that the form hh, iia∗ depends on the choice of a prime element $ F . Let H(a∗ ) be the Heisen berg group associated to the symplectic space a∗ , hh, iia∗ . Let ψ¯ be the character of f ¯ O (t)) := ψ(t$ λ F −1 ), where t ∈ O F . It is clear that ψ¯ is nontrivial. defined by ψ(5 F F Let (ω¯ ψ¯ , S) be the Weil representation of the finite symplectic group Sp(a∗ ) associ ated to the triple a∗ , hh, iia∗ , ψ¯ . Although the skew-symmetric form hh, iia∗ and the character ψ¯ depend on the choice of $ F , the Weil representation ω¯ ψ¯ does not. This can be seen easily from the Schr¨odinger model of (ω¯ ψ¯ , S). Let ρ¯ψ¯ denote the representation of H(a∗ ) corresponding to the character ψ¯ on the space S given by the Stone–von Neumann theorem. Now H(A∗ ) := A∗ × pλFF −1 is a subgroup of the Heisenberg group H(W ). We have a homomorphism 5H(A∗ ) : H(A∗ ) → H(a∗ )
by (a, t) 7 → 5 A∗ (a), 5O F (t$ F1−λ F )
for a ∈ A∗ and t ∈ pλFF −1 . Let K A be the stabilizer in Sp(W ) of the good lattice A, and let K A0 be the subgroup of K A defined by K A0 := g ∈ K A (g − 1) · A∗ ⊆ A . (5.1.2) It is clear that K A0 is a normal subgroup of K A and that the quotient K A /K A0 is isomorphic to Sp(a∗ ). Let ρ eψ be the representation of H(A∗ ) inflated from ρ¯ψ¯ by the projection 5H(A∗ ) , and let e ωψ be the representation of K A inflated from ω¯ ψ¯ by the 0 projection K A → K A /K A . Let S (A) be the space of locally constant, compactly supported maps f : H(W ) → S such that f ι(t)h 1 h 2 = ψ(t)e ρψ (h 1 ) · f (h 2 ), (5.1.3) where t ∈ F, h 1 ∈ H(A∗ ), h 2 ∈ H(W ), and ι denotes the embedding F → H(W ) by sending t to (0, t). Now H(W ) acts on S (A) by the right translation; that is, (ρψ (h) · f )(h 0 ) := f (h 0 h), where h, h 0 ∈ H(W ), f ∈ S (A), and ρψ denotes the action. This realization of the Weil representation (ωψ , S (A)) is called a generalized lattice model.
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To simplify the notation, we identify W with the subset W ×{0} of H(W ). Hence, A∗ is identified with A∗ × {0} ⊂ H(A∗ ). Then we have ρ eψ (a) · f (w) = f (aw) = f ((a, 0)(w, 0)) by (5.1.3) for a ∈ A∗ and w ∈ W . Now 1 1 (a, 0)(w, 0) = a + w, hha, wii = ι hha, wii (a + w, 0) 2 2 by the group law of the Heisenberg group defined in (2.2.1). Finally, we have f ι((1/2)hha, wii)(a + w, 0) = ψ (1/2)hha, wii f (a + w) by (5.1.3) again. Therefore, we can identify S (A) with the space of locally constant, compactly supported maps f : W → S such that f (a + w) = ψ
1 2
hhw, aii ρ eψ (a) · f (w)
(5.1.4)
for any a ∈ A∗ , w ∈ W . For a union of A∗ -cosets R, we define S (A) R := f ∈ S (A) the support of f is in R , S (A)w := S (A) A∗ +w
for w ∈ W .
(5.1.5)
Clearly, S (A)w is a vector space of dimension equal to the dimension of S. From now on, let S (A) be the space of locally constant, compactly supported maps f : W → S satisfying (5.1.4). For g ∈ Sp(W ), we define M[g] ∈ Aut(S (A)) by Z 1 (M[g] · f )(w) := ψ hha, wii ρ eψ (a −1 ) · f g −1 · (a + w) da, (5.1.6) 2 A∗ where g ∈ Sp(W ), f ∈ S (A), w ∈ W , and da is a Haar measure on A∗ . It is easy to ^ check that (g, M[g]) belongs to Sp( W ); that is, it satisfies the identity M[g]◦ρψ (h) = ρψ (g · h) ◦ M[g]. Then we can normalize the Haar measure da so that (M[k] · f )(w) = e ωψ (k) · f (k −1 · w)
(5.1.7)
eA be the inverse image of K A in for k ∈ K A , f ∈ S (A), and w ∈ W . Let K ^ ^ eA given by Sp( W ) under the extension Sp( W ) → Sp(W ). The map K A → K e k 7→ (k, M[k]) defines a splitting of the extension K A → K A . Therefore, if we ^ identify K A as a subgroup of Sp( W ) by the splitting k 7 → (k, M[k]), then ωψ (k) is equal to ωψ (k, M[k]) = M[k]; that is, the action of the Weil representation restricted K A is just given by M[k]. 5.2. Basic setting of reductive dual pairs Let (D, O , $, τ ) be as defined in Section 1.1. Let κ (resp., κ 0 ) be the integer κ in (4.1.1) defining the duality of lattices in V (resp., V 0 ). Let ψ be a fixed character of F
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of conductor pλFF for an integer λ F . Then ψ ◦ Trd D/F is a character of D of conductor pλ , where λ = λ F if D = F or an unramified extension of F, and λ = 2λ F − 1 otherwise. We assume that λ = κ + κ0 (5.2.1) throughout this paper. As in Section 5.1, we also assume that the duality of lattices in W is defined with respect to λ F . ^ Let H denote the Hecke algebra of U (V ); that is, let H be the space of com^ pactly supported locally constant functions f : U (V ) → C such that f (α(z)g) ˜ = −1 × × ^ ^ z f (g), ˜ where z ∈ C , α : C → U (V ) (cf. (2.2.3)), g˜ ∈ U (V ) with the multiplication defined by convolution (cf. [MVW, chap. 5, sec. I.1] ). We note here that the category of nondegenerate left H -modules is equivalent to the category of admissible ^ representations of U (V ). If M (resp., M 0 ) is a lattice in V (resp., V 0 ), then M ⊗O M 0 is a lattice in W . From now on, for simplicity, we write M ⊗ M 0 instead of M ⊗O M 0 to denote the tensor product of M and M 0 as O -modules. Let 0 (resp., 0 0 ) be a fixed maximal good lattice in V (resp., V 0 ). Define A := 0 ∗ ⊗ 0 0 ∩ 0 ⊗ 0 0∗ or, equivalently, A = 0 ∗ ⊗ 0 0∗ $ + 0 ⊗ 0 0 . Then A is a lattice in W and A∗ $ F ⊆ A ⊆ A∗ ; that is, A is a good lattice. We can check that A∗ = 0 ∗ ⊗ 0 0 + 0 ⊗ 0 0∗ = 0 ∗ ⊗ 0 0∗ ∩ 0 ⊗ 0 0 $ −1 . Let R (0) denote the set of pairs of lattices (M, N ) in V such that N$ |∩ 0∗$
M ⊆ N |∩ |∩ ⊆ 0 ⊆ 0∗ ⊆
Suppose that a pair of lattices (M, N ) is in R (0). We define J M,N := g ∈ U (V ) (g − 1) · N ∗ ⊆ M .
(5.2.2)
(5.2.3)
We can check that J M,N is an open compact subgroup of U (V ). Define B M,N := M ∗ ⊗ 0 0 + N ∗ ⊗ 0 0∗ = M ∗ ⊗ 0 0∗ ∩ N ∗ ⊗ 0 0 $ −1 .
(5.2.4)
Then B M,N is a lattice in W and contains A∗ . For convenience we freely identify the following pairs of isomorphic objects: W ' Hom D (V , V 0 ),
A ' HomO (0, 0 0 $ κ ) ∩ HomO (0 ∗ , 0 0∗ $ κ ), A∗ ' HomO (0, 0 0∗ $ κ ) ∩ HomO (0 ∗ , 0 0 $ κ−1 ), B M,N ' HomO (M, 0 0∗ $ κ ) ∩ HomO (N , 0 0 $ κ−1 ).
(5.2.5)
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5.3. A result of Waldspurger In this section we record a deep result by Waldspurger which plays a crucial role for ^ our main result. Suppose that K is a compact subgroup of U (V ) and that (π, V ) is a representation of K . Suppose that (η, E) is an irreducible representation of K . Let V (η) denote the isotypic component of π of type η. If η is the trivial representation, then V (η) is the subspace fixed by K and is also denoted by V K . The following important result is in [W, cor. III.2]. PROPOSITION
Let (U (V ), U (V 0 )) be a (type I) reductive dual pair, let 0 (resp., 0 0 ) be a maximal good lattice in V (resp., V 0 ), let A be the lattice defined in Section 5.2, let (M, N ) ^ be in R (0), let K be a compact subgroup of U (V ), and let η be an irreducible representation of K . Suppose that J M,N is contained in K , S (A) B M,N is stable by K , and S (A) B M,N (η) 6= {0}. Then S (A)(η) = ωψ (H 0 ) · S (A) B M,N (η),
(5.3.1)
^ where H 0 is the Hecke algebra of U (V 0 ). 6. The main theorem The statement of Proposition 6.3 was modified from a suggestion of Jiu-Kang Yu. Let (G, G 0 ) := (U (V ), U (V 0 )) be a reductive dual pair. 6.1. A lattice in the symplectic space Let L := {L i | i ∈ Z} (resp., L 0 := {L 0j | j ∈ Z}) be an admissible lattice chain in V (resp., V 0 ) with numerical invariant (n, n 0 ) (resp., (n, n 00 )). For any integer s, we define \ ] \ 0] Bs (L , L 0 ) := L i ⊗ L 0j ∩ Li ⊗ L j . (6.1.1) i+ j=s
i+ j=s
It is clear that Bs (L , L 0 ) is a lattice in W . For any nonnegative integer d such that the number n + n 0 + n 00 + d is even, we define d B L , L 0, := B(−n−n 0 −n 0 −d)/2 (L , L 0 ). (6.1.2) 0 n LEMMA
Let L (resp., L 0 ) be an admissible lattice chain in V (resp., V 0 ) with numerical invariant (n, n 0 ) (resp., (n, n 00 )). (i) If s ≥ (−n − n 0 − n 00 + 1)/2, then Bs (L , L 0 ) ⊆ Bs (L , L 0 )∗ . (ii) If s ≤ (−n − n 0 − n 00 )/2, then Bs (L , L 0 )∗ ⊆ Bs (L , L 0 ). In particular, d ∗ d B L , L 0, ⊆ B L , L 0, n n
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for any nonnegative integer d. The proof of this lemma is postponed to Section 9. If d = 0 and L is a regular small admissible lattice generated by a single good lattice L, then B(L , L 0 , 0/n) is denoted by B(L , L 0 ), where L 0 is the good lattice in V 0 which generates L 0 . Hence, B(L , L 0 ) := L ∗ ⊗ L 0 ∩ L ⊗ L 0∗ .
(6.1.3)
It is not difficult to check that B(L , L 0 ) = L ⊗ L 0 + L ∗ $ ⊗ L 0∗ . Hence, we have B(L , L 0 )∗ = L ∗ ⊗ L 0 + L ⊗ L 0∗ = L$ −1 ⊗ L 0 ∩ L ∗ ⊗ L 0∗ . It is also easy to see that B(L , L 0 ) is a good lattice in W ; that is, B(L , L 0 )∗ $ F ⊆ B(L , L 0 ) ⊆ B(L , L 0 )∗ . 6.2. The vectors fixed by open compact subgroups Let (ρψ , S ) be an irreducible smooth representation of the Heisenberg group H(W ) associated to the character ψ of F of conductor pλFF . Let B be a lattice in W such that B ⊆ B ∗ . Then we have hhb, b0 ii ∈ pλFF for any b, b0 ∈ B. Therefore, B × pλFF is a subgroup of H(W ). The representation (ρψ | B×pλ F , S ) factors through the projection F
B×pλFF → B because pλFF is in the kernel of ψ. The action of B on the space S is also denoted by ρψ . By Lemma 6.1 and the above remark, we know that B(L , L 0 , d/n)∗ acts on S when d is nonnegative. PROPOSITION
^ Let (ωψ , S ) be the Weil representation of Sp( W ). (i) Suppose that d is a positive integer, L is a small admissible lattice chain in V with numerical invariant (n, n 0 ), and L 0 is a small admissible lattice chain in V 0 with numerical invariant (n, n 00 ) such that n + n 0 + n 00 + d is even. Then 0 ∗ the subspace S B(L ,L ,d/n) is fixed pointwise by G L ,(d/n)+ and G 0L 0 ,(d/n)+ . (ii) Suppose that L (resp., L 0 ) is a good lattice in V (resp., V 0 ). Then the subspace 0 S B(L ,L ) is fixed pointwise by G L ,0+ and G 0L 0 ,0+ . We postpone the proof of this proposition to Section 9. 6.3. Key proposition for positive depths Let L be a small admissible lattice chain in V with numerical invariant (n, n 0 ). Let Q (d) denote the set of small admissible lattice chains L 0 in V 0 with numerical invariant (n, n 00 ) such that n + n 0 + n 00 + d is even. The following proposition, which may be regarded as a generalization of Proposition 5.3, plays a crucial role for our main theorem.
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PROPOSITION
Let (U (V ), U (V 0 )) be a reductive dual pair in Sp(W ). Let d be a positive integer, and let L be a regular small admissible lattice chain in V with numerical invariant ^ (n, n 0 ). Let (ωψ , S ) be a model of the Weil representation of Sp( W ). Then X 0 ∗ S G L ,(d/n)+ = ωψ (H 0 ) · S B(L ,L ,d/n) , (6.3.1) L 0 ∈Q (d)
^ where H 0 is the Hecke algebra of U (V 0 ). The proof of this proposition is postponed to Section 10. 6.4. The case when the Witt index is large COROLLARY
Keep the assumptions in the previous proposition. If we also assume that the Witt P index of V 0 is large enough, then the sum L 0 ∈Q(d) in Proposition 6.3 can be taken over all regular small admissible lattice chains in Q (d). The proof of this corollary is also in Section 10. 6.5. Key proposition for depth zero The following proposition, whose proof is postponed to Section 11, is an analogue of Proposition 6.3 for depth zero. PROPOSITION
Let (G, G 0 ) := (U (V ), U (V 0 )) be a reductive dual pair, and let ψ be a nontrivial character of F. Suppose that L is a good lattice in V . Then X 0 S G L ,0+ = ωψ (H 0 ) · S B(L ,L ) , (6.5.1) L0
^ where L 0 is a good lattice in V 0 , H 0 is the Hecke algebra of U (V 0 ), and (ωψ , S ) is ^ the Weil representation of Sp( W ). 6.6. The main theorem THEOREM
Let (U (V ), U (V 0 )) be a reductive dual pair. Let (π, V ) (resp., (π 0 , V 0 )) be an ir^ ^ reducible admissible representation of U (V ) (resp., U (V 0 )) such that π and π 0 are paired in the theta correspondence. Then the depth of π is equal to the depth of π 0 .
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SHU-YEN PAN
Proof Let G := U (V ) and G 0 := U (V 0 ). First, we suppose that the depth of π is zero. As ^ in Section 4.9, we know that an irreducible admissible representation (π, V ) of U (V ) G L ,0+ is of depth zero if and only if V is nontrivial for some good lattice L in V . Let 5 be the projection S → V ⊗C V 0 , where S is the Weil representation. We have a nontrivial surjective map S G L ,0+ → V G L ,0+ ⊗C V 0 . Because we assume that the 0 space V G L ,0+ is nontrivial, by Proposition 6.5 there is an element f ∈ S B(L ,L ) for some good lattice L 0 in V 0 such that 5( f ) is not zero. But 5( f ) is fixed by G L ,0+ and G 0L 0 ,0+ by Proposition 6.2(ii). Therefore, 5( f ) belongs to V G L ,0+ ⊗C V
0G 0
L 0 ,0+
. Hence,
0G 0
V L 0 ,0+ is also nontrivial. Therefore, the depth of π 0 is also zero. By symmetry, if π 0 is of depth zero, then π is also of depth zero. Next, we suppose that the depth of π is positive. From Section 4.9 we may assume that the depth of π is d/n for some positive integers d, n and that (π, V ) has a minimal K -type (G L ,d/n , ζ ) for some regular small admissible lattice chain L in V with numerical invariant (n, n 0 ). Therefore, we know that V G L ,(d/n)+ is nontrivial. Let V 00 be an 0 -hermitian space in the same Witt tower of V 0 with a large Witt index, and let G 00 := U (V 00 ). Then (π, V ) still occurs in the theta correspondence for the reductive dual pair (U (V ), U (V 00 )) by the induction principle of the theta correspondence (see Proposition 2.4). Suppose that (π, V ) is paired with an irreducible admissible representation (π 00 , V 00 ) of U^ (V 00 ). Let (ωψ , S ) be the Weil representa^ tion of Sp(V ⊗ V 00 ). By Proposition 6.3, we have S G L ,(d/n)+ = ωψ (H 00 ) ·
X
S B(L ,L
0 ,d/n)∗
,
(6.6.1)
L 0 ∈Q (d)
where H 00 is the Hecke algebra of U^ (V 00 ). Because we have the map 5 : S G L ,(d/n)+ −→ V G L ,(d/n)+ ⊗C V 00 and V G L ,(d/n)+ is nontrivial, there must be a small admissible lattice chain L 0 in V 00 0 ∗ such that n + n 0 + n 00 + d is even and 5 S B(L ,L ,d/n) is nontrivial. Because the Witt index of V 00 is very large, we can arrange L 0 to be a regular small admissible lattice chain in V 00 by Corollary 6.4. Then V 00
G 00
L 0 ,(d/n)+
is nontrivial by Proposition
00 00 G L 0 ,(d/n)+
6.2(i). Therefore, V is also nontrivial. Hence, by Proposition 3.2 and the remark in Section 4.9, the depth of π 00 is less than or equal to d/n, which is the depth of π. By the induction principle (see Proposition 2.4), we know that π 00 is a ^ U (V 00 )
(π 0 ⊗ ξ ), where P is a parabolic subquotient of the induced representation Ind Pe subgroup of U (V 00 ) whose Levi component is isomorphic to U (V 0 ) × T for a split e is the inverse image of P in U^ torus T in U (V 00 ), P (V 00 ), and ξ is an unramified
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555
character of T . Then by the result of Moy and Prasad (see [MP2, Th. 5.2]) and the remark in Section 3.7, we know that the depths are preserved by parabolic induction; that is, the depth of π 0 ⊗ ξ is equal to the depth of π 00 . It is obvious that the depth of π 0 ⊗ ξ is equal to the depth of π 0 since ξ is an unramified character. Hence, we have proved that the depth of π is greater than or equal to the depth of π 0 . But the depth of π 0 is not zero; otherwise, the depth of π is also zero, as proved in the first paragraph. Hence, by symmetry, the depth of π 0 is also greater than or equal to the depth of π . Therefore, depths of π and π 0 must be equal when one of the representations has positive depth. The proof is complete.
7. Cayley transform 7.1. Cayley transforms, I As usual, let U (V ) denote the group of isometries of an -hermitian space V , h, i . Let u(V ) be the Lie algebra of U (V ), that is, the space of elements c ∈ End D (V ) such that hc · v, v 0 i + hv, c · v 0 i = 0 for all v, v 0 ∈ V . If c belongs to u(V ) and 1 + c is invertible, then we define u(c) := (1 − c)(1 + c)−1 . Similarly, if u belongs to U (V ) and 1 + u is invertible, then we define c(u) := (1 − u)(1 + u)−1 . It is easy to check that u(c) (resp., c(u)) belongs to U (V ) (resp., u(V )) when it is defined. For x, y ∈ V , define an element cx,y in End D (V ) by the formula cx,y · v := xhy, vi − yhx, vi.
(7.1.1)
It is easy to check that cx,y belongs to u(V ). Let u x,y := u(cx,y ) if 1 + cx,y is invertible. If u x,y is defined for some x, y, then it is easy to check that 1 + u x,y is invertible and c(u x,y ) = cx,y . LEMMA
We have cx,ya = cxτ (a),y and cx,y = c y,−x for any x, y ∈ V and any a ∈ D. Proof From the definition in (7.1.1), we have cx,ya · v = xhya, vi − yahx, vi for any v, x, y ∈ V . Hence, cx,ya · v is equal to xτ (a)hy, vi − yhxτ (a), vi, which is just cxτ (a),y · v. This proves the first equality. For the second equality, we have cx,y · v = xhy, vi − yhx, vi = yh−x, vi − (−x)hy, vi = c y,−x · v.
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7.2. Cayley transforms, II Let L be a regular small admissible lattice chain in V with numerical invariant (n, n 0 ). LEMMA
Suppose that d is a nonnegative integer and that g is an element in G L ,(d+1)/n (resp., G L ,(d/n)+ ). Then the element c(g) is well defined and belongs to gL ,(d+1)/n (resp., gL ,(d/n)+ ). Proof Suppose that g is an element in G L ,(d+1)/n . Then (g − 1) · L i ⊆ L i+d+1 and (g − ] ] 1) · L i ⊆ L i+d+1 for all i. Therefore, it is clear that g + 1 is invertible. Hence, the ]
]
element c(g) is well defined. It is clear that (1 + g) · L i = L i and (1 + g) · L i = L i , so ] ] (1− g)(1+ g)−1 · L i = (1− g)· L i and (1− g)(1+ g)−1 · L i = (1− g)· L i . Therefore, ] ] we have (1 − g)(1 + g)−1 · L i ⊆ L i+d+1 and (1 − g)(1 + g)−1 · L i ⊆ L i+d+1 . Hence, c(g) belongs to gL ,(d+1)/n . The proof for the other case is similar, so we omit it. 7.3. A valuation In the remaining part of this section, we want to express an element g ∈ G L ,(d/n)+ in terms of u x,y with x, y satisfying certain conditions determined by L and d. This result is used in Section 10. Suppose that L is a lattice in V . Define a valuation ord L : V → Z
(7.3.1) ]
by ord L (v) := m if v ∈ L$ m − L$ m+1 . Recall that L i := L ∗−i−n 0 from Section 4.5. PROPOSITION
Let d be a nonnegative integer, let L := {L i }i∈Z be a regular small admissible lattice chain in V of period n, and let κ be the integer in (5.2.1). Suppose that x, y are elements in V satisfying the conditions ord L ∗j (x) + ord L j+d (y) ≥ −κ, ord(L ] )∗ (x) + ord L ] (y) ≥ −κ, j
j+d
ord L ∗j (y) + ord L j+d (x) ≥ −κ, ord(L ] )∗ (y) + ord L ] (x) ≥ −κ, j
(7.3.2)
j+d
for all j ∈ Z. Then cx,y is in gL ,d/n , and gL ,d/n is generated by those elements cx,y where x, y satisfy the above conditions and are multiples of elements in a given L -admissible basis of V . Proof Let {vi }i∈I , for some finite index set I , be an L -admissible basis of V given by
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Lemma 4.3, and let l be an element in L j . Because of the assumption in (7.3.2), we have ord L j+d (yhx, li) ≥ 0 and ord L j+d (xhy, li) ≥ 0, for all j; that is, yhx, li ∈ L j+d and xhy, li ∈ L j+d , for all j. Therefore, cx,y (l) belongs to L j+d for all j because cx,y · l is just xhy, li − yhx, li. Then cx,y · L j is contained in L j+d for all j. By this ] ] argument we can prove that cx,y · L j ⊆ L j+d for all j. Therefore, cx,y is an element of gL ,d/n . This proves the first statement of the proposition. On the other hand, suppose that c is an element in gL ,d/n . Then we can find βi, j ∈ D for i, j ∈ I such that c is equal to the map c0 ∈ End D (V ) defined by P c0 · v := i, j∈I v j βi, j hvi , vi for any v ∈ V . Define c00 ∈ End D (V ) by the formula P c00 · v := − i, j∈I vi τ (βi, j )hv j , vi for any v ∈ V . Then it is not difficult to check that hc0 · v, v 0 i + hv, c00 · v 0 i is equal to zero for any v, v 0 ∈ V . Because c0 ∈ u(V ), that is, hc0 · v, v 0 i + hv, c0 · v 0 i = 0 for all v, v 0 ∈ V , we know that c0 = c00 , c00 ∈ u(V ), and c = (1/2)(c0 + c00 ). Therefore, it is easy to check that c · v = (1/2)(c0 + c00 ) · v = P i, j∈I cv j βi, j ,vi /2 · v for v ∈ V , that is, that c=
X
cv j βi, j ,vi /2 .
i, j∈I
Now the only thing left is to prove that each pair (v j βi, j , vi /2) does satisfy the conditions in (7.3.2) for any i, j. But this is an easy consequence of the assumption that the basis {vi }i∈I is L -admissible. 7.4 PROPOSITION
Let L := {L i }i∈Z be a regular small admissible lattice chain in V of period n, and let d be a positive integer. Suppose that x, y are elements in V satisfying (7.3.2) for all j ∈ Z. Then u x,y is defined and u x,y ∈ G L ,d/n . Moreover, G L ,d/n is topologically generated by those u x,y where x, y satisfy the above conditions and are multiples of elements in a given L -admissible basis of V . Proof Let {vi }i∈I be an L -admissible basis of V as given by Lemma 4.3. From (7.3.2) and ] the proof of Proposition 7.3, we know that cx,y · L j is contained in L j+d for all j. P k Therefore, we know that limk→∞ ckx,y = 0. Then ∞ k=0 (−cx,y ) converges and the limit is the inverse of 1 + cx,y . Therefore, 1 + cx,y is invertible and u x,y is defined. ] It is easy to check that (u x,y − 1) = −2cx,y (1 + cx,y )−1 . Now (u x,y − 1) · L j = ]
]
]
−2cx,y (1 + cx,y )−1 · L j is equal to −2cx,y · L j . Finally, −2cx,y · L j is contained in L j+d for all j from the proof of the previous proposition. Therefore, we have u x,y ∈ G L ,d/n . Now we begin to prove the second statement. Let g be an element in G L ,d/n for
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]
some d ≥ 1. Then we have (g − 1) · L j ⊆ L j+d and (g − 1) · L j ⊆ L j+d for all j from (4.8.1). Therefore, it is easy to see that 1 + g is invertible. Let c := c(g) be as defined in Section 7.1. Then c belongs to gL ,d/n by Lemma 7.2. As in the proof of P Proposition 7.3, we can find βi, j ∈ D for i, j ∈ I such that c = i, j∈I cv j βi, j ,vi /2 and each pair (v j βi, j , vi /2) also satisfies (7.3.2). Hence, u v j βi, j ,vi /2 is defined and belongs to G L ,d/n for each (i, j) ∈ I × I by the first part of Proposition 7.4. Define an element Y g1 := u v j βi, j ,vi /2 , i, j∈I
where the product can be taken in any order. Then it is not difficult to see that (c(g) − c(g1 )) · l is contained in L j+2d for any l ∈ L j and that (c(g) − c(g1 )) · l ] is contained ] ] in L j+2d for any l ] ∈ L j . Therefore, we have ((g − 1) + (g1−1 − 1)) · L j ⊆ L j+2d and ]
]
((g − 1) + (g1−1 − 1)) · L j ⊆ L j+2d for all j. Let g2 denote the element gg1−1 . Then g2 belongs to G L ,d/n , and g2 − 1 is equal to (g − 1)(g1−1 − 1) + (g1−1 − 1) + (g − 1). Therefore, (g2 − 1) · L j ⊆ (g − 1) · L j+d + L j+2d , which is contained in L j+2d ] ] for all j and also (g2 − 1) · L j ⊆ L j+2d for all j. Therefore, we have proved that for any element g ∈ G L ,d/n , we can find an element g1 which is generated by those u x,y for x, y satisfying (7.3.2) such that gg1−1 belongs to G L ,2d/n . By mathematical induction, we can prove that for any element g ∈ G L ,d/n and any positive integer i, there exists an element gi generated by those u x,y for x, y satisfying (7.3.2) such that ggi−1 belongs to G L ,id/n . But we know that the set {G L ,id/n }i∈N forms a system of open neighborhoods of the identity. Therefore, g can be topologically generated by those u x,y for x, y satisfying (7.3.2). Note that x, y can always be chosen from multiples of elements in a given L -admissible basis. 7.5 COROLLARY
Let L := {L i }i∈Z be a regular small admissible lattice chain in V of period n, let d be a positive integer, and let κ be the integer in (5.2.1). Suppose that x, y ∈ V satisfy the conditions ord L ∗j (x) + ord L ]
j+d+1
(y) ≥ −κ,
ord(L ] )∗ (x) + ord L j+d (y) ≥ −κ, j
ord L ∗j (y) + ord L ]
j+d+1
(x) ≥ −κ,
ord(L ] )∗ (y) + ord L j+d (x) ≥ −κ,
(7.5.1)
j
for all j ∈ Z. Then u x,y is defined and u x,y ∈ G L ,(d/n)+ . Moreover, G L ,(d/n)+ is topologically generated by those u x,y where x, y satisfy (7.5.1) and are multiples of elements in a given L -admissible basis. Proof The proof is almost the same as that of Proposition 7.4. We therefore omit it.
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7.6 COROLLARY
Suppose that L is a good lattice in V and that x, y are elements in V . (i) If ord L (x) + ord L (y) ≥ −κ and ord L ∗ (x) + ord L ∗ (y) ≥ 1 − κ, then u x,y is defined and belongs to G L ,0+ . (ii) If ord L (x) + ord L ∗ (y) ≥ 1 − κ and ord L (y) + ord L ∗ (x) ≥ 1 − κ, then u x,y is defined and belongs to G L ,1 . Proof The proof is again similar to the proof of Proposition 7.4. In fact, (i) is [W, lem. I.17].
7.7 COROLLARY
Let L be a regular small admissible lattice chain of period n in V , let d be a positive integer, and let κ be the integer in (5.2.1). (i) The group G L ,d/n is topologically generated by those u x,y with x, y ∈ V satisfying the following two conditions: ] (1) x ∈ L i , y ∈ L j for some i, j such that i + j ≥ d − 1 + n − n 0 − κn; ] (2) x ∈ L i 0 , y ∈ L j 0 for some i 0 , j 0 such that i 0 + j 0 ≥ d − 1 + n − n 0 − κn. (ii) The group G L ,(d/n)+ is topologically generated by those u x,y with with x, y ∈ V satisfying the following two conditions: ] ] (1) x ∈ L i , y ∈ L j for some i, j such that i + j ≥ d + n − n 0 − κn; (2) x ∈ L i 0 , y ∈ L j 0 for some i 0 , j 0 such that i 0 + j 0 ≥ d − 1 + n − n 0 − κn. Proof We prove the statement only for the group G L ,d/n because the proof of (ii) is similar. By Proposition 7.4, it suffices to prove that x, y satisfy (7.3.2); that is, ord L ∗j (x) + ord L j+d (y) ≥ −κ, ord L ∗j (y) + ord L j+d (x) ≥ −κ, ord(L ] )∗ (x) + ord L ] (y) ≥ −κ, j
j+d
and ord(L ] )∗ (y) + ord L ] (x) ≥ −κ, for all j if and only if there exist i 0 , j0 , i 1 , j1 j
j+d
]
]
such that x ∈ L i0 , y ∈ L j0 with i 0 + j0 ≥ d − 1 + n − n 0 − r n and x ∈ L i1 , y ∈ L j1 with i 1 + j1 ≥ d − 1 + n − n 0 − r n. ] ] Note that (L j )∗ = (L ∗− j−n 0 )∗ = L − j−n 0 and L ∗j = L − j−n 0 . Therefore, if x is in ]
L i0 and y is in L j0 for some i 0 , j0 such that i 0 + j0 ≥ d − 1 + n − n 0 − κn, and if x ]
is in L i1 and y is in L j1 for some i 1 , j1 such that i 1 + j1 ≥ d − 1 + n − n 0 − κn, then it is clear that x, y satisfy (7.3.2). On the other hand, suppose that x, y satisfy (7.3.2). Suppose also that x is in ] ] L i0 − L i0 +1 for some i 0 and in L i1 − L i1 +1 for some i 1 . Then ord L ] (x) is zero − j−n 0
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for j such that i 0 − n < − j − n 0 ≤ i 0 ; that is, ord L ]
− j−n 0
(x) is zero for j such that ]
−i 0 − n 0 + n + d > j + d ≥ −i 0 − n 0 + d. Since L ∗j = L − j−n 0 and ord L ∗j (x) + ord L j+d (y) ≥ −κ, we have ord L j+d (y) ≥ −κ for any j such that j + d < −i 0 − n 0 +n +d. In particular, y belongs to L −i0 −n 0 +n+d−1 $ −κ = L −i0 −n 0 +n+d−1−κn . Let j0 := −i 0 −n 0 +n+d−1−κn. Then we have y ∈ L j0 and i 0 + j0 ≥ d−1+n−n 0 −κn. Also, we have that ord L − j−n0 (x) is zero for j such that i 1 − n < − j − n 0 ≤ i 1 ; that is, ord L − j−n0 (x) is zero for j such that −i 1 − n 0 + n + d > j + d ≥ −i 1 − n 0 + d. ]
Since (L j )∗ = L − j−n 0 and ord(L ] )∗ (x) + ord L ] (y) ≥ −κ, we have ord L ] (y) ≥ j
j+d
j+d
−κ for any j such that j + d < −i 1 − n 0 + n + d. In particular, y belongs to ] ] L −i1 −n 0 +n+d−1 $ −κ = L −i1 −n 0 +n+d−1−κn . Let j1 := −i 1 − n 0 + n + d − 1 − κn. ]
Then we have y ∈ L j1 and i 1 + j1 ≥ d − 1 + n − n 0 − κn. So (i) is proved. 8. A few lemmas In this section, we provide several lemmas needed for the proof of our major result. Statements of Lemmas 8.3 and 8.4 were due to Jiu-Kang Yu. Material in Sections 8.1 and 8.6 is from [MVW]. 8.1 Let A be a good lattice in W , and let K A0 be as defined in (5.1.2). If g is an element in K A0 , then it is clear that 1 + g is invertible. Hence, c(g) (cf. Section 7.1) is well defined. As in Section 5.1, let (ωψ , S (A)) denote the generalized lattice model of the Weil representation with respect to a good lattice A in W . The following lemma is [W, lem. II.4]. LEMMA
Let g be an element in K A0 , let w be in W , and let c := c(g) belong to sp(W ). (1) Suppose that c · w is in A∗ . Then ωψ (g) · f (w) = ψ hhw, c · wii ρ eψ (2c · w) · f (w) (2)
for f ∈ S (A). Suppose that c = cx,y for some x, y ∈ V and that c · w is in A∗ . Then hhw, cx,y · wii = −2 Trd D/F hw · x, w · yi0 , where Trd D/F denotes the reduced trace from D to F.
8.2 Suppose that Q is a lattice in W and is contained in the good lattice A. Then we have Q ⊆ Q ∗ . Hence, Q acts on S (A), as remarked in Section 6.2.
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
561
LEMMA
Let Q be a lattice in W contained in a good lattice A. Then S (A) Q = S (A) Q ∗ . Proof Let a be an element in Q, let f be an element in S (A), and let ρψ denote the action of Q on S (A). Then we have (ρψ (a)· f )(w) = f (w +a) = f (a +w). We know that f (a + w) = ψ (1/2)hhw, aii ρ eψ (a) · f (w) from (5.1.4). But ρ eψ (a) is trivial because Q is contained in A. Therefore, we have 1 ρψ (a) · f (w) = ψ hhw, aii f (w) 2 for w ∈ W . Then f is fixed by Q if and only if ψ (1/2)hhw, aii is equal to 1 for all a ∈ Q and w ∈ W with f (w) 6 = 0. Therefore, f is fixed by Q if and only if f has support in Q ∗ . By this lemma, for K = G L ,(d/n)+ such that J M,N ⊆ G L ,(d/n)+ and η trivial, (5.3.1) can be rewritten as G ∗ S (A)G L ,(d/n)+ = ωψ (H 0 ) · S (A) B M,N L ,(d/n)+ ∗ because B M,N ⊆ A ⊆ B M,N . Clearly, the lattice A in the above expression is not essential. Hence, we have ∗
S G L ,(d/n)+ = ωψ (H 0 ) · (S B M,N )G L ,(d/n)+ ,
where S is any model of the Weil representation. In particular, we have G S (A0 )G L ,(d/n)+ = ωψ (H 0 ) · S (A0 ) B M,N L ,(d/n)+
(8.2.1)
(8.2.2)
for any good lattice A0 in W such that A0 ⊆ B M,N . 8.3 Let A be a good lattice in W , and let K be a compact subgroup of Sp(W ) such that K is contained in K A , the stabilizer of A in Sp(W ). We know that the action M[g] of K on S (A) is a representation (not just a projective representation). LEMMA
Let A be a good lattice in W , and let B be another lattice so that A∗ ⊆ B. Let K be a compact subgroup of Sp(W ) such that K ⊆ K A . If K stabilizes B, then K also stabilizes the space S (A) B . Proof Let w be an element in B, and let g be an element in the subgroup K . Then
562
SHU-YEN PAN
M[g](S (A)w ) is clearly contained in S (A)g·(A∗ +w) because of (5.1.7) and the assumption that K is contained in the stabilizer of B. Now A is contained in B, and w belongs to B. Hence, g · (A∗ + w) is contained in g · B = B. Because S (A) B is a sum of the subspaces S (A)w for w ∈ B, we see that S (A) B is stabilized by K . 8.4 Let (U (V ), U (V 0 )) be a reductive dual pair in Sp(W ), and let (ωψ , S ) be a model of ^ the Weil representation of Sp( W ). Suppose that K is a compact subgroup of U (V ). ^ We can regard K as a subgroup of Sp( W ) via some splitting. Hence, K acts on S . LEMMA
Let H ⊆ K be compact subgroups of U (V ), and let S0 be a subspace of S such that ^ S H = ωψ (H 0 ) · S0H , where H 0 is the Hecke algebra of U (V 0 ). Suppose that S0 K K 0 is K -stable. Then S = ωψ (H ) · S0 . Proof ^ ^ It is clear that ωψ (H 0 ) · S0K is contained in S K because U (V ) and U (V 0 ) commute ^ e be the inverse image of K under the extension U with each other. Let K (V ) → × e with K × C . Let χ K be the element in H U (V ). As a set, we can identify K e. Then we know that with constant value 1 on K × {1} and with value 0 outside K
ωψ (χ K ) is a projection of S onto S K . Now S K = ωψ (χ K ) · S K is contained in ωψ (χ K )ωψ (H 0 ) · S0H because S K is contained in S H = ωψ (H 0 ) · S0H . Then ωψ (χ K )ωψ (H 0 ) · S0H is equal to ωψ (H 0 )ωψ (χ K ) · S0H by the commutativity of ^ ^ U (V ) and U (V 0 ). Now ωψ (χ K ) · S H is equal to (S H ) K = S K because S0 is 0
0
0
K -stable. Therefore, we obtain the inequality S K ⊆ ωψ (H 0 ) · S0K . Hence, we conclude that S K = ωψ (H 0 ) · S0K . 8.5 LEMMA
Let A be a good lattice in W , and let w be an element in W . Let K be the subgroup of K A0 of elements g such that (g − 1) · w belongs to A. Then the map ψw : K → C× defined by g 7→ ψ((1/2)hh(g − 1) · w, wii) is a character of K . Proof Let g1 , g2 be two elements in K . Then both (g1−1 − 1) · w and (g2 − 1) · w belong to A. It is clear that hhg · w, wii = hh(g − 1) · w, wii because hhw, wii = 0. Now g1 g2 − 1
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
563
is equal to (g1 − 1)(g2 − 1) + (g1 − 1) + (g2 − 1). Therefore, ψw (g1 g2 ) = ψ
1 2 1
hh(g1 g2 − 1) · w, wii
1 hh(g1 − 1)(g2 − 1) · w, wii ψ hh(g1 − 1) · w, wii 2 2 1 · ψ hh(g2 − 1) · w, wii . 2
=ψ
It is easy to compute that hh(g1 − 1)(g2 − 1) · w, wii = hh(g2 − 1) · w, (g1−1 − 1) · wii. Hence, ψ (1/2)hh(g1 − 1)(g2 − 1) · w, wii = 1 because both elements (g2 − 1) · w, (g1−1 − 1) · w are in A and A is a good lattice. Therefore, we have ψ
1
1 1 hh(g1 g2 − 1) · w, wii = ψ hh(g1 − 1) · w, wii ψ hh(g2 − 1) · w, wii , 2 2 2
that is, ψw (g1 g2 ) = ψw (g1 )ψw (g2 ). It is clear that the map ψw is continuous. Hence, ψw is a character of K . 8.6 Let K be a compact subgroup of Sp(W ) contained in K A0 for some good lattice A in W . If w is an element in W and f is a vector in S (A)w , then we define Z f [w, K ] := ωψ (k) · f dk, (8.6.1) K
where dk is a Haar measure on K . Then it is clear that f [w, K ] belongs to S (A) A∗ +K ·w . If f [w, K ] is not the zero vector, then f [w, K ] is fixed by K . Moreover, those f [w, K ] when f runs over a basis of S (A)w span the subspace of S (A) K of functions with support in A∗ + K · w; that is, we have X K S (A) A∗ +K ·w = C f [w, K ] (8.6.2) f ∈S (A)w
(cf. [MVW, chap. 5, sec. III.1]). The following lemma, which is from [MVW, chap. 5, sec. III.3], plays an important role in the proofs of the main results in Section 11. LEMMA
Let w be an element in W , and let K be a compact subgroup of K A0 . Suppose that f is a nonzero vector in S (A)w . Then f [w, K ] is nonzero if and only if f is fixed by the subgroup K 1 := {g ∈ K | g −1 · w ∈ A + w}. Proof First, we prove that f [w, K ] is nonzero if and only if f [w, K ](w) is nonzero. Now
564
SHU-YEN PAN
f [w, K ] is a mapping with support in A∗ + K · w. Suppose that f [w, K ] is nonzero, that is, that f [w, K ](w0 ) 6= 0 for some element w0 in A∗ + K ·w. Write w0 = a +k ·w for some a ∈ A∗ , k ∈ K . Now we have 1 f [w, K ](a + k · w) = ψ hhk · w, aii ρ eψ (a) · f [w, K ](k · w) 2 by (5.1.4) and (5.1.7). Hence, f [w, K ](w0 ) 6 = 0 if and only if f [w, K ](k · w) 6 = 0. Moreover, from (8.6.1) it is clear that f [w, K ](k·w) = f [w, K ](w). Hence, f [w, K ] is nonzero if and only if f [w, K ](w) is nonzero. R R Now f [w, K ](w) = K (ωψ (k) · f )(w) dk is equal to K f (k −1 · w) dk because K is contained in K A0 . Thus, we have Z Z f (k −1 · w) dk = f (k −1 · w) dk K
K1
because f is supported in A∗ + w. Now f (k −1 · w) = ψ (1/2)hh(k − 1) · w, wii f (w) for k ∈ K 1 because (k − 1) · w belongs to A. Therefore, we conclude that Z Z f [w, K ](w) = ψw (k) f (w) dk = ψw (k) dk f (w), (8.6.3) K1
K1
where ψw is defined as in Lemma 8.5. By Lemma 8.5, we know that ψw is a character R of K 1 . The integral K 1 ψw (k) dk in (8.6.3) is essentially equal to the sum of values of a character over all elements of a finite group. Therefore, the last integral in (8.6.3) is nonzero if and only if ψw is trivial on K 1 . Hence, f [w, K ] 6 = 0 if and only if ψw is trivial on K 1 .
9. Proofs of Lemma 6.1 and Proposition 6.2 9.1 Here we have an easy lemma describing some basic properties of the lattice Bs (L , L 0 ) defined in Section 6.1, where L := {L i | i ∈ Z}, L 0 := {L j | j ∈ Z} are admissible lattice chains of period n in V , V 0 , respectively. Recall that \ ] \ 0] Bs (L , L 0 ) := L i ⊗ L 0j ∩ Li ⊗ L j . i+ j=s
i+ j=s
Part (ii) of the following lemma is due to Jiu-Kang Yu. LEMMA
Let s, s1 , s2 be integers:
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
(i) (ii) (iii)
565
B (L , L 0 ) ⊆ Bs2 (L , L 0 ) if s2 ≤ s1 ; Ts1 P 0 0 i+ j=s L i ⊗ L j = i 0 + j 0 =s+n−1 L i 0 ⊗ L j 0 ; T T ] 0] Bs (L , L 0 )∗ = i+ j=−s−n 0 −n 0 −n+1 L i ⊗L 0j + i+ j=−s−n 0 −n 0 −n+1 L i ⊗L j . 0
0
Proof Part (i) is trivial. Part (ii) is easy to check by Lemma 4.3. Now we prove (iii). The dual lattice Bs (L , L 0 )∗ of Bs (L , L 0 ) is \ \ X ] X ] 0] ∗ 0] L i ⊗ L 0j ∩ Li ⊗ L j = (L i )∗ ⊗ L 0∗j + L i∗ ⊗ (L j )∗ . i+ j=s
i+ j=s
i+ j=s
]
i+ j=s ]
0]
0]
We know that (L i )∗ = L −i−n 0 , L 0∗j = L − j−n 0 , L i∗ = L −i−n 0 , and (L j )∗ = L 0− j−n 0 0 0 from Section 4.5. Therefore, X X ] 0] Bs (L , L 0 )∗ = L −i−n 0 ⊗ L − j−n 0 + L −i−n 0 ⊗ L 0− j−n 0 0
i+ j=s
X
=
0]
]
X
Li ⊗ L j +
i+ j=−s−n 0 −n 00
0
i+ j=s
L i ⊗ L 0j .
(9.1.1)
i+ j=−s−n 0 −n 00
Similarly to (ii) of this lemma, we have X 0] Li ⊗ L j = i+ j=−s−n 0 −n 00
0]
\
Li ⊗ L j ,
i+ j=−s−n 0 −n 00 −n+1 ]
X i+ j=−s−n 0 −n 00
]
\
L i ⊗ L 0j =
L i ⊗ L 0j .
(9.1.2)
i+ j=−s−n 0 −n 00 −n+1
Hence, we conclude that Bs (L , L 0 )∗ =
\ i+ j=−s−n 0 −n 00 −n+1
]
L i ⊗ L 0j +
\
0]
Li ⊗ L j .
i+ j=−s−n 0 −n 00 −n+1
9.2. Proof of Lemma 6.1 By Lemma 9.1(iii) it is clear that if s ≥ −s − n 0 − n 00 − n + 1 i.e., if s ≥ (−n − n 0 − n 00 + 1)/2 , then Bs (L , L 0 ) is contained in Bs (L , L 0 )∗ . Hence, (i) is proved. Because of the assumption s ≤ (−n − n 0 − n 00 )/2, we have s ≤ −s −n−n 0 −n 00 . Therefore, we have Bs (L , L 0 )∗ ⊆ Bs (L , L 0 ) for s ≤ (−n − n 0 − n 00 )/2. Hence, (ii) is proved.
566
SHU-YEN PAN
9.3 LEMMA
Let s be an integer. Then \ \ L i ⊗ L 0j ∩ i+ j=s
]
0]
i+ j=s+1
]
\
Li ⊗ L j ⊆
L i ⊗ L 0j +
i+ j=s+1
0]
\
Li ⊗ L j .
i+ j=s+1
Proof ] 0] ] 0] 0] ] 0] ] If L i = L i or L 0j = L j , then it is clear that L i ⊗L j = L i ⊗L j or L i ⊗L j = L i ⊗L 0j . ]
0]
We know that L i ⊆ L i ⊆ L i−1 and L 0j ⊆ L j ⊆ L 0j−1 . Hence, it suffices to prove that ]
0]
]
0]
L i−1 ⊗ L 0j ∩ L i ⊗ L 0j−1 ∩ L i ⊗ L j ⊆ L i ⊗ L 0j + L i ⊗ L j ]
(9.3.1)
]
0]
when L i 6= L i and L 0j 6= L j . Then from Section 4.5 we have L i = L i∗ $ k and 0]
L j = L 0∗j $ l for some k, l. Because L i+n = L i $ and L 0j+n = L 0j $ for any i, j, we only need to consider the cases for k = 0, 1 and l = 0, 1. Define L := L i when k = 0, L := L i∗ $ when k = 1; and define L 0 := L 0j when l = 0, L 0 := L 0∗j $ when l = 1. It is clear that L , L 0 are good lattices in V , V 0 , respectively. First, suppose that k = l = 0. Clearly, L i−1 ⊆ L$ −1 and L 0j−1 ⊆ L 0 $ −1 . Then we have ]
0]
L i−1 ⊗ L 0j ∩ L i ⊗ L 0j−1 ∩ L i ⊗ L j ⊆ L ⊗ L 0 $ −1 ∩ L ∗ ⊗ L 0∗ , ]
0]
L i ⊗ L 0j + L i ⊗ L j = L ∗ ⊗ L 0 + L ⊗ L 0∗ .
(9.3.2)
]
Next, suppose that k = 1 and l = 0. Clearly, L i = L ∗ $ , L i = L i∗ $ = L, L i−1 ⊆ L ∗ , and L 0j−1 ⊆ L 0 $ −1 . Then we have ]
0]
L i−1 ⊗ L 0j ∩ L i ⊗ L 0j−1 ∩ L i ⊗ L j ⊆ L ∗ ⊗ L 0 ∩ L ⊗ L 0∗ , ]
0]
L i ⊗ L 0j + L i ⊗ L j = L ⊗ L 0 + L ∗ ⊗ L 0∗ $.
(9.3.3)
The equality of the two terms in the right-hand side of (9.3.2) or (9.3.3) follows easily from Lemma 9.1(ii). Hence, (9.3.1) holds for the two cases. The proofs for the cases (k, l) = (0, 1) or (1, 1) are similar and omitted. 9.4 Let L (resp., L 0 ) be a small admissible lattice chain in V (resp., V 0 ) with numerical invariant (n, n 0 ) (resp., (n, n 00 )). Define As (L , L 0 ) :=
\ i+ j=s
]
L i ⊗ L 0j +
\ i+ j=s
0]
Li ⊗ L j .
(9.4.1)
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
As in the proof of Lemma 9.1, it is easy to check that \ ] As (L , L 0 )∗ = L i ⊗ L 0j ∩ i+ j=−s−n−n 0 −n 00 +1
We define 0 L0 ⊗ L0 A := L −1 ⊗ L 0−1 $ Aν (L , L 0 )
567
0]
\
L i ⊗ L j . (9.4.2)
i+ j=−s−n−n 0 −n 00 +1
if n = 1, L is self-dual, and n 0 = n 00 = 0; if n = 1, L is self-dual, and n 0 = n 00 = 1; if either n ≥ 2 or n = 1,
n 0 + n 00
(9.4.3)
= 1,
where ν := d(−n − n 0 − n 00 + 2)/2e. LEMMA
The lattice A defined in (9.4.3) is a good lattice in W . Proof First, we suppose that n = 1, L is self-dual, and n 0 = n 00 = 0. Now A∗ = L ∗0 ⊗ L 0∗ 0. We know that L 0 = L ∗0 because n 0 = 0 and L is self-dual. We also know that 0 0∗ 0 0 L 0∗ 0 $ ⊆ L 0 ⊆ L 0 from Section 4.5(iii), that is, that L 0 is a good lattice in V . Hence, A is a good lattice in W . Next, suppose that n = 1, L is self-dual, and n 0 = n 00 = 1. Now A∗ = L ∗−1 ⊗ 0∗ L −1 $ −1 . Now L ∗−1 = L 0 = L −1 $ because L is self-dual and n 0 = 1. So A∗ = 0∗ 0 0∗ −1 from Section 4.5(iii). Hence, A is a L −1 ⊗ L 0∗ −1 . We have L −1 ⊆ L −1 ⊆ L −1 $ good lattice in W . ] 0] Now we consider the remaining cases. We know that L i ⊆ L i−1 and L j ⊆ L 0j−1 T ] for any i, j. So As (L , L 0 ) ⊆ i+ j=s−1 L i ⊗ L 0j . We know that L i ⊆ L i and 0]
L 0j ⊆ L j for any i, j. Hence, \ \ L i ⊗ L 0j ⊆
]
i+ j=−s−n−n 0 −n 00 +1
i+ j=s−1
0]
\
L i ⊗ L 0j ∩
Li ⊗ L j
i+ j=−s−n−n 0 −n 00 +1
if s − 1 ≥ −s − n − n 0 − n 00 + 1. Then we have, by (9.4.2), As (L , L 0 ) ⊆ As (L , L )∗ for s ≥ (−n − n 0 − n 00 + 2)/2. In particular, Aν (L , L 0 ) ⊆ Aν (L , L )∗ . Moreover, we have Aν (L , L 0 )∗ $ F ⊆ Aν (L , L 0 )∗ $ \ =
]
i+ j=−ν−n 0 −n 00 −n+1
=
\ i+ j=−ν−n 0 −n 00 +1
]
L i ⊗ L 0j ∩
0]
\
L i ⊗ L 0j $ ∩
Li ⊗ L j $
i+ j=−ν−n 0 −n 00 −n+1
\ i+ j=−ν−n 0 −n 00 +1
0]
Li ⊗ L j .
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SHU-YEN PAN
Because now either n ≥ 2 or n = 1 and n 0 + n 00 = 1, we have ν ≤ −ν − n 0 − n 00 + 1. So Aν (L , L 0 )∗ $ F ⊆ Aν (L , L 0 ). Hence, Aν (L , L 0 ) is a good lattice in W . 9.5 LEMMA A∗ ⊆ B(L , L 0 , d/n)
for any nonnegative integer d.
Proof First, we suppose that n = 1, L is self-dual, and n 0 = n 00 = 0. Then A∗ = L 0 ⊗ L 0∗ 0. 0 0 Now d must be odd, and it is not difficult to see that B(L , L , 1/1) = L 0 ⊗ L −1 = 0 −1 , as in the proof of Lemma 9.4. L 0 ⊗ L 00 $ −1 . We know that L 0 ⊗ L 0∗ 0 ⊆ L 0 ⊗ L 0$ Next, we suppose that n = 1, L is self-dual, and n 0 = n 00 = 1. Then A∗ = L ∗−1 ⊗ 0∗ L −1 $ −1 . Now d is odd again, and it is not difficult to check that B(L , L 0 , 1/1) = L −1 ⊗ L 0−1 . We know that now L ∗−1 = L 0 = L −1 $ . Moreover, we also know that 0 ∗ 0 L 0∗ −1 ⊆ L −1 from the proof of Lemma 9.4. Then it is clear that A ⊆ B(L , L , 1/1). Next, we consider the other situation. Recall that now \ \ ] 0] A∗ = L i ⊗ L 0j ∩ Li ⊗ L j i+ j=−ν−n−n 0 −n 00 +1
i+ j=−ν−n−n 0 −n 00 +1
from (9.4.2), where ν = d(−n − n 0 − n 00 + 2)/2e, and that \ ] \ d 0] B L , L 0, = L i ⊗ L 0j ∩ Li ⊗ L j n i+ j=µ
i+ j=µ
from (6.1.2), where µ = (−n − n 0 − n 00 − d)/2. Suppose that −n − n 0 − n 00 is even. Then −ν − n 0 − n 00 − n + 1 = (−n − n 0 − n 00 )/2 ≥ (−n − n 0 − n 00 − d)/2 for any nonnegative integer d. Suppose that −n − n 0 − n 00 is odd. In this case, d has to be odd because we always assume that −n −n 0 −n 00 −d is even. Then −ν −n 0 −n 00 −n +1 = (−n − n 0 − n 00 − 1)/2 ≥ (−n − n 0 − n 00 − d)/2 for any nonnegative odd integer d. Hence, the lemma is proved. 9.6 Let A be as defined in Section 9.4. Recall that K A0 := {g ∈ Sp(W ) | (g −1)· A∗ ⊆ A} from Section 5.1. LEMMA
Let d be a positive integer, and let g be an element in G L ,(d/n)+ . Then d ∗ d (g − 1) · B L , L 0 , ⊆ B L , L 0, . n n In particular, G L ,(d/n)+ is a subgroup of K A0 .
(9.6.1)
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
569
Proof Recall that we identify G L ,d/n and ιV 0 (G L ,d/n ); so G L ,d/n is regarded as a subgroup of Sp(W ). Let g be an element in G L ,(d/n)+ . Then \ \ d 0] ] (g − 1) · B L , L 0 , = (g − 1) · L i ⊗ L j ∩ (g − 1) · L i ⊗ L 0j . n i+ j=µ
i+ j=µ
]
]
But we know that (g − 1) · L i ⊆ L i+d+1 and (g − 1) · L i ⊆ L i+d from (4.8.1). Therefore, we have \ ] \ d 0] ⊆ L i+d+1 ⊗ L j ∩ L i+d ⊗ L 0j (g − 1) · B L , L 0 , n i+ j=µ i+ j=µ \ \ ] 0] Li ⊗ L j ∩ ⊆ L i ⊗ L 0j . i+ j=µ+d
i+ j=µ+d+1
From Lemma 9.3, we know that \ \ ] 0] L i ⊗ L 0j ∩ Li ⊗ L j ⊆ i+ j=µ+d
i+ j=µ+d+1
\
]
L i ⊗ L 0j +
i+ j=µ+d+1
\
0]
Li ⊗ L j .
i+ j=µ+d+1
Now µ + d + 1 = −µ − n 0 − n 00 − n + 1. So from Lemma 9.1(iii) we have proved d ∗ d (g − 1) · B L , L 0 , ⊆ B L , L 0, n n
(9.6.2)
for any g ∈ G L ,(d/n)+ . We know that A∗ ⊆ B(L , L 0 , d/n). Then B(L , L 0 , d/n)∗ ⊆ A. Therefore, we have (g − 1) · A∗ ⊆ A for any g ∈ G L ,(d/n)+ . Hence, G L ,(d/n)+ is a subgroup of K A0 . 9.7 Let L (resp., L 0 ) be a good lattice in V (resp., V 0 ), and let B := B(L , L 0 ) as in (6.1.3). We know that B is a good lattice in W . LEMMA
G L ,0+ is a subgroup of K B0 . Proof Recall that B = L ∗ ⊗ L 0 ∩ L ⊗ L 0∗ , B ∗ = L ∗ ⊗ L 0∗ ∩ L$ −1 ⊗ L 0 , and G L ,0+ = {g ∈ G | (g − 1) · L ∗ ⊆ L , (g − 1) · L ⊆ L ∗ $ }. Let g be an element in G L ,0+ . Then (g − 1) · B ∗ = (g − 1) · L ∗ ⊗ L 0∗ ∩ (g − 1) · L$ −1 ⊗ L 0 ⊆ L ⊗ L 0∗ ∩ L ∗ $ ⊗ L 0 = B. Hence, G L ,0+ is a subgroup of K B0 .
570
SHU-YEN PAN
9.8. Proof of Proposition 6.2. Let A be the lattice defined in (9.4.3). From Lemma 9.4 we know that A is a good lattice in W . Let (ωψ , S (A)) be the generalized lattice model of the Weil representation with respect to the good lattice A. Let φ : S → S (A) be an equivalence of the two models of the Weil representation. Clearly, we have φ(S G L ,(d/n)+ ) = S (A)G L ,(d/n)+ ,
φ(S B(L ,L
0 ,d/n)∗
) = S (A) B(L ,L
B(L ,L 0 ,d/n)∗
Therefore, it suffices to prove that the subspace S (A) by G L ,(d/n)+ . Let g be an element in G L ,(d/n)+ . Then we have
d ∗ d ⊆ B L , L 0, (g − 1) · B L , L 0 , n n
0 ,d/n)∗
.
is fixed pointwise
(9.8.1)
from Lemma 9.6. Since A∗ ⊆ B(L , L 0 , d/n) for any positive integer d from 0 ∗ Lemma 9.5, we know that S (A) B(L ,L ,d/n) = S (A) B(L ,L 0 ,d/n) by Lemma 8.2. From (9.8.1) it is clear that B(L , L 0 , d/n) is stabilized by G L ,(d/n)+ . Hence, S (A) B(L ,L 0 ,d/n) is stabilized by G L ,(d/n)+ by Lemma 8.3. Suppose that w is an element in B(L , L 0 , d/n) and that f is a nonzero element in S (A)w . Then ωψ (g) · f (w) = f (g −1 · w) = f (g −1 − 1) · w + w 1 = ψ hhw, (g −1 − 1) · wii f (w) 2 because f satisfies (5.1.4) and (g −1 − 1) · w is in A. But (g −1 − 1) · w is also in B(L , L 0 , d/n)∗ , so ψ (1/2)hhw, (g −1 − 1) · wii is equal to 1. Hence, we have proved that (ωψ (g) · f )(w) = f (w) for any w ∈ B(L , L 0 , d/n). Therefore, S (A) B(L ,L 0 ,d/n) is fixed pointwisely by G L ,(d/n)+ . Similarly, S (A) B(L ,L 0 ,d/n) is also fixed pointwisely by G 0L 0 ,(d/n)+ . This is the proof of (i). Now we prove (ii) of the proposition. Let B denote the lattice B(L , L 0 ). Suppose f ∈ S (B) B ∗ and g ∈ G L ,0+ . We know that g is in K B0 by Lemma 9.6. We have (ωψ (g) · f )(x) = (M B [g] · f )(x) = e ωψ (g) · ( f (g −1 · x)) from (5.1.7) for any x ∈ W . Now e ωψ (g) is trivial because g is in K B0 . Hence, (ωψ (g) · f )(x) is not zero only if −1 g · x belongs to B ∗ . But B ∗ is stable by g, so ωψ (g) · f belongs to S (B) B ∗ . Now suppose that w is in B ∗ . Hence, (g −1 − 1) · w is in B. Then ωψ (g) · f (w) = f (g −1 − 1) · w + w 1 e (g −1 − 1) · w · f (w). = ψ hhw, (g −1 − 1) · wii ρ 2 Because (g −1 − 1) · w is in B, ρ e((g −1 − 1) · w) becomes trivial. Moreover, w ∈ B ∗ and (g −1 − 1) · w ∈ B imply that ψ (1/2)hh(g − 1) · w, wii = 1. Hence,
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(ωψ (g) · f )(w) = f (w) for any w ∈ B ∗ . Therefore, S (B) B ∗ is fixed pointwise 0 by G L ,0+ . By Lemma 8.2, we conclude that S B(L ,L ) is fixed pointwise by G L ,0+ . 0 Similarly, S B(L ,L ) is also fixed pointwise by G 0L 0 ,0+ . 10. Proofs of Proposition 6.3 and Corollary 6.4 We prove Proposition 6.3 and Corollary 6.4 in this section. Jiu-Kang Yu provided several ideas to simplify the original proof. In particular, the statements of Lemma 10.1 and Propositions 10.5 and 10.7 are modified from his suggestion. 10.1 We fix a maximal good lattice 0 (resp., 0 0 ) in V (resp., V 0 ). LEMMA
Let n be a positive integer, let n 00 be 0 or 1, and let {Mi }i∈Z and {Ni }i∈Z be two decreasing chains of free O -modules in V 0 satisfying the following conditions: (i) Mi+n = Mi $ and Ni+n = Ni $ for all i; (ii) Mi ⊆ Ni ⊆ Mi−1 for all i; 0 0 (iii) hMi , N j i0 ⊆ pκ +b(i+ j+n 0 )/nc for all i, j. Then there exists a small admissible lattice chain {L i0 }i∈Z in V 0 with numerical in0] variant (n, n 00 ) such that Mi ⊂ L i0 and Ni ⊂ L i for all i. Proof We can always find two decreasing chains {Mi0 }i∈Z and {Ni0 }i∈Z of lattices in V 0 satisfying conditions (i), (ii), (iii) in the lemma and Mi ⊆ Mi0 , Ni ⊆ Ni0 for all i. Therefore, without loss of generality, we may assume that all Mi and Ni are in 0 fact lattices in V 0 . If n 00 = 1, then hN0 , N0 i0 ⊆ hM−1 , N0 i0 ⊆ pκ from (iii). Then N0 ⊆ N0∗ . Hence, N0 is contained in some maximal good lattice in V 0 , as explained 0 in Section 4.1. If n 00 = 0, then hM0 , M0 i0 ⊆ pκ . Hence, M0 is contained in some maximal good lattice in V 0 . Let 0 00 be a maximal good lattice in V 0 such that N0 ⊆ 0 00 if n 00 = 1 and M0 ⊆ 0 00 if n 00 = 0. Define L 0−k 0 := N−k 0 + 0 00∗ , ( N−b(n+n 0 )/2c + 0 00∗ 0 0 L −b(n+n 0 )/2c := 0 M−b(n+n 0 )/2c + 0 00∗ 0
L 0k := (N−k−n 0 + 0 00∗ )∗ , 0
if n + n 00 is odd, if n + n 00 is even, (10.1.1)
for k 0 = 1, . . . , b(n + n 00 )/2c − 1 and k = 0, . . . , b(n − 1 − n 00 )/2c. Note that b(n + n 00 )/2c + b(n − 1 − n 00 )/2c = n − 1. Therefore, the lattices L i0 for i such that −b(n + n 00 )/2c ≤ i ≤ −b(n + n 00 )/2c + n − 1 have been defined. Let L 0 := {L i0 }i∈Z
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be the chain of lattices generated by the set {L i0 }−b(n+n 0 )/2c≤i≤−b(n+n 0 )/2c+n−1 via 0 0 0 the formula L i+nl = L i0 $ l for any l. Now we want to check that L 0 satisfies all requirements in the lemma. It is clear from (10.1.1) that M−k 0 ⊆ L 0−k 0 for k 0 = 1, . . . , b(n + n 00 )/2c. Suppose that k = 0, . . . , b(n − 1 − n 00 )/2c. Because hMk , N−k−n 0 i0 ⊆ pκ +b(k−k−n 0 +n 0 )/nc ⊆ 0
0
0
0
∗ 00 pκ , we have Mk ⊆ N−k−n 0 . We also know that Mk ⊆ M0 ⊆ 0 . Therefore, Mk ⊆ 0
0
∗ 0 0 0 l 00 00∗ ∗ l N−k−n 0 ∩ 0 = (N−k−n 0 + 0 ) = L k . Since Mi+nl = Mi $ and L i+nl = L i $ , 0
0
we have Mi ⊆ L i0 for all i. 0] We know that L −i−n 0 := L i0∗ . Suppose that k = 0, . . . , b(n − 1 − n 00 )/2c. Then 0
0]
0]
L −k−n 0 = N−k−n 0 + 0 00∗ from (10.1.1). Hence, it is clear that N−k−n 0 ⊆ L −k−n 0 0
0
0
0
for k = 0, . . . , b(n − 1 − n 00 )/2c. Suppose that k 0 = 1, . . . , b(n + n 00 )/2c − 1. Then 0] L k 0 −n 0 = (Nk 0 −n 0 + 0 00∗ )∗ = Nk∗0 −n 0 ∩ 0 00 from (10.1.1). It is clear that Nk 0 −n 0 ⊆ 0
0
0
0
Nk∗0 −n 0 . If n 0 = 0, then k 0 − n 00 ≥ 1. Hence, Nk 0 −n 0 ⊆ M0 ⊆ 0 00 . If n 0 = 1, 0
0
then
k0
−
n 00
≥ 0. Hence, Nk 0 −n 0 ⊆ N0 ⊆ 0
0]
0 00 .
Hence, Nk 0 −n 0 ⊆ L k 0 −n 0 for 0
0
k 0 = 1, . . . , b(n + n 00 )/2c − 1. Suppose now that α := b(n + n 00 )/2c ≥ 1. We have 0] ∗ 00 ⊆ M ∗ Nα−n 0 ⊆ Nα−n ∩ 0 00 = L α−n 0 . Therefore, we conclude that 0 ∩ 0 α−n 0 0
0
0]
0
0
Ni ⊆ L i for all i. Now we want to check that L 0 is a small admissible lattice chain with numerical invariant (n, n 00 ). From (10.1.1), it is clear that L 0−1 ⊆ L 0−2 ⊆ · · · ⊆ L 0−b(n+n 0 )/2c , 0
L 0b(n−1−n 0 )/2c ⊆ L 0b(n−1−n 0 )/2c−1 ⊆ · · · ⊆ L 00 . 0
(10.1.2)
0
∗ 00 and L 0 00∗ from (10.1.1). Therefore, it Now L 00 = N−n 0 ∩ 0 −1 = N−1 + 0 0
is clear that L 00 ⊆ L 0−1 . Let α := b(n + n 00 )/2c and β := b(n − 1 − n 00 )/2c. Then we want to check that L 0−α $ ⊆ L 0β . Now L 0−α $ = M−α $ + 0 00∗ $ and ∗ 00 from (10.1.1). It is clear that 0 00∗ $ ⊆ 0 00 and M $ = L 0β = N−β−n 0 ∩ 0 −α 0
Md(n−n 0 )/2e ⊆ M0 ⊆ 0 00 . And N−β−n 0 ⊆ M−β−n 0 −1 ⊆ M−n ⊆ 0 00 $ −1 . Then 0 0 0 ∗ ∗ 0 00∗ $ = (0 00 $ −1 )∗ ⊆ N−β−n 0 . From Lemma 10.1(iii), we know that Mi ⊆ N−i−n 0 . 0
0
∗ ∗ ∗ ∗ So M−α+n ⊆ Nα−n−n 0 = N−d(n+n 0 )/2e ⊆ N−b(n−1+n 0 )/2c = N−β−n 0 . Hence, we 0 0 0 0 have proved that L 0−b(n+n 0 )/2c $ ⊆ L 0b(n−1−n 0 )/2c . 0
L0
0
Therefore, is a decreasing chain of lattices in V It is obvious that the period of L is n. In this paragraph we assume that n + n 00 is even. Let α := b(n + n 00 )/2c. Then we have −α ≥ −n. Therefore, M−α ⊆ M−n = M0 $ −1 ⊆ 0 00 $ −1 . Hence, M−α + 0.
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
573
0 00∗ ⊆ 0 00 $ −1 . Therefore, we may regard M−α /(M−α ∩ 0 00∗ ) ' (M−α + 0 00∗ )/ 0 00∗
(10.1.3)
as a d-subspace of 0 00 $ −1/ 0 00∗ , where d := O /p. Let {z i }i∈I , for some finite index set I , be a subset of M−α such that the images of these z i in M−α /(M−α ∩ 0 00∗ ) are linearly independent over d. We have −2α + n 00 ≥ −n. Thus, hM−α , M−α i0 ⊆ pκ +b(−2α+n 0 )/nc ⊆ pκ +b−n/nc ⊆ pκ −1 . 0
0
0
0
(10.1.4)
Therefore, hz i $, z j $ i0 ⊆ pκ +1 for any i, j ∈ I . Then the set {z i $ }i∈I as subset of 0 00 satisfies the conditions in Lemma 4.1(iii) with l = 1. Then by Lemma 4.1, there exist a 0 00 -admissible decomposition V 0 = X ⊕ V 0◦ ⊕ Y , where X, Y are totally isotropic and in duality, and a basis {vi }i∈I of X such that vi − z i $ ∈ 0 00 $ for every i ∈ I . Note that a decomposition of V 0 is 0 00 -admissible if and only if it is 0 00∗ -admissible. Therefore, from the choice of the set {z i }i∈I , we have 0
M−α + 0 00∗ = 0 00∗ + (0 00∗ ∩ X )$ −1 = (0 00∗ ∩ X ) ⊕ (0 00∗ ∩ V 0◦ ) ⊕ (0 00∗ ∩ Y ) + (0 00∗ ∩ X )$ −1 = (0 00∗ ∩ X )$ −1 ⊕ (0 00∗ ∩ V 0◦ ) ⊕ (0 00∗ ∩ Y ).
(10.1.5)
Therefore, we have (M−α + 0 00∗ )∗ = (0 00 ∩ X ) ⊕ (0 00 ∩ V 0◦ ) ⊕ (0 00 ∩ Y )$.
(10.1.6)
Note that 0 00 ∩ X = 0 00∗ ∩ X and 0 00 ∩ Y = 0 00∗ ∩ Y . Therefore, (M−α + 0 00∗ )∗ is a good lattice in V 0 . Now (M−α + 0 00∗ )∗ ⊆ L 0k ⊆ 0 00 for k = 0, . . . , b(n − 1 − n 00 )/2c. Hence, L 0k is a good lattice for k = 0, . . . , b(n − 1 − n 00 )/2c. Thus, each L i0 is similar to a good lattice or the dual lattice of a good lattice. The proof for the case when n + n 00 is odd is similar. It is not difficult to see from (10.1.1) that L i0∗ = L 0−i−n 0 for i 6≡ 0
0 0 or bn/2c (mod n). If n 00 = 1, then we also have L 0∗ 0 = L −1 . If n +n 0 is odd, then we also have b(n + n 00 )/2c = b(n − 1 − n 00 )/2c + n 00 and L 0∗ = L 0b(n−1−n 0 )/2c . −b(n+n 0 )/2c 0
0
Note that −b(n + n 00 )/2c ≡ bn/2c (mod n) when n + n 00 is odd. So Section 4.5(ii) is satisfied. Now we show that Section 4.5(iii) is also satisfied. If β ≥ 0, it is easy to check 0 0 0∗ that L 0β and L 00 are good lattices in V 0 . So we have L 0∗ β $ ⊆ L β and L 0 ⊆ L 0 . From 0∗ 0 0∗ 00∗ 00∗ (10.1.1.) we know that L 0 = N−n 0 + 0 and L −1 = N−1 + 0 . Hence, L 0 ⊆ L 0−1 . 0 −1 = (M ∗ ∩ Suppose that n + n 00 is even. Now L 0−α = M−α + 0 00∗ and L 0∗ −α $ −α 00 −1 00∗ 00 −1 0 )$ . It is obvious that 0 ⊆ 0 $ . We already know that M−α ⊆ 0 00 $ −1 , so ∗ $ −1 . From (10.1.4), we know that M ∗ −1 . Hence, we have 0 00∗ ⊆ M−α −α ⊆ M−α $
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0 0 −1 . Now L 0 −1 = (N ∗ L 0−α ⊆ L 0∗ ∩ 0 00 )$ −1 . Since −α $ −α−1 = L β−n = L β $ −β−n 0 0
now n + n 00 is even, we have −β − n 00 = −α + 1. So we have N−β−n 0 ⊆ M−α . 0 0 ∗ ⊆ N∗ −1 ⊆ L 0 Then M−α . Hence, L 0∗ −α $ −α−1 . The proof when n + n 0 is odd is −β−n 0 0
similar. Hence, Section 4.5(iii) is satisfied. Therefore, we conclude that L 0 is a small admissible lattice chain in V 0 with numerical invariant (n, n 00 ). 10.2 LEMMA
Let 0 0 be a maximal good lattice in V 0 . Let {Mi }i∈Z and {Ni }i∈Z be two decreasing chains of free O -modules in V 0 satisfying all conditions in Lemma 10.1. Suppose also that M0 ⊆ 0 0∗ when n 00 = 0, N0 ⊆ 0 0∗ when n 00 = 1. Then we can also require that L 00 ⊆ 0 0 . Proof 0 Suppose that n 0 = 0. Because M0 ⊆ 0 0∗ and hM0 , M0 i0 ⊆ pκ , we have hM0 + 0 0 , M0 + 0 0 i0 ⊆ pκ . 0
(10.2.1)
If z+0 0 is a nontrivial coset in (M0 +0 0 )/ 0 0 , then z+0 0 must be isotropic by (10.2.1). But we know that 0 0∗/ 0 0 is anisotropic because 0 0 is maximal. Hence, M0 + 0 0 must be equal to 0 0 ; that is, M0 is contained in 0 0 . Similarly, if n 00 = 1, then we have N0 ⊆ 0 0 . Hence, the maximal good lattice 0 0 can be used as 0 00 in (10.1.1). Then from (10.1.1) we see that L 00 ⊆ 0 0 . 10.3 Let L := {L i | i ∈ Z} be a small admissible lattice chain in V with numerical invariant (n, n 0 ), and let d be a positive integer. Put n 00 := 0 or 1, so that n+n 0 +n 00 +d is even. Let µ denote the integer (−n − n 0 − n 00 − d)/2. Fix a maximal good lattice 0 (resp., 0 0 ) in V (resp., V 0 ). We assume that every good lattice in L is contained in 0. Define ( L −µ−n 0 if n 00 = 0, M := ] L −µ−n 0 = L ∗µ if n 00 = 1, L] = L ∗µ+n 0 if n 00 = 0, −µ−n 0 −n 00 0 N := (10.3.1) L 0 if n 00 = 1. −µ−n 0 −n 0
Clearly, (M, N ) belongs to the set R (0) defined in Section 5.2. LEMMA
Let M and N be defined as above. Then the group JM,N defined in (5.2.3) is contained
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575
in G L ,(d/n)+ . Proof Suppose that n 00 = 0. Then we know that N ∗ = L µ+n 0 and M = L −µ−n 0 . Therefore, 0 we have J M,N = g ∈ U (V ) (g − 1) · L µ+n 0 ⊆ L −µ−n 0 . 0
Let g be an element in J M,N . Because −µ − n 0 − (µ + n 00 ) = n + d, we have (g − 1) · L µ+n 0 ⊆ L µ+n 0 +d+n . Let i be an integer such that µ + n 00 ≤ i < µ + n 00 + n. 0
0
]
Then L i ⊆ L µ+n 0 and L µ+n 0 +d+n ⊆ L i+d+1 ⊆ L i+d+1 . Hence, (g − 1) · L i ⊆ 0
]
0
]
]
L i+d+1 for any µ+n 00 ≤ i < µ+n 00 +n. Because L i+kn = L i $ k and L i+kn = L i $ k , we have (g − 1) · L i ⊆ ] (g−1)· L i+1
] L i+d+1
for all i. We also have
] L i+1
⊆ L i . Thus, we have
⊆ L i+d+1 for all i. Hence, J M,N is contained in G L ,(d/n)+ from (4.8.1). The proof for n 00 = 1 is similar, so we omit it. 10.4 LEMMA
Suppose that d ≥ 3n + 2 and that L 0 is a small admissible lattice chain in V 0 such that every good lattice in L 0 is contained in a fixed maximal good lattice 0 0 . Then (i) (0 ⊗ 0 0∗ ∩ 0 ∗ ⊗ 0 0 )∗ ⊆ B M,N , (ii) (g − 1) · B M,N ⊆ 0 ⊗ 0 0∗ ∩ 0 ∗ ⊗ 0 0 for any g ∈ G L ,(d/n)+ , (iii) (0 ⊗ 0 0∗ ∩ 0 ∗ ⊗ 0 0 )∗ ⊆ B(L , L 0 , d/n), (iv) G L ,(d/n)+ ⊆ K A0 , where A := 0 ⊗ 0 0∗ ∩ 0 ∗ ⊗ 0 0 . Proof As in Section 6.1, we know that (0 ⊗ 0 0∗ ∩ 0 ∗ ⊗ 0 0 )∗ = 0$ −1 ⊗ 0 0 ∩ 0 ∗ ⊗ 0 0∗ . Suppose first that n 00 = 0. Then M = L −µ−n 0 and N = L ∗µ+n 0 . Hence, M ∗ = ]
0
]
L µ , N ∗ = L µ+n 0 , and B M,N = L µ ⊗ 0 0∗ ∩ L µ+n 0 ⊗ 0 0 $ −1 . Recall that µ := 0
0
]
(−n − n 0 − n 00 − d)/2; so µ < −2n. Hence, 0 ∗ ⊆ L µ and 0$ −1 ⊆ L µ+n 0 $ −1 . 0 Therefore, 0$ −1 ⊗ 0 0 ∩ 0 ∗ ⊗ 0 0∗ ⊆ B M,N . The proof for n 00 = 1 is similar. Let g be an element in G L ,(d/n)+ . Suppose that n 00 = 0. Then we know that ] B M,N = L µ ⊗ 0 0∗ ∩ L µ+n 0 ⊗ 0 0 $ −1 . Therefore, (g − 1) · B M,N ⊆ L µ+d ⊗ 0 0∗ ∩ 0
]
L µ+n 0 +d+1 $ −1 ⊗ 0 0 . Clearly, if d is large enough (in particular, if d ≥ 3n + 2), 0
]
then we have L µ+d ⊆ 0 and L µ+n 0 +d+1 $ −1 ⊆ 0 ∗ . Hence, (g − 1) · B M,N ⊆ 0
0 ⊗ 0 0∗ ∩ 0 ∗ ⊗ 0 0 . The proof for n 00 = 1 is similar.
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Recall that \ ] \ d 0] B L , L 0, = L i ⊗ L 0j ∩ Li ⊗ L j . n i+ j=µ
i+ j=µ
]
0]
Now µ < −2n. We know that 0 ∗ ⊆ L i ⊆ L i and 0 0∗ ⊆ L 0j ⊆ L j when i < 0 and j < 0. Hence, (iii) is obvious. From (i) we have A∗ ⊆ B M,N . From (ii) we have (g − 1) · B M,N ⊆ A for any g ∈ G L ,(d/n)+ . Hence, (g − 1) · A∗ ⊆ (g − 1) · B M,N ⊆ A. Therefore, G L ,(d/n)+ is a subgroup of K A0 . 10.5 Recall that for a positive integer d and a small admissible lattice chain L in V , we let Q (d) denote the set of small admissible lattice chains L 0 in V 0 with numerical invariant (n, n 00 ) such that −n − n 0 − n 00 − d is even. PROPOSITION
Let L be a regular small admissible lattice chain in V of period n, and let d be a positive integer. Assume that d ≥ 3n + 2. Then we have X 0 ∗ S G L ,(d/n)+ ⊆ ωψ (H 0 ) · S B(L ,L ,d/n) . (10.5.1) L 0 ∈Q (d)
Proof Let (n, n 0 ) denote the numerical invariant of L . Put n 00 := 0 or 1 so that n+n 0 +n 00 +d is even. Let Q (0 0 , d) denote the subset of small admissible lattice chains L 0 in Q (d) such that every good lattice L 0 ∈ L 0 is contained in a fixed maximal good lattice 0 0 . Obviously, it suffices to prove that X 0 ∗ S G L ,(d/n)+ ⊆ ωψ (H 0 ) · S B(L ,L ,d/n) . L 0 ∈Q (0 0 ,d)
Let M, N be defined as in Section 10.2. Let A be the good lattice 0 ⊗ 0 0∗ ∩ 0 ∗ ⊗ 0 0 in W . By Lemma 10.4 we have A∗ ⊆ B M,N and A∗ ⊆ B(L , L 0 , d/n) for any L 0 ∈ Q (0 0 , d). Hence, by Lemma 8.2, it suffices to prove that X S (A)G L ,(d/n)+ ⊆ ωψ (H 0 ) · S (A) B(L ,L 0 ,d/n) . L 0 ∈Q (0 0 ,d)
Now we identify B M,N with HomO (L −µ−n 0 $ −κ , 0 0∗ ) ∩ HomO (L ] −µ−n
0 0 −n 0
$ 1−κ , 0 0 ) if n 00 = 0,
−κ Hom (L ] , 0 0∗ ) ∩ HomO (L −µ−n 0 −n 0 $ 1−κ , 0 0 ) if n 00 = 1 O −µ−n 0 $ 0 (10.5.2)
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as in (5.2.5), where µ := (−n − n 0 − n 00 − d)/2. Assume that w is an element in B M,N and that f [w, G L ,(d/n)+ ] is not zero, where f [w, G L ,(d/n)+ ] is defined in (8.6.1) with a nonzero element f ∈ S (A)w . By Lemma 10.4(iv), we know that G L ,(d/n)+ is a subgroup of K A0 , so f [w, G L ,(d/n)+ ] is defined. ] Let x 0 be an element in L i , let y 0 be an element in L j , and let x be x 0 , y be y 0 $ 1−κ for some i, j. Assume that i, j are two integers such that d − n 0 ≤ i + j < d + n − n 0 . ] ] We know that L i ⊆ L i and L j ⊆ L j ⊆ L j−1 . Then the pair x, y satisfies the conditions in Corollary 7.7(ii). Therefore, u x,y is defined and belongs to G L ,(d/n)+ . By Lemma 10.4(ii), we have (u x,y − 1) · B M,N ⊆ A. Therefore, by Lemma 8.1(i), we have ωψ (u x,y )· f = ψ hhw, cx,y ·wii f . Since we assume that f [w, G L ,(d/n)+ ] 6= 0, ψ hhw, cx,y · wii must be 1 by Lemma 8.6. Therefore, hhw, cx,y · wii is in pλFF , which is the kernel of ψ, so we have Trd D/F (hw · x, w · yi0 ) ∈ pλFF by Lemma 8.1(ii). Then 0 we have hw · x, w · yi0 ∈ pλ = pκ+κ from (5.2.1). Hence, we have hw · x 0 $ −κ , w · y 0 $ −κ i0 ∈ pκ −1 . 0
]
Because x 0 and y 0 can be arbitrary in L i and L j , respectively, we have ]
hw · L i $ −κ , w · L j $ −κ i0 ⊆ pκ −1 . 0
(10.5.3)
Define Mi := w(L −µ−n 0 +i $ −κ ), ]
N j := w(L −µ−n 0 + j $ −κ ),
(10.5.4)
for any i, j. Therefore, (10.5.3) becomes hMi+µ+n 0 , N j+µ+n 0 i0 ⊆ pκ −1 . 0
Let i 0 := i + µ + n 0 , j 0 := j + µ + n 0 . The assumption d − n 0 ≤ i + j < d + n − n 0 implies −n −n 00 ≤ i + j −(n −n 0 +n 00 +d) < −n 00 . Then we have −n ≤ i 0 + j 0 +n 00 < 0. Hence, we have 0 0 0 0 hMi 0 , N j 0 i0 ⊆ pκ +b(i + j +n 0 )/nc for all i 0 , j 0 such that −n ≤ i 0 + j 0 + n 00 < 0. Because Mi+nk = Mi $ k and N j+nk = N j $ k for any k, we have hMi , N j i0 ⊆ pκ +b(i+ j+n 0 )/nc 0
0
for all i, j. It is clear that M j and N j are free O -modules in V 0 , M j1 (resp., N j1 ) is contained in M j2 (resp., N j2 ) whenever j2 ≤ j1 , and M j+n (resp., N j+n ) is equal to M j $ (resp., N j $ ) for each j. Moreover, we have N j+1 ⊆ M j ⊆ N j because
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]
]
L −µ−n 0 + j+1 ⊆ L −µ−n 0 + j ⊆ L −µ−n 0 + j . Then by Lemma 10.1 there is a small admissible lattice chain L 0 := {L 0j } j∈Z in V 0 with numerical invariant (n, n 00 ) such 0]
that M j (resp., N j ) is contained in L 0j (resp., L j ) for all j. From the definition, we T ] know that w is an element in i+ j=µ L i ⊗ M j ∩ L i ⊗ N j under the identification ] ] L i ⊗ M j = HomO (L i )∗ $ −κ , M j = HomO (L −i−n 0 $ −κ , M j ), ]
L i ⊗ N j = HomO (L i∗ $ −κ , N j ) = HomO (L −i−n 0 $ −κ , N j ). T T ] 0] Therefore, w is in the lattice i+ j=µ L i ⊗ L 0j ∩ i+ j=µ L i ⊗ L j ; that is, w is in B(L , L 0 , d/n). By (10.5.2) and (10.5.4), we know that M0 is contained in 0 0∗ if n 00 = 0 and that N0 is contained in 0 0∗ if n 00 = 1. So we can require that L 00 be contained in 0 0 by Lemma 10.2. Therefore, all good lattices in L 0 are contained in 0 0 . Then we have A∗ ⊆ B(L , L 0 , d/n) by Lemma 10.4(iii). We know that X X G S (A) B M,N L ,(d/n)+ = C f [w, G L ,(d/n)+ ] w∈B M,N f ∈S (A)w
from (8.6.2). The support of f [w, G L ,(d/n)+ ] is contained in B(L , L 0 , d/n) because A∗ + w ⊆ B(L , L 0 , d/n) and B(L , L 0 , d/n) is stabilized by G L ,(d/n)+ . Therefore, we have proved that if f [w, G L ,(d/n)+ ] is nonzero, then f [w, G L ,(d/n)+ ] must be in S (A) B(L ,L 0 ,d/n) for some small admissible lattice chain L 0 ∈ Q (0 0 , d). Hence, we have X G S (A) B M,N L ,(d/n)+ ⊆ S (A) B(L ,L 0 ,d/n) . (10.5.5) L 0 ∈Q (0 0 ,d)
It is clear that G L ,(d/n)+ stabilizes B M,N . Hence, G L ,(d/n)+ stabilizes S (A) B M,N by Lemma 8.3. It is also obvious that (S (A) B M,N )G L ,(d/n)+ is not trivial. Therefore, we have G S (A)G L ,(d/n)+ = ωψ (H 0 ) · S (A) B M,N L ,(d/n)+ (10.5.6) by (8.2.2). Combining (10.5.5) and (10.5.6), we have proved that G L ,(d/n)+ X S (A)G L ,(d/n)+ ⊆ ωψ (H 0 ) · S (A) B(L ,L 0 ,d/n) .
(10.5.7)
L 0 ∈Q (0 0 ,d)
We know that each space S (A) B(L ,L 0 ,d/n) is fixed by G L ,(d/n)+ by Proposition 6.2(i). So we have X S (A)G L ,(d/n)+ ⊆ ωψ (H 0 ) · S (A) B(L ,L 0 ,d/n) . (10.5.8) L 0 ∈Q (0 0 ,d)
Hence, the proposition is proved.
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
579
10.6 Let A be the good lattice defined in (9.4.3). LEMMA
Let d be an integer greater than 1. Then d (g − 1) · B L , L 0 , ⊆A n for any g ∈ G L ,((d−1)/n)+ .
G L ,((d−1)/n)+ . In particular, B(L , L 0 , d/n) is stabilized by
Proof Let g be an element in G L ,((d−1)/n)+ . We have \ \ d ] 0] (g − 1) · B L , L 0 , = (g − 1) · L i ⊗ L 0j ∩ (g − 1) · L i ⊗ L j . n i+ j=µ
i+ j=µ
]
]
We know that (g − 1) · L i ⊆ L i+d and (g − 1) · L i ⊆ L i+d−1 by (4.8.1). Hence, we have \ \ d ] 0] (g − 1) · B L , L 0 , ⊆ L i ⊗ L 0j ∩ Li ⊗ L j . n i+ j=µ+d
i+ j=µ+d−1
From Lemma 9.3, we know that d (g − 1) · B L , L 0 , ⊆ n
]
\
L i ⊗ L 0j +
i+ j=µ+d
\
0]
Li ⊗ L j .
i+ j=µ+d
We have µ + d = (−n − n 0 − n 00 + d)/2. First, suppose that n = 1, L is self-dual, and n 0 = n 00 = 0. Now µ + d = (−1 + d)/2 ≥ 0. So it is clear that \ i+ j=µ+d
]
\
L i ⊗ L 0j +
0]
L i ⊗ L j ⊆ L 0 ⊗ L 00 = A.
i+ j=µ+d
Next, we suppose that n = 1, L is self-dual, and n 0 = n 00 = 1. Now µ + d = (−3 + d)/2 ≥ −1. So \ \ ] 0] L i ⊗ L 0j + L i ⊗ L j ⊆ L 0 ⊗ L 0−1 = L −1 ⊗ L 0−1 $. i+ j=µ+d
i+ j=µ+d
0 We know that L 0−1 $ ⊆ L 0∗ −1 from the proof of Lemma 9.4. Hence, L −1 ⊗ L −1 $ ⊆ L −1 ⊗ L 0∗ −1 = A. Now we consider the remaining cases. Because we now assume that
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SHU-YEN PAN
d is greater than 1 and −n − n 0 − n 00 + d is even, we have (−n − n 0 − n 00 + d)/2 ≥ d(−n − n 0 − n 00 + 2)/2e = ν. Hence, we conclude that \ \ ] 0] L i ⊗ L 0j + L i ⊗ L j ⊆ Aν (L , L 0 ). i+ j=µ+d
i+ j=µ+d
10.7 PROPOSITION
Suppose that t ≥ 2 and that L (resp., L 0 ) is a small admissible lattice chain in V (resp., V 0 ) with numerical invariant (n, n 0 ) (resp., (n, n 00 )) such that n + n 0 + n 00 + t is even. Then X 0 ∗ 00 ∗ (S B(L ,L ,t/n) )G L ,((t−1)/n)+ ⊆ S B(L ,L ,(t−1)/n) . (10.7.1) L 00 ∈Q (t−1)
Proof Let A be the good lattice defined in Section 9.4 for given L and L 0 . From Lemma 9.5 we know that A∗ ⊆ B(L , L 0 , t/n) for any nonnegative t, so the subspace S (A) B(L ,L 0 ,t/n) is defined. Let Q (A, t − 1) denote the subset of small admissible lattice chains in Q (t − 1) such that A∗ ⊆ B(L , L 00 , (t − 1)/n). Obviously, it suffices to prove that X 00 ∗ 0 ∗ (S B(L ,L ,t/n) )G L ,((t−1)/n)+ ⊆ S B(L ,L ,(t−1)/n) . L 00 ∈Q (A,t−1)
By the remark in Section 8.2, we need to prove that X G S (A) B(L ,L 0 ,t/n) L ,((t−1)/n)+ ⊆
S (A) B(L ,L 00 ,(t−1)/n) .
L 00 ∈Q (A,t−1)
Let w be an element in B(L , L 0 , t/n), and let µ be the integer (−n − n 0 − n 00 − t)/2. As in Section 10.5, we define Mi := w(L −µ−n 0 +i $ −κ ), ]
N j := w(L −µ−n 0 + j $ −κ ),
(10.7.2)
for any i, j. By an argument similar to the proof of Proposition 10.5, we can show that 0 0 hMi , N j i0 ⊆ pκ +b(i+ j+1+n 0 )/nc (10.7.3) for all i, j. (Note that G L ,(d/n)+ in the proof of Proposition 10.5 is replaced by G L ,((t−1)/n)+ here.) Let Mi0 := Mi−n 0 and N 0j := N j−n 0 . Then (10.7.3) becomes 0
0
hMi0 , N 0j i0 ⊆ pκ +b(i+ j+1−n 0 )/nc 0
0
(10.7.4)
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
581
for all i, j. Then by Lemma 10.1 there is a small admissible lattice chain L 00 := {L 00j } j∈Z in V 0 with numerical invariant (n, 1 − n 00 ) such that M 0j is contained in L 00j 00]
and N 0j is contained in L j for all j. Now we have \ ] \ ] \ ] Li ⊗ M j = L i ⊗ M 0j+n 0 ⊆ L i ⊗ L 00j+n 0 = i+ j=µ
and \
0
i+ j=µ
\
Li ⊗ N j =
i+ j=µ
i+ j=µ
\
L i ⊗ N 0j+n 0 ⊆ 0
i+ j=µ
0
00]
i+ j=µ
L i ⊗ L 00j
i+ j=µ+n 00
00]
\
L i ⊗ L j+n 0 = 0
]
\
Li ⊗ L j .
i+ j=µ+n 00
But we know that µ + n 00 =
−n − t − n 0 − n 00 −n − (t − 1) − n 0 − (1 − n 00 ) + n 00 = . 2 2
Hence, ]
\
\
L i ⊗ L 00j ∩
i+ j=µ+n 00
i+ j=µ+n 00
t − 1 00] L i ⊗ L j = B L , L 00 , . n
(10.7.5)
T ] From the definition in (10.7.2), we know that w belongs to i+ j=µ L i ⊗ M j ∩ T 00 i+ j=µ L i ⊗ N j . So we have shown that w is in B(L , L , (t − 1)/n) for some L 00 . From (10.7.4) we have hM 0j , N−0 j−1+n 0 i0 ⊆ pκ . 0
0
Identify B(L , L 0 , t/n) with \ \ ] 0] Hom(L −i−n 0 $ −κ , L 0j ) ∩ Hom(L −i−n 0 $ −κ , L j ). i+ j=µ
(10.7.6)
i+ j=µ 0]
Then we know that N−0 j−1+n 0 is contained in L − j−1 . Thus, we may choose L 00j to 0
0]
contain (L − j−1 )∗ = L 0j+1−n 0 . Therefore, 0
\
] Li
⊗
L 00j
⊇
i+ j=µ+n 00
]
\
0
i+ j=µ+n 00
]
\
L i ⊗ L 0j+1−n 0 =
L i ⊗ L 0j .
i+ j=µ+1
Moreover, since M 0j ⊆ L 0j−n 0 , we may also choose L 00j to be contained in L 0j−n 0 . Then 0
\
00]
Li ⊗ L j =
i+ j=µ+n 00
0
\
L i ⊗ L 00∗ − j−(1−n 0 ) ⊇ 0
i+ j=µ+n 00
=
\ i+ j=µ+n 00
\
0]
L i ⊗ L j+1−n 0 = 0
L i ⊗ L 0∗ − j−1
i+ j=µ+n 00
\ i+ j=µ+1
0]
Li ⊗ L j .
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SHU-YEN PAN
We know that t − 2 B L , L 0, = n
]
\
L i ⊗ L 0j ∩
i+ j=µ+1
\
0]
Li ⊗ L j .
(10.7.7)
i+ j=µ+1
Hence, the lattice B(L , L 00 , (t − 1)/n) contains the lattice B(L , L 0 , (t − 2)/n). Hence, B(L , L 00 , (t − 1)/n) contains the lattice A∗ by Lemma 9.5 because now t ≥ 2. Then the support of f [w, G L ,((t−1)/n)+ ] is contained in B(L , L 00 , (t − 1)/n). Therefore, we have proved that if w ∈ B(L , L 0 , t/n) and f [w, G L ,((t−1)/n)+ ] is nonzero, then f [w, G L ,((t−1)/n)+ ] must be in S (A) B(L ,L 00 ,(t−1)/n) for some small admissible lattice chain L 00 with numerical invariant (n, 1 − n 00 ). Therefore, we conclude that X G S (A) B(L ,L 0 ,t/n) L ,((t−1)/n)+ ⊆ S (A) B(L ,L 00 ,(t−1)/n) . (10.7.8) L 00 ∈Q (t−1)
10.8. Proof of Proposition 6.3. Let k be an integer such that d + k ≥ 3n + 2. Then we have X ∗ S G L ,((d+k)/n)+ ⊆ ωψ (H 0 ) · S B(L ,M ,(d+k)/n)
(10.8.1)
M ∈Q (d+k)
by Proposition 10.5. Then we have S G L ,((d+k−1)/n)+ = (S G L ,((d+k)/n)+ )G L ,((d+k−1)/n)+ X ∗ G L ,((d+k−1)/n)+ = ωψ (H 0 ) · S B(L ,M ,(d+k)/n) . M ∈Q (d+k)
By Lemma 10.6, each B(L , M , (d + k)/n) is stabilized by G L ,((d+k−1)/n)+ . Hence, ∗ each space S B(L ,M ,(d+k)/n) is stabilized by G L ,((d+k−1)/n)+ by Lemmas 8.2 and 8.3. Hence, X ∗ G L ,((d+k−1)/n)+ S G L ,((d+k−1)/n)+ = ωψ (H 0 ) · S B(L ,M ,(d+k)/n) M ∈Q (d+k)
by Lemma 8.4. Then we have
X
S B(L ,M ,(d+k)/n)
∗
G L ,((d+k−1)/n)+
M ∈Q (d+k)
=
X M ∈Q (d+k)
(S B(L ,M ,(d+k)/n) )G L ,((d+k−1)/n)+ . ∗
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
583
By Proposition 10.7, we have (S B(L ,M ,(d+k)/n) )G L ,((d+k−1)/n)+ ⊆ ∗
S B(L ,M
X
0 ,(d+k−1)/n)∗
.
M 0 ∈Q (d+k−1)
(10.8.2) Therefore, we have S G L ,((d+k−1)/n)+ ⊆ ωψ (H 0 ) ·
(S B(L ,M
X
0 ,(d+k)/n)∗
)G L ,((d+k−1)/n)+
M 0 ∈Q (d+k)
S B(L ,M
X
⊆ ωψ (H 0 ) ·
0 ,(d+k−1)/n)∗
.
M 0 ∈Q (d+k−1)
Repeating the same process, we conclude that X S G L ,(d/n)+ ⊆ ωψ (H 0 ) ·
S B(L ,L
0 ,d/n)∗
.
(10.8.3)
L 0 ∈Q (d)
The opposite inclusion ωψ (H 0 ) ·
X
S B(L ,L
0 ,d/n)∗
⊆ S G L ,(d/n)+
(10.8.4)
L 0 ∈Q (d)
^ ^ follows from Proposition 6.2 and the commutativity of U (V ) and U (V 0 ). The proof is complete. 10.9. Proof of Corollary 6.4. Let Q 0 (d) denote the subset of regular small admissible lattice chains in Q (d). It is clear that X 0 ∗ S G L ,(d/n)+ ⊃ ωψ (H 0 ) · S B(L ,L ,d/n) . L 0 ∈Q 0 (d)
So now we prove the opposite inclusion. Because the Witt index of V 0 is now large, we may assume that we have a decomposition V 0 = V10 ⊕ X ⊕ Y such that X, Y are totally isotropic and dual to each other, M j ⊆ V10 , N j ⊆ V10 for all j, and dim D (X ) = n 0 + 1 for some number n 0 ≥ n − 1, where M j , N j are defined in (10.5.3) or (10.7.2). Moreover, we can assume that the decomposition V 0 = V10 ⊕ X ⊕ Y satisfies the condition M M xk O ⊕ yk O , 0 0 = (0 0 ∩ V10 ) ⊕ 0≤k≤n 0
0≤k≤n 0
where {xi , yi }0≤i≤n 0 is a self-dual basis of X ⊕ Y (i.e., hxi , x j i = hyi , y j i = 0 and 0 hxi , y j i = δi j $ κ ). Then 0 0 ∩ V10 is a maximal good lattice in V10 . From the proofs of Proposition 10.5 and Proposition 10.7, we know that M j , N j are free O -modules
584
SHU-YEN PAN
in V10 , and that M j+n = M j $ , N j+n = N j $ , M j ⊆ N j ⊆ M j−1 , and hMi , N j i0 ⊆ 0 0 pκ +b(i+ j+n 0 )/nc for all i, j. Suppose that L i00 are the lattices L i0 in V10 constructed in the proof of Lemma 10.1, and let L 00 be the small admissible lattice chain {L i00 }i∈Z (in V10 ). Two lattices in L 00 are not necessarily distinct. Let M M M L i0 := L i00 ⊕ xk p ⊕ xk O ⊕ yk O 0≤k≤i
i+1≤k≤n 0
0≤k≤n 0
for i = 1 − n 0 , . . . , b(n − n 0 − 1)/2c. We regard as a subset of L i0 for each i. Let L 0 be the lattice chain as constructed in the proof of Lemma 10.1 for the new L i0 . Then clearly L 0 is a regular small admissible lattice chain in V 0 . As in Sections 10.5 and 10.7, we show that if f [w, G L ,(d/n)+ ] is nonzero, then w is in B(L , L 00 , d/n) for some L 00 given as above. Hence, w is in B(L , L 0 , d/n) because L i00 ⊂ L i0 for each i. Thus, by an argument similar to that in Section 10.5, we prove that X 0 ∗ S G L ,(d/n)+ ⊂ ωψ (H 0 ) · S B(L ,L ,d/n) . L i00
L 0 ∈Q 0 (d)
The proof is complete.
11. Proof of Proposition 6.5 11.1 0 in V 0 such that We fix a maximal good lattice 0 0 and a minimal good lattice 0m 0 ⊆ 00. 0m LEMMA 0∗ is a good lattice in W . Let L be a good lattice in V . Then L ∗ ⊗ 0 0 ∩ L ⊗ 0m
Proof From [W, sec. I.15] we know that there exists a decomposition V = X 1 ⊕ X 2 such that L = (L ∩ X 1 ) ⊕ (L ∩ X 2 ) and L ∗ = (L ∩ X 1 ) ⊕ (L ∩ X 2 )$ −1 . There also exists a decomposition V 0 = Y1 ⊕ Y2 ⊕ Y3 ⊕ Y4 such that 0 0m = (0 0 ∩ Y1 )$ ⊕ (0 0 ∩ Y2 )
⊕ (0 0 ∩ Y3 )
⊕ (0 0 ∩ Y4 ),
⊕ (0 0 ∩ Y3 )
⊕ (0 0 ∩ Y4 ),
0 0∗ = (0 0 ∩ Y1 )
⊕ (0 0 ∩ Y2 )$ −1 ⊕ (0 0 ∩ Y3 )
⊕ (0 0 ∩ Y4 ),
0∗ 0m = (0 0 ∩ Y1 )
⊕ (0 0 ∩ Y2 )$ −1 ⊕ (0 0 ∩ Y3 )$ −1 ⊕ (0 0 ∩ Y4 ).
0 0 = (0 0 ∩ Y1 )
⊕ (0 0 ∩ Y2 )
From the above decompositions it is easy to check that 0∗ 0 0∗ L ⊗ 0m ∩ L ∗ ⊗ 0 0 ⊆ L ∗ ⊗ 0m + L ⊗ 0 0∗ = (L ⊗ 0m ∩ L ∗ ⊗ 0)0∗ .
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
585
Moreover, we have 0 L ∗ ⊗ 0m + L ⊗ 0 0∗ ⊆ L ∗ ⊗ 0 0 + L ⊗ 0 0∗ = L$ −1 ⊗ 0 0 ∩ L ∗ ⊗ 0 0∗ 0∗ ⊆ (L ⊗ 0m ∩ L ∗ ⊗ 0 0 )$ −1 . 0∗ is a good lattice in W . Hence, L ∗ ⊗ 0 0 ∩ L ⊗ 0m
From the decompositions in the proof of the lemma, it is easy to check that 0∗ 0∗ L ∗ ⊗ 0 0 ∩ L ⊗ 0m = L ⊗ 0 0 + L ∗ $ ⊗ 0m .
(11.1.1)
11.2 LEMMA Let 0 0 be
(i)
(ii)
(iii)
a fixed maximal good lattice in V 0 . Suppose that M0 , M−1 are two O -modules in V 0 such that M0 ⊆ M−1 ⊆ 0 M0 $ −1 , M0 ⊆ 0 0 , and hM0 , M−1 i0 ⊆ pκ . Then there exists a good lattice L 0 in V 0 such that L 0 ⊆ 0 0 , M0 ⊆ L 0 , and M−1 ⊆ L 0∗ . 0 be a minimal good lattice in V 0 such that 0 0 ⊆ 0 0 . Suppose that Let 0m m M0 , M−1 are two O -modules in V 0 such that M0 ⊆ M−1 ⊆ M0 $ −1 , M0 ⊆ 0∗ , and hM , M i0 ⊆ pκ 0 . Then there exists a good lattice L 0 in 0 0 , M−1 ⊆ 0m 0 −1 0 ⊆ L 0 ⊆ 0 0 , M ⊆ L 0 , and M 0∗ V 0 such that 0m 0 −1 ⊆ L . 0 Suppose that M is an O -module in V 0 such that hM, Mi0 ⊆ pκ and M ⊆ 0 0∗ . Then M is contained in 0 0 .
Proof Without loss of generality, we may assume that M0 , M−1 are lattices in V 0 . From the assumption, we have M−1 + 0 0∗ ⊆ M0 $ −1 + 0 0∗ ⊆ 0 0 $ −1 . So we may regard M−1 /(M−1 ∩ 0 0∗ ) ' (M−1 + 0 0∗ )/ 0 0∗ as a d-subspace of 0 0 $ −1/ 0 0∗ . Let {z i }i∈J (for some finite index set J ) be a subset of M−1 such that the images of these z i in M−1 /(M−1 ∩ 0 0∗ ) are linearly independent over d. From the assumption, we have 0 0 hM−1 , M−1 i0 ⊆ pκ −1 . Therefore, hz i $, z j $ i0 ⊆ pκ +1 for any i, j ∈ J . Therefore, the set {z i $ }i∈J as subset of 0 0 satisfies the condition in Lemma 4.1(iii) with l = 1. Thus, there exists a 0 0 -admissible decomposition V 0 = X ⊕ V 0◦ ⊕ Y , where X , Y are totally isotropic and in duality, and a basis {vi }i∈J of X such that vi − z i $ ∈ 0 0 $ for every i ∈ J . Note that a decomposition of V 0 is 0 0 -admissible if and only if it is 0 0∗ -admissible. Therefore, from the choice of the set {z i }i∈J , we have M−1 + 0 0∗ = (0 0∗ ∩ X )$ −1 ⊕ (0 0∗ ∩ V 0◦ ) ⊕ (0 0∗ ∩ Y ).
(11.2.1)
From (11.2.1), we see that (M−1 + 0 0∗ )∗ is a good lattice in V 0 . Define L 0 := (M−1 + ∗ ∩ 0 0 . Therefore, L 0∗ = M 0∗ 0 0 0 0∗ )∗ = M−1 −1 + 0 . Hence, we have L ⊆ 0 and
586
SHU-YEN PAN
∗ . Therefore, we have M ∗ 0∗ 0 L 0 ⊆ M−1 −1 ⊆ L . Since M0 ⊆ M−1 and M0 ⊆ 0 , we have 0 0 M0 ⊆ L . Then L satisfies all requirements. 0 ⊆ L 0. For (ii), let L 0 be given as in the proof of (i). We only need to check that 0m 0∗ because As in the previous paragraph, we have L 0∗ = M−1 + 0 0∗ . Hence, L 0∗ ⊆ 0m 0∗ 0∗ 0∗ 0 0 M−1 ⊆ 0m and 0 ⊆ 0m . Therefore, we conclude that 0m ⊆ L . Part (iii) has already been proved in Lemma 10.2.
11.3 LEMMA 0∗ . Then G Let L be a good lattice in V , and let A := L ∗ ⊗0 0 ∩ L ⊗0m L ,0+ is contained 0 in K A .
Proof From (11.1.1) it is easy to see that 0 A∗ = L ∗ ⊗ 0 0∗ ∩ L ⊗ 0m $ −1 .
(11.3.1)
0 ⊆ A. Hence, Let g be an element in G L ,0+ . Then (g − 1) · A∗ ⊆ L ⊗ 0 0∗ ∩ L ∗ ⊗ 0m 0 g is in K A .
11.4 PROPOSITION 0∗ ∩ L$ −1 ⊗ 0 0 and that Let L be a good lattice in V . Suppose that w is in L ∗ ⊗ 0m 0∗ . Then f [w, G L ,0+ ] is nonzero for some f ∈ S (A)w , where A := L ∗ ⊗ 0 0 ∩ L ⊗ 0m 0 ∗ 0 0 0 0 w belongs to B(L , L ) for some good lattice L such that 0m ⊆ L ⊆ 0 .
Proof We know that A is a good lattice in W by Lemma 11.1. By Lemma 11.3, G L ,0+ 0∗ ∩ L$ −1 ⊗ 0 0 is a subgroup of K A0 , so f [w, G L ,0+ ] is defined. Identify L ∗ ⊗ 0m −κ 0∗ ∗ 1−κ 0 with HomO (L$ , 0m ) ∩ HomO (L $ , 0 ). Let x be an element in L, and let y be an element in L ∗ $ 1−κ . Then ord L ∗ (x) ≥ ord L (x) ≥ 0, ord L (y) ≥ −κ, and ord L ∗ (y) ≥ 1 − κ. Then x, y satisfy the condition in Corollary 7.6(i). Hence, u x,y is defined and belongs to G L ,0+ . Now cx,y · w ⊆ A as in the proof of Lemma 11.3, so ωψ (u x,y ) · f (w) = ψ hhw, cx,y · wii f (w) by Lemma 8.1(i). Because we have (u −1 −1)·w ∈ A and we assume that f [w, G L ,0+ ] x,y is nonzero, we have ψ hhw, cx,y · wii = 1 by Lemma 8.6. Hence, we have hhw, cx,y · wii ∈ pλFF . Now Trd D/F hw · x, w · yi0 is in pλFF by Lemma 8.1(ii). Therefore, 0 hw · x, w · yi0 is in pλ = pκ+κ from the definition in Section 5.2. Therefore, hw · 0 x$ −κ , w · yi0 ∈ pκ . Because x (resp., y) is arbitrary in L (resp., L ∗ $ 1−κ ), we have hw · L$ −κ , w · L ∗ $ 1−κ i0 ⊆ pκ . 0
DEPTH PRESERVATION IN LOCAL THETA CORRESPONDENCE
587
Now we have w · L ∗ $ 1−κ ⊆ w · L$ −κ ⊆ w · L ∗ $ −κ since L is a good lattice. 0∗ ∩ L$ −1 ⊗ 0 0 , we have w · L ∗ $ 1−κ ⊆ 0 0 and w · Because w is in L ∗ ⊗ 0m −κ 0∗ L$ ⊆ 0m . Hence, by Lemma 11.2(ii) there exists a good lattice L 0 in V 0 such 0 ⊆ L 0 ⊆ 0 0 , w · L ∗ $ 1−κ ⊆ L 0 , and w · L$ −κ ⊆ L 0∗ . Therefore, w is in that 0m HomO (L$ −κ , L 0∗ )∩HomO (L ∗ $ 1−κ , L 0 ), which is exactly L ∗ ⊗ L 0∗ ∩ L ⊗ L 0 $ −1 = B(L , L 0 )∗ . 11.5. Proof of Proposition 6.5. Now we begin to prove Proposition 6.5. The inclusion X 0 ωψ (H 0 ) · S B(L ,L ) ⊆ S G L ,0+ L0
^ is an easy consequence of Proposition 6.2(ii) and the commutativity of U (V ) and 0 ^ ^ U (V ) in Sp(W ). We prove the opposite inclusion by discussion according to the following three separate cases: (1) L = L ∗, (2) L = L ∗$ , (3) L ∗ $ 6= L 6= L ∗ . First, suppose that we are in the first case, that is, that L is self-dual. Hence, L is a maximal good lattice in V . Then G L ,0+ = {g ∈ G | (g − 1) · L ⊆ L$ }. Let M := L$ , N := L, and let R (L), B M,N , J M,N be as defined in Section 5.2. Clearly, the pair (M, N ) belongs to R (L). So B M,N = L$ −1 ⊗ 0 0 + L ⊗ 0 0∗ = L$ −1 ⊗ 0 0 for a fixed maximal good lattice 0 0 in V 0 . It is easy to check that J M,N = G L ,0+ in this case. Now we have (g − 1) · B M,N ⊆ L ⊗ 0 0 for all g ∈ G L ,0+ . Clearly, A := L ⊗ 0 0 = B(L , 0 0 ) is a good lattice in W , and G L ,0+ is a subgroup of K A0 . Let x be an element in L, and let y be in L$ 1−κ . Then we have ord L (x) ≥ 0 and ord L (y) ≥ 1 − κ. Therefore, x, y satisfy the condition in Corollary 7.6(i). Hence, u x,y is defined and belongs G L ,0+ . Let w be an element in B M,N . Then cx,y · w belongs to A. Therefore, ωψ (u x,y ) · f = ψ hhw, cx,y · wii f by Lemma 8.1(i) for f ∈ S (A)w . Suppose that f [w, G L ,0+ ] is nonzero. Then ψ hhw, cx,y · wii = 1 by Lemma 8.6. Hence, we have hhw, cx,y · wii ∈ pλFF . Now Trd D/F hw · x, w · yi0 ∈ pλFF 0 by Lemma 8.1(ii). From Section 5.2 we know that hw · x, w · yi0 ∈ pλ = pκ+κ . 0 Hence, we have hw · x$ −κ , w · yi0 ∈ pκ . Therefore, hw · L$ −κ , w · L$ 1−κ i0 is 0 contained in pκ . Since w is in L$ −1 ⊗ 0 0 , we have w · L$ 1−κ ⊆ 0 0 . Hence, by Lemma 11.2(i), we know that there exists a good lattice L 0 in V 0 such that L 0 ⊆ 0 0 and w · L$ −κ ⊆ L 0∗ . Hence, w ∈ L ⊗ L 0∗ = B(L , L 0 )∗ because L = L ∗ in this case. Therefore, we have proved that if w is in B M,N and f [w, G L ,0+ ] is nonzero for some f ∈ S (A)w , then w must be in B(L , L 0 )∗ for some good lattice L 0 ⊆ 0 0 . So
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we conclude that S (A) B M,N
G L ,0+
=
X
X
f [w, G L ,0+ ] ⊆
w∈B M,N f ∈S (A)w
X
S (A) B(L ,L 0 )∗ .
L0
Therefore, by (8.2.1) and Proposition 5.3, we have X ∗ 0 S G L ,0+ = ωψ (H 0 ) · (S B M,N )G L ,0+ ⊆ ωψ (H 0 ) · S B(L ,L ) . L0
Secondly, suppose that L = L ∗ $ . The proof for this case is very similar to the proof for the first case. Now again G L ,0+ = {g ∈ G | (g − 1) · L ⊆ L$ } in this case. Let M = N := L. We have (M, N ) ∈ R (0), where 0 is a fixed maximal good lattice in V containing L. Now we have B M,N = L ∗ ⊗ 0 0 + L ∗ ⊗ 0 0∗ = L ∗ ⊗ 0 0∗ for a fixed maximal good lattice 0 0 in V 0 . And we also have J M,N = G L ,0+ . Clearly, A := L ⊗ 0 0∗ is a good lattice in W . Then G L ,0+ is a subgroup of K A0 . Let x be an element in L, and let y be an element in L$ −κ . Then x, y satisfy the conditions in Corollary 7.6(i). Hence, u x,y is defined and belongs to G L ,0+ . By the same argument 0 as in the first case, we can prove that hw · L$ −κ , w · L$ −κ i0 ⊆ pκ . Now w · L$ −κ is contained in 0 0∗ because w ∈ B M,N . Therefore, we get w · L$ −κ ⊆ 0 0 by Lemma 11.2(iii). Let L 0 := 0 0 , which is of course a good lattice in V 0 . Hence, w is in L ∗ ⊗ L 0 = B(L , L 0 )∗ because L ∗ = L$ −1 in this case. Therefore, we have P B(L ,L 0 ) by the same argument as in the first case. S G L ,0+ ⊆ ωψ (H 0 ) · L0 S Finally, we suppose that L ∗ $ 6 = L 6 = L ∗ . Now we let M := L ∗ $ , N := L. Then we know that (M, N ) ∈ R (0), B M,N = L$ −1 ⊗ 0 0∗ ∩ L ∗ ⊗ 0 0 $ −1 ,
(11.5.1)
and J M,N = G L ,1 . Suppose that g is an element in G L ,1 . Then from (11.5.1) we have (g − 1) · B M,N ⊆ L ⊗ 0 0∗ ∩ L ∗ ⊗ 0 0 . Let A be the good lattice L ⊗ 0 0∗ ∩ L ∗ ⊗ 0 0 . Hence, (g − 1) · B M,N ⊆ A ⊆ A∗ ⊆ B M,N for all g ∈ G L ,1 . Thus, G L ,1 is a subgroup of K A0 . Let w be an element in B M,N . We identify B M,N with HomO (L ∗ $ 1−κ , 0 0∗ ) ∩ HomO (L$ 1−κ , 0 0 ). Let x be an element in L, and let y be an element in L$ 1−κ . Then x, y satisfy the condition in Corollary 7.6(ii). Hence, u x,y is defined and belongs to G L ,1 . By the 0 same argument as in the first case, we can prove that hw · L$ 1−κ , w · L$ −κ i0 ⊆ pκ . We know that w · L$ 1−κ ⊆ 0 0 . Therefore, by Lemma 11.2(i), we have that w · L$ −κ 0 be a is contained in L 00∗ for some good lattice L 00 in V 0 such that L 00 ⊆ 0 0 . Let 0m
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0 ⊆ L 00 . Hence, w · L$ −κ is contained in minimal good lattice in V 0 such that 0m 0∗ ∗ 0m . Let x be an element in L $ , and let y be an element in L ∗ $ 1−κ . Then again x, y satisfy the condition in Corollary 7.6(ii). Hence, u x,y is defined and belongs to G L ,1 . By the same argument as in the first case, we can prove that hw · L ∗ $ 1−κ , w · 0 L ∗ $ 1−κ i0 ⊆ pκ . Since w is in B M,N , we have w · L ∗ $ 1−κ ⊆ 0 0∗ . Therefore, we have w · L ∗ $ 1−κ ⊆ 0 0 by Lemma 11.2(iii). Therefore, we have proved that if w ∈ B M,N and f [w, G L ,1 ] is nonzero for some f ∈ S (A)w , then w must be in 0∗ ∩ L ⊗ 0 0 $ −1 for some minimal good lattice 0 0 ⊆ 0 0 . Therefore, we have L ∗ ⊗ 0m m G L ,1 X ⊆ S (A) R , S (A) B M,N 0 0m
0∗ ∩ L ⊗ 0 0 $ −1 . Hence, by Lemma 8.2, we have where R := L ∗ ⊗ 0m X ∗ ∗ (S B M,N )G L ,1 ⊆ SR , 0 0m
^ where S is any model of the Weil representation of Sp( W ). Define A0 := L ∗ ⊗ 0 0 ∩ 0∗ . Now by Proposition 11.4 we know that if w ∈ R and f [w, G L ⊗0m L ,0+ ] is nonzero for some f ∈ S (A0 )w , then w ∈ B(L , L 0 )∗ for some good lattice L 0 in V 0 such that L 0 ⊆ 0 0 . Hence, we have proved X X 0 ∗ G L ,0+ SR ⊆ S B(L ,L ) . 0 0m
L0
Therefore, by (8.2.1) and Proposition 5.3, we have ∗
S G L ,0+ ⊆ ωψ (H 0 ) · (S B M,N )G L ,0+ ⊆ ωψ (H 0 ) ·
X
0 S B(L ,L ) .
L0
The proof is complete.
12. Admissible splitting and depth preservation 12.1. Reductive dual pairs and splittings ^ A reductive dual pair (U (V ), U (V 0 )) is said to be split if both extensions U (V ) → 0 0 ^ U (V ) and U (V ) → U (V ) split. It is known that all reductive dual pairs are split except the cases when D = F and the orthogonal space is odd dimensional. Suppose ^ now that (U (V ), U (V 0 )) is a split reductive dual pair. Fix splittings U (V ) → U (V ) 0 0 ^ ^ and U (V ) → U (V ). Then an irreducible admissible representation of U (V ) (resp., ^ U (V 0 )) can be regarded as an irreducible admissible representation of U (V ) (resp.,
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U (V 0 )) via the splittings. Hence, we establish a one-to-one correspondence between some irreducible admissible representations of U (V ) and some irreducible admissible representations of U (V 0 ). We emphasize here that this correspondence depends on the choice of splittings as well as the forms h, i, h, i0 and the character ψ. An explicit formula of a splitting for each metaplectic cover is given in [K2] when the splitting exists. Another splitting of the metaplectic cover is given in [P1] whenever D is com^ eL 0 : U (V ) → U mutative. From now on we assume that D is commutative. Let β (V ) V L e be the splitting given in [P1]. As indicated in the notation, the splitting βV 0 depends on a good lattice L in V . Recall that when we define the minimal K -types for the ^ metaplectic group U (V ), we need to choose an Iwahori subgroup I of U (V ). We choose a good lattice L such that I ⊆ G L , where G L is the stabilizer of L in U (V ). Then we have the following result. PROPOSITION
^ Let π be an irreducible admissible representation of U (V ). Then the depth of π is L e equal to the depth of the representation π ◦ βV 0 of U (V ). Proof eL 0 has nontrivial fixed points by G x,r + for some point x in the Suppose that π ◦ β V building of G. There exists an element g ∈ G such that gG x,r + g −1 = G g·x,r + ⊆ I . eL 0 has nontrivial fixed points by G g·x,r + . Therefore, π has nontrivial Hence, π ◦ β V eL 0 (G g·x,r + ). Because the depth of π (resp., π ◦ β eL 0 ) is the minimal fixed points by β V V eL 0 ) has nontrivial points fixed by β eL 0 (G y,r + ) real number r such that π (resp., π ◦ β V V (resp., G y,r + ) for all y such that G y,r + ⊆ I , the proposition is proved. 12.2. Depth preservation for split reductive dual pairs From Proposition 12.1 and Theorem 6.6, we have the following result on depth preservation for split reductive dual pairs. THEOREM
Let (U (V ), U (V 0 )) be a split reductive dual pair such that D is commutative. Fix an Iwahori subgroup I (resp., I 0 ) of U (V ) (resp., U (V 0 )) and a good lattice L (resp., L 0 ) in V (resp., V 0 ) such that I ⊆ G L (resp., I 0 ⊆ G 0L 0 ). Let π (resp., π 0 ) be an ^ ^ irreducible admissible representation of U (V ) (resp., U (V 0 )) such that π and π 0 eL 0 is equal to correspond in the local theta correspondence. Then the depth of π ◦ β V 0 L 0 e . the depth of π ◦ β V
Acknowledgments. This work is the major part of the author’s dissertation at Cor-
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nell University. The author would like to thank his advisor, Professor Dan Barbasch, wholeheartedly for his help, encouragement, and constant support during these years. Professor Jiu-Kang Yu carefully read the early drafts, eliminated many mistakes, and provided numerous improvements. The details of his contributions are in the main body of this paper. Finally, the author would like to thank the referee for pointing out a few errors in the first draft of the paper. References [A]
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[BT]
F. BRUHAT and J. TITS, Sch´emas en groupes et immeubles des groupes classiques sur
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J.-K. YU, Descent mapping in Bruhat-Tits theory, preprint, 1998. 533, 539, 546
, Unrefined minimal K -types for p-adic classical groups, preprint, 1998. 533, 537, 540, 546
Department of Mathematics, National Cheng Kung University, 1 Ta Hsuen Road, Tainan City 701, Taiwan, Republic of China; [email protected]
