DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1,
TODA VERSUS PFAFF LATTICE AND RELATED POLYNOMIALS M. ADLER and P. VAN MOERBEKE
Abstract The Pfaff lattice was introduced by us in the context of a Lie algebra splitting of gl (∞) into sp (∞) and an algebra of lower-triangular matrices. The Pfaff lattice is equivalent to a set of bilinear identities for the wave functions, which yield the existence of a sequence of “τ -functions”. The latter satisfy their own set of bilinear identities, which moreover characterize them. In the semi-infinite case, the τ -functions are Pfaffians, in the same way that for the Toda lattice the τ -functions are H¨ankel determinants; interesting examples occur in the theory of random matrices, where one considers symmetric and symplectic matrix integrals for the Pfaff lattice and Hermitian matrix integrals for the Toda lattice. There is a striking parallel between the Pfaff lattice and the Toda lattice, and even more striking, there is a map from one to the other, mapping skew-orthogonal to orthogonal polynomials. In particular, we exhibit two maps, dual to each other, mapping Hermitian matrix integrals to either symmetric matrix integrals or symplectic matrix integrals. Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Splitting theorems, as applied to the Toda and Pfaff lattices . . . . . . . . 2. Wave functions and their bilinear equations for the Pfaff lattice . . . . . . 3. Existence of the Pfaff τ -function . . . . . . . . . . . . . . . . . . . . . 4. Semi-infinite matrices m ∞ , (skew-)orthogonal polynomials, and matrix integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. ∂m ∞ /∂tk = 3k m ∞ , orthogonal polynomials, and Hermitian matrix integrals (α = 0) . . . . . . . . . . . . . . . . . . . . . . . . . .
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DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1, Received 9 September 1999. Revision received 22 December 2000. 2000 Mathematics Subject Classification. Primary 37K10, 70H06; Secondary 70F45, 82C23. Adler’s work supported by National Science Foundation grant number DMS-98-4-50790. Van Moerbeke’s work supported by National Science Foundation grant number DMS-98-4-50790, a North Atlantic Treaty Organization grant, a Fonds National de la Recherche Scientifique grant, and a Francqui Foundation grant. 1
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ADLER and VAN MOERBEKE
∂m ∞ /∂tk = 3k m ∞ + m ∞ 3>k , skew-orthogonal polynomials, and symmetric and symplectic matrix integrals (α = ±1) . . . . . . . . 5. A map from the Toda to the Pfaff lattice . . . . . . . . . . . . . . . . . 6. Example 1: From Hermitian to symmetric matrix integrals . . . . . . . . 7. Example 2: From Hermitian to symplectic matrix integrals . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Free parameter in the skew-Borel decomposition . . . . . . . . . . . B. Simultaneous (skew-)symmetrization of L and N . . . . . . . . . . . C. Proof of Lemma 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.
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0. Introduction Consider a weight on R, depending on t = (t1 , t2 , . . .) ∈ C∞ , ρt (z) dz = e
P∞ 1
ti z i
P∞
ρ(z) dz = e−V (z)+
1
ti z i
dz, with −
ρ 0 (z) g(z) = V 0 (z) = , ρ(z) f (z) (0.1)
with rational g’s and f ’s, and with ρ(z) decaying fast enough at ∞. The Toda lattice, its τ -functions and Hermitian matrix integrals (revisited) This weight leads to a t-dependent moment matrix Z m n (t) = µk+` (t) 0≤k,`≤n−1 = z k+` ρt (z) dz , R
0≤k,`≤n−1
with the semi-infinite moment matrix m ∞ , satisfying the commuting equations ∂m ∞ = 3k m ∞ = m ∞ 3k . ∂tk
(0.2)
3 is the customary shift matrix, with zeros everywhere, except for 1’s just above the diagonal; that is, (3v)n = vn+1 . Consider the Borel decomposition into a lowertriangular and an upper-triangular matrix, m ∞ = S −1 S >−1 , and the following t-dependent matrix integrals (n ≥ 0): Z P∞ i τn (t) := eTr(−V (X )+ 1 ti X ) d X = det m n and Hn
(0.3)
τ0 = 1,
(0.4)
where d X is Haar measure on the ensemble Hn = {(n × n)–Hermitian matrices}. As is well known (e.g., see E. Witten [18] or M. Adler and P. van Moerbeke [6]), integral (0.4) is a solution to the following two systems:
TODA VERSUS PFAFF LATTICE
3
(i) the KP-hierarchy∗ 1 ∂2 ˜ sk+4 (∂) − τn ◦ τn = 0 2 ∂t1 ∂tk+3
for k, n = 0, 1, 2, . . . ;
(ii) the Toda lattice, that is, the tridiagonal matrix 1/2 τ0 τ2 τ1 ∂ ∂t1 log τ0 τ12 1/2 τ0 τ2 τ2 ∂ 2 ∂t1 log τ1 −1 τ 1 L(t) := S3S = 1/2 τ1 τ3 0 τ22 .. .
(0.5)
0 1/2
τ1 τ3 τ22
∂ ∂t1
τ3 τ2
log
..
.
, .. .
(0.6)
satisfying the commuting Toda equations h1 i ∂L = (L n )sk , L , ∂tn 2 where (A)sk denotes the skew part of the matrix A for the Lie algebra splitting into skew and lower-triangular matrices. Moreover, the following t-dependent polynomials in z are defined by the S-matrix obtained from the Borel decomposition (0.3); it is also given, on the one hand, in terms of the functions τn (t) and, on the other hand, by a classic determinantal formula (for a ∈ C, define [a] := (a, a 2 /2, a 3 /3, . . .) ∈ C∞ ): pn (t; z) :=
n X i=0
Sni (t)z i = z n
τn (t − [z −1 ]) √ τn τn+1
1 =√ τn τn+1
det
1 z .. .
m n (t) µn,0 (t)
...
µn,n−1 (t) z n
.
The pn (t; z)’s are orthonormal with respect to the (symmetric) inner product h , isy , defined by hz i , z j isy = µi j , which is a restatement of the Borel decomposition (0.3) (see [6]). The vector p(t; z) = ( pn (t; z))n≥0 is an eigenvector of the matrix L(t) in (0.6): L(t) p(t; z) = zp(t; z). P∞ i P ˜ := s` (∂/∂t1 , (1/2)(∂/∂t2 ), s` ’s are the elementary Schur polynomials e 1 ti z := i≥0 si (t)z i and s` (∂) . . .). Given a polynomial p(t1 , t2 , . . .), define the customary Hirota symbol p(∂t ) f ◦ g := p(∂/∂ y1 , ∂/∂ y2 , . . .) f (t + y)g(t − y) .
∗ The
y=0
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ADLER and VAN MOERBEKE
The Pfaff lattice and its τ -functions For use throughout this paper, define the skew-symmetric matrix J =
..
. 0 1 −1 0
0 0 1 −1 0
0
0 1 −1 0 ..
.
,
with J 2 = −I,
(0.7)
and the involution on the space D := gl∞ of infinite matrices, J : D −→ D : a 7 −→ J (a) := J a > J.
(0.8)
Also, consider the splitting of D = k + n into two Lie subalgebras k and n, with the corresponding projections denoted πk and πn , where k is the Lie algebra of lowertriangular matrices with some special feature (see (1.17)) and where n := {a ∈ D such that J a > J = a} = sp(∞). Given a skew-symmetric semi-infinite matrix m ∞ , consider the commuting differential equations ∂m ∞ = 3i m ∞ + m ∞ 3>i ; (0.9) ∂ti they maintain the skew-symmetry of m ∞ . The Borel decomposition of m ∞ into lower-triangular times upper-triangular matrices requires the insertion of the skewsymmetric matrix J : m ∞ (t) = Q −1 (t)J Q −1> (t). (0.10) Dressing up the shift 3 with the lower-triangular matrix Q(t) leads to the commuting equations (0.11). THEOREM 0.1 The Pfaff lattice equations
∂L = [−πk L i , L] = [πn L i , L] ∂ti
(0.11)
TODA VERSUS PFAFF LATTICE
5
maintain the locus of semi-infinite matrices of the form (ai 6= 0): 0 1 −d1 a1 O d1 1 . L= −d2 a2 ∗ d 2 .. .
(0.12)
The solutions L to (0.11) of the form (0.12) are given by L(t) = Q(t)3Q −1 (t), where Q is a lower-triangular matrix, whose entries are given by the coefficients of the polynomials, obtained by the finite Taylor expansion in z −1 of τ2n (t −[z −1 ]) below (h 2n := τ2n+2 (t)/τ2n (t)): q2n (t; z) :=
2n X
−1/2 τ2n (t
− [z −1 ]) , τ2n (t)
Q 2n, j (t)z j = z 2n h 2n
j=0
q2n+1 (t; z) :=
2n+1 X j=0
−1/2 (z
Q 2n+1, j (t)z j = z 2n h 2n
+ ∂/∂t1 )τ2n (t − [z −1 ]) , τ2n (t)
(0.13)
with τ0 , τ2 , τ4 , a sequence of functions of t1 , t2 , . . . , characterized by the following bilinear identities for all n, m ≥ 0 (τ0 = 1): I P∞ dz 0 i τ2n t − [z −1 ] τ2m+2 t 0 + [z −1 ] e 1 (ti −ti )z z 2n−2m−2 2πi z=∞ I P∞ 0 dz −i + τ2n+2 t + [z] τ2m t 0 − [z] e 1 (ti −ti )z z 2n−2m = 0. (0.14) 2πi z=0 Remark. Theorem 0.1 is robust and remains valid for the bi-infinite matrix L. In that case, the summations in the expressions q2n and q2n+1 run from j = −∞ instead of running from j = 0. The τ -functions are given by Pfaffians pf m 2n (t) and satisfy, as a consequence of the bilinear relations (0.14), the Pfaffian KP-hierarchy for k, n = 0, 1, 2, . . . , 2 ˜ τ2n+2 ◦ τ2n−2 . ˜ −1 ∂ sk+4 (∂) τ2n ◦ τ2n = sk (∂) 2 ∂t1 ∂tk+3
(0.15)
The t-dependent polynomials qn (t; z) = (Q(t) (1, z, z 2 , z 3 , . . .)> )n in z, obtained in (0.13), are “skew-orthonormal” with respect to the skew-inner product h , isk , defined
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ADLER and VAN MOERBEKE
by hy i , z j isk = µi j (t), namely, hqi , q j isk
0≤i, j<∞
= J,
and are eigenvectors for the matrix L: L(t)q(t; z) = zq(t; z).
(0.16)
Explicit representations of L, in terms of the τ2n ’s, are as follows: Lˆ 00 Lˆ 01 0 0 Lˆ 0 10 Lˆ 11 Lˆ 12 1/2 ˆ ˆ ˆ −1 −1/2 h , L 21 L 22 L 23 L = Q3Q = h ˆ 32 Lˆ 33 . . . ∗ L .. . with the entries Lˆ i j and the entries of h being (2 × 2)-matrices h = diag(h 0 I2 , h 2 I2 , h 4 I2 , . . . ), and (· = ∂/∂t1 ),
−(log τ2n )·
Lˆ nn := ˜ 2n − − s2 (τ∂)τ 2n
˜ 2n+2 s2 (−∂)τ τ2n+2
∗ (log τ2n+2 )·· Lˆ n+1,n := . ∗ ∗
h 2n = τ2n+2 /τ2n ,
1 (log τ2n+2 )·
,
0 0 Lˆ n,n+1 := , 1 0
(0.17)
TODA VERSUS PFAFF LATTICE
7
Also, the t-dependent polynomials qn (t, z) in z have Pfaffian expressions, in contrast to the determinantal expressions in the Hermitian case, mentioned earlier: 1 z 1 .. q2n (t; z) = √ pf . , m 2n+1 (t) τ2n τ2n+2 z 2n −1 ... −z 2n 0 q2n+1 (t; z) 1 =√ τ2n τ2n+2
pf
m 2n (t) −1 −µ0,2n+1
−z −µ1,2n+1
1 z .. . z 2n−1 ... 0 2n+1 ... z
µ0,2n+1 µ1,2n+1 .. .
.
µ2n−1,2n+1 −z 2n+1 0 (0.18)
Theorem 0.1 and the subsequent statements are established in Sections 2 and 3. We show how a general skew-symmetric infinite matrix flowing according to (0.11) and its skew-Borel decomposition (0.10) lead to wave vectors 9, satisfying bilinear relations and differential equations. Section 3 deals with the existence, in the general setting, of a so-called Pfaffian τ -function, satisfying bilinear equations and the socalled Pfaff-KP hierarchy. In [4], these results were obtained by embedding the system in 2-Toda theory, while in this paper they are obtained in an intrinsic fashion. For k = 0, the Pfaff-KP equation (0.15) has already appeared in the context of the charged BKP hierarchy, studied by V. Kac and J. van de Leur [13]; the precise relationship between the charged BKP hierarchy of Kac and van de Leur and the Pfaff lattice, introduced here, deserves further investigation. (See the recent paper by van de Leur [16].)
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ADLER and VAN MOERBEKE
Examples: “Symmetric” and “symplectic” matrix integrals An important example is given by the skew-symmetric matrix m ∞ = (µi j )i, j≥0 of moments∗ defined by (α = ±1) (see [4]): ZZ P∞ i i hy i , z j isk := y i z j e 1 ti (y +z ) 2D α δ(y − z)ρ(y)ρ(z) dy dz (0.19) R2 Z Z P∞ i i (1) i j 1 ti (y +z ) ε(y − z)ρ(y)ρ(z) dy dz, µ (t) = y z e ij R2 for α = −1, = Z P∞ µ(2) (t) = {y i , y j }(y)e 1 2ti y i ρ(y) ˜ 2 dy, for α = +1. ij R (1)
(2)
The associated moment matrices m 2n and m 2n satisfy the differential equations (0.9) and lead to “symmetric” matrix integrals Z P∞ 1 i (1) (1) τ2n (t) := eTr(−V (X )+ 1 ti X ) d X = pf(m 2n ) (2n)! S2n and “symplectic” matrix integrals Z P∞ 1 i (1) (2) τ2n (t) := e2 Tr(−V (X )+ 1 ti X ) d X = pf(m 2n ), n! T2n both expressed in terms of the Pfaffian of the upper-left-hand principal minors of the (i) “moment” matrix m ∞ , where (1) for i = 1, d X denotes Haar measure on the space S2n of symmetric matrices, and (2) for i = 2, d X denotes Haar measure on the (2n × 2n)-matrix realization T2n of the space of self-dual (n×n)–Hermitian matrices, with quaternionic entries. A remarkable map from Toda to Pfaff lattice Remembering the notation (0.1), we act with the z-operator, s P∞ k d f d p f 0 + g 1/2 P∞ tk z k 1 nt := fρt = e−(1/2) 1 tk z f (z) − (z) e ρt dz dz 2
(0.20)
on the t-dependent orthonormal polynomials pn (t, z) in z; in [6], we showed that the matrix N defined by nt p(t, z) = ∗ We
f (L)M −
f0+g (L) p(t, z) =: N p(t, z) 2
have ε(x) = 1 for x ≥ 0, ε(x) = −1 for x < 0, and { f, g} = f 0 g − f g 0 .
(0.21)
TODA VERSUS PFAFF LATTICE
9
is skew-symmetric. The t-dependent matrix N is expressed in terms of L and a new matrix M, defined by zp = L p
e−(1/2)
and
P
tk z k
d (1/2) P tk z k e p = M p. dz
(0.22)
Consider now the skew-Borel decomposition of N (2t) and its inverse∗ N (2t)−1 , in terms of lower-triangular matrices O(+) (t) and O(−) (t), respectively:† −1 >−1 N (2t)±1 = −O(±) (t)J O(±) (t).
(0.23)
Then, the lower-triangular matrices O(±) (t) map orthonormal polynomials into skew‡ ˜ orthonormal polynomials, and the tridiagonal L-matrix into an L-matrix: pn (t; z) 7−→ qn(±) (t; z) = O(±) (t) p(t; z) n , ˜ L(t) 7−→ L(t) = O(±) (t)L(2t)O(±) (t)−1 (Toda lattice)
(0.24)
(Pfaff lattice).
It also maps the weight into a new weight ρ(z) = e−V (z) 7−→ ρ˜± (z) = e−V (z) := e−(1/2)(V (z)∓log f (z)) , ˜
and the corresponding string of τ -functions into a new string of Pfaffian τ -functions P i (remember Vt (z) = V (z) − ∞ 1 ti z ): ( (+) R Z ˜ τ2n (t) := T2n eTr 2(−Vt (X )) d X (β = 4), Tr(−Vt (X )) τk (t) = e d X 7 −→ R ˜ (−) τ2n (t) := S2n eTr(−Vt (X )) d X (β = 1). Hk For the classical orthogonal polynomials pn (z), we have shown in [6] that N (0) is not only skew-symmetric but also tridiagonal; that is,
b0 a0 L=
a0 b1 a1
a1 b2 .. .
.. ..
. .
,
0 −c0 −N =
c0 0 −c1
c1 0 .. .
..
.
. (0.25)
∗ See
Appendix B. upper signs (resp., lower signs) correspond to each other throughout this section. ‡ We have p(t; z) := ( p (t; z), p (t; z), . . .)> . 0 1 † The
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ADLER and VAN MOERBEKE
In Sections 6 and 7, we show that the maps O(−) and O(+) , as in (0.24), involve only three steps, in the following sense: for β = 1, r c2n (−) q2n (0; z) = p2n (0; z), a2n r a2n (−) q2n+1 (0; z) = c2n 2n c2n X × − c2n−1 p2n−1 (0, z) + bi p2n (0; z) + c2n p2n+1 (0; z) ; a2n 0
(0.26) for β = 4, r √ a2n−2 (+) (+) p2n (0; z) = −c2n−1 q2n−2 (0; z) + a2n c2n q2n (0; z), c2n−2 r a2n−2 (+) p2n+1 (0; z) = −c2n q (0; z) c2n−2 2n−2 r 2n X r c c2n (+) 2n (+) − bi q (0; z) + q (0; z). a2n 2n a2n 2n+1
(0.27)
0
The abstract map O(−) for t = 0 has already appeared in the work of E. Br´ezin and H. Neuberger [9]. This has been applied recently in [1] to a problem in the theory of random matrices. 1. Splitting theorems, as applied to the Toda and Pfaff lattices In this section, we show how the equations ∂m ∞ = 3i m ∞ ∂ti
and
∂m ∞ = 3i m ∞ + m ∞ 3>i ∂ti
(1.1)
both lead to commuting Hamiltonian vector fields related to a Lie algebra splitting. First recall the splitting theorem due to Adler, B. Kostant, and W. Symes in [5], and later recall the R-version due to A. Reyman and M. Semenov-Tian-Shansky [15]. The R-version allows for more general initial conditions. PROPOSITION 1.1 Let g = k + n be a (vector space) direct sum of a Lie algebra g in terms of Lie subalgebras k and n, with g paired with itself via a nondegenerate ad-invariant inner product∗ h , i; this in turn induces a decomposition g = k⊥ + n⊥ and isomorphisms ∗ hAd
g
X ; Y i = hX, Adg−1 Y i, g ∈ G, and thus h[z, x], yi = hx, −[z, y]i.
TODA VERSUS PFAFF LATTICE
11
g ' g∗ , k⊥ ' n∗ , n⊥ ' k∗ . Let πk and πn be projections onto k and n, respectively. Let G , Gk , and Gn be the groups associated with the Lie algebras g, k, and n. Let I (g) be the Ad∗ ' Ad-invariant functions on g∗ ' g. (i) Then, given an element ε ∈ g : [ε, k] ⊂ k⊥
and [ε, n] ⊂ n⊥ ,
the functions ϕ(ε + ξ 0 )|k⊥ ,
with ϕ ∈ I (g) and ξ 0 ∈ k⊥ ,
(1.2)
respectively, Poisson commute for the respective Kostant-Kirillov symplectic structures of n ∗ ' k⊥ ; the associated Hamiltonian flows are expressed in terms of the Lax pairs∗ ξ˙ = [−πk ∇ϕ(ξ ), ξ ] = [πn ∇ϕ(ξ ), ξ ]
for ξ ≡ ε + ξ 0 , ξ 0 ∈ k⊥ .
(1.3)
(ii) The splitting also leads to a second Lie algebra g R , derived from g, such that g∗R ' g R , namely, g R : [x, y] R =
1 1 [Rx, y] + [x, Ry] = [πk x, πk y] − [πn x, πn y], 2 2
(1.4)
with R = πk − πn . The functions ϕ(ξ )|g R ,
with ϕ ∈ I (g) and ξ ∈ g R ,
respectively, Poisson commute for the respective Kostant-Kirillov symplectic structures of g∗R ' g R , with the same associated (Hamiltonian) Lax pairs ξ˙ = [−πk ∇ϕ(ξ ), ξ ] = [πn ∇ϕ(ξ ), ξ ]
for ξ ∈ g R .
(1.5)
Each of the equations (1.3) and (1.5) has the same solution expressible in two different ways:† ξ(t) = Ad K (t) ξ0 = Ad S −1 (t) ξ0 , (1.6) with‡
K (t) = πGk et∇ϕ(ξ0 )
and
S(t) = πGn et∇ϕ(ξ0 ) .
is defined as the element in g∗ such that dϕ(ξ ) = h∇ϕ, dξ i, ξ ∈ g. naively write Ad K (t) ξ0 = K (t)ξ0 K (t)−1 , Ad S −1 ξ0 = S −1 (t)ξ0 S(t). ‡ Consider the group factorization A = π Gk A πGn A. ∗ ∇ϕ
† We
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ADLER and VAN MOERBEKE
Example 1 (The standard Toda lattice and the equations ∂m/∂ti = 3i m for the H¨ankel matrix m ∞ ) Since, in particular, the matrix m ∞ is symmetric, the Borel decomposition into lowertriangular times upper-triangular matrix must be done with the same lower-triangular matrix S: m ∞ = S −1 S >−1 . (1.7) In turn, the matrix S defines a wave vector 9, and operators∗ L and M, the same as the ones defined in (0.22): P∞
9(t, z) := e(1/2) 1 ti z Sχ, L := S3S −1 , ∞ 1X M := S ∂ + iti 3i−1 S −1 , 2 i
(1.8)
1
satisfying the following well-known equations:† L9 = z9,
∂ 9, with [L , M] = 1, ∂z ∂9 1 = (L n )sk 9, ∂tn 2 ∂M 1 = (L n )sk , M . ∂tn 2
M9 =
∂S 1 = − (L n )bo S, ∂tn 2 1 ∂L = (L n )sk , L ∂tn 2
(1.9)
The wave vector 9 can then be expressed in terms of a sequence of τ -functions τn (t) = det m n (t), but it also has a simple expression in terms of orthonormal polynomials, with respect to the moment matrix m ∞ : 9(t, z) = e(1/2)
P
ti z i
= e(1/2)
P
ti z i
τn (t − [z −1 ]) zn p τn (t)τn+1 (t) n≥0 pn (t, z) n≥0 .
(1.10)
The vector fields (1.9) on L are commuting Hamiltonian vector fields, in view of the Adler-Kostant-Symes (AKS) splitting theorem (version (i)), ∂L = [−πk ∇ Hi , L] = [πn ∇ Hi , L], ∂ti
Hi =
tr L i+1 , ∇ Hi = L i , i +1
(1.11)
with L = 3> a + b + a3,
a and b diagonal matrices,
(1.12)
the formulas below, χ (z) = (z 0 , z, z 2 , . . .)> and ∂ is the matrix such that (d/dz)χ (z) = ∂χ (z). notations ( )sk and ( )bo refer to the skew part and the lower-triangular (Borel) part, respectively, that is, projection onto k and n, respectively.
∗ In
† The
TODA VERSUS PFAFF LATTICE
13
for the splitting of the Lie algebra of semi-infinite matrices D = gl∞ = k + n := {skew-symmetric} + {lower-triangular}
= k⊥ + n⊥ := {symmetric} + {strictly upper-triangular}, (1.13) with the form (1.12) of L being preserved in time. Note that the solution (1.6) to the differential equation (1.5) in the AKS theorem can also be exressed by the factorization of m ∞ followed by the dressing up of 3. i
Example 2 (The Pfaff lattice and the equations ∂m/∂ti = 3i m + m3> ) Throughout this paper the Lie algebra D = gl∞ of semi-infinite matrices is viewed as being composed of (2 × 2)-blocks. It admits the natural decomposition into subalgebras: D = D− ⊕ D0 ⊕ D+ = D− ⊕ D0− ⊕ D0+ ⊕ D+ , (1.14) where D0 has (2×2)-blocks along the diagonal with zeros everywhere else and where D+ (resp., D− ) is the subalgebra of upper-triangular (resp., lower-triangular) matrices with (2 × 2)-zero matrices along D0 and zero below (resp., above). As pointed out in (1.14), D0 can further be decomposed into two Lie subalgebras: D0− = {all (2 × 2)-blocks ∈ D0 are proportional to Id}, D0+ = {all (2 × 2)-blocks ∈ D0 have trace zero}.
(1.15)
Remember from (0.7) and (0.8) in the introduction that the matrix J and the associated Lie algebra involution J . The splitting into two Lie subalgebras∗ D = k + n,
(1.16)
with k = D− + D0− = algebra of
..
.
0 Q 2n,2n 0
0 Q 2n,2n
∗
Q 2n+2,2n+2 0
0 Q 2n+2,2n+2
n = {a ∈ D , such thatJ a = a} = {b + J b, b ∈ D } = sp(∞), ∗ Note
that n is the fixed point set of J .
..
.
, (1.17)
14
ADLER and VAN MOERBEKE
with corresponding Lie groups∗ Gk and Gn = Sp(∞), plays a crucial role here. Let πk and πn be the projections onto k and n. Notice that n = sp(∞) and Gn = Sp(∞) stand for the infinite-rank affine symplectic algebra and group (e.g., see [12]). Any element a ∈ D decomposes uniquely into its projections onto k and n, as follows: a = πk a + πn a 1 1 = a− − J a+ + a0 − J a0 + a+ + J a+ + a0 + J a0 . 2 2 (1.18) The following splitting, with k+ = D+ + D0−
and
n+ = n,
is also used in Section 2; the projections take on the following form: a = πk+ a + πn+ a n 1 o n 1 o = a+ − J a− + a0 − J a0 + a− + J a− + a0 + J a0 . 2 2 (1.19) Note that J intertwines πk and πk+ : J πk = πk+ J .
(1.20)
For a skew-symmetric semi-infinite matrix m ∞ , the skew-Borel decomposition m ∞ := Q −1 J Q −1> ,
with Q ∈ Gk ,
(1.21)
is unique, as was shown in [2]. Here we may assume m ∞ to be bi-infinite, as long as factorization (1.21) is unique, upon imposing a suitable normalization. Then we use Q to dress up 3: L = Q3Q −1 . i
Then letting m ∞ run according to the equations ∂m/∂ti = 3i m + m3> , we show in the next proposition and corollary that L evolves according to a system of commuting equations, which by virtue of the AKS theorem are Hamiltonian vector fields (for details, see [2]). ∗G k
is the group of invertible elements in k, that is, invertible lower-triangular matrices, with nonzero (2 × 2)-blocks proportional to Id along the diagonal.
TODA VERSUS PFAFF LATTICE
15
PROPOSITION 1.2 For the matrices
m ∞ := Q −1 J Q −1>
and
L := Q3Q −1 ,
with Q ∈ Gk ,
the following three statements are equivalent: ∂ Q −1 (i) Q = −πk L i , ∂ti ∂ Q −1 (ii) Li + Q ∈ n, ∂ti ∂m ∞ i (iii) = 3i m ∞ + m ∞ 3> . ∂ti Whenever the vector fields on Q or m satisfy (i), (ii), or (iii), the matrix L = Q3Q −1 is a solution of the AKS-Lax pair ∂L = [−πk L i , L] = [πn L i , L]. ∂ti Proof Written out and using (1.18), Proposition 1.2 amounts to showing the equivalence of the three formulas: 1 ∂ Q −1 (I) Q + (L i )− − J (L i+ )> J + (L i )0 − J ((L i )0 )> J = 0, ∂ti 2 ∂ Q −1 ∂ Q −1 > (II) Li + Q − J Li + Q J = 0, ∂ti ∂ti ∂m ∞ (III) 3i m ∞ + m ∞ 3>i − = 0. ∂ti The point is to show that (I)+ = 0,
(I)− = (II)− = −J (II+ )> J,
(I)0 =
1 (II)0 , 2
>
Q −1 (II)J Q −1 = (III).
(1.22)
The details of this proof are found in [2]. 2. Wave functions and their bilinear equations for the Pfaff lattice Consider the commuting vector fields ∂m ∞ /∂ti = 3i m ∞ + m ∞ 3>i
(2.1)
on the skew-symmetric matrix m ∞ (t) and the skew-Borel decomposition m ∞ (t) = Q −1 (t)J Q >−1 (t),
Q(t) ∈ Gk ;
(2.2)
16
ADLER and VAN MOERBEKE
remember from (1.17) that Q(t) ∈ Gk means that Q(t) is lower-triangular, with (2 × 2)-matrices c2n I along the “diagonal,” with c2n 6 = 0. In this section, we give the properties of the wave vectors and their bilinear relations. In this and the next section, the matrices are assumed to be bi-infinite; the semi-infinite case is dealt with by specialization. Upon setting Q 1 = Q(t)
and
Q 2 = J Q >−1 (t),
(2.3)
the matrix Q(t) defines wave operators W1 (t) := Q 1 (t)e
P∞ 1
ti 3i
,
W2 (t) := Q 2 (t)e−
P∞ 1
ti 3>i
= J W1−1> (t),
(2.4)
L-matrices L := L 1 := Q 1 3Q −1 1 ,
L 2 := −J (L 1 ) = Q 2 3> Q −1 2 ,
(2.5)
and wave and dual wave vectors 91 (t, z) = W1 (t)χ(z),
91∗ (t, z) = W1−1 (t)> χ(z −1 ) = −J 92 (t, z −1 ),
92 (t, z) = W2 (t)χ(z),
92∗ (t, z) = W2−1 (t)> χ(z −1 ) = J 91 (t, z −1 ).
(2.6)
From the definition, it follows that the wave functions 91 have the following asymptotics: ( P k 91,2n (t, z) := e tk z z 2n c2n (t)ψ1,2n (t, z), ψ1,2n = 1 + O(z −1 ), P k 91,2n+1 (t, z) = e tk z z 2n+1 c2n (t)ψ1,2n+1 (t, z), ψ1,2n+1 = 1 + O(z −2 ), ( P −k −1 92,2n (t, z) = e− tk z z 2n+1 c2n (t)ψ2,2n (t, z), ψ2,2n = 1 + O(z), P −k −1 92,2n+1 (t, z) = e− tk z z 2n (−c2n (t))ψ2,2n+1 (t, z), ψ2,2n+1 = 1 + O(z 2 ), (2.7) where the ci are the elements of the diagonal part of Q. THEOREM 2.1 The following statements are equivalent: (i) m ∞ satisfies ∂m ∞ /∂ti = 3i m ∞ + m ∞ 3>i , (ii) Q 1 satisfies the hierarchy of equations (with L i defined in (2.5))
∂ Q1 = −(πk L i1 )Q 1 , ∂ti (iii)
Q 2 = J Q >−1 satisfies 1 ∂ Q2 = − J (πk L i1 ) Q 2 = (πk+ L i2 )Q 2 , ∂ti
(2.8)
TODA VERSUS PFAFF LATTICE
(iv)
91 = e
P∞ 1
ti z i Q
1 (t)χ(z)
17
satisfies ∂91 = (πn L i1 )91 , ∂ti
(v)
92 = e−
P∞ 1
ti z −i
(2.9)
J Q >−1 (t)χ(z) satisfies 1
∂92 = −(L i2 − πk+ L i2 )92 = −(πn+ L i2 ), 92 , ∂ti 91 and 92 satisfy the bilinear identity for all n, m ∈ Z, I I dz 0 −1 dz 91,n (t, z)92,m (t , z ) + 92,n (t, z)91,m (t 0 , z −1 ) = 0. 2πi z 2πi z ∞ 0 (2.10) If any one of these six conditions is satisfied, then (vi)
∂ L1 = [−πk L i1 , L 1 ], ∂ti
∂ L2 = [πk+ L i2 , L 2 ], ∂ti
(2.11)
and L 1 91 = z91 ,
L 2 92 = z −1 92 ,
? ? L> 1 91 = z91 ,
? −1 ? L> 2 92 = z 92 .
(2.12)
For later use, we also consider the “monic” wave functions, with the factors c2n (t) removed, that is, ˆ 1 (t, z) := Q −1 91 9 0
and
ˆ 2 (t, z) := Q 0 92 9
(2.13)
and the matrix Lˆ 1 , normalized so as to have 1’s above the main diagonal, with Qˆ := Q −1 0 Q, −1 −1 −1 ˆ Qˆ −1 , Lˆ 1 = Q −1 = Q3 0 L 1 Q 0 = (Q 0 Q)3(Q 0 Q) −1 ˆ Lˆ 2 = Q 0 L 2 Q −1 0 = −Q 0 J (L 1 )Q 0 = −J ( L 1 ).
(2.14)
Then, in terms of the elements qˆi j of the matrix Qˆ := Q −1 0 Q, one easily computes by conjugation that Lˆ 1 has the following block structure: −1 −1 −1 Lˆ 1 = Q −1 0 L 1 Q 0 = (Q 0 Q)3(Q 0 Q) .. . . . . Lˆ ˆ 01 0 00 L ˆ ˆ ˆ L 10 L 11 L 12 = ∗ Lˆ 21 Lˆ 22 ∗ ∗ Lˆ 32
0 0 Lˆ 23 Lˆ 33 .. .
, . . .
18
ADLER and VAN MOERBEKE
with Lˆ ii :=
0 0 Lˆ i,i+1 := , −qˆ2i+2,2i+1 1 0 ! 2 − qˆ2i+3,2i+1 + qˆ2i+2,2i ∗ −qˆ2i+2,2i+1 . (2.15) := ∗ ∗
qˆ2i,2i−1 qˆ2i+1,2i−1 − qˆ2i+2,2i
Lˆ i+1,i
1
,
These definitions lead to a new statement that is equivalent to Theorem 2.1. 2.2 ˆ 9 ˆ 1, 9 ˆ 2 satisfy the following equations: Lˆ i , Q,
THEOREM
∂ Qˆ = − ( Lˆ n1 )− − Q −2 J (( Lˆ n1 )+ )> J Q 20 Qˆ 0 ∂tn
(2.16)
and ˆ 1 = z9 ˆ 1, Lˆ 1 9
ˆ 2 = z −1 9 ˆ 2, Lˆ 2 9
(2.17)
with ∂ ˆ 1 (t, z) = ( Lˆ n )+ + ( Lˆ n )0 + Q −2 J (( Lˆ n )+ )Q 20 9 ˆ 1 (t, z), 9 1 1 1 0 ∂tn ∂ ˆ 2 (t, z) = J ( Lˆ n )+ + ( Lˆ n )0 + Q −2 J (( Lˆ n )+ )Q 20 9 ˆ 2 (t, z) 9 1 1 1 0 ∂tn ˆ 2 (t, z). = − ( Lˆ n )− + ( Lˆ n )0 + Q 20 J (( Lˆ n )− )Q −2 9 2
2
2
0
The proof of Theorem 2.1 hinges on the following proposition. PROPOSITION 2.3 The following three statements are equivalent: (i) ∂m ∞ /∂ti = 3i m ∞ + m ∞ 3>i , (ii) the matrices W1 (t) and W2 (t) (defined in (2.4)) satisfy
W1 (t)W1 (t 0 )−1 = W2 (t)W2 (t 0 )−1 , (iii)
(2.18)
the 9i (t, z) = Wi (t)χ(z) satisfy the bilinear identity I I dz 0 −1 dz 91,n (t, z)92,m (t , z ) + 92,n (t, z)91,m (t 0 , z −1 ) = 0. 2πi z 2πi z ∞ 0
Proof The solution to (2.1) is given by m ∞ (t) = e
P∞ 1
tk 3k
m ∞ (0)e
P∞ 1
tk 3>k
.
TODA VERSUS PFAFF LATTICE
19
Therefore, skew-Borel decomposing m ∞ (t) and m ∞ (0), we find Q −1 (0)J Q >−1 (0) = e−
P
ti 3i
Q −1 (t)J Q >−1 (t)e−
P
ti 3>i
,
(2.19)
and so, from the definition of W1 and W2 , W1−1 (0)W2 (0) = Q −1 (0)J Q >−1 (0) P∞ P i −1 i >−1 = Q(t)e 1 ti 3 J Q(t)e ti 3 = W1 (t)
−1
(using (2.19))
J W1 (t)
>−1
= W1 (t)−1 W2 (t),
(2.20)
implying the independence in t of the right-hand side of (2.20). Therefore, we have W1 (t)−1 W2 (t) = W1 (t 0 )−1 W2 (t 0 ) for all t, t 0 ∈ C∞ , and so W1 (t)W1−1 (t 0 ) = W2 (t)W2−1 (t 0 ), thus yielding (ii). Reversing the steps yields the differential equation (i). Finally, the proof of the bilinear identity (iii) proceeds as follows. Using the wellknown formula (see [3, Prop. 4.1]), I dz 0 −1 91 (t, z) ⊗ 91∗ (t 0 , z) W1 (t)W1 (t ) = , ∞ 2πi z 2 2 2 2 0 statement (ii) becomes I I dz dz 91 (t, z) ⊗ 91∗ (t 0 , z) = 92 (t, z) ⊗ 92∗ (t 0 , z) , 2πi z 2πi z ∞ 0 whose (m, n)th component is I I dz dz ∗ 0 ∗ 91,n (t, z)91,m (t , z) − 92,n (t, z)92,m (t 0 , z) = 0. 2πi z 2πi z ∞ 0 Next we use the relations 91∗ (t, z) = −J 92 (t, z −1 ) and 92∗ (t, z) = J 91 (t, z −1 ) to yield I I dz dz 91 (t, z) ⊗ J 92 (t 0 , z −1 ) + 92 (t, z) ⊗ J 91 (t 0 , z −1 ) = 0, 2πi z 2πi z ∞ 0 which again leads to (iii). That (iii) implies (ii) is obtained by reversing the arguments.
20
ADLER and VAN MOERBEKE
Proof of Theorem 2.1 The proof of statement (ii) for Q 1 , namely, ∂ Q1 = −(πk L i1 )Q 1 , ∂ti follows at once from Proposition 1.2. The proof of (iii) for Q 2 = J Q >−1 is based on the identity J πk a = πk+ J a. 1 Indeed, we compute ∂ Q> ∂ Q 2 −1 1 Q 2 = −J Q >−1 Q >−1 Q −1 1 1 2 ∂ti ∂ti i > >−1 > = −J Q >−1 Q> Q1 J 1 (πk L 1 ) Q 1 1
= −J (πk L i1 )> J = −J (πk L i1 ) = −πk+ J L i1 = −πk+ J (−J L 2 )i
(using (2.5))
= −πk+ J (−1) (J L 2 )i i
= −πk+ J (−1)i (−1)i−1 J L i2 = πk+ L i2 . Statements (iv) and (v) for 91 , 92 are straightforwardly equivalent to (ii) and (iii), respectively. According to Propositions 1.2 and 2.3 combined, the bilinear identity (2.10) in (vi) is equivalent to statement (i), (ii), or (iii). The hierarchy concerning the L i ’s follows at once from (ii) and (iii), thus ending the proof of Theorem 2.1. Proof of Theorem 2.2 To prove (2.16), remember from Theorem 2.1 that 1 ∂ Q −1 Q = −πk L n = − (L n )− − J (L n+ )> J − (L n )0 − J ((L n )0 )> J ; ∂tn 2 hence, taking the ( )0 -part of this expression yields ∂Q 1 1 ∂ log Q 0 = Q −1 = −πk (L n )0 = − (L n )0 + J (L n )> 0 J. 0 ∂tn ∂tn 2 2 ˙ Using the fact that Q 0 , Q −1 0 , Q 0 ∈ Gk ∩D0 commute among themselves and commute ˆ with J and the fact that D0 D± , D± D0 ⊂ D± , we compute for Qˆ = Q −1 0 Q, L 1
TODA VERSUS PFAFF LATTICE
21
= Q −1 0 L 1 Q 0 (see (2.14)) ∂ Qˆ ˆ −1 −1 ˙ −1 ˙ −1 Q 0 Q = −Q −1 Q 0 + Q −1 0 Q0 Q0 Q Q 0 QQ ∂tn = −Q −1 Q˙ 0 + Q −1 Q˙ Q −1 Q 0 0
0
˙ −1 ˙ −1 = Q −1 0 (− Q 0 Q 0 + Q Q )Q 0 = Q −1 − (L n1 )− + J (L n1+ )> J Q 0 0 −1 −1 n −1 n = −(Q −1 0 L 1 Q 0 )− + Q 0 J Q 0 (Q 0 L 1 Q 0 )+ Q 0 > ˆn = −( Lˆ n1 )− + Q −2 J Q 20 . 0 J ( L 1 )+
>
J Q0
ˆ 1 (t, z) = z 9 ˆ 1 (t, z), we find Using this result and Lˆ 1 9 ˆ 1 (t, z) ∂9 ∂ P ti z i ˆ = e Qχ(z) ∂tn ∂tn P∞ i P∞ i 2 ˆ ˆ ˆn > = z n e 1 ti z Qχ(z) + e 1 ti z − ( Lˆ n1 )− + Q −2 0 J (( L 1 )+ ) J Q 0 Qχ(z) 2 ˆ ˆn > = Lˆ n1 − ( Lˆ n1 )− + Q −2 0 J ((L1 )+ ) J Q 0 91 (t, z) 2 ˆ ˆn = ( Lˆ n1 )+ + ( Lˆ n1 )0 + Q −2 (2.21) 0 (J ( L 1 )+ )Q 0 91 (t, z). ˆ 1 = Q −1 91 (t, z) and 9 ˆ 2 = Q 0 92 (t, z) satisfy, using W2 = But we also have that 9 0 −1> J W1 , ˆ 1 (t, z) ∂9 −1 −1 · · −1 ˆ = (Q −1 0 W1 ) χ(z) = (Q 0 W1 ) (Q 0 W1 ) 91 (t, z), ∂tn ˆ 2 (t, z) ∂9 = (Q 0 W2 )· χ(z) = (Q 0 W2 )· (Q 0 W2 )−1 (Q 0 92 ) ∂tn = ( Q˙ 0 W2 + Q 0 W˙ 2 )W −1 Q −1 (Q 0 92 ) 2
(2.22)
0
−1 −1 ˙ = ( Q˙ 0 Q −1 0 + Q 0 W2 W2 Q 0 )Q 0 92 −1 −1 ˙ = Q˙ 0 Q −1 0 + Q 0 J ( W1 W1 )Q 0 Q 0 92 −1 > −1 ˙ = Q˙ 0 Q −1 0 + Q 0 J ( W 1 W 1 ) J Q 0 Q 0 92 −1 ˙ −1 > = − J Q˙ 0 Q −1 0 + J (Q 0 W1 W1 Q 0 ) J Q 0 92 −1 ˙ −1 > ˙ = J (−Q −1 0 Q 0 + Q 0 W1 W1 Q 0 ) J Q 0 92 −1 · −1 = J (Q −1 (Q 0 92 ). (2.23) 0 W1 ) (Q 0 W1 )
Comparing (2.21), (2.22), and (2.23), and invoking (2.14), we have −J ( Lˆ n1 ) = Lˆ n2 ,
22
ADLER and VAN MOERBEKE
and so, in particular, −J (( Lˆ n1 )± ) = ( Lˆ n2 )∓
and
− J (( Lˆ n1 )0 ) = ( Lˆ n2 )0 ,
ˆ 2 (t, z) ∂9 ˆ 2 (t, z), = − ( Lˆ n2 )− + ( Lˆ n2 )0 + Q 20 (J (L n2 )− )Q −2 9 0 ∂tn which establishes Theorem 2.2. 3. Existence of the Pfaff τ -function The point of this section is to show that the solution of the Pfaff lattice can be expressed in terms of a sequence of functions τ , which are not τ -functions in the usual sense but which enjoy a different set of bilinear identities and partial differential equations. PROPOSITION 3.1 There exist functions τ2n (t) such that
ψ1,2n (t, z) =
τ2n (t − [z −1 ]) τ2n (t)
and
ψ2,2n (t, z) =
τ2n+2 (t + [z]) . τ2n+2 (t)
(3.1)
The proof of Proposition 3.1 is postponed until later. For future use, we define the diagonal matrix τ2n+2 . (3.2) h = diag(. . . , h −2 , h −2 , h 0 , h 0 , h 2 , h 2 , . . .) ∈ D0− , with h 2n = τ2n 3.2 91 and 92 have the following τ -function representation: THEOREM
91,2n (t, z) = e
P
−1 ti z i 2n −1/2 τ2n (t − [z ]) , z h 2n τ2n (t)
+ ∂/∂t1 )τ2n (t − [z −1 ]) , τ2n (t) P −i −1/2 τ2n+2 (t + [z]) 92,2n (t, z) = e− ti z z 2n+1 h 2n , τ2n (t) −1 − ∂/∂t )τ P −i 1 2n+2 (t + [z]) −1/2 (z , 92,2n+1 (t, z) = e− ti z z 2n+1 h 2n τ2n (t) 91,2n+1 (t, z) = e
P
ti z i 2n −1/2 (z z h 2n
(3.3)
with the τ2n (t) satisfying the following bilinear identity for all n, m ∈ Z: I P dz 0 i τ2n t − [z −1 ] τ2m+2 t 0 + [z −1 ] e (ti −ti )z z 2n−2m−2 2πi z=∞ I P 0 dz −i + τ2n+2 t + [z] τ2m t 0 − [z] e (ti −ti )z z 2n−2m = 0. (3.4) 2πi z=0
TODA VERSUS PFAFF LATTICE
23
Conversely, this bilinear relation characterizes the τ -function for the Pfaff lattice. Remark. L then has the following representation in terms of the Pfaffian τ -functions: .. . . . . Lˆ ˆ 01 0 0 00 L ˆ ˆ ˆ L L L 0 10 11 12 h 1/2 Lh −1/2 = , ˆ ˆ ˆ ∗ L 21 L 22 L 23 ˆ 32 Lˆ 33 . . . ∗ ∗ L .. . with (· = ∂/∂t1 ),
−(log τ2n )·
Lˆ nn := ˜ 2n − s2 (τ∂)τ − 2n
1
˜ 2n+2 s2 (−∂)τ τ2n+2
(log τ2n+2 )·
,
ˆL n,n+1 := 0 0 , 1 0
∗ (log τ2n+2 )·· Lˆ n+1,n := . ∗ ∗
(3.5)
The following bilinear relations follow from (3.4) and are due to [4]. 3.3 The functions τ2n (t) satisfy the “differential Fay identity”∗ τ2n (t − [u]), τ2n (t − [v]) COROLLARY
+ (u −1 − v −1 ) τ2n (t − [u])τ2n (t − [v]) − τ2n (t)τ2n (t − [u] − [v]) = uv(u − v)τ2n−2 (t − [u] − [v])τ2n+2 (t) (3.6) and Hirota-type bilinear equations, always involving nearest neighbors, 1 ∂2 ˜ ˜ τ2n+2 ◦ τ2n−2 , k, n = 0, 1, 2, . . . . pk+4 (∂) − τ2n ◦ τ2n = pk (∂) 2 ∂t1 ∂tk+3 (3.7) LEMMA 3.4 Consider an arbitrary function ϕ(t, z) depending on t ∈ C∞ , z ∈ C, having the asymptotics ϕ(t, z) = 1 + O(1/z) for z % ∞ and satisfying the functional relation
ϕ(t − [z 1−1 ], z 2 ) ϕ(t − [z 2−1 ], z 1 ) = , ϕ(t, z 1 ) ϕ(t, z 2 ) ∗ We
define { f, g} := f 0 g − f g 0 , where 0 = ∂/∂t1 .
t ∈ C∞ , z ∈ C.
24
ADLER and VAN MOERBEKE
Then there exists a function τ (t) such that ϕ(t, z) =
τ (t − [z −1 ]) . τ (t)
Proof See Appendix C. LEMMA 3.5 The following holds for the Pfaffian wave functions 91 and 92 , as in (2.7):
ψ1,2n (t − [z 2−1 ], z 1 ) ψ1,2n (t − [z 1−1 ], z 2 ) = ψ1,2n (t, z 1 ) ψ1,2n (t, z 2 )
(3.8)
ψ2,2n−2 t − [z −1 ], z −1 ψ1,2n (t, z) = 1.
(3.9)
and
Proof Setting (2.7) in the bilinear equation (2.12), with n 7 → 2n, m 7 → 2n − 2, yields I P c2n (t) dz 0 i e (ti −ti )z ψ1,2n (t, z)ψ2,2n−2 (t 0 , z −1 ) c2n−2 (t) ∞ 2πi I P c2n−2 (t) z 2 dz 0 −i + e (ti −ti )z ψ2,2n (t, z)ψ1,2n−2 (t 0 , z −1 ) = 0. c2n (t) 0 2πi Setting t − t 0 = [z 1−1 ] + [z 2−1 ] in the above and using e
P∞ 1
x i /i
= 1/(1 − x) yields
ψ1,2n (t, z)ψ2,2n−2 (t 0 , z −1 ) dz c2n−2 ∞ (1 − z/z 1 )(1 − z/z 2 ) 2πi I 1 1 dz c2n−2 =− z2 1 − 1− ψ2,2n (t, z)ψ1,2n−2 (t 0 , z −1 ) = 0, c2n 0 zz 1 zz 2 2πi c2n
I
the latter being equal to zero, because the integrand on the right-hand side is holomorphic. The integral on the left-hand side can be viewed as an integral along a contour encompassing ∞ and the points z 1 and z 2 , thus leading to ψ1,2n (t, z 1 )ψ2,2n−2 t − [z 1−1 ] − [z 2−1 ], z 1−1
= ψ1,2n (t, z 2 )ψ2,2n−2 t − [z 1−1 ] − [z 2−1 ], z 2−1 , (3.10) with ψ1,2n (t, z) = 1 + O(z −1 ),
ψ2,2n−2 t − [z 1−1 ] − [z 2−1 ], z −1 = 1 + O(z −1 ).
TODA VERSUS PFAFF LATTICE
25
Therefore, letting z 2 % ∞, one finds ψ1,2n (t, z 1 )ψ2,2n−2 t − [z 1−1 ], z 1−1 = 1,
(3.11)
yielding (3.9), and so, upon shifting t 7 → t − [z 2−1 ], ψ2,2n−2 t − [z 1−1 ] − [z 2−1 ], z 1−1 =
1 ψ1,2n (t − [z 2−1 ], z 1 )
;
similarly, ψ2,2n−2 t − [z 1−1 ] − [z 2−1 ], z 2−1 =
1 ψ1,2n (t − [z 1−1 ], z 2 )
.
(3.12)
Setting the two expressions (3.12) in (3.10) yields ψ1,2n (t − [z 2−1 ], z 1 ) ψ1,2n (t − [z 1−1 ], z 2 ) = . ψ1,2n (t, z 1 ) ψ1,2n (t, z 2 ) Proof of Proposition 3.1 From Lemmas 3.4 and 3.5 there exists, for each 2n, a function τ2n such that the first relation of (3.1) is satisfied; that is, ψ1,2n (t, z) =
τ2n (t − [z −1 ]) , τ2n (t)
and so from (3.9) ψ2,2n−2 t − [z −1 ], z −1 =
1 τ2n (t) = , ψ1,2n (t, z) τ2n (t − [z −1 ])
thus leading to ψ2,2n−2 (t, z) =
τ2n (t + [z]) , τ2n (t)
which is the second relation of (3.1). Proof of Theorem 3.2 At first, remembering that Qˆ = Q −1 0 Q, observe that P i ˆ e ti z Qχ(z) = Q −1 0 91 (t, z) 2n 2n =e
P
ti z i 2n
z ψ1,2n (t, z)
ti z i 2n τ2n (t
− [z −1 ]) τ2n (t) ∞ P i X ˜ 2n (t) sk (−∂)τ = e ti z z 2n 1 + , τ2n (t)
=e
P
z
n=1
26
ADLER and VAN MOERBEKE
showing that a few subdiagonals of the matrix Qˆ are given by Qˆ =
..
. 1 0
0 1
qˆ2n,2n−2 qˆ2n+1,2n−2
qˆ2n,2n−1 qˆ2n+1,2n−1
1 0
..
with qˆ2n,2n−1 = −
∂ log τ2n , ∂t1
qˆ2n,2n−2 =
Remembering that (2.7), normalized, becomes ( P ˆ 1,2n (t, z) = e tk z k z 2n ψ1,2n (t, z), 9 P ˆ 1,2n+1 (t, z) = e tk z k z 2n+1 ψ1,2n+1 (t, z), 9 (
0 1
ˆ 2,2n (t, z) = e− tk z −k z 2n+1 ψ2,2n (t, z), 9 P ˆ 2,2n+1 (t, z) = −e− tk z −k z 2n ψ2,2n+1 (t, z), 9 P
.
˜ 2n s2 (−∂)τ . τ2n
(3.13)
ψ1,2n = 1 + O(z −1 ), ψ1,2n+1 = 1 + O(z −2 ),
(3.14)
ψ2,2n = 1 + O(z), ψ2,2n+1 = 1 + O(z 2 ), (3.15)
we now show (3.3). Compute, using Theorem 2.2, e
P
ti z i
∂ + z z 2n ψ1,2n (t, z) ∂t1 ∂ ˆ 1 (t, z) = 9 2n ∂t1 ˆ 1 (t, z) = ( Lˆ 1 )+ + ( Lˆ 1 )0 + Q −2 J ( Lˆ 1+ )> J Q 20 9 0
2n
(3.16)
and e−
P
ti z −i
∂ 1 2n+1 − z ψ2,2n (t, z) ∂t1 z ∂ ˆ 2 (t, z) = 9 2n ∂t1 > 2 ˆ ˆ ˆ = J ( L 1 )+ + ( Lˆ 1 )0 + Q −2 0 J ( L 1+ ) J Q 0 92 (t, z) 2n .
(3.17)
TODA VERSUS PFAFF LATTICE
27
In expression (3.16), the matrix equals, according to (2.15), > 2 ˆ ( Lˆ 1 )+ + ( Lˆ 1 )0 + Q −2 0 J ( L 1+ ) J Q 0 .. . . . . qˆ0,−1 1 0 q ˆ − q ˆ − q ˆ 1 1,−1 20 21 0 0 q ˆ 21 = c02 /c22 0 qˆ31 − qˆ42 0 0 0 2 0 0 c2 /c42
0 0 1 −qˆ43 0 0
and, acting with J on this matrix, ˆ 1+ )> J Q 20 J ( Lˆ 1 )+ + ( Lˆ 1 )0 + Q −2 J ( L 0 .. . . . . q ˆ 1 0 21 2 /c2 q ˆ − q ˆ − q ˆ c 1,−1 20 0,−1 0 2 0 0 qˆ43 = qˆ31 − qˆ42 1 0 0 0 0 0 0 1
0 0 0 1 qˆ43 qˆ53 − qˆ64
b d
−d = c
0 0 1 −qˆ21 0 0
0 0 0 c22 /c42 qˆ65 qˆ53 − qˆ64
b . −a
Therefore the 2nth rows of both matrices, respectively, have the form (0, . . . , 0, qˆ2n,2n−1 (t), 1, 0, 0, . . . ), ↑
2n (0, . . . , 0, qˆ2n+2,2n+1 (t), 1, 0, 0, . . . ), ↑
2n
, . . .
using the fact that a J c
0 0 0 0 1 −qˆ65 .. .
0 0 0 0 1 −qˆ43 .. .
, . . .
28
ADLER and VAN MOERBEKE
and thus from (3.16) and (3.17), and expansions (3.14) and (3.15), we have ∂ + z z 2n ψ1,2n (t, z) = qˆ2n,2n−1 (t)z 2n ψ1,2n + z 2n+1 ψ1,2n+1 , ∂t1 ∂ − z −1 z 2n+1 ψ2,2n (t, z) = qˆ2n+2,2n+1 (t)z 2n+1 ψ2,2n + z 2n ψ2,2n+1 . ∂t1
(3.18)
So, using the expression (3.13) for qˆ2n,2n−1 and the first expression of (3.1), z 2n+1 ψ1,2n+1 (t, z) ∂ 2n = z+ z ψ1,2n (t, z) − qˆ2n,2n−1 (t)z 2n ψ1,2n (t, z) ∂t1 ∂ ∂ 2n = z+ z ψ1,2n (t, z) + log τ2n (t) z 2n ψ1,2n (t, z) ∂t1 ∂t1 τ (t − [z −1 ]) −1 ∂ τ2n (t − [z ]) ∂ 2n = z+ z 2n + τ2n (t) z 2n 2 (t) ∂t1 τ2n (t) ∂t1 τ2n = z 2n
(z + ∂/∂t1 )τ2n (t − [z −1 ]) , τ2n (t)
(3.19)
and similarly, using the second relation (3.18), z 2n ψ2,2n+1 (t, z) = z 2n+1
(−z −1 + ∂/∂t1 )τ2n+2 (t + [z]) . τ2n+2 (t)
(3.20)
This establishes (3.3) modulo the denominators. Therefore, we also have 1 1 ∂ ∂τ2n −1 ˜ 2n z −2 + · · · ψ1,2n+1 (t, z) = z+ τ2n (t) − z + s2 (−∂)τ z τ2n (t) ∂t1 ∂t1 2 1 ∂ ˜ τ2n z −2 + O(z −3 ); =1+ − 2 + s2 (−∂) τ2n (t) ∂t1 thus, referring to the matrix Qˆ just preceding (3.13), qˆ2n+1,2n = 0,
qˆ2n+1,2n−1 =
2 ˜ 1 ˜ − ∂ τ2n = −s2 (∂)τ2n . s2 (−∂) 2 τ2n τ2n ∂t1
(3.21)
To show (3.4), setting n 7 → 2n and m 7→ 2n in bilinear relation (2.10) and substituting, using (2.7) and the expressions for ψ1,2n (t, z) and ψ2,2n (t, z) in the proof of Proposition 3.1, 91,2n (t, z) = e
P
and 92,2n (t 0 , z) = e−
P
tk z k 2n
z c2n (t)
τ2n (t − [z −1 ]) τ2n (t)
0 tk0 z −k 2n+1 −1 0 τ2n+2 (t + [z]) z c2n (t ) τ2n+2 (t 0 )
TODA VERSUS PFAFF LATTICE
29
into I
91,2n (t, z)92,2n (t , z 0
∞
−1
dz ) + 2πi z
I
92,2n (t, z)91,2n (t 0 , z −1 ) 0
dz =0 2πi z
yields c2n (t) c2n (t 0 )
I e
P (tk −tk0 )z k τ2n (t
− [z −1 ])τ2n+2 (t 0 + [z −1 ]) dz τ2n (t)τ2n+2 (t 0 ) 2πi z 2 I P 0 0 c2n (t ) 0 −k τ2n+2 (t + [z])τ2n (t − [z]) dz + e (tk −tk )z . c2n (t) 0 τ2n+2 (t)τ2n (t 0 ) 2πi
∞
Setting t 0 = t + [α] amounts to replacing the exponential: e
P (tk −tk0 )z k
= 1 − αz,
e
P 0 (tk −tk )z −k
=
1 , 1 − α/z
so that the first integral has a simple pole at z = ∞ and the second integral has one at z = α. Evaluating the integrals yields −α
2 (t)(τ c2n 2n+2 (t)/τ2n (t)) 2 (t 0 )τ 0 0 c2n 2n+2 (t )/τ2n (t )
that is, e
P i (α /i)(∂/∂ti )
+ α = 0;
τ2n+2 (t) − 1 cn2 (t) =0 τ2n (t)
yields the following relation, which involves a constant cn , independent of time: 2 c2n (t) = cn
τ2n (t) = cn · h −1 2n (t). τ2n+2 (t)
(3.22)
Rescaling τ2n 7→ τ2n /(c1 c2 · · · cn−1 ), in effect, sets cn = 1, and then (3.22), (2.7), (3.1), (3.19), and (3.20) yield (3.3); substituting (3.3) into (2.10) yields (3.4). Finally, identity (3.22) actually says Q 0 = h −1/2 . To derive the form (3.5) of the matrix L, set (3.13) and (3.21) in the elements just below the main diagonal of matrix (2.15), to yield (· = ∂/∂t1 ) 2 −qˆ2n,2n−1 −qˆ2n+1,2n−1 + qˆ2n,2n−2 τ˙ 2 (s (−∂) ˜ − (∂ 2 /∂t 2 ))τ2n ˜ 2n s2 (−∂)τ 2 2n 1 =− − + τ2n τ2n τ2n τ¨2n τ˙2n 2 = − τ2n τ2n = (log τ2n )··
30
ADLER and VAN MOERBEKE
and ˜ − ∂ 2 /∂t 2 )τ2n ˜ 2n+2 (s2 (−∂) s2 (−∂)τ 1 − τ2n τ2n+2 ˜ 2n ˜ 2n+2 s2 (∂)τ s2 (−∂)τ =− − , τ2n τ2n+2
qˆ2n+1,2n−1 − qˆ2n+2,2n =
concluding the proof of Theorem 3.2 and the remark following it, upon substituting the relations (3.13) and (3.14) and also Q 0 = h −1/2 into (2.15). 4. Semi-infinite matrices m ∞ , (skew-)orthogonal polynomials, and matrix integrals In this section, consider the following inner product∗ for α = 0, ∓1: ZZ P i i h f, git = f (y)g(z)e ti (y +z ) 2D α δ(y − z)ρ(y) ˜ ρ(z) ˜ dy dz R2 Z P 2ti y i f (y)g(y)e 2ρ(y) ˜ 2 dy for α = 0, R Z Z P∞ i i f (y)g(z)e 1 ti (y +z ) ε(y − z)ρ(y) ˜ ρ(z) ˜ dy dz for α = −1, = 2 R Z P∞ i { f, g}(y)e 1 2ti y ρ(y) ˜ 2 dy for α = +1. R
(4.1) Each type of inner product is discussed in Sections 4.1 and 4.2. 4.1. ∂m ∞ /∂tk = 3k m ∞ , orthogonal polynomials, and Hermitian matrix integrals (α = 0) The inner product above, with α = 0, corresponds to Hermitian matrix integrals; this theory is sketched here for the sake of completeness and analogy; it mainly summarizes [6]. Consider a t-dependent weight ρt (dz) := e
P
ti z i
ρ(dz) = e−V (z)+
P
ti z i
dz
on R, as in (0.1), and the induced t-dependent measure eTr(−V (X )+
P
ti X i )
dX
(4.2)
on the ensemble Hn of Hermitian matrices, with Haar measure d X ; the latter can be decomposed into a spectral part (radial part) and an angular part: d X :=
n Y 1
d X ii
Y
(d<X i j d=X i j ) = 12 (z) dz 1 · · · dz n dU,
(4.3)
1≤i< j≤n
have ε(x) = sign x, having the property ε0 = 2δ(x). Also, consider the Wronskian { f, g} := (∂ f /∂ y)g − f (∂g/∂ y).
∗ We
TODA VERSUS PFAFF LATTICE
31
Q where 1(z) = 1≤i< j≤n (z i − z j ) is the Vandermonde determinant. Here we form the matrix integral Z Z n P P Y ∞ i i eTr(−V (X )+ ti X ) d X = cn 12 (z) e 1 ti z k ρ(dz k ). (4.4) Hn
Rn
k=1
The weight ρt (dz) defines a (symmetric) t-dependent inner product of the type (4.1) for α = 0: Z P∞ i sy h f, git = f (z)g(z)e 1 ti z ρ(dz), with moments µi j (t) := hz
i
sy , z j it
Z
z i+ j e
=
P
tk z k
ρ(dz),
R
satisfying
Z P k ∂µi j = z i+ j+` e tk z ρ(dz) = µi+`, j (t). ∂t` R Therefore the semi-infinite moment matrix m ∞ (t) = (µi j (t))i, j≥0 satisfies ∂m ∞ i = 3i m ∞ = m ∞ 3> . ∂ti
(4.5)
The point now is that the following integral can be expressed as a determinant of moments; namely, Z Z n P Y Tr(−V (X )+ ∞ ti X i ) 1 e dX = 12 (z) ρt (dz k ) Hn
Rn
Z
k=1
X
=
Rn σ ∈S n
Z
X
=
Rn σ ∈S n
=
X
det
k−1 det(z σ`−1 (k) z σ (k) )1≤`,k≤n
ρt (dz k )
k=1
det(z σ`+k−2 (k) )1≤`,k≤n
n Y
ρt (dz σ (k) )
k=1
Z
σ ∈Sn
= n! det
n Y
R
Z
z σ`+k−2 ρ (dz ) t σ (k) (k)
z `+k−2 ρt (dz) R
1≤`,k≤n
1≤`,k≤n
= n! det(µi j )0≤i, j≤n−1 = n!τn (t) is a τ -function for the KP-equation; also, in view of (4.5) and the upper-lower Borel decomposition (0.3) of m ∞ , the integrals form a vector of τ -functions for the Toda lattice. The polynomials pn (t; z) defined by the Borel decomposition m ∞ (t) = S −1 S >−1 and p(t; z) = Sχ(z) are orthonormal with regard to the inner product sy hz i , z j it = µi j (t).
32
ADLER and VAN MOERBEKE
4.2. ∂m ∞ /∂tk = 3k m ∞ + m ∞ 3>k , skew-orthogonal polynomials, and symmetric and symplectic matrix integrals (α = ±1) Consider a skew-symmetric semi-infinite matrix m ∞ (t) = µi j (t) i, j≥0 , with m n (t) = µi j (t) 0≤i, j≤n−1 , satisfying ∂m ∞ /∂tk = 3k m ∞ + m ∞ 3>n .
(4.6)
Then we have shown in Sections 2 and 3 that, upon skew-Borel decomposing m ∞ , these equations ultimately imply the existence of functions τ (t) satisfying bilinear equations (3.4). Remember also that h(t) = diag(h 0 , h 0 , h 2 , h 2 , . . .) ∈ D0− ,
with h 2n (t) =
τ2n+2 (t) . τ2n (t)
Here we need the Pfaffian pf(A) of a skew-symmetric matrix A = (ai j )0≤i, j≤n−1 for∗ even n: pf(A) d x0 ∧ · · · ∧ d xn−1 = =
1 2n/2 (n/2)!
X
1 (n/2)!
X
ai j d xi ∧ d x j
n/2
0≤i< j≤n−1
ε(σ )ai0 ,i1 ai2 ,i3 · · · ain−2 ,in−1 d x0 ∧ · · · ∧ d xn−1 , (4.7)
σ
so that pf(A)2 = det A. We now state the following theorem due to Adler, E. Horozov, and van Moerbeke [2], in complete analogy with the discussion of the Hermitian case. THEOREM 4.1 Consider a semi-infinite skew-symmetric matrix m ∞ , evolving according to (4.6); set
τ2n (t) = pf m 2n (t)
and
h 2n =
pf(m 2n+2 (t)) . pf(m 2n (t))
(4.8)
Then, modulo the exponential, the wave vector 91 (defined by (3.3)) is a sequence of polynomials, P i 91,k (t, z) = e ti z qk (t, z), (4.9) where the qk ’s are skew-orthonormal polynomials of the forms (0.13) and (0.18), satisfying hqi , q j isk 0≤i, j<∞ = J, with hy i , z j isk := µi j . (4.10) ∗ In the formula below, (i , i , . . . , i 0 1 n−2 , i n−1 ) = σ (0, 1, . . . , n − 1), where σ is a permutation and ε(σ ) its parity.
TODA VERSUS PFAFF LATTICE
33
The matrix Q defined by q(z) = Qχ(z) is the unique solution (modulo signs) to the skew-Borel decomposition of m ∞ : m ∞ (t) = Q −1 J Q >−1 ,
with Q ∈ k.
(4.11)
The matrix L = Q3Q −1 , also defined by zq(t, z) = Lq(t, z), and the diagonal matrix h satisfy the equations ∂L = [−πk L i , L] ∂ti
and
h −1
∂h = 2πk (L i )0 . ∂ti
(4.12)
Sketch of proof At first, note that looking for skew-orthogonal polynomials is tantamount to the skewBorel decomposition of m ∞ , so that (4.10) and (4.11) are equivalent. The skeworthogonality of the polynomials (0.18) follows from expanding the Pfaffians explicitly in terms of z-columns, upon using the expression for the Pfaffian in terms of a column X ˆ . . . , ` − 1) = pf(0, . . . , ` − 1, i). (−1)k aki pf(0, . . . , k, 0≤k≤`−1
(For details, see [2].) On the other hand, Theorem 3.2 gives 9(t, z) and hence Q in terms of τn (t) = pf m 2n (t) of (4.8). By the uniqueness of decomposition (4.11), the two ways of arriving at Q, (0.18) and (3.3), must coincide. Important remark. The polynomials (0.18) provide an explicit algorithm to perform the skew-Borel decomposition of the skew-symmetric matrix m ∞ . Namely, the coefficients of the polynomials qi provide the entries of the matrix Q. This fact is used later in the examples. Symmetric matrix integrals (α = −1) Here we focus on integrals over the space S2n of symmetric matrices of the type Z P∞ i eTr(−V (X )+ 1 ti X ) d X, (4.13) S2n
where d X denotes Haar measure for X = U diag(z 1 , . . . , z n )U > , UU > = I , Y d X := d X i j = |1(z)| dz 1 · · · dz n dU. (4.14) 1≤i≤ j≤n
34
ADLER and VAN MOERBEKE
As appears below, integral (4.13)Pleads to a skew-inner product of the type (4.1) with P i i α = −1, with weight ρt (z) := e ti z ρ(z) = e−V (z)+ ti z , ZZ
f (x), g(y) := f (x)g(y)ε(x − y)ρt (x)ρt (y) d x dy, (4.15) R2
leading to skew-symmetric moments∗ ZZ µi j (t) = x i y j ε(x − y)ρt (x)ρt (y) d x dy 2 Z ZR = (x i y j − x j y i )ρt (x)ρt (y) d x dy x≥y Z = F j (x)G i (x) − Fi (x)G j (x) d x,
(4.16)
R
where (0 = d/d x) Z x P k Fi (x) := y i e tk y ρ(y) dy
and
−∞
G i (x) := Fi0 (x) = x i e
P
tk x k
ρ(x).
By simple inspection, the moments µk` (t) satisfy ZZ P ∂µk` n n = (x k+i y ` + x k y `+i )ε(x − y)e tn (x +y ) ρ(x)ρ(y) d x dy ∂ti R2 = µk+i,` + µk,`+i , and so m ∞ satisfies (4.6). According to M. Mehta [14], the symmetric matrix integral can now be expressed in terms of the Pfaffian, as follows, taking into account a constant c2n , coming from ∗ We
have ε(x) = 1 for x ≥ 0, and ε(x) = −1 for x < 0.
TODA VERSUS PFAFF LATTICE
35
the integration of the orthogonal group: 1 (2n)!
Z S2n (E)
e
P Tr(−V (X )+ ti X i ) d X
Z = −∞
= ...,
z 2n z 2n−2
R2n
|12n (z)|
n Y
=
e
P
tk z ik
ρ(z i ) dz i
i=1
ρt (z 2k ) dz 2k det
−∞
Z
z2
−∞
i i z 2n−1 ρt (z 2n−1 ) dz 2n−1 , z 2n
Z
2n Y
i=1
−∞
Z
Z
2n Y det z ij+1 ρt (z j+1 ) 0≤i, j≤2n−1 dz i n Y
Z
1 = (2n)!
z 1i ρt (z 1 ) dz 1 , z 2i ,
0≤i≤2n−1
ρt (z 2k ) dz 2k
× det Fi (z 2 ), z 2i , Fi (z 4 ) − Fi (z 2 ), z 4i , i . . . , Fi (z 2n ) − Fi (z 2n−2 ), z 2n 0≤i≤2n−1 Z n Y = dz 2i −∞
× det Fi (z 2 ), G i (z 2 ), . . . , Fi (z 2n ), G i (z 2n ) 0≤i≤2n−1 Z Y n 1 = dyi det Fi (y1 ), G i (y1 ), . . . , Fi (yn ), G i (yn ) 0≤i≤2n−1 n! Rn 1 Z 1/2 = det G i (y)F j (y) − Fi (y)G j (y) dy 0≤i, j≤2n−1
R
(using de Bruijn’s lemma (see [14, p. 446])) P∞ i i = pf y k z ` ε(y − z)e 1 ti (y +z ) ρ(y)ρ(z) dy dz 0≤k,`≤2n−1 R2 = pf µi j (t) 0≤i, j≤2n−1 = τ2n (t), ZZ
(4.17)
which is a Pfaffian τ -function. Symplectic matrix integrals: (α = +1) Here we concentrate on integrals of the type Z P∞ i e2 Tr(−V (X )+ 1 ti X ) d X, T2n
(4.18)
36
ADLER and VAN MOERBEKE
where d X denotes Haar measure∗ dX =
N Y
d Xk
Y
(0) ¯(0) (1) ¯(1) d X k` d X k` d X k` d X k`
k<`
1
on the space T2N of self-dual (N × N )–Hermitian matrices, with real quaternionic entries; the latter can be realized as the space of (2N × 2N )-matrices with entries (i) X `k ∈ C, (0) (1) X k` X k` > T2N = X = (X k` )1≤k,`≤N , X k` = with X `k = X¯ k` . (1) (0) − X¯ X¯ k`
k`
Another skew-symmetric moment matrix m satisfying P (4.6) is given by inner P ∞ α α product (4.1) for α = 1, with ρt (y) = ρ(y)e tα y = e−V (y)+ tα y , Z µi j (t) = {y i , y j }ρt (y)2 I E (y) dy ZR i = y ρt (y), y j ρt (y) I E (y) dy ZR = G i (y)F j (y) − Fi (y)G j (y) dy, (4.19) R
upon setting (0 = d/d x) F j (x) = x j ρt (x)
and
0 G j (x) := F j0 (x) = x j ρt (x) .
That m ∞ satisfies (4.6) follows at once from expression (4.18): Z Z k ` 2 µk` (t) = {y , y }ρt (y) dy = (k − `)y k+`−1 ρt (y)2 dy Z ∂µk` = 2 {y k , y ` }y i ρt (y)2 dy ∂ti Z = (k + i − `)y k+i+`−1 + (k − ` − i)y k+i+`−1 ρt (y)2 dy = µk+i,` + µk,`+i , thus leading to (4.6). Using the relation Y (xi − x j )4 = det x1i (x1i )0 x2i
(x2i )0
···
xni
(xni )0
0≤i≤2n−1
,
1≤i, j≤n
means the usual complex conjugate. The condition on the (2×2)-matrices X k` implies that X kk = X k I , with X k ∈ R and I the identity. ∗X ¯
TODA VERSUS PFAFF LATTICE
37
one computes, using again de Bruijn’s lemma, Z Z n P Y Y 1 1 i e2 Tr(−V (X )+ ti X ) d X = (xi − x j )4 ρt (xi )2 d xi (n)! T2n n! Rn 1≤i, j≤n
=
=
1 n!
n Y
Z
Rn k=1 × det x1i Z Y n
1 n!
Rn 1
= det1/2
i=1
ρt (xk )2 d xk (x1i )0
x2i
(x2i )0
dyi det Fi (y1 )
Z
···
G i (y1 )
xni ···
(xni )0
0≤i≤2n−1
Fi (yn )
G i (y)F j (y) − Fi (y)G j (y) dy
R
G i (yn ) 0≤i≤2n−1
0≤i, j≤2n−1
= pf µi j (t) 0≤i, j≤2n−1 = τ2n (t),
(4.20)
which is a Pfaffian τ -function as well. 5. A map from the Toda to the Pfaff lattice P k Remember from (0.1) the notation ρt (z) = ρ(z)e tk z and ρ 0 /ρ = −g/ f . Assuming, in addition, that f (z)ρ(z) vanishes at the endpoints of the interval under consideration (which could be finite, infinite, or semi-infinite), one checks that the t-dependent operator in z, s f d p fρt nt : = ρt dz P P k d f0+g (−1/2) tk z k =e f (z) − (z) e1/2 tk z dz 2 ∞ 0 X d f + gt = f (z) − (z), with gt (z) = g(z) − f (z) ktk z k−1 , (5.1) dz 2 1
{1, z, z 2 , . . .} and is skew-symmetric with respect to the t-dependent
maintains H+ = sy inner product h , it , defined by the weight ρt (z) dz, Z Z sy sy hnt ϕ, ψit = (nt ϕ)(z)ψ(z)ρt (z) dz = − ϕ(nt ψ)ρt dz = −hϕ, nt ψit . E
E
The orthonormality of the t-dependent polynomials pn (t, z) in z implies that
sy pn (t, z), pm (t, z) t = δmn . The matrices L and M are defined by zp = L p
and
e−(1/2)
P
tk z k
d 1/2 P tk z k e p = M p. dz
38
ADLER and VAN MOERBEKE
The skewness of nt implies the skew-symmetry of the matrix N (t) = f (L)M −
f0+g (L) such that nt p(t, z) = N p(t, z); 2
(5.2)
so N (t) can be viewed as the operator nt , expressed in the polynomial basis ( p0 (t, z), p1 (t, z), . . .). In the next theorem, we consider functions F of two (noncommutative) variables z and nt such that the (pseudo-)differential operator ut := F(z, nt ) in z and the matrix U := F(L , N ), related by∗ F(z, nt ) p(t, z) = F(L , N ) p(t, z), are skew-symmetric as well. Examples of F’s are† ` 2k+1 † F(z, nt ) := nt , n−1 } , t , or {z , nt
corresponding to F(L , N ) = N , N
−1
, or {N
2k+1
, L ` }† .
5.1 Any H¨ankel matrix m ∞ evolving according to the vector fields THEOREM
∂m ∞ (t) = 3k m ∞ ∂tk leads to matrices L and M, evolving according to the Toda lattice equations ∂ L/∂tn = (1/2)[(L n )sk , L]
and
∂ M/∂tn = (1/2)[(L n )sk , M]
(see (1.9)). Consider a function F of two variables such that the operator ut := sy F(z, nt ) is skew-symmetric with respect to h , it and so the matrix U (t) = F L(t), N (t) , defined by ut p(t, z) = U p(t, z), is skew-symmetric. This induces a natural lower-triangular matrix O(t), mapping the Toda lattice into the Pfaff lattice (for notation πbo , πk , Gk , etc., see (1.9), (1.18), (1.17)): pn (t, z) = (S(t)χ(z))n orthonormal with respect to sy m ∞ (t) = hz i , z j it 0≤i, j≤∞ = S −1 S >−1 , Toda lattice h 1 i ∂L = − πbo L j , L , j = 1, 2, . . . L(t) = S3S −1 satisfies ∂t j 2 ∗ It
is to be understood that F(L , N ) reverses the order of z, u in F(z, u). define {A, B}† := AB + B A.
† We
TODA VERSUS PFAFF LATTICE
39
−U (2t) = O −1 (2t)J O >−1 (2t), map O(2t) such that O(2t) is lower-triangular, O(2t)S(2t) ∈ Gk y q (t, z) = O(2t) p(2t, z) , skew-orthonormal with regard to n n m˜ ∞ (t) := −S −1 (2t)U (2t)S >−1 (2t) = Q −1 (t)J Q >−1 (t) = hz i , z j isk t 0≤i, j≤∞ Pfaff lattice sy = hz i , u2t z j i2t 0≤i, j≤∞ , ˜ L(t) := O(2t)L(2t)O(2t)−1 satisfies ∂ L˜ ˜ j = 1, . . . . = [−πk L˜ j , L], ∂t j Proof Since U (t) is skew-symmetric, it admits a skew-Borel decomposition −U (t) = O −1 (t)J O >−1 (t),
with lower-triangular O(t).
(5.3)
But the new matrix, defined by m˜ ∞ (t) := −S −1 (2t)U (2t)S >−1 (2t),
(5.4)
is skew-symmetric and thus admits a unique skew-Borel decomposition ˜ >−1 , m˜ ∞ (t) = Q˜ −1 (t)J Q(t)
˜ with Q(t) ∈ Gk .
(5.5)
Comparing (5.3), (5.4), and (5.5) leads to a unique choice of matrix O(t), skew-Borel decomposing −U (2t), as in (5.3), such that ˜ O(2t)S(2t) = Q(t) ∈ Gk . Using, as a consequence of (5.2) and (1.9), ∂U (2t) = πsy L k (2t), U (2t) ∂tk and
we compute
∂S (2t) = − πbo L k (2t) S(2t), ∂tk
(5.6)
40
ADLER and VAN MOERBEKE
∂ ∂ m˜ ∞ ∂S (t) = S −1 (2t)S −1 U (2t)S >−1 (2t) − S −1 (2t) U (2t) S >−1 (2t) ∂tk ∂tk ∂tk > ∂S + S −1 (2t)U (2t)S >−1 (2t)S >−1 ∂tk = −S −1 πbo L k (2t) U S >−1 − S −1 [πsy L k , U ]S >−1 − S −1 U (πbo L k )> S >−1 = −S −1 (πbo L k + πsy L k )U S >−1 − S −1 U (πbo L k )> − πsy L k S >−1 = −S −1 L k U S >−1 − S −1 U L >k S >−1 (using (5.7)) = −3k S −1 U S >−1 − S −1 U S >−1 3>k S > S >−1 = 3k m˜ ∞ (t) + m˜ ∞ (t)3>k
(using L k = S3k S −1 )
(by (5.4)).
For an arbitrary matrix A, we have A = A> ⇐⇒ A = (Abo )> − Asy .
(5.7)
Indeed, remembering that∗ Abo = 2A− + A0 and Asy = A+ − A− , one checks that (Abo )> − Asy − A = 2(A− )> + A0 −(A+ − A− )− A− − A+ − A0 = −2(A+ −(A− )> ), so that the left-hand side vanishes if the right-hand side does; the latter means A is symmetric. ˜ We now define L(t) by conjugation of L(2t) by O(2t): ˜ ˜ L(t) := O(2t)L(2t)O(2t)−1 = O(2t)S(2t)3S −1 (2t)O(2t)−1 = Q(t)3 Q˜ −1 (t); ˜ satisfies the Pfaff Lax equation. Therefore the sequence thus, by Proposition 1.2, L(t) of polynomials ˜ q(t, z) := O(2t) p(2t, z) = O(2t)S(2t)χ(z) = Q(t)χ(z) is skew-orthonormal,
sk qi (t, z), q j (t, z) = Ji j , with regard to the skew-inner product specified by the matrix m˜ ∞ : hz i , z j isk ˜ i j (t). t =µ sy
In the last step, we show that hϕ, ψisk t = hϕ, uψi2t . Since U (2t) = −O −1 (2t)J O >−1 (2t) = −U > (2t), ∗A
(5.8)
± means the usual strictly upper-(lower-)triangular part, and A0 means the diagonal part in the common sense.
TODA VERSUS PFAFF LATTICE
41
we compute
sy
sy qi (t, z), (u2t q) j (t, z) 2t = (O p)i (2t), (uO p) j (2t) 2t
sy = (O p)i (2t), (Ou p) j (2t) 2t
sy = (O p)i (2t), (O U p) j (2t) 2t
sy = O(2t) pk (2t), p` (2t) k,`≥0 (O U )> (2t) i j = O(2t)I (O U )> (2t) i j = O(2t)U > (2t)O > (2t) i j = − O(2t)U (2t)O > (2t) i j (using (5.8)).
= Ji j
(5.9)
0
Therefore, defining a new skew-inner product h , isk , 0
sy
hϕ, ψisk := hϕ, uψi2t , we have shown that 0
sk hqi , q j isk t = hqi , q j it = Ji j ,
and so by completeness of the basis qi , we have 0
sk h , isk t = h , it ,
thus ending the proof of Theorem 4.1. 6. Example 1: From Hermitian to symmetric matrix integrals Striking examples are given by using the map O(t) obtained from skew-Borel decomposing N −1 (t) and N (t) (see (5.2)). This section deals with N −1 (t), whereas Section 7 deals with N (t). PROPOSITION 6.1 The special transformation
U (t) = N
−1
(t) =
f (L)M −
−1 f0+g (L) (t) 2
maps the Toda lattice τ -functions with initial weight ρ = e−V , V 0 = −g/ f (Hermitian matrix integral) to the Pfaff lattice τ -functions (symmetric matrix integral), with initial weight ρ (z) 1/2 P∞ i 2t ρ˜t (z) := = e−(1/2)(V (z)+log f (z)−2 1 ti z ) f (z) =: e−V (z)+ ˜
P∞ 1
ti z i
= ρ(z)e ˜
P
ti z i
.
42
ADLER and VAN MOERBEKE
To be precise,
Toda lattice
pn (t, z) orthonormal polynomialsZ in z for the inner product P i sy hϕ, ψit = ϕ(z)ψ(z)e ti z ρ(z) dz, sy
µi j (t) = hz i , z j it and j≤n−1 , Z m n = (µi j )0≤i, P∞ 1 i eTr(−V (X )+ 1 ti X ) d X τn (t) = det m n = n! Hn −N −1 (2t) = O −1 (2t)J O >−1 (2t), map O(2t) such that O(2t) is lower-triangular, O(2t)S(2t) ∈ Gk y qn (t, z) = O(2t) p(2t, z) n skew-orthonormal polynomials in z for the skew-inner product (weight ρ), ˜ sy −1 sk hϕ, ψit := hϕ, n2t ψi2t ZZ P 1 ti (x i +y i ) = ϕ(x)ψ(y)ε(x − y)e 2 R2 r r ρ ρ Pfaff lattice (y) d x dy, × (x) f f µ˜ i j (t) = hx i , y j isk ˜ n = (µ˜ i j )0≤i, j≤n−1 , t and m Z P 1 i Tr(−V˜ (X )+ ∞ 1 ti X ) d X , e τ ˜ (t) = pf( m ˜ ) = 2n 2n n (2n)! 2 S2n 1 with V˜ (z) = V (z) + log f (z) . 2
In the first integral defining τn (t), d X denotes Haar measure on Hermitian matrices (see Section 4.1), whereas the second integral τ˜2n (t) involves Haar measure on symmetric matrices (see Section 4.2). Proof At first, check that Z d −1 1 ϕ(x) = ε(x − y)ϕ(y) dy. dx 2
(6.1)
TODA VERSUS PFAFF LATTICE
43
Indeed, d d −1 ϕ(x) = dx dx
Z
1 ∂ ε(x − y)ϕ(y) dy 2 ∂x
Z
δ(x − y)ϕ(y) dy
=
(using
∂ ε(x) = 2δ(x)) ∂x
= ϕ(x). Consider now the operator s −1 f d p −1 ut = nt = fρt , ρt dz
so that ut p = n−1 t p =N
according to (5.2). Let it act on a function ϕ(x): s d −1 ρ (x) 1 t n−1 ϕ(x) t ϕ(x) = √ f (x) f (x)ρt (x) d x s Z 1 ε(x − y) ρt (y) = ϕ(y) dy √ 2 f (y) f (x)ρt (x) R
−1
p,
(using (6.1)).
One computes sy
hϕ, ψisk t = hϕ, u2t ψi2t
sy
= hϕ, n−1 2t ψi2t s ZZ s 1 ρ2t (x) ρ2t (y) = ε(x − y) ϕ(x)ψ(y) d x dy 2 R2 f (x) f (y) ZZ P∞ 1 k k 1 tk (x +y ) ε(x − y)ϕ(x)ψ(y) d x dy. = ρ(x) ˜ ρ(y)e ˜ 2 2 R So, finally setting V˜ (x) = (1/2)(V (x) + log f (x)) yields by (4.17) Z P∞ 1 i ˜ τ˜2n (t) = pf(m˜ 2n ) = eTr(−V (X )+ 1 ti X ) d X. (2n)! S2n The map O for the classical orthogonal polynomials at t = 0 CLAIM
The matrix O, mapping orthonormal pk into skew-orthonormal polynomials qk , is
44
ADLER and VAN MOERBEKE
given by a lower-triangular three-step relation: r c2n q2n (0, z) = p2n (0, z), a2n r a2n q2n+1 (0, z) = c2n 2n c2n X bi p2n (0, z) + c2n p2n+1 (0, z) , × − c2n−1 p2n−1 (0, z) + a2n 0
(6.2) where the ai and bi are the entries in the tridiagonal matrix defining the orthonormal polynomials, and the ci ’s are the entries of the skew-symmetric matrix N . In [6], we showed that, in the classical cases below, N is tridiagonal at the same time as L (see Appendix B):
b0 a0 L=
a0 b1 a1
a1 b2 .. .
.. , . .. .
0 −c0 −N =
c0 0 −c1
c1 0 .. .
.. , .
(6.3)
with the following precise entries: 2
Hermite: ρ(z) = e−z , an−1 =
p n/2, bn = 0, cn = an ; p Laguerre: ρ(z) = e−z z α I[0,∞) (z), an−1 = n(n + α), bn = 2n + α + 1, cn = an /2; Jacobi: ρ(z) = (1 − z)α (1 + z)ρ I[−1,1] (z); 1/2 4n(n + α + β)(n + α)(n + β) , (2n + α + β)2 (2n + α + β + 1)(2n + α + β − 1) α2 − β 2 bn = , (2n + α + β)(2n + α + β + 2) α + β cn = an +n+1 . 2
an−1 =
If the skew-symmetric matrix N has the tridiagonal form above, then one checks that
TODA VERSUS PFAFF LATTICE
its inverse has the following form: 0 − c10 0 1 0 0 c0 0 0 0 c1 1 0 c0 c2 c2 0 0 0 −N −1 = c1 c3 c 3 0 c2 c4 c0 c2 c4 0 0 0 c1 c3 c5 c3 c5 cccc 0 c2 c4 c6 0246
45
−c1 c0 c2
0 − c12 0 0 0 0 0
−c1 c3 c0 c2 c4
0 0 0 0 0
0 −c3 c2 c4
1 c4
0 c5 c4 c6
0 − c14 0 0 0
0 0 0 0 0 0 0 1 c6
−c1 c3 c5 c0 c2 c4 c6
0
−c3 c5 c2 c4 c6
0 −c5 c4 c6
0 − c16 0 ..
.
(6.4) In order to find the matrix O, we must perform the skew-Borel decomposition of the matrix −U : −U = −N −1 = O −1 J O >−1 . The recipe for doing so is given in Theorem 4.1 (see also the important remark following that theorem). It suffices to form the Pfaffians (0.18) by appropriately bordering the matrix −N −1 , as in (0.18), with rows and columns of powers of z, yielding skeworthonormal polynomials; we choose to call them r ’s, with Oχ(z) = r (z). They turn out to be the following simple polynomials, with 1/τ˜˜2n = c0 c2 c4 · · · c2n−2 : r2n (z) = q r2n+1 (z) = q
1 τ˜˜2n τ˜˜2n+2 1 τ˜˜2n τ˜˜2n+2
c2n z 2n 1 = √ c2n z 2n , c2n c0 c2 · · · c2n c2n z 2n+1 − c2n−1 z 2n−1 1 = √ (c2n z 2n+1 − c2n−1 z 2n−1 ). c0 c2 · · · c2n c2n
Then, also from Appendix A, in order to get O → Oˆ in the correct form, we compute ˆ the skew-orthonormal polynomials rˆk , with Oχ(z) = rˆ (z): r c2n 2n 1 z , rˆ2n (z) = √ r2n (z) = a2n a2n P2n bi √ rˆ2n+1 (z) = √0 r2n (z) + a2n r2n+1 (z) a2n r 2n a2n c2n X 2n = − c2n−1 z 2n−1 + bi z + c2n z 2n+1 . (6.5) c2n a2n 0
From the coefficients of the polynomial rˆk , one reads off the transformation matrix from orthonormal to skew-orthonormal polynomials; it is given by the matrix
46
ADLER and VAN MOERBEKE
ˆ ˆ Oˆ such that Oχ(z) = rˆ (z). Therefore q(t, z) = O(2t) p(2t, z) yields, after setting t = 0, r c2n q2n (0, z) = p2n (0, z), a2n r a2n q2n+1 (0, z) = c2n 2n c2n X × − c2n−1 p2n−1 (0, z) + bi p2n (0, z) + c2n p2n+1 (0, z) , a2n 0
(6.6) confirming (6.2).
7. Example 2: From Hermitian to symplectic matrix integrals PROPOSITION 7.1 The matrix transformation
N = f (L)M −
f0+g (L) 2
maps the Toda lattice τ -functions with t-dependent weight P∞
ρt (z) = e−V (z)+
1
ti z i
,
V 0 = g/ f
(Hermitian matrix integral), to the Pfaff lattice τ -functions (symplectic matrix integral), with t-dependent weight P∞ i 1/2 ρ˜t (z) := ρ2t (z) f (z) = e−(1/2)(V (z)−log f (z)−2 1 ti z ) =: e−V (z)+ ˜
P
ti z i
= ρ(z)e ˜
P
ti z i
.
To be precise, pn (t, z) orthonormal polynomials in z for the inner product P i R sy hϕ, ψit = ϕ(z)ψ(z)e ti z ρ(z) dz, sy Toda lattice µi j (t) = hz i , z j it and Zm n = (µi j )0≤i, j≤n−1 , P i τ (t) = det m (t) = 1 eTr(−V (X )+ ti X ) d X n n n! Hn
TODA VERSUS PFAFF LATTICE
47
−N (2t) = O −1 (2t)J O >−1 (2t), map O(2t) such that O(2t) is lower-triangular, O(2t)S(2t) ∈ Gk y qn (t, z) = O(2t) p(2t, z) n skew-orthonormal polynomials in z for the skew-inner product (weight ρ˜t ), sy sk hϕ, ψit := hϕ, n2t ψi2t ZZ P i 1 =− ϕ(z), ψ(z) e2 ti z ρ(z) f (z) dz, 2 R2 Pfaff lattice i , z j isk and m µ ˜ (t) = hz ˜ n = det(µ˜ i j )0≤i, j≤n−1 , ij t Z P 1 2 Tr(−V˜ (X )+ ti X i ) τ ˜ (t) = pf( m ˜ (t)) = e d X, 2n 2n (−2)n n! T2n 1 with V˜ (z) = V (z) − log f (z) . 2 Proof Representing d/d x as an integral operator Z Z Z ∂ d ϕ(x) = δ(x − y)ϕ 0 (y) dy = − δ(x − y)ϕ(y) dy = δ 0 (x − y)ϕ(y) dy, dx R R ∂y R compute s ut = nt =
f d p fρt , ρt dz
so that nt p(t, z) = N p(t, z);
remember N from (5.2). Let it act on a function ϕ(x): s f d p fρt ϕ(x) ut ϕ(x) = ρt d x s Z p f (x) 0 = δ (x − y) f (y)ρt (y)ϕ(y) dy. ρt (x) R
48
ADLER and VAN MOERBEKE
Then sy
sy
hϕ, ψisk t = hϕ, u2t ψi2t = hϕ, n2t ψi2t s ZZ p f (x) 0 δ (x − y) f (y)ρ2t (y)ψ(y) d x dy = ρ2t (x)ϕ(x) ρ2t (x) R2 ZZ p p = f (x)ρ2t (x)ϕ(x)δ 0 (x − y) f (y)ρ2t (y)ψ(y) d x dy 2 ZRZ p p ∂ =− f (x)ρ2t (x)ϕ(x) δ(x − y) f (y)ρ2t (y)ψ(y) d x dy 2 ∂x Z R p p ∂ =− f (x)ρ2t (x)ϕ(x) f (x)ρ2t (x)ψ(x) d x R ∂x Z p p 1 ∂ =− f (x)ρ2t (x)ϕ(x) f (x)ρ2t (x)ψ(x) d x 2 R ∂x Z p ∂ p 1 + f (x)ρ2t (x)ϕ(x) f (x)ρ2t (x)ψ(x) d x 2 R ∂x Z p p 1 =− f (x)ρ2t (x)ϕ(x), f (x)ρ2t (x)ψ(x) d x 2 R Z P∞ i 1 =− ϕ(x), ψ(x) ρ˜02 (x)e2 1 ti x d x, 2 R using the notation in the statement of this proposition. Setting ρ(x) ˜ = e−V (x) , with V˜ (x) = (1/2)(V (x) − log f ), Z P∞ i 1 hx i , x j isk = − {x i , x j }ρ˜ 2 (x)e2 1 ti x d x 2 R Z P i 1 ˜ {x i , x j }e−2(V (x)− ti x ) d x, =− 2 R ˜
and so τ2n (t) = pf m˜ 2n (t) =
1 (−2)n n!
Z
e2 Tr(−V (x)+ ˜
T2n
P
ti x i )
d x.
The map O −1 for the classical orthogonal polynomials at t = 0 CLAIM
The matrix O, mapping orthonormal pk into skew-orthonormal polynomials qk , is
TODA VERSUS PFAFF LATTICE
49
given by a lower-triangular three-step relation: r √ a2n−2 p2n (0, z) = −c2n−1 q2n−2 (0, z) + a2n c2n q2n (0, z), c2n−2 r a2n−2 p2n+1 (0, z) = −c2n q2n−2 (0, z) c2n−2 r 2n X r c c2n 2n q2n+1 (0, z), − q2n (0, z) + bi a2n a2n 0
(7.1) where the ai and bi are the entries in the tridiagonal matrix defining the orthonormal polynomials, and the ci are the entries in the skew-symmetric matrix. In this case, we need to perform the following skew-Borel decomposition at t = 0: −U = −N = O −1 J O >−1 , where N is the matrix (6.3). Here again, in order to find O, we use the recipe given in Theorem 4.1, namely, writing down the corresponding skew-orthogonal polynomials (0.18), but where the µi j are the entries of −U = −N . Consider the Pfaffians of the bordered matrices (0.18); they have the leading term τ˜˜2n =
n−1 Y
c2 j .
0
Then one computes n i−1 n−i−1 Y X Y 1 r2n = q z 2n−2i c2 j c2n−2 j−1 , 0 0 τ˜˜2n τ˜˜2n+2 i=0 n−1 n i−1 n−i−1 Y Y X Y 1 z 2n+1 c2 j + z 2n−2i c2 j c2n−2 j−1 , r2n+1 = q 0 i=1 0 0 τ˜˜2n τ˜˜2n+2
(7.2) with
q
√ τ˜˜2n τ˜˜2n+2 = c0 c2 · · · c2n−2 c2n ,
q
√ τ˜˜0 τ˜˜2 = c0 .
Setting q q q q ˜ ˜ ˜ ˜ ˜ ˜ D := diag( τ˜0 τ˜2 , τ˜0 τ˜2 , τ˜2 τ˜4 , τ˜˜2 τ˜˜4 , . . .), the matrix O is the set of coefficients of the polynomials above; that is,
50
ADLER and VAN MOERBEKE
1 0 c 1 c 2 −1 c1 c3 O=D c1 c4 c1 c3 c5 c1 c3 c6 =: D
−1
0 0 1 0 0 c0 0 0 0 c0 c3 0 c0 c4 0 c0 c3 c5 0 c0 c3 c6
0 0 0 c0 0 0 0 0
0 0 0 0 c0 c2 0 c0 c2 c5 c0 c2 c6
0 0 0 0 0 c0 c2 0 0
R.
0 0 0 0 0 0 c0 c2 c4 0
0 0 0 0 0 0 0 c0 c2 c4
..
.
(7.3)
As before, in order to get the skew-symmetric polynomials in the right form, from the orthogonal ones, one needs to multiply to the left with the matrix E, defined in (A.2): Oˆ = E O = E D −1 R,
(7.4)
Oˆ −1 = R −1 D E −1 ;
(7.5)
and so it turns out that the matrix Oˆ is complicated, but its inverse is simple. Namely, compute 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 c1 1 − c0 0 0 0 0 0 0 c0 c 1 − 2 0 0 0 0 0 0 c0 c0 c3 0 1 0 − c0 c2 0 0 0 0 c c 0 2 −1 1 R = 0 0 0 0 0 − c0c4c2 0 c0 c2 c5 1 0 0 0 0 − c0 c2 c4 0 0 c0 c2 c4 0 c6 1 0 0 0 − 0 0 c0 c2 c4 c0 c2 c4 .. .
TODA VERSUS PFAFF LATTICE
and
E −1
=
51
α0 −β0
0 1 α0
α2 −β2
0 1 α2
α4 −β4
0 1 α4
0
0 , .. .
(7.6)
with α2n and β2n as in (A.5). Carrying out the multiplication (7.5) leads to the matrix Oˆ −1 , with a few nonzero bands, yielding the map (7.1), by the recipe of Proposition 6.1 inverted.
Appendices A. Free parameter in the skew-Borel decomposition If the Borel decomposition of −H = O −1 J O >−1 is given by a matrix O ∈ Gk , with the diagonal part of O being σ0 0 0 σ0 0 σ2 0 0 σ 2 , (A.1) (O)0 = σ4 0 0 σ4 .. . 0 then the new matrix Oˆ := E O :=
1/α0 β0
0 α0 1/α2 β2
0 α2 1/α4 β4
0
0 α4
0 , .. .
(A.2)
52
ADLER and VAN MOERBEKE
with free parameters α2n , β2n , is a solution of the Borel decomposition −H = Oˆ −1 J Oˆ >−1 , as well. The diagonal part of Oˆ consists of (2 × 2)-blocks σ /α 0 1/α2n 0 σ2n 0 = 2n 2n . β2n α2n 0 σ2n β2n σ2n α2n σ2n Imposing the condition that X
qi (z) =
Oˆ i j p j (z),
with pk (z) =
0≤ j≤i
k X
pki z i ,
i=0
has the required form, that is, the same leading term for q2n and q2n+1 and no z 2n -term in q2n+1 , q2n (z) = q2n,2n z 2n + · · · , q2n+1 (z) = q2n,2n z 2n+1 + q2n,2n−1 z 2n−1 + · · ·
(A.3)
implies σ2n p2n,2n = σ2n α2n p2n+1,2n+1 , α2n σ2n β2n p2n,2n + σ2n α2n p2n+1,2n = 0 yielding, upon using the explicit form of the coefficients pk` of the polynomials pk , associated with three-step relations (see Lemma A.1), 2 α2n =
p2n,2n p2n+1,2n+1
= a2n ,
p2n+1,2n β2n =− = α2n p2n,2n
P2n 0
a2n
bi
.
(A.4)
Hence α2n =
√ a2n
and
2n 1 X β2n = √ bi . a2n 0
So, if r (z) = Oχ(z),
(A.5)
TODA VERSUS PFAFF LATTICE
53
then (A.2) yields ˆ rˆ (z) := Oχ(z) =
1/α0 β0
0 α0 1/α2 β2
0 α2 1/α4 β4
0 α4
0
0 , .. .
r (z) = Er (z), and thus 1 rˆ2n (z) = √ r2n (z), a2n P2n bi √ rˆ2n+1 (z) = √0 r2n + a2n r2n+1 (z). a2n
(A.6) (A.7)
LEMMA A .1 Pn i A sequence of polynomials pn (z) = i=0 pni z of degree n satisfying three-step recursion relation∗
zpn = an−1 pn−1 + bn pn + an pn+1 ,
n = 0, 1, . . . ,
has the form pn+1 (z) =
n pn,n n+1 X n z − bi z + · · · . an 0
Proof Equating the z n+1 and z n coefficients of (A.8) divided by pn,n yields 1 pn+1,n+1 = pn,n an and
pn,n−1 pn+1,n = an + bn . pn,n pn,n Combining both equations leads to an ∗ We
set a−1 = 0.
pn+1,n pn,n−1 − an−1 = −bn , pn,n pn−1,n−1
(A.8)
54
ADLER and VAN MOERBEKE
yielding n
an
X pn+1,n =− bi pn,n
(using a−1 = 0).
0
B. Simultaneous (skew-)symmetrization of L and N CLAIM
For the classical polynomials, the matrices L and N can be simultaneously symmetrized and skew-symmetrized. Sketch of proof This statement was established by us in [6]. Given the monic orthogonal polynomials p˜ n with respect to the weight ρ, with ρ 0 /ρ = −g/ f , we have that the operators z and s d f0−g f d p fρ = f + n= ρ dz dz 2 acting on the polynomials p˜ n have the form 2 z p˜ n = an−1 p˜ n−1 + bn p˜ n + p˜ n+1 ,
n p˜ n = . . . − γn p˜ n+1 ,
(B.1)
in view of the fact that for the classical orthogonal polynomials,∗ d n = dz − z, Hermite: d Laguerre: n = z dz − 12 (z − α − 1), d Jacobi: − 12 ((α + β + 2)z + (α − β)). n = (1 − z 2 ) dz For the orthonormal polynomials, the matrices L and −N are symmetric and skewsymmetric, respectively. Therefore the right-hand side of these expressions must have the form 2 z p˜ n = an−1 p˜ n−1 + bn p˜ n + p˜ n+1 , 2 n p˜ n = an−1 γn−1 p˜ n−1 − γn p˜ n+1 .
Therefore, upon rescaling the p˜ n ’s, to make them orthonormal, we have zpn = (L p)n = an−1 pn−1 + bn pn + an pn+1 , n pn = (N p)n = an−1 γn−1 p˜ n−1 − an γn p˜ n+1 , ∗ They
have the respective weights ρ = e−z , ρ = e−z z α , ρ = (1 − z)α (1 + z)β . 2
TODA VERSUS PFAFF LATTICE
from which it follows that
0 −c0 −N =
55
c0 0 −c1
c1 0 .. .
.. , .
with cn = an γn ,
where −γn is the leading term in expression (B.1).
C. Proof of Lemma 3.4 For future use, consider the first-order differential operators η(t, z) =
∞ −j X z ∂ j ∂t j
∞
and
B(z) = −
j=1
X ∂ ∂ + z − j−1 ∂z ∂t j
(C.1)
j=1
having the property B(z)e−η(z) f (t) = B(z) f t − [z −1 ] = 0.
(C.2)
LEMMA C .1 Consider an arbitrary function ϕ(t, z) depending on t ∈ C∞ , z ∈ C, having the asymptotics ϕ(t, z) = 1 + O(1/z) for z % ∞ and satisfying the functional relation
ϕ(t − [z 2−1 ], z 1 ) ϕ(t − [z 1−1 ], z 2 ) = , ϕ(t, z 1 ) ϕ(t, z 2 )
t ∈ C∞ , z ∈ C.
(C.3)
Then there exists a function τ (t) such that ϕ(t, z) =
τ (t − [z −1 ]) . τ (t)
(C.4)
Proof Applying B1 := B(z 1 ) to the logarithm of (C.3) and using (C.1) and (C.2) yields (e−η(z 2 ) − 1)B1 log ϕ(t; z 1 ) = −B1 log ϕ(t, z 2 ) =−
∞ X j=1
− j−1
z1
∂ log ϕ(t, z 2 ), ∂t j
which, upon setting j
f j (t) = Resz 1 =∞ z 1 B1 log ϕ(t, z 1 ),
56
ADLER and VAN MOERBEKE
yields termwise in z 1 , (e−η(z 2 ) − 1) f j (t) = −
∂ log ϕ(t, z 2 ). ∂t j
(C.5)
Acting with ∂/∂ti on the latter expression and with ∂/∂t j on the same expression with j replaced by i, and subtracting,∗ one finds ∂f ∂fj i (e−η(z 2 ) − 1) − = 0, ∂t j ∂ti yielding
∂fj ∂ fi − = 0; ∂t j ∂ti
the constant vanishes because ∂ f i /∂t j never contains constant terms. Therefore, there exists a function log τ (t1 , t2 , . . .) such that −
∂ log τ = f j (t) = Resz=∞ z j B log ϕ, ∂t j
and hence, using (C.5), ∂ ∂ log ϕ(t, z) = (e−η(z) − 1) log τ, ∂t j ∂t j or, what is the same, ∂ log ϕ − (e−η − 1) log τ = 0, ∂t j from which it follows that log ϕ − (e−η − 1) log τ = −
∞ X bi 1
i
z −i
is, at worst, a holomorphic series in z −1 with constant coefficients, which we call −bi /i. Hence τ (t − [z −1 ])e− ϕ(t, z) = τ (t) = ∗ It
is obvious that [∂/∂ti , e−η(z) ] = 0.
τ (t − [z −1 ])e τ (t)e
P∞ 1
P∞ 1
P∞ 1
(bi /i)z −i
bi (ti −z −i /i)
bi ti
;
TODA VERSUS PFAFF LATTICE
57
that is, ϕ(t, z) =
τ˜ (t − [z −1 ]) , τ˜ (t)
where τ˜ = τ (t)e
P∞ 1
bi ti
.
Thus Lemma 3.4 is proved. References [1]
[2]
[3]
[4] [5] [6]
[7] [8]
[9] [10]
[11] [12] [13]
M. ADLER, P. J. FORRESTER, T. NAGAO, and P. VAN MOERBEKE, Classical skew
orthogonal polynomials and random matrices, J. Statist. Phys. 99 (2000), 141–170. MR 2001k:82046 10 M. ADLER, E. HOROZOV, and P. VAN MOERBEKE, The Pfaff lattice and skew-orthogonal polynomials, Internat. Math. Res. Notices 1999, 569–588. MR 2000f:37112 14, 15, 32, 33 M. ADLER, T. SHIOTA, and P. VAN MOERBEKE, A Lax representation for the vertex operator and the central extension, Comm. Math. Phys. 171 (1995), 547–588. MR 97a:58072 19 , Pfaff tau-functions, Math. Ann. 322 (2002), 423–476. 7, 8, 23 M. ADLER and P. VAN MOERBEKE, Completely integrable systems, Euclidean Lie algebras, and curves, Adv. in Math. 38 (1980), 267–317. MR 83m:58041 10 , Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials, Duke Math. J. 80 (3) (1995), 863–911. MR 96k:58101 2, 3, 8, 9, 30, 44, 54 , Group factorization, moment matrices, and Toda lattices, Internat. Math. Res. Notices 1997, 555–572. MR 98e:58089 , Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum, Ann. of Math. (2) 153 (2001), 149–189. MR 1 826 412 ´ BREZIN ´ E. and H. NEUBERGER, Multicritical points of unoriented random surfaces, Nuclear Phys. B 350 (1991), 513–553. MR 91m:81179 10 E. DATE, M. KASHIWARA, M. JIMBO, and T. MIWA, “Transformation groups for soliton equations” in Nonlinear Integrable Systems—Classical Theory and Quantum Theory (Kyoto, 1981), World Sci., Singapore, 1983, 39–119. MR 86a:58093 L. DICKEY, Soliton Equations and Hamiltonian Systems, Adv. Ser. Math. Phys. 12, World Sci., River Edge, N.J., 1991. MR 93d:58067 V. G. KAC, Infinite-dimensional Lie algebras, 3d ed., Cambridge Univ. Press, Cambridge, 1990. MR 92k:17038 14 V. G. KAC and J. VAN DE LEUR, “The geometry of spinors and the multicomponent BKP and DKP hierarchies” in The Bispectral Problem (Montreal, 1997), CRM
58
[14] [15]
[16] [17]
[18]
ADLER and VAN MOERBEKE
Proc. Lecture Notes 14, Amer. Math. Soc., Providence, 1998, 159–202. MR 99a:17026 7 M. L. MEHTA, Random Matrices, 2d ed., Academic Press, Boston, 1991. MR 92f:82002 34, 35 A. G. REYMAN and M. A. SEMENOV-TIAN-SHANSKY, Reduction of Hamiltonian systems, affine Lie algebras and Lax equations, Invent. Math. 54 (1979), 81–100. MR 81b:58021 10 J. VAN DE LEUR, Matrix integrals and the geometry of spinors, J. Nonlinear Math. Phys. 8 (2001), 288–310. MR 1 839 189 7 P. VAN MOERBEKE, “Integrable lattices: Random matrices and random permutations” in Random Matrix Models and Their Applications, ed. P. Bleher and A. Its, Math. Sci. Res. Inst. Publ. 40, Cambridge Univ. Press, Cambridge, 2001, 321–406. MR 1 842 794 E. WITTEN, “Two-dimensional gravity and intersection theory on moduli space” in Surveys in Differential Geometry (Cambridge, Mass., 1990), Lehigh Univ., Bethlehem, Pa., 1991, 243–310. MR 93e:32028
2 Adler Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454, USA;
[email protected] van Moerbeke Department of Mathematics, Universit´e de Louvain, 1348 Louvain-la-Neuve, Belgium and Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454, USA;
[email protected] and
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1,
THE CRAMER-RAO INEQUALITY FOR STAR BODIES ERWIN LUTWAK, DEANE YANG, and GAOYONG ZHANG
Abstract Associated with each body K in Euclidean n-space Rn is an ellipsoid 02 K called the Legendre ellipsoid of K . It can be defined as the unique ellipsoid centered at the body’s center of mass such that the ellipsoid’s moment of inertia about any axis passing through the center of mass is the same as that of the body. In an earlier paper the authors showed that corresponding to each convex body K ⊂ Rn is a new ellipsoid 0−2 K that is in some sense dual to the Legendre ellipsoid. The Legendre ellipsoid is an object of the dual Brunn-Minkowski theory, while the new ellipsoid 0−2 K is the corresponding object of the Brunn-Minkowski theory. The present paper has two aims. The first is to show that the domain of 0−2 can be extended to star-shaped sets. The second is to prove that the following relationship exists between the two ellipsoids: If K is a star-shaped set, then 0−2 K ⊂ 02 K with equality if and only if K is an ellipsoid centered at the origin. This inclusion is the geometric analogue of one of the basic inequalities of information theory—the Cramer-Rao inequality. Associated with each body K in Euclidean n-space Rn is an ellipsoid 02 K called the Legendre ellipsoid of K . The Legendre ellipsoid is a basic concept from classical mechanics. It can be defined as the unique ellipsoid centered at the body’s center of mass such that the ellipsoid’s moment of inertia about any axis passing through the center of mass is the same as that of the body. In [26] the authors showed that corresponding to each convex body K ⊂ Rn is a new ellipsoid 0−2 K . The results in this paper hint at a remarkable duality between this new ellipsoid and the Legendre ellipsoid. DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1, Received 12 September 2000. Revision received 31 March 2001. 2000 Mathematics Subject Classification. Primary 52A40; Secondary 94A17. Author’s work supported in part by National Science Foundation grant numbers DMS-9803261 and DMS0104363 59
60
LUTWAK, YANG, and ZHANG
The present paper has two aims. The first is to show that the domain of 0−2 can be extended to star-shaped sets. The second is to prove that the following relationship exists between the two ellipsoids: If K is a star-shaped set, then 0−2 K ⊂ 02 K
(1)
with equality if and only if K is an ellipsoid centered at the origin. This inclusion is the geometric analogue of one of the basic inequalities of information theory—the Cramer-Rao inequality. The Brunn-Minkowski theory (often called the theory of mixed volumes) is the heart of analytic convex geometry. Many of the fundamental ingredients of the theory were developed by H. Minkowski a century ago. R. Schneider’s book [28] is the classical reference for the subject. Over the years the tools of the Brunn-Minkowski theory have proven to be remarkably effective in solving inverse problems for which the data involve projections of convex bodies. A quarter of a century ago, the elements of a dual Brunn-Minkowski theory were introduced in [22] (and related papers). The basic idea of the dual theory is to replace the projections of the Brunn-Minkowski theory with intersections. In [23] it was shown that there is in fact a “dictionary” between the theories. Not only do concepts like the “elementary mixed volumes” of the classical theory become the “elementary dual mixed volumes” of the dual theory, but even objects such as the “projection bodies” of the Brunn-Minkowski theory have dual counterparts, “intersection bodies,” in the dual theory. In fact, it was this dual notion of “intersection body” that played a key role in the ultimate solution of the Busemann-Petty problem (see, e.g., R. Gardner [11], [12], [13]; G. Zhang [29], [30]; A. Koldobsky [17], [18], [19], [20]; Gardner, Koldobsky, and T. Schlumprecht [15]). Gardner’s book on geometric tomography [14] is an excellent reference for the interplay between the classical and dual theories. Minkowski showed that what was to become known as the Brunn-Minkowski theory could be developed naturally by combining the notion of volume with an addition of convex bodies now known as Minkowski addition. In the early 1960s, W. Firey [9] introduced and studied an L p -generalization of Minkowski addition. In the 1990s, in [24] and [25], these Minkowski-Firey L p -sums were combined with the notion of volume to form embryonic L p -versions of the Brunn-Minkowski theory. It is easily seen that the classical Legendre ellipsoid belongs to the dual BrunnMinkowski theory. This observation led the authors to the obvious question: What is the dual analog of the Legendre ellipsoid in the Brunn-Minkowski theory? The answer was given by the new ellipsoid introduced in [26]. This new ellipsoid actually belongs to the L 2 -Brunn-Minkowski theory. The nature of the duality between the Brunn-Minkowski theory and the dual Brunn-Minkowski theory is not understood. The objects of study of the Brunn-
THE CRAMER-RAO INEQUALITY FOR STAR BODIES
61
Minkowski theory are convex bodies, while the objects of study of the dual BrunnMinkowski theory are star bodies. The basic functionals of the Brunn-Minkowski theory are often expressed as integrals involving the support and curvature functions. The basic functionals of the dual Brunn-Minkowski theory are integrals involving radial functions. A first step in understanding the nature of the duality between the Brunn-Minkowski theory and its dual is to extend some of the functionals of the Brunn-Minkowski theory so that they are defined for star bodies (rather than convex bodies) and to provide new definitions of these functionals that involve only radial functions (rather than support and curvature functions). In this article, this first step is accomplished for one object of the Brunn-Minkowski theory: the 0−2 -ellipsoid. One of the central problems in information theory is how to extract useful information from noisy signals. Let x0 ∈ Rn be the transmitted signal. A simple model for the received signal is a random vector x ∈ Rn with a probability distribution p(x − x0 ) d x on Rn , where the probability measure p(x) d x has mean zero. Suppose the same signal is transmitted repeatedly and x1 , . . . , x N are the received signals. What is the best estimate for the transmitted signal, and what is the error of this estimate? One possible estimate is the mean x=
x1 + · · · + x N . N
By the central limit theorem, as N becomes large, the distribution of the √ random variable x approaches a Gaussian with mean x0 and covariance matrix C/ N , where the matrix C is given by Z Ci j =
Rn
xi x j p(x) d x.
Another estimate is the maximum likelihood estimate x M , which is obtained by maximizing the log-likelihood function L(x) =
N X
log p(xi − x).
i=1
R. Fisher [10] and J. Doob [7] (see also [1, Appendix I]) showed that as N becomes large, the distribution of the random variable x M approaches a Gaussian with mean √ x0 and covariance matrix F −1 / N , where the matrix F is known as the Fisher information matrix and is given by Z ∂(log p) ∂(log p) p d x. Fi j = ∂xi ∂x j Rn A fundamental result in information theory is the Cramer-Rao inequality (see, e.g., [5]), which states that the covariance and Fisher information matrices satisfy the
62
LUTWAK, YANG, and ZHANG
inequality v · Cv ≥ v · F −1 v
(2)
for all v ∈ Rn . Moreover, equality holds for all v ∈ Rn if and only if the distribution p is Gaussian. The Cramer-Rao inequality is important because it shows that the error in the maximum likelihood estimate is smaller than the mean estimate and that the mean estimate is optimal only if the distribution is Gaussian. It has previously been observed (see [4], [5], [6]) that there exists some connection between the subject of information theory and the Brunn-Minkowski theory. The authors believe that the true connection is between information theory and the L 2 Brunn-Minkowski theory. The authors have found small bits of an embryonic “dictionary” connecting the subject of information theory and the L 2 -Brunn-Minkowski theory. In this dictionary a probability distribution corresponds to a convex body and the entropy power of the distribution to the volume of the body. Associated with the covariance matrix C of a probability distribution is the ellipsoid E C = {v ∈ Rn : v · C −1 v ≤ 1}. In our dictionary this ellipsoid corresponds to the Legendre ellipsoid 02 K of a convex body K . Associated with the Fisher information matrix F is the ellipsoid E F = {v ∈ Rn : v · Fv ≤ 1}, which corresponds to the ellipsoid 0−2 K . The Cramer-Rao inequality (2) is equivalent to the statement E F ⊂ EC
(3)
with equality holding if and only if the probability distribution is Gaussian. Using the dictionary, the main result of this paper, (1), corresponds to (3). The definition of the 0−2 -ellipsoid uses the derivative of the radial function of the star body and therefore appears to require some differentiability assumptions for the star body. We show that, remarkably, no such assumptions are necessary for either the definition or the results of this paper. 1. Notation and overview A bounded set K ⊂ Rn which is star-shaped about the origin is uniquely determined by its radial function, ρ K : Rn \{0} → R, where ρ K (x) = sup{λ ≥ 0 : λx ∈ K }. A star body is a bounded set that is star-shaped about the origin and whose radial function is positive and continuous.
THE CRAMER-RAO INEQUALITY FOR STAR BODIES
63
Throughout this paper the boundary of K is denoted ∂ K , and dy denotes the density associated with the (n − 1)-dimensional Hausdorff measure on ∂ K . The unit sphere in Rn is denoted S n−1 , and the density associated with the (n − 1)-dimensional Hausdorff measure on S n−1 is denoted du. Given a convex body K ⊂ Rn containing the origin in its interior, the ellipsoid 0−2 K was defined in [26] as the body whose radial function is given by Z 1 (x · ν(y))2 −2 ρ0−2 K (x) = dy, (4) V (K ) ∂ K y · ν(y) where · denotes the standard inner product on Rn and where ν(y) ∈ S n−1 is the outer unit normal at y ∈ ∂ K . This formula can be used to define 0−2 K for any body K ⊂ Rn with sufficiently smooth boundary. In §2 we use polar coordinates to obtain the following new formula for 0−2 K . PROPOSITION 3 Given a convex body K ⊂ Rn and x ∈ Rn \{0}, Z 2 1 −2 ρ0−2 K (x) = x · ∇ρ K (u) ρ K (u)n−4 du. V (K ) S n−1
In §§3–5 we present preliminary definitions and lemmas that are needed for the remainder of the paper. We show in §6 how to define for a star body K ⊂ Rn the set 0−2 K ⊂ Rn . In §10 it is also shown that this set is an ellipsoid that is possibly degenerate. If the boundary of K is Lipschitz, then 0−2 K is a nondegenerate ellipsoid. On the other hand, if the boundary of K is sufficiently singular, then 0−2 K is just a single point, namely, the origin. The following extension of [26, Lemma 1*] is proved in §7. LEMMA 13 If K ⊂ Rn is a star body and φ ∈ GL(n),
0−2 φ K = φ0−2 K . In §8 we derive a formula for the volume of K , and in §9 we use the following formula to establish our main result. THEOREM
If K ⊂ Rn is a star body, then 0−2 K ⊂ 02 K
64
LUTWAK, YANG, and ZHANG
with equality holding if and only if K is an ellipsoid centered at the origin. Recall that a set E ⊂ Rn is an ellipsoid centered at the origin if there exists an (n × n)–positive definite symmetric matrix A such that E = {x : x · Ax ≤ 1}. A set E ⊂ Rn is a degenerate ellipsoid centered at the origin if there exists a proper subspace L ⊂ Rn such that E ⊂ L and E is an ellipsoid in L. 2. A new formula for the 0−2 -ellipsoid Given a convex body K ⊂ Rn containing the origin in its interior, there is a natural parameterization of the boundary ∂ K given by the map φ K : S n−1 → ∂ K , where S n−1 ⊂ Rn is the unit sphere and φ K (u) = ρ K (u)u. Moreover, the radial function ρ K is Lipschitz, and for almost every y ∈ ∂ K , there exists a unique outer unit normal ν(y) to ∂ K at y (see, e.g., [28, p. 53] for a proof of these facts). When the meaning is clear, the subscript K in ρ K and φ K may be suppressed. To change the variable of integration in (4) from y ∈ ∂ K to u ∈ S n−1 , the following two lemmas are needed. LEMMA 1 If K ⊂ Rn is a convex body containing the origin in its interior, then for almost every u ∈ S n−1 , ν(φ(u)) ∇ρ(u) =− , φ(u) · ν(φ(u)) ρ(u)2 where ∇ρ denotes the gradient of ρ in Rn .
Proof Since ρ is constant along the boundary ∂ K , its gradient is normal to the boundary. In other words, given y ∈ ∂ K , there exists λ(y) ∈ R such that ∇ρ(y) = λ(y)ν(y). On the other hand, since ρ is homogeneous of degree −1, 1 = ρ(y) = −y · ∇ρ(y) = −λ(y)y · ν(y).
THE CRAMER-RAO INEQUALITY FOR STAR BODIES
65
Therefore, λ(y) = −
1 y · ν(y)
and ∇ρ(y) = −
ν(y) . y · ν(y)
Substituting in y = φ(u) and observing that ∇ρ is homogeneous of degree −2 yields the desired formula. The formula for the surface area measure of ∂ K in polar coordinates is given in the following lemma. 2 If K ⊂ Rn is a convex body containing the origin in its interior, then LEMMA
dy =
ρ K (u)n du. φ(u) · ν(φ(u))
The parameterization φ : S n−1 → ∂ K is Lipschitz and therefore differentiable almost everywhere. By [8, Theorem 3.2.3], the change of measure is given by the determinant of the Jacobian. A straightforward computation yields the formula above. PROPOSITION 3 Given a convex body K ⊂ Rn and x ∈ Rn \{0}, Z 2 1 ρ0−2 K (x)−2 = x · ∇ρ K (u) ρ K (u)n−4 du, V (K ) S n−1
(5)
where ∇ρ K is the gradient of ρ K in Rn . Proof By Lemmas 1 and 2, Z 1 (x · ν(y))2 dy V (K ) ∂ K y · ν(y) Z 1 (x · ν(φ(u)))2 = ρ K (u)n du V (K ) S n−1 (φ(u) · ν(φ(u)))2 Z 1 = (x · ∇ρ K (u))2 ρ K (u)n−4 du. V (K ) S n−1
ρ0−2 K (x)−2 =
The operator 0−2 can be extended immediately to the class of star bodies with Lipschitz radial function using either formula (4) or formula (5). To extend it to the entire
66
LUTWAK, YANG, and ZHANG
class of star bodies requires a little more work. The next three sections contain preliminary definitions and results that are needed to this end. 3. A projection of the sphere onto the cylinder The projection of a vector x ∈ Rn into each tangent space of S n−1 defines a vector field x(u) ˆ = x − (x · u)u. (6) We want to use the integral curves of xˆ to define coordinates on S n−1 . This can be done as follows. Given x ∈ Rn \{0}, let x ⊥ = {v ∈ Rn : v · x = 0}, and let hxi =
x . |x|
There is a natural projection of S n−1 onto the cylinder Rx + (S n−1 ∩ x ⊥ ) obtained by mapping the unit vector u to the intersection of the cylinder with the ray containing u. The inverse of this projection is given by Mx : R × (S n−1 ∩ x ⊥ ) → S n−1 \ h−xi, hxi , φ + hxi sinh |x|t (t, φ) 7 → u = . cosh |x|t A straightforward calculation shows that x · u = |x| tanh |x|t
(7)
and ∂u x − φ|x| sinh |x|t = ∂t (cosh |x|t)2 = x − (x · u)u. This implies that for each f ∈ C 1 (S n−1 ), ∂( f ◦ Mx ) ∇x(M (t, φ), ˆ x (t,φ)) f M x (t, φ) = ∂t
(8)
where ∇v f denotes the directional derivative of f in direction v. Also, du =
|x| dt dφ , (cosh |x|t)n−1
(9)
where dφ is the density corresponding to the (n − 2)-dimensional Hausdorff measure on S n−1 ∩ x ⊥ .
THE CRAMER-RAO INEQUALITY FOR STAR BODIES
67
4. Differentiability along a vector field In this section we make precise the notions of continuous differentiability and L 2 differentiability along the vector field xˆ of a function on the unit sphere, without assuming any regularity in other directions. Given x ∈ Rn , let C 0 (S n−1 ∩ x ⊥ , C 1 (R)) denote the space of continuous functions f : R × (S n−1 ∩ x ⊥ ) → R such that f (·, φ) ∈ C 1 (R) for each φ ∈ S n−1 ∩ x ⊥ . Define C x1 (S n−1 ) = f ∈ C 0 (S n−1 ) : f ◦ Mx ∈ C 0 S n−1 ∩ x ⊥ , C 1 (R) . The restriction of any f ∈ C x1 (S n−1 ) to an integral curve of the vector field xˆ is C 1 . Therefore, ∇xˆ f is well defined and given explicitly by ∂ f Mx (t, φ) . ∇xˆ f Mx (t, φ) = ∂t
(10)
The following chain rule holds. LEMMA 4 Given a C 1 -function φ and f ∈ C x1 (S n−1 ), the function φ ◦ f ∈ C x1 (S n−1 ). Moreover,
∇xˆ (φ ◦ f ) = (φ 0 ◦ f )∇xˆ f. To define the notion of L 2 -differentiability, we need the following integration by parts formulas. LEMMA 5 Given x ∈ Rn and f, g ∈ C x1 (S n−1 ), Z Z f ∇xˆ g du = g −∇xˆ f + (n − 1)(x · u) f du S n−1
(11)
S n−1
and Z
Z S n−1
f ∇xˆ g du = −
Proof By (9) and (10), Z S n−1
R×(S n−1 ∩x ⊥ )
(g ◦ Mx )
Z f ∇xˆ g du =
R×(S n−1 ∩x ⊥ )
∂ h |x|( f ◦ Mx ) i dt dφ. ∂t (cosh |x|t)n−1
( f ◦ Mx )
∂(g ◦ Mx ) |x| dt dφ . ∂t (cosh |x|t)n−1
(12)
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Equation (12) now follows by integrating by parts. By (9), (8), and (7), Z Z ∂ h |x|( f ◦ Mx ) i f ∇xˆ g du = − (g ◦ Mx ) dt dφ ∂t (cosh |x|t)n−1 S n−1 R×(S n−1 ∩x ⊥ ) Z i h ∂( f ◦ M ) x − (n − 1)( f ◦ Mx )|x| tanh |x|t =− (g ◦ Mx ) ∂t R×(S n−1 ∩x ⊥ ) |x| dt dφ × (cosh |x|t)n−1 Z = g −∇xˆ f + (n − 1)(x · u) f du. S n−1
Equation (11) can be used to define for any g ∈ C 0 (S n−1 ) the directional derivative ∇xˆ g as a distribution. This motivates the following definition. Definition Given x ∈ Rn and g ∈ C 0 (S n−1 ), we say that ∇xˆ g ∈ L 2 (S n−1 ) if there exists a function g 0 ∈ L 2 (S n−1 ) such that for any f ∈ C x1 (S n−1 ), Z Z g −∇xˆ f + (n − 1)(x · u) f du = g 0 f du. (13) S n−1
S n−1
If such a function g 0 ∈ L 2 (S n−1 ) exists, it will be denoted ∇xˆ g. Since L 2 (S n−1 ) is its own dual, we have the following lemma. LEMMA
6
If x ∈ and g ∈ C 0 (S n−1 ), then ∇xˆ g ∈ L 2 (S n−1 ) if and only if there exists c > 0 such that Z g −∇xˆ f + (n − 1)(x · u) f du ≤ ck f k2 (14) Rn
S n−1
for each f ∈ C x1 (S n−1 ). Lemma 5 now gives the following lemma. LEMMA
7
If g ∈ C x1 (S n−1 ), the pointwise definition (10) of ∇xˆ g agrees with the distributional definition (13). 5. Smoothing a function along a vector field Fix x ∈ Rn \{0}. We want to define a smoothing operator that smooths only along the integral curves of the vector field x. ˆ
THE CRAMER-RAO INEQUALITY FOR STAR BODIES
69
Given τ > 0, let χτ be a smooth nonnegative function on R which is supported in the interval (−τ, τ ) and satisfies Z ∞ χτ (t) dt = 1. −∞
Using the map Mx defined in §3, we define the function gτ ∈ C 0 (S n−1 ) by Z ∞ gτ Mx (t, φ) = g Mx (t − s, φ) χτ (s) ds
(15)
for each g ∈ C 0 (S n−1 ) and τ > 0. Observe that Z ∞ g Mx (s, φ) χτ (t − s) ds gτ Mx (t, φ) =
(16)
−∞
−∞
and that gτ extends continuously across hxi and h−xi ∈ S n−1 . Before stating and proving the main proposition about gτ , we begin with two lemmas. The first is an analogue of the standard (and trivial) fact that convolution on Rn commutes with partial differentiation. 8 Given g ∈ C 0 (S n−1 ) and x ∈ Rn , if ∇xˆ g ∈ L 2 (S n−1 ), then LEMMA
∇xˆ gτ = (∇xˆ g)τ . Proof It suffices to prove that
∂(gτ ◦ Mx ) = (∇xˆ g)τ ◦ Mx . ∂t By (16) and (10), given any f ∈ C 0 (S n−1 ∩ x ⊥ ), Z ∂(gτ ◦ Mx ) f (φ) (t, φ) dφ n−1 ⊥ ∂t S ∩x Z = f (φ)g Mx (s, φ) χτ0 (t − s) ds dφ R×(S n−1 ∩x ⊥ ) Z ∂ χτ (t − s) f (φ) ds dφ =− g Mx (s, φ) ∂s R×(S n−1 ∩x ⊥ ) Z = f (φ)χτ (t − s)(∇x(u) ˆ g) M x (s, φ) ds dφ n−1 ⊥ R×(S ∩x ) Z = f (φ)(∇x(u) ˆ g)τ M x (t, φ) dφ. S n−1 ∩x ⊥
This holds for each f ∈ C 0 (S n−1 ∩ x ⊥ ) and thus yields (17).
(17)
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The next lemma is a weak form of the Young inequality. 9 There exists T > 0 such that, for each h ∈ L 2 (S n−1 ) and 0 < τ < T , LEMMA
kh τ k2 < 2khk2 . Proof Observe that
cosh |x|(t − s) ≤ e|sx| cosh |x|t
for every t ∈ R. Therefore, if 0<τ <
log 4 , (n − 1)|x|
it follows by the Minkowski inequality (see, e.g., [21, Theorem 2.4]) that Z Z ∞ 2 |x| dt dφ 1/2 kh τ k2 = h Mx (t − s, φ) χτ (s) ds (cosh |x|t)n−1 R×(S n−1 ∩x ⊥ ) −∞ Z ∞ Z 2 |x| dt dφ 1/2 ≤ χτ (s) h Mx (t − s, φ) ds (cosh |x|t)n−1 −∞ R×(S n−1 ∩x ⊥ ) Z Z ∞ 2 cosh |x|(t − s) n−1 = χτ (s) h Mx (t − s, φ) cosh |x|t −∞ R×(S n−1 ∩x ⊥ ) 1/2 |x| dt dφ × ds (cosh |x|(t − s))n−1 ≤ 2khk2 . The main result of this section is the following proposition. PROPOSITION 10 Given x ∈ Rn \{0} and g ∈ C 0 (S n−1 ) such that ∇xˆ g ∈ L 2 (S n−1 ), there exists a 1-parameter family of functions gτ ∈ C x1 (S n−1 ), for τ > 0, such that the following hold: lim sup gτ (u) − g(u) = 0 (18) τ →0 u∈S n−1
and lim k∇xˆ gτ − ∇xˆ gk2 = 0.
τ →0
(19)
Proof Since g is uniformly continuous on S n−1 , there exist for each > 0 quantities δ > 0
THE CRAMER-RAO INEQUALITY FOR STAR BODIES
71
and T > 0 such that g Mx (t1 , φ) − g Mx (t2 , φ) < , g Mx (t, φ) − g hxi < , 2 g Mx (−t, φ) − g h−xi < , 2 for every φ ∈ S n−1 ∩ x ⊥ , t ∈ (T −δ, ∞), and t1 , t2 ∈ [−T, T ] satisfying |t1 −t2 | < δ. It follows that, given any τ ∈ (0, δ/2), Z ∞ gτ Mx (t, φ) − g Mx (t, φ) ≤ g Mx (s, φ) − g Mx (t, φ) χτ (t − s) ds −∞
< for every (t, φ) ∈ R × (S n−1 ∩ x ⊥ ). This proves (18). Given > 0, there exists a function f ∈ C 0 (S n−1 ) such that k f − ∇xˆ gk2 < . By (18), f τ converges uniformly to f . Therefore, there exists T > 0 such that k f τ − f k2 < for any τ ∈ (0, T ). By Lemmas 8 and 9, for any τ ∈ (0, T ), k∇xˆ gτ − ∇xˆ gk2 ≤ k∇xˆ gτ − f τ k2 + k f τ − f k2 + k f − ∇xˆ gk2
= (∇xˆ g − f )τ 2 + k f τ − f k2 + k f − ∇xˆ gk2 ≤ 2k∇xˆ g − f k2 + k f τ − f k2 + k f − ∇xˆ gk2 ≤ 4. This proves (19). We can now prove the following analogue of Lemma 4. 11 Suppose that x ∈ Rn , φ is a C 1 function, and g ∈ C 0 (S n−1 ). If ∇xˆ g ∈ L 2 (S n−1 ), then ∇xˆ (φ ◦g) ∈ L 2 (S n−1 ) and LEMMA
∇xˆ (φ ◦g) = (φ 0 ◦g)∇xˆ g. Proof By (18), as τ → 0, φ 0 ◦gτ converges uniformly to φ 0 ◦g, and φ ◦gτ to φ ◦g. Therefore,
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(φ 0 ◦gτ )∇xˆ gτ converges in L 2 to (φ 0 ◦g)∇xˆ g, and, given any f ∈ C x1 (S n−1 ), Z (φ ◦g) −∇xˆ f + (n − 1)(x · u) f du S n−1 Z (φ ◦gτ ) −∇xˆ f + (n − 1)(x · u) f du = lim τ →0 S n−1 Z 0 = lim (φ ◦gτ )∇xˆ gτ f du τ →0 S n−1 Z 0 = (φ ◦g)∇xˆ g f du. S n−1
The lemma now follows by (13). 6. The 0−2 -ellipsoid for star bodies Given a star body K ⊂ Rn , define g K : Rn \{0} → R to be ρ (x)(n/2)−1 K if n > 2, (n/2) − 1 g K (x) = log ρ (x) if n = 2.
(20)
K
If ρ K : Rn \{0} → R is C 1 , then formula (5) can be rewritten as Z 2 1 ρ0−2 K (x)−2 = ∇x g K (u) du, V (K ) S n−1
(21)
where ∇x g K denotes the directional derivative of g K (viewed as a function on Rn \{0}) in direction u. In general, g K is known only to be C 0 , and its directional derivative does not necessarily exist. Nevertheless, it is still possible to define 0−2 K . Obviously, given p ∈ R, any function g ∈ C 1 (S n−1 ) has a unique extension to Rn \{0} which is homogeneous of degree p. Euler’s equation says that, for each y ∈ Rn \{0}, ∇ y g(y) = pg(y). Therefore, given x ∈ Rn , ∇x g(u) = ∇x(u) ˆ g(u) + p(x · u)g(u)
(22)
for any u ∈ S n−1 . If we assume only that g ∈ C 0 (S n−1 ) and ∇xˆ g ∈ L 2 (S n−1 ), then the function ∇x g ∈ L 2 (S n−1 ) can be defined using (22). In particular, if g K ∈ C 1 (S n−1 ), then the homogeneity of ρ K and Euler’s equation imply that, for any u ∈ S n−1 , ∇x g K (u) = ∇xˆ g K (u) − (x · u)ρ K (u)(n/2)−1 .
(23)
THE CRAMER-RAO INEQUALITY FOR STAR BODIES
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If we assume only that ∇xˆ g K ∈ L 2 (S n−1 ), then equation (23) can be used to define ∇x g K ∈ L 2 (S n−1 ). Lemma 11 ensures that (23) is consistent with the definition of ∇x ρ K , as given by (22). We can now define the set 0−2 K . Definition Given a star body K ⊂ Rn , define 0−2 K = x ∈ Rn : ∇x g K ∈ L 2 (S n−1 ) and k∇x g K k22 ≤ V (K ) .
(24)
If the body K is convex, then this definition agrees with formula (5) and therefore with the original definition given in [26]. The following integration by parts formulas are needed later. LEMMA 12 Suppose that K ⊂ Rn is a star body, and suppose that x ∈ Rn such that ∇xˆ g K ∈ L 2 (S n−1 ). Then for every f ∈ C x1 (S n−1 ), Z g K ∇x f du if n > 2, − Z S n−1 f ∇x g K du = (25) Z S n−1 − (∇x f ) g K − log | f | du if n = 2, S n−1
where f has been extended to be homogeneous of degree −(n/2) on Rn \{0}. Proof First, assume that g K ∈ C x1 (S n−1 ). The case n > 2 follows directly from equation (22) and Lemmas 4 and 5. The case n = 2 requires the additional observations that f ∇x (log | f |) ∈ L ∞ (S n−1 ) and that f (u)∇x log | f (u)| = ∇x f (u) for almost every u ∈ S n−1 . If ∇xˆ g K ∈ L 2 (S n−1 ), then equation (25) holds for (g K )τ , as defined by Proposition 10. The lemma now follows by taking the limit τ → 0.
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7. Invariance under GL(n) 13 If K ⊂ Rn is a star body and φ ∈ GL(n), LEMMA
0−2 φ K = φ0−2 K . Proof Given a star body K ⊂ Rn , let g K be as defined by (20). Given a star body L ⊂ Rn , let n/2 f L = ρL . Recall that hxi = x/|x| for any x ∈ Rn . Let φ ∈ SL(n), and let v = hφ −1 ui. Using the fact that ρφ K (u) = ρ K (φ −1 u) and (25), we have for n > 2, Z Z f L ∇x gφ K du = − gφ K (u)∇x f L (u) du n−1 S n−1 ZS =− g K (φ −1 u)∇x f L (u) du n−1 ZS =− |φ −1 u|1−(n/2) g K hφ −1 ui ∇x f L hφvi du n−1 ZS =− g K (v)∇x f L (φv)|φ −1 u|−n du S n−1 Z =− g K (v)∇φ −1 x f φ −1 L (v) dv. S n−1
If n = 2, then Z S n−1
ρφ K ρ L ∇x log du ρL S n−1 Z ρφ K (u) = −∇x ρ L (u) log du ρ L (u) S n−1 Z ρ K (φ −1 u) = −∇x ρ L hφvi log du ρφ −1 L (φ −1 u) S n−1 Z ρ K (v) = −∇x ρ L (φv) log |φ −1 u|−2 du ρφ −1 L (v) S n−1 Z
f L ∇x gφ K du =
THE CRAMER-RAO INEQUALITY FOR STAR BODIES
75
ρ K (v) dv ρφ −1 L (v) Z ρ K (v) dv = ρφ −1 L (v)∇φ −1 x log ρφ −1 L (v) S n−1 Z = f φ −1 L ∇φ −1 x g K dv. Z
=
S n−1
−∇φ −1 x ρφ −1 L (v) log
S n−1
Thus, for each dimension n ≥ 2 and star body L with a C 1 –radial function, Z Z ∇x gφ K f L du = ∇φ −1 x g K f φ −1 L du. S n−1
S n−1
It follows by linearity that, for any f ∈ C 1 (S n−1 ), Z Z ∇x gφ K f du = ∇φ −1 x g K f ◦φ du. S n−1
S n−1
Therefore, ∇x gφ K ∈ L 2 (S n−1 ) if and only if ∇φ −1 x g K ∈ L 2 (S n−1 ). Given any f ∈ C 1 (S n−1 ), extend it to Rn \{0} as a function homogeneous of degree −(n/2). There exists a star-shaped set L such that n/2
ρL
= | f |.
Observe that, for any φ ∈ SL(n), n/2
ρφ −1 L = | f ◦φ|, and therefore k f ◦φk22 = nV (φ −1 L) = nV (L) = k f k22 . It follows that nZ o k∇x gφ K k2 = sup ∇x gφ K f du : k f k2 = 1 S n−1 nZ o = sup ∇φ −1 x g K f ◦φ du : k f ◦φk2 = 1 S n−1
= k∇φ −1 x g K k2
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LUTWAK, YANG, and ZHANG
and 0−2 φ K = x ∈ Rn : ∇x gφ K ∈ L 2 (S n−1 ) and k∇x gφ K k22 ≤ V (φ K ) = x ∈ Rn : ∇φ −1 x g K ∈ L 2 (S n−1 ) and k∇φ −1 x g K k22 ≤ V (K ) = φw ∈ Rn : ∇w g K ∈ L 2 (S n−1 ) and k∇w g K k22 ≤ V (K ) = φ0−2 K . For the case of dilation, let λ > 0. It is easily seen that ∇x gλK = ∇λ(n/2)−1 x g K = λn/2 ∇λ−1 x g K . Therefore, 0−2 λK = x ∈ Rn : ∇x gλK ∈ L 2 (S n−1 ) and k∇x gλK k22 ≤ V (λK ) = x ∈ Rn : ∇λ−1 x g K ∈ L 2 (S n−1 ) and k∇λ−1 x g K k22 ≤ V (K ) = λw ∈ Rn : ∇w g K ∈ L 2 (S n−1 ) and k∇w g K k22 ≤ V (K ) = λ0−2 K . 8. A formula for volume The following formula for the volume of a star body K is needed to prove that the Legendre ellipsoid contains the 0−2 -ellipsoid. LEMMA 14 Given x ∈ S n−1 and a star body K ⊂ Rn , if ∇xˆ g K ∈ L 2 (S n−1 ), then Z V (K ) = (x · u)ρ K (u)(n/2)+1 −∇x g K (u) du. S n−1
Proof For convenience we omit the subscript K , denoting ρ K by ρ and g K by g. By (23), Lemma 11, and (22), the following holds in L 2 (S n−1 ): ρ (n/2)+1 ∇x g = ρ (n/2)+1 ∇xˆ g − (x · u)ρ (n/2)−1 = ρ n−1 ∇xˆ ρ − (x · u)ρ n 1 = ∇xˆ (ρ n ) − (x · u)ρ n . n
THE CRAMER-RAO INEQUALITY FOR STAR BODIES
77
Since ∇xˆ (ρ n ) ∈ L 2 (S n−1 ), it follows by (13) that Z Z i h 1 (x · u)ρ (n/2)+1 (−∇x g) du = (x · u) − ∇xˆ (ρ n ) + (x · u)ρ n du n−1 n S n−1 ZS 1 ∇xˆ (x · u) + (x · u)2 ρ n du = S n−1 n Z 1 = ρ n du n S n−1 = V (K ).
9. Inclusion and equality THEOREM
If K ⊂ Rn is a star body, then 0−2 K ⊂ 02 K with equality holding if and only if K is an ellipsoid centered at the origin. Proof By applying a linear transformation to K , we can assume that its Legendre ellipsoid 02 K is the unit ball. This means that Z 1 (x · u)2 ρ K (u)n+2 du = |x|2 (26) V (K ) S n−1 for every x ∈ Rn . Let x ∈ 0−2 K . First, it follows from the definition of 0−2 K that ∇xˆ g K ∈ L 2 (S n−1 ). By Lemma 14, the H¨older inequality, (26), and (24), Z 1 2 |x| = (x · u)ρ(u)(n/2)+1 −∇x g K (u) du V (K ) S n−1 1 Z 1/2 1 Z 2 1/2 ≤ (x · u)2 ρ K (u)n+2 du ∇x g K (u) du V (K ) S n−1 V (K ) S n−1 ≤ |x|. (27) Thus, |x| ≤ 1, and therefore, 0−2 K is contained in the unit ball. This proves that 0−2 K ⊂ 02 K .
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Suppose that 0−2 K = 02 K . Since we are assuming that 02 K is the unit ball, so is 0−2 K . This implies that, for each x ∈ S n−1 , we have ∇x g K ∈ L 2 (S n−1 ), and equality holds in all of the inequalities in (27). From the equality conditions of the H¨older inequality, there exists for each x ∈ S n−1 a constant c(x) such that (x · u)ρ K (u)(n/2)+1 = c(x)∇x g K (u)
(28)
for almost every u ∈ S n−1 . Integrating the square of both sides with respect to u ∈ S n−1 and using (24) and (26) shows that c(x) = ±1. Since (28) holds for all x ∈ S n−1 , it follows by (13) that, for each f ∈ C 1 (Rn \{0}) homogeneous of degree −(n/2) − 1, if n > 2, Z S n−1
(n/2)+1 ρK
f du =
n Z X i=1
=±
=∓
=∓
S n−1
(u · ei )(ei · u)ρ K (u)(n/2)+1 f (u) du
n Z X i=1 S n Z X
n−1
i=1 S n Z X
n−1
i=1
Z =∓ S n−1
Z
(u · ei ) f (u)∇ei g K (u) du ∇ei (u · ei ) f g K du
S n−1
(u · ei )∇ei f + f g K du
(∇u f + n f )g K du fρ (n/2)−1 du.
=∓ S n−1
If n = 2, then for each f ∈ C 1 (Rn \{0}) homogeneous of degree −(n/2) − 1, Z (n/2)+1 ρK f du S n−1
=
n Z X i=1
=±
=±
S n−1
n Z X i=1 S n Z X i=1
=∓
(u · ei )(ei · u)ρ K (u)(n/2)+1 f (u) du
n−1
h i 1 (u · ei ) f ∇ei g K − log | f |1/2 + (u · ei )∇ei f du 2 S n−1
n Z X i=1
(u · ei ) f (u)∇ei g K (u) du
h S n−1
i 1 g − log | f |1/2 ∇ei (u · ei ) f + f ∇ei (u · ei ) du 2
THE CRAMER-RAO INEQUALITY FOR STAR BODIES
79
Z f du.
=∓ S n−1
Therefore,
ρ (n/2)+1 = ±ρ (n/2)−1 .
Since ρ K is positive and continuous, it must be the constant function 1. In other words, K is the unit ball centered at the origin.
10. The set 0−2 K is an ellipsoid COROLLARY
If K ⊂ Rn is a star body, then the set 0−2 K ⊂ Rn is an ellipsoid that is possibly degenerate. Proof Suppose that K is a star body and that g K is as defined by (20). Using (13), it is easily seen that, for all x, v ∈ Rn and λ ∈ R, if ∇x g K , ∇v g K ∈ L 2 (S n−1 ), then ∇λx g K , ∇x+v g K ∈ L 2 (S n−1 ). It follows that the set L = x ∈ Rn : ∇x g K ∈ L 2 (S n−1 ) is a linear subspace of Rn . Moreover, 0−2 K ⊂ L. Let dim L = m ≤ n. Choose a basis e1 , . . . , en of Rn such that e1 , . . . , em is a basis of L. Given x ∈ L, write m X x= xi ei . i=1
Then x ∈ 0−2 K if and only if x · Ax ≤ V (K ), where A is the nonnegative symmetric matrix whose (i, j)th entry is Z Ai j = (∇ei g K )(∇e j g K ) du, S n−1
where 1 ≤ i, j ≤ m. Observe that since ei , e j ∈ L, it follows from the H¨older inequality that the integral is bounded. Now suppose that A has a zero eigenvalue. Then the set 0−2 K is unbounded. However, the Legendre ellipsoid 02 K is bounded and, by the theorem in §9, 0−2 K ⊂ 02 K . Therefore, 0−2 K is bounded. The contradiction shows that A is positive definite and that 0−2 K is a nondegenerate ellipsoid in L.
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derivative information, I: Fisher information and convex duality, Math. Oper. Res. 21 (1996), 442–468. MR 97e:90114 61 H. J. BRASCAMP and E. H. LIEB, Best constants in Young’s inequality, its converse, and its generalization to more than three functions, Adv. Math. 20 (1976), 151–173. MR 54:492 , On extensions of the Brunn-Minkowski and Pr´ekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal. 22 (1976), 366–389. MR 56:8774 M. H. M. COSTA and T. M. COVER, On the similarity of the entropy power inequality and the Brunn-Minkowski inequality, IEEE Trans. Inform. Theory 30 (1984), 837–839. MR 86d:26029 62 T. M. COVER and J. A. THOMAS, Elements of Information Theory, Wiley Ser. Telecom., Wiley-Interscience, New York, 1991. MR 92g:94001 61, 62 A. DEMBO, T. M. COVER, and J. A. THOMAS, Information-theoretic inequalities, IEEE Trans. Inform. Theory 37 (1991), 1501–1518. MR 92h:94005 62 J. L. DOOB, Probability and statistics, Trans. Amer. Math. Soc. 36 (1934), 759–775. MR 1 501 765 61 H. FEDERER, Geometric Measure Theory, Grundlehren Math. Wiss. 153, Springer, New York, 1969. MR 41:1976 65 W. J. FIREY, p-means of convex bodies, Math. Scand. 10 (1962), 17–24. MR 25:4416 60 R. A. FISHER, Theory of statistical estimation, Philos. Trans. Roy. Soc. London Ser. A 222 (1930), 309–368. 61 R. J. GARDNER, On the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies, Bull. Amer. Math. Soc. (N.S.) 30 (1994), 222–226. MR 94m:52007 60 , A positive answer to the Busemann-Petty problem in three dimensions, Ann. of Math. (2) 140 (1994), 435–447. MR 95i:52005 60 , Geometric tomography, Notices Amer. Math. Soc. 42 (1995), 422–429. MR 97b:52003 60 , Geometric Tomography, Encyclopedia Math. Appl. 58, Cambridge Univ. Press, Cambridge, 1995. MR 96j:52006 60 R. J. GARDNER, A. KOLDOBSKY, and T. SCHLUMPRECHT, An analytic solution to the Busemann-Petty problem on sections of convex bodies, Ann. of Math. (2) 149 (1999), 691–703. MR 2001b:52011 60 O. G. GULERYUZ, E. LUTWAK, D. YANG, AND G. ZHANG, Information theoretic inequalities for contoured distributions, preprint, 2001, http://eeweb.poly.edu/∼ onur/online pub.html A. KOLDOBSKY, Intersection bodies and the Busemann-Petty problem, C. R. Acad. Sci. Paris S´er. I Math. 325 (1997), 1181–1186. MR 98k:52007 60 , Intersection bodies in R4 , Adv. Math. 136 (1998), 1–14. MR 99j:52013 60
THE CRAMER-RAO INEQUALITY FOR STAR BODIES
[19] [20] [21] [22] [23] [24] [25] [26] [27]
[28] [29] [30]
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, Intersection bodies, positive definite distributions, and the Busemann-Petty problem, Amer. J. Math. 120 (1998), 827–840. MR 99i:52005 60 , Second derivative test for intersection bodies, Adv. Math. 136 (1998), 15–25. MR 99j:52014 60 E. H. LIEB and M. LOSS, Analysis, Grad. Stud. Math. 14, Amer. Math. Soc., Providence, 1997. MR 98b:00004 70 E. LUTWAK, Dual mixed volumes, Pacific J. Math. 58 (1975), 531–538. MR 52:1528 60 , Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), 232–261. MR 90a:52023 60 , The Brunn-Minkowski-Firey theory, I: Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), 131–150. MR 94g:52008 60 , The Brunn-Minkowski-Firey theory, II: Affine and geominimal surface area, Adv. Math. 118 (1996), 244–294. MR 97f:52014 60 E. LUTWAK, D. YANG, and G. ZHANG, A new ellipsoid associated with convex bodies, Duke Math. J. 104 (2000), 375–390. MR 2001j:52011 59, 60, 63, 73 V. D. MILMAN and A. PAJOR, “Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space” in Geometric Aspects of Functional Analysis: Israel Seminar (1987– 88), Lecture Notes in Math 1376, Springer, Berlin, 1989, 64–104. MR 90g:52003 R. SCHNEIDER, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia Math. Appl. 44, Cambridge Univ. Press, Cambridge, 1993. MR 94d:52007 60, 64 G. ZHANG, Centered bodies and dual mixed volumes, Trans. Amer. Math. Soc. 345 (1994), 777–801. MR 95d:52008 60 , A positive solution to the Busemann-Petty problem in R4 , Ann. of Math. (2) 149 (1999), 535–543. MR 2001b:52010 60
Lutwak Department of Mathematics, Polytechnic University, Six Metrotech Center, Brooklyn, New York 11201, USA;
[email protected] Yang Department of Mathematics, Polytechnic University, Six Metrotech Center, Brooklyn, New York 11201, USA;
[email protected] Zhang Department of Mathematics, Polytechnic University, Six Metrotech Center, Brooklyn, New York 11201, USA;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1,
THE HIGHEST LOWEST ZERO AND OTHER APPLICATIONS OF POSITIVITY STEPHEN D. MILLER
In memory of Teresa Bowen Abstract The first nontrivial zeros of the Riemann ζ -function are ≈ 1/2 ± 14.13472i. We investigate the question of whether or not any other L-function has a higher lowest zero. To do so, we try to quantify the notion that the L-function of a “small” automorphic representation (i.e., one with small level and archimedean type) does not have small zeros, and vice-versa. We prove that many types of automorphic L-functions have a lower first zero than ζ ’s (see Theorems 1.1 and 1.2). This is done using Weil’s explicit formula with carefully chosen test functions. When this method does not immediately show that L-functions of a certain type have low zeros, we then attempt to turn the tables and show that no L-functions of that type exist. Thus, the argument is a combination of proving that low zeros exist and that certain cusp forms do not. Consequently, we are able to prove vanishing theorems and improve upon existing bounds on the Laplace spectrum on L 2 (SLn (Z)\ SLn (R)/ SOn (R)). These in turn can be used to show that SL68 (Z)\ SL68 (R)/ SO68 (R) has a discrete, nonconstant, noncuspidal eigenvalue outside the range of the continuous spectrum on L 2 (SL68 (R)/ SO68 (R)), but that this never happens for SLn (Z)\ SLn (R)/ SOn (R) in lower rank. Another application is to cuspidal cohomology: we show there are no cuspidal harmonic forms on SLn (Z)\ SLn (R)/ SOn (R) for n < 27. 1. Introduction The Riemann ζ -function’s first critical zeros are surprisingly large—about 1/2 ± 14.13472i. Our main interest in this paper is the following question: Does any other automorphic L-function have a larger first zero? DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1, Received 25 September 2000. Revision received 11 April 2001. 2000 Mathematics Subject Classification. Primary 11F66; Secondary 11M26, 11M41, 11F55, 11F72, 11F75. Author’s work supported by National Science Foundation graduate and postdoctoral fellowships and by a Yale Hellman Fellowship.
83
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STEPHEN D. MILLER
This question was raised by A. Odlyzko [O], who proved that the Dedekind ζ -function of any number field has a zero whose imaginary part is less than 14. Odlyzko also proved related conditional results for Artin L-functions. Every automorphic L-function conjecturally factors into products of standard Lfunctions of cusp forms on GLn over the rationals, and we shall be content to discuss these.∗ In fact, by twisting a cuspidal automorphic representation of GLn /Q by a power of the determinant, it is possible to shift the zeros any amount vertically, so we restrict ourselves to studying cuspidal automorphic representations π = ⊗ p≤∞ π p of GLn /Q whose central character is normalized to have finite order. In most examples coming from number theory, the archimedean type π∞ is real; that is, the gamma factors multiplying L(s, π) have real shifts. The following result answers the above question for such cusp forms. 1.1 N
THEOREM
Let π = p≤∞ π p be a cuspidal automorphic representation of GLn over Q with a real archimedean type π∞ and a unitary central character. Then L(s, π) has a low zero that either (i) is on the critical axis between 1/2 ± 14.13472i or (ii) violates the generalized Riemann hypothesis (GRH) in an effective range. When we speak of a zero violating GRH “in an effective range,” we mean that should conclusion (i) fail, then one could theoretically find an effective constant T > 0 such that the box (1/2, 1) × [−T, T ]i contains a zero. For brevity we use the following terminology. Definition. An L-function has a low zero if it either vanishes on the critical axis between 1/2 ± 14.13472i or violates GRH in an effectively bounded range (see Section 2.2). We use this definition to state unconditional results, but not much is actually gained philosophically or numerically in this problem by assuming GRH. The L-functions in Theorem 1.1 include those of Dirichlet characters, rational elliptic curves, and, conjecturally, all rational abelian varieties. Of course, they are also expected to include all Artin L-functions, for example, L-functions of Galois representations. We have been unable to squeeze our technique to answer Odlyzko’s question in full generality, but we can prove many cases. For example, we have the following theorem. ∗ Nevertheless,
our arguments work under various wider assumptions, as they are mostly sensitive to the analytic properties of the L-function.
THE HIGHEST LOWEST ZERO
85
THEOREM 1.2 Let π be a cuspidal automorphic representation of GL2 over Q with a central character of finite order. Then L(s, π) has a low zero (that is on the critical axis between 1/2 ± 14.13472i or else violates GRH in an effective range).
This includes modular form and Maass form L-functions. Other results can be proven about low zeros. For example, every L-function that is related to itself by an odd functional equation automatically vanishes at s = 1/2. For a fixed degree n, most cuspidal automorphic representations of GLn over Q have low zeros. In fact, the possible exceptions all lie in a bounded subset of the unitary dual and have bounded level. This subset tends to be devoid of cusp forms, which is why our method is successful. Thus Odlyzko’s question is related to vanishing theorems about automorphic forms. Our technique uses Weil’s explicit formula relating the coefficients and zeros of automorphic L-functions. It is a variation on the Stark-Odlyzko positivity technique, as formulated by J.-P. Serre, G. Poitou, J.-F. Mestre, and others (see [O] for a survey). In particular, one can compute an exact formula for sum of certain test functions over the critical zeros. If we use a test function that is positive only in a certain range, then finding that this sum is positive ensures a zero in that range. On the other hand, if this sum is negative, then we can often construct another test function that is positive in the critical strip yet whose sum over the zeros is negative. This contradiction shows that the L-function actually could not have existed to begin with. Our main difficulty is that it is often very difficult to construct this second test function given the failure of the first. The latter contradiction, of positive terms yielding a negative sum, can be used to prove vanishing theorems about automorphic forms since they cannot exist when their L-functions do not. Independent of our interest in low zeros, this leads to applications in group cohomology and spectral theory. Other applications One of the consequences of the Ramanujan-Selberg temperedness conjecture is that the discrete cuspidal spectrum of the laplacian 1 on L 2 (SLn (Z)\ SLn (R)/ SOn (R)) is contained in the continuous spectrum of 1 on L 2 (SLn (R)/ SOn (R)). (We always normalize 1 so that this continuous spectrum is the interval [(n 3 − n)/24, ∞)). This consequence should be true more generally for congruence covers of SLn (Z)\ SLn (R)/ SOn (R), but in the following particular case slightly more was proven in [M].
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THEOREM 1.3 ([M]) There exists a constant c > 0 such that the Laplace eigenvalue of every cusp form φ on SLn (Z)\ SLn (R)/ SOn (R) satisfies λ(φ) > λ1 SLn (R)/ SOn (R) + cn.
The following new result is superior for small n. 1.4 Let φ be a cuspidal eigenfunction of the noneuclidean laplacian 1 on SLn (Z)\ SLn (R)/ SOn (R). Then φ’s Laplace eigenvalue satisfies THEOREM
λ(φ) >
1 n 3 − 4n + 25.92 1 + . 24 n−1
(1)
It can be applied to answer a question of A. Lubotzky: When does the eigenvalue of a noncuspidal, square-integrable eigenfunction of the laplacian on SLn (Z)\ SLn (R)/ SOn (R) lie outside [(n 3 − n)/24, ∞)? 1.5 There exists a discrete Laplace eigenfunction THEOREM
φ ∈ L 2 SL68 (Z)\ SL68 (R)/ SO68 (R) such that 1φ = λφ φ,
λφ ≈ 12916.6 <
Yet for n ≤ 67, the bound λ≥
683 − 68 = 13098.5. 24
n3 − n 24
is valid for every nonzero discrete eigenvalue of 1 on L 2 (SLn (Z)\ SLn (R)/ SOn (R)).
Finally, we can apply our technique to cuspidal cohomology and extend a result in [M], where it was shown that SLn (Z)\ SLn (R)/ SOn (R) has no harmonic cuspidal automorphic forms for n < 23.
THE HIGHEST LOWEST ZERO
87
THEOREM 1.6 For 1 < n < 27, the constant-coefficients cuspidal cohomology of SL n (Z) vanishes: · Hcusp SLn (Z); C = 0.
The technique used to prove this theorem is related to the one in [M]. S. Fermigier [F] obtained a similar, but weaker, result using positivity with a different L-function. Here we combine both methods to go further. 2. L-functions and positivity By conjectures of R. Langlands, the most general automorphic L-function is a product of standard L-functions of cuspidal automorphic representations π = ⊗ p≤∞ π p on GLm over the rational adeles AQ . These “primitive” L-functions are degree m Euler products m Y Y (1 − α p, j p −s )−1 , α p, j ∈ C, L(s, π) = p prime j=1 and have completions 3(s, π) =
m Y
π (−s+η j )/2 0
j=1
s + η j
2
L(s, π),
η j ∈ C,
which are entire unless m = 1 and L(s, π) = ζ (s). We have used the duplication property of the gamma function in writing the gamma factors in this way. The conductor is D, and for π p unramified, the α p, j are Hecke eigenvalue parameters and the η j are related to the archimedean parameters of π∞ . With this normalization, 3(s, π) has the functional equation √ 3(s, π) = τπ D −s 3(1 − s, π˜ ), τπ ∈ C, |τπ | = D, D > 0, where π˜ is the contragredient representation to π. The Jacquet-Shalika (see [JS]) bounds imply that 1 Re η j > − . (2) 2 2.1. The Weil formula The explicit formula of A. Weil equates a sum over the zeros of an L-function with a sum over its coefficients and gamma factors: X
3((1/2)+iγ ,π)=0
h(γ ) = 2 Re
m nX j=1
l(η j ) −
∞ o X cn √ g(log n) + g(0) log D, n n=1
(3)
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STEPHEN D. MILLER
where g is an even, differentiable real function, Z g(r ˆ ) = h(r ) = g(x)eir x d x, R
0R (s) = π −s/2 0(s/2), and Z 00 1 1 h(r ) R + η + ir dr 2π R 0R 2 Z Z − log π 1 1 0 0 1 η ir dr + dr = h(r ) h(r ) + + 2π R 2 2π R 20 4 2 2 Z ∞ log π g(0) 1 g(x/2)e−(1/4+η/2)x =− − g(0) − d x. 2 2 0 1 − e−x ex x
l(η) =
Here we have made use of the fact that L(s, π) is entire; for ζ (s) and Rankin-Selberg L-functions, there is a polar term that is introduced when needed later on (see [RS] for a proof of (3)). If g is supported in the interval [− log 2, log 2], then the formula can be viewed as giving the value of the sum over the zeros from the gamma factors: X
h(γ ) = 2 Re
m X j=1
l(η j ) + g(0) log D.
(4)
The basis of the positivity technique is the observation that if h(γ ) ≥ 0 for each zero, then the sum on the right-hand side of (4) must also be positive. This immediately gives a lower bound on the conductor D, which is the original application of the positivity technique. Fortunately, the sum on the right-hand side of (4) is explicitly computable in terms of the η j ’s and D; if it is negative, then the L-function L(s, π) cannot exist, and hence neither can the original cusp form π. Upon assuming GRH, let · · · ≤ γ−2 ≤ γ−1 ≤ 0 ≤ γ1 ≤ γ2 ≤ · · · be the imaginary parts of the zeros of L(s, π). Let g and h = gˆ be chosen so that h ≥ 0 on R, and let c > 0 be a cutoff parameter. Then the function h m (r ) = h(r )(c2 − r 2 ) is positive exactly when |r | < c and it is the Fourier transform of gm = c2 g + g 00 . The support of gm is of course also contained in [− log 2, log 2], provided g is suitably P regular. If the sum 2 Re mj=1 lm (η j ) + gm (0) log D in (4) is positive, then γ1 < c or γ−1 > −c; that is, L(s, π) has a small zero. We summarize as follows.
THE HIGHEST LOWEST ZERO 0.4 0.2 0 −0.2 −0.4
0
89
5
10
15
20
Figure 1. A contour plot of the function Re h 1m (x + i y). The positive set is white and the negative one black.
Criteria 2.1 Our strategy is then, for given archimedean parameters η j and conductor D, to find a function g of support contained in [− log 2, log 2] and for which either 2 Re
m X j=1
l(η j ) + g(0) log D < 0
(which shows the L-function does not exist) or 2 Re
m X j=1
lm (η j ) + gm (0) log D > 0
(which shows that it must have a low zero or violate GRH in an effective range, as discussed below). 2.2. What low zeros mean without GRH Even if we do not assume GRH, we may still conclude from 2 Re
m X j=1
that the sum
lm (η j ) + gm (0) log D > 0 X
h m (γ ) > 0.
Thus, there are zeros ρ = (1/2) + iγ in the region where h m (γ ) > 0. We can explicitly compute the functions h m for our choices of g and examine where they are positive and negative within the critical strip. Since the density of zeros increases only logarithmically with their height (with an effective constant), and our functions h m (z) decay polynomially as z → ∞ in the critical strip, the zero must be contained in an effectively bounded region of the critical strip. As an example, Figure 1 is a contour plot of the function h 1m defined at the end of Section 3. The white regions are where Re h 1m > 0, the black where Re h 1m < 0. Figures 2 and 3 contain plots for the other functions we use.
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STEPHEN D. MILLER
3. A library of functions The main functions we use in this paper are π|x| πx π 1 − |x| cos + sin p p p / cosh(x/2), g1, p (x) = π |x| 2π x + 3 sin 2π|x| 4π 1 − |x| p + 2π 1 − p cos p p / cosh(x/2), g2, p (x) = 6π and g3, p (x) =
54π 1− |x| p
cos
πx p
+6π 1− |x| p
cos
3π x p
60π
+27 sin
π|x| p
cosh(x/2)
+11 sin
3π|x| p
.
We have normalized g j, p (0) = 1, and we often write g j (x) = g j,log(2) (x). Ignoring the cosh(x/2)’s temporarily, the functions g j, p are rescalings of the convolutions of (cos(π x/2)) j with itself. Without the cosh(x/2) term, they would thus have a positive Fourier transform on the real line, and the cosh(x/2) term spreads the positivity into the critical strip. Were we to assume GRH, we would not need it. LEMMA 3.1 If the Fourier transform of an even function g(x) is positive on the real line, then the Fourier transform of g(x)/ cosh(x/2) is positive in the strip −(1/2) < Im r < 1/2.
Proof The Fourier transform of sech(x) is Z ∞ πr 2 eir x d x = π sech . x −x 2 −∞ e + e This has positive real part for −(1/2) < Im r < 1/2, and the Fourier transform converts multiplication into convolution, so the smeared g\ · sech remains positive in this strip. We define modified functions gm =
c2 g(x) + g 00 (x) , c2 g(0) + g 00 (0)
which also have gm (0) = 1. (Of course, we multiplicatively normalize g(0) = gm (0) = 1 to compare the explicit formulas from various test functions.) These are
THE HIGHEST LOWEST ZERO
91
used for showing the presence of low zeros, and since we do not assume GRH for this, we actually use g = g1, p (x) cosh(x/2), g2, p (x) cosh(x/2), or g3, p (x) cosh(x/2). Thus, g1m, p,c (x) π 2 −1+ |x| p cos
πx p
p2
=
+
π sin
π|x| p 2 p
−
c2 − π 1− |x| p cos
πx p
π
π|x| p
−sin
π2 p2
c2 −
,
g2m, p,c (x)
c2 4π
=
1− |x| p
+2π 1− xp cos 6π
2π x p
+3 sin
c2
2π|x| p
−
4π 2 3 p2
cos
−
8π 3 1− xp cos p2
2π x p
6π
+
4π 2 sin
2π|x| p p2
,
and g3m, p,c (x) =
c2 54π 1 −
|x| p
cos
πx p
+ 6π 1 −
|x| p
3π x p
+ 27 sin
π |x| p
+ 11 sin
3π |x| p
2 60π c2 − 9π 5 p2 πx 3π x + 54π 3 − 1 + |x| + 81π 2 sin π |x| − 63π 2 sin 3πp|x| 54π 3 − 1 + |x| p cos p p cos p p . + 2 60π p 2 c2 − 9π 5 p2
Since we are interested in finding zeros in the range from 1/2 ± 14.13472i, we now take c = 14.13472 and write g1m (x) = g1m,log(2),14.13472 (x),
g2m (x) = g2m,log(2),14.13472 (x), and g3m (x) = g3m,log(2),14.13472 (x). The Fourier transforms of these functions are h 1m (r ) =
−8 p 3 π 2 (c2 − x 2 )cos
px 2 2
(−c2 p 2 + π 2 )(π − px)2 (π + px)2 2 128 pπ 4 (c2 − x 2 )sin px 2 h 2m (r ) = , 2 (3c2 p 2 − 4π 2 )(−4π 2 x + p 2 x 3 )
,
92
STEPHEN D. MILLER 0.4 0.2 0 −0.2 −0.4
0
10
20
30
40
Figure 2. A contour plot of the function Re h 2m (x + i y). The positive set is white and the negative one black. 0.4 0.2 0 -0.2 -0.4 0
10
20
30
40
50
Figure 3. A contour plot of the function Re h 3m (x + i y). The positive set is white and the negative set black.
and h 3m (r ) =
px 2 2 . 2 2 2 2 4 2 2 2 (5c p − 9π )(9π − 10 p π x + p 4 x 4 )
2304 p 3 π 6 (c2 − x 2 )cos
We show the contour plots of the functions h 2m and h 3m in Figures 2 and 3. The plot of h 1m is presented in Figure 1.
4. The highest lowest zero for π∞ real THEOREM 1.1 N Let π = p≤∞ π p be a cuspidal automorphic representation of GLn over Q with a real archimedean type π∞ and a unitary central character. Then L(s, π) has a low zero that either (i) is on the critical axis between 1/2 ± 14.13472i or (ii) violates the generalized Riemann hypothesis (GRH) in an effective range.
First we note that for a fixed degree m, L-functions with large η j ’s or large conductor D must have low zeros. This is because the Stirling formula implies that Z 00 1 1 l(η) = h(r ) R + η + ir dr 2π R 0R 2 has a positive real part for η large. Thus, the lowest zero is only an issue for “small” archimedean parameters η j and small conductor—partly because l(η) is bounded from below in Re η > −1/2 (which we may assume by (2)). We present two different proofs of Theorem 1.1.
THE HIGHEST LOWEST ZERO
93
0.8 0.6 0.4 0.2 5
10
15
−0.2
20
25
30
η
l1 (η)
−0.4
l3m (η)
−0.6
Figure 4. The functions l1 (η) and l3m (η)
Picture proof of Theorem 1.1 Figures 4, 5, and 6 indicate that l1 (η) < l3m (η) for η ≥ −(1/2), so the theorem follows from Criteria 2.1.
Less-pictorial proof of Theorem 1.1 This proof also relies on numerical computation but demonstrates how a proof can be made even if the function l is not strictly less than the modified l m . It uses l2m instead of l3m . We noted before in Criteria 2.1 that if 2
m X j=1
l2m (η j ) + log D ≥ 0,
then there is a indeed a low zero, while if 2
m X j=1
l1 (η j ) + log D ≤ 0,
the L-function actually cannot exist to begin with. Thus, we are reduced to dismissing the situation where m X l2m (η j ) < 0 j=1
94
STEPHEN D. MILLER
20
40
60
80
100
−0.002 −0.004 −0.006 −0.008 −0.01 −0.012 −0.014 Figure 5. The difference between l 1 (η) and l3m (η)
η 60
70
80
90
−0.00002 −0.00004 −0.00006 −0.00008 −0.0001 Figure 6. The difference between l 1 (η) and l3m (η), magnified
100
η
THE HIGHEST LOWEST ZERO
and
95
m X l1 (η j ) − l2m (η j ) > 0 j=1
hold simultaneously. Partition the η j ∈ (−1/2, ∞) into three sets: i 1 N = η j | l2m (η j ) ≤ 0, l1 (η j ) − l2m (η j ) ≤ 0 = − , 5.4471 · · · , 2 S = η j | l2m (η j ) > 0, l1 (η j ) − l2m (η j ) ≤ 0 = (5.4471 · · · , 8.6553 · · · ], and P = η j | l2m (η j ) > 0, l1 (η j ) − l2m (η j ) > 0 = (8.6553 · · ·, ∞).
Of course, if
m X
l2m (η j ) < 0,
j=1
then also
X
l2m (η j ) < 0,
η j ∈N ∪P
and if
m X l1 (η j ) − l2m (η j ) > 0, j=1
then
X l1 (η j ) − l2m (η j ) > 0
η j ∈N ∪P
as well. Thus, we need only consider the case when S is empty. From computer investigations (see Figures 7 and 8) on the functions l 1 (η) and l2m (η), we can determine the following very precise information: η j ∈ N =⇒ −.628291 ≤ l 1 (η j ) ≤ 0,
l1 (η j ) − l2m (η j ) ≤ −.001201
and η j ∈ P =⇒ l1 (η j ) ≥ .187484,
0 ≤ l1 (η j ) − l2m (η j ) ≤ .0005801.
Thus, 0>
m X j=1
l2m (η j ) =
X
η j ∈N
l2m (η j ) +
X
η j ∈P
≥ |N |(−.628291) + |P|(.187484),
l2m (η j )
96
STEPHEN D. MILLER
η 10
20
30
40
50
−0.002 −0.004 −0.006 −0.008 −0.01 −0.012 −0.014 Figure 7. The difference between l 1 (η) and l2m (η)
0.2
η 2
4
6
8
−0.2
l1 (η) −0.4
l2m (η) −0.6
Figure 8. The graphs of l1 (η) and l2m (η)
10
THE HIGHEST LOWEST ZERO
97
which implies
On the other hand,
|N | .187484 < = .298403. |P| .628291
m X X X 0< l1 (η j ) − l2m (η j ) = l1 (η j ) − l2m (η j ) + l1 (η j ) − l2m (η j ) j=1
η j ∈N
η j ∈P
≤ |N |(−.001201) + |P|(.005801) forces
a contradiction.
|N | .0005801 > = .483104, |P| .001201
5. Low zeros for modular form L-functions In this section we prove that L-functions of cusp forms on GL2 over Q have low zeros. THEOREM 1.2 Let π be a cuspidal automorphic representation of GL2 over Q with a central character of finite order. Then L(s, π) has a low zero (that is on the critical axis between 1/2 ± 14.13472i or else violates (GRH) in an effective range).
Before giving the proof, we give some background on the hardest case—Maass form L-functions. In particular, we precisely describe their completions, analytic continuations, and functional equations in some important cases. 5.1. Background on Maass forms on 00 ( p)\H It is known that if D = p is a prime and π is a cuspidal automorphic representation with a trivial central character and not corresponding to a holomorphic or antiholomorphic modular form, then π instead corresponds to a Maass form φ on 00 ( p)\H. The Laplace operator 1 and the Hecke operators Tn , n ≥ 0, as well as the involutions (T−1 f )(x + i y) = f (−x + i y), −1 (W p f )(x + i y) = f , p(x + i y) all commute. Thus, after diagonalizing, we may take a basis of Maass cusp forms on 00 ( p)\H which are joint eigenfunctions of 1, Tn , T−1 , and W p . Writing 1φ = λφ,
λ=
1 − ν2, 4
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STEPHEN D. MILLER
φ has the Fourier expansion φ(x + i y) =
X n∈Z
where K ν = K −ν =
√ cn y K ν 2π|n|y e2πinx ,
1 2
Z
∞
e−y(t+t
−1 )/2
tν
0
dt t
is the K -Bessel function of order ν. The cuspidality condition forces c0 = 0; the involution T−1 interchanges cn and c−n . There are four symmetry classes of Maass forms under the action of the involutions T−1 and W p . The standard argument of Hecke and Maass to prove that the L-functions of cusp forms are entire also describes the functional equations of Lfunctions of Maass forms having various symmetries. PROPOSITION 5.1 Suppose φ is a Maass form on 00 ( p)\H with
T−1 φ = (−1)τ φ and W p φ = (−1)ω φ,
τ, ω = 0 or 1.
Multiplicatively normalize the coefficients of φ so that a 1 = 1 and P∞ √ an y K ν (2πny) cos(2πnx), τ = 0, φ(x + i y) = Pn=1 √ ∞ τ = 1. n=1 an y K ν (2πny) sin(2πnx), Then
3(s, φ) = 0R (s + τ + ν)0R (s + τ − ν)
∞ X an n=1
ns
satisfies the functional equation 3(s, φ) = (−1)τ +ω p 1/2−s 3(1 − s, φ).
(5)
THE HIGHEST LOWEST ZERO
99
Proof First, consider the case τ = 0. Then Z ∞ Z ∞ ∞ X dy dy = φ(i y)y s−1/2 an K ν (2πny)y s y y 0 0 n=1 ∞ hZ ∞ X dy i = an (2πn)−s K ν (y)y s y 0 = =
n=1 ∞ X n=1
h s + ν s − ν i an (2πn)−s 2s−2 0 0 2 2
1 3(s, φ). 4
The transformation property φ(i y) = (−1)ω φ
i py
gives Z ∞ dy dy i y s−1/2 = 4(−1)ω φ(i y)y s−1/2 φ y py y 0 0 Z ∞ dy i y φ y 1/2−s = 4(−1)ω p y 0 Z ∞ dy = 4 p 1/2−s (−1)ω φ(i y)y 1/2−s y 0
3(s, φ) = 4
Z
∞
= (−1)ω p 1/2−s 3(1 − s, φ). If τ = 1, then actually φ(i y) = 0 and we instead consider the derivative d φ 0 (x + i y) := φ(x + i y) dx ∞ X √ = (2πn)an y K ν (2πny) cos(2πnx). n=1
The action under W p now reads φ 0 (i y) = (−1)ω φ 0
i −1 . py 2 py 2
We also have Z ∞ Z ∞ ∞ X dy dy φ 0 (i y)y s+1/2 (2πn)an K ν (2πny)y s+1 = y y 0 0 =
n=1 ∞ X n=1
h s + 1 + ν s + 1 − ν i an (2πn)−s 2s−1 0 0 , 2 2
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STEPHEN D. MILLER
and the functional equation for 3(s, φ) follows as before. 5.2. Low zeros for Maass form L-functions We first prove Theorem 1.2 for Maass forms through a series of propositions. 5.2 Every Maass form L-function whose conductor satisfies PROPOSITION
D≥3
if T−1 φ = φ
or D≥2
if T−1 φ = −φ
has a low zero. Proof In these two symmetry classes, the gamma factors of 3(s, φ) are either 0R (s + ν)0R (s − ν) or 0R (s + 1 + ν)0R (s + 1 − ν), depending on whether φ is even or odd under T−1 . In each case we may assume the parameter ν is not real, and hence purely imaginary, because Theorem 1.1 already covers the case of real archimedean type. In the first case, we have Re l3m (ir ) − l1 (ir ) > 0 if − 5.1 < r < 5.1, a range in which Re l1 (ir ) < −(log 3/4) ≈ −0.274653 (see Figure 9). In the second case, Re l3m (1 + ir ) − l1 (1 + ir ) > 0 if − 5.5 < r < 5.5, where Re l1 (1 + ir ) and Re l3m (1 + ir ) are both less than −(log 2/4) (see Figure 10). Criteria 2.1 thus shows that there are low zeros in either case. The following proposition handles the case of Maass forms at full level (i.e., unramified for all primes p < ∞). PROPOSITION 5.3 If φ is a Maass form on SL2 (Z)\H, then L(s, φ) has a low zero.
THE HIGHEST LOWEST ZERO
101
1
2
3
4
−0.1
Re(l3m (iη) − l1 )
−0.2
Re(l1 (iη))
5
η
−0.3 −0.4 −0.5
Figure 9. The functions l3m (η) − l1 (η) and l1 (η): A plot showing that Re l1 (ir ) + (log 3/4) < 0 and Re (l3m (ir ) − l1 (ir )) > 0 for −5.1 < r < 5.1
0.02 0.015
Re(l1 (1 + iη)) +
0.01
log(2) 4
Re(l3m (1 + iη)) +
log(2) 4
0.005 5.3
5.4
5.5
5.6
−0.005
Figure 10. The functions l1 (η) and l3m (η): A plot showing that Re l1 (1 + ir ) + (log 2/4) < 0 and Re (l3m (1 + ir ) − l1 (1 + ir )) > 0 for −5.5 < r < 5.5
η
102
STEPHEN D. MILLER
0.1 2
4
6
8
η
−0.1 −0.2 −0.3
Re(l3 (iη))
−0.4
Re(l1m (iη))
−0.5
−0.6 Figure 11. The functions l3 (η) and l1m (η): A plot showing that Re l3 (ν) is negative when Re l 1m (ν) is
Proof We again break the proof up into two cases according to whether φ is even or odd under T−1 . By Theorem 1.1 we need only consider the case Re ν = 0. If φ is even, then the gamma factors of L(s, φ) are 0R (s + ν)0R (s − ν). Figure 11 shows that Re l3 (ν) is negative when Re l 1m (ν) is, which by Criteria 2.1 proves the proposition in this case. If instead φ is odd, the gamma factors are 0R (s + 1 + ν)0R (s + 1 − ν), and, similarly, Re l3 (1 + ν) is negative when Re l 1m (1 + ν) is (see Figure 12). The proposition follows by invoking Criteria 2.1. To handle the remaining case, that of even Maass forms on 00 (2)\H, we use a result about the smallest even eigenvalue of the laplacian there. Perhaps Proposition 5.5 can be proven without such explicit information. PROPOSITION 5.4 If φ is a Maass form on 00 (2)\H which is even under both T−1 and W2 , then its Laplace eigenvalue exceeds (1/4) + 6.142 .
D. Hejhal [H] has numerically computed that the first such eigenvalue is ≈ (1/4) + 8.9222 . We present the following argument to demonstrate a technique.
THE HIGHEST LOWEST ZERO
103
0.1 2
4
6
8
η
−0.1 −0.2
Re(l3 (1 + iη))
−0.3
Re(l1m (1 + iη))
−0.4
Figure 12. The functions l3 (η) and l1m (η): A plot showing that Re l3 (1 + ν) is negative when Re l 1m (1 + ν) is
Proof First, Figure 13 shows that log 2 <0 4 for −6.07 ≤ r ≤ 6.07, so we need only consider the range 6.07 ≤ r ≤ 6.14. Using the symmetries, set Re l1 (ir ) +
∞ n i X an f (t) = φ √ et = √ Wir √ et = f (−t), n 2 2 n=1 √ Wir (y) = y K ir (2π y).
Thus, f is an even function in t, and so f 0 (0) =
∞ n n X an √ Wir0 √ √ = 0 n 2 2
(6)
n=1
and f 000 (0) = =:
∞ n n3 n 3n 2 n n i X an h + Wir0 √ √ √ Wir000 √ √ + Wir00 √ n 2 2 2 2 2 2 2 n=1 ∞ X n=1
an √ Vir (n) = 0. n
(7)
The terms in the Fourier expansion decay rapidly with n, and so we use the first three terms as an approximation. Recall that we are focusing on the range 6.07 ≤ r ≤ 6.14.
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STEPHEN D. MILLER
We may assume that φ is a Hecke eigenform with a1 = 1, and [BDHI] has proven that their coefficients satisfy the bound |an | ≤ τ (n)n 5/28 ,
√ where τ (n) is the number of divisors of n. Using the crude bound τ (n) ≤ 2 n, we can bound the tails ∞ X n n an √ Wir0 √ √ ≤ 1.14 · 10−7 n 2 2
(8)
n=4
and
∞ X an √ Vir (n) ≤ 2.7 · 10−5 . n
(9)
n=4
Thus, (6) and (8) show that 1 2 2 3 3 a2 a3 0 1 Wir √ √ + √ Wir0 √ √ + √ Wir0 √ √ ≤ 1.14 · 10−7 , 3 2 2 2 2 2 2 2
while (7) and (9) show that a2 a3 Vir (1) + √ Vir (2) + √ Vir (3) ≤ 2.7 · 10−5 . 3 2 √ √ Now |W√ir0 (3/ √2)| ≤ 1.2 · 10−6 in the range 6.07 ≤ r ≤ 6.14. Yet Wir (1/ 2) and Wir (2/ 2)2/ 2 are much larger, never smaller than 5.7 · 10−5 in magnitude. The ratio of Wir0 √1 √1 2 2 Wir0 √2 √2 2 2 √ is smallest at r = 6.07, where it is ≈ 1.475 > 1. Thus, we must have a 2 / 2 > 1 for (6) to be valid. √ At the same time, such a value of a2 / 2 is too large to achieve equality in (7). This is because it makes the second term much larger than the first and third terms could possibly be with the constraint that a3 ≤ 2 · 35/28 : 35/28 Vir (1) + 2 ·√ Vir (3) < Vir (2) , 3
6.07 ≤ r ≤ 6.14.
So (6) and (7) cannot hold simultaneously. This contradiction shows every Maass form on 00 (2)\H which is even under both T−1 and W2 has Laplace eigenvalue greater than (1/4) + 6.142 .
THE HIGHEST LOWEST ZERO
105
0.01 0.005 6.05
6.1
−0.005
6.15
6.2
η Re(l1 (iη)) +
−0.01
log(2) 4
Re(l1m (iη)) +
log(2) 4
Figure 13. The functions l1 (η) and l1m (η): A plot showing that Re l1m (ir ) > −(log 2/4) for r > 6.135, while Re l1 (ir ) < −(log 2/4) for r < 6.07 PROPOSITION 5.5 Maass form L-functions with conductor D = 2 (which correspond to Maass forms on 00 (2)\H) have low zeros.
Proof By Proposition 5.2 we need only consider the even Maass forms, where the gamma factors are 0R (s + ν)0R (s − ν). In fact, by Proposition 5.1 we can assume that φ is even under both W2 and T−1 ; otherwise, (5) dictates that 3
1
1 , φ = −3 , φ = 0. 2 2
The function Re l1m (ir ) > −(log 2/4) for r > 6.135 (see Figure 13), and Proposition 5.4 shows that all even eigenvalues are in that range. Proof of Theorem 1.2 Every cuspidal automorphic representation on GL2 over Q comes from either a Maass form or a holomorphic modular form. Both holomorphic modular forms and nontempered Maass forms (i.e., λ < 1/4) have real archimedean type and are thus covered under Theorem 1.1. The rest of the Maass forms (the tempered ones) are covered by Propositions 5.2, 5.3, and 5.5.
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STEPHEN D. MILLER
6. Cuspidal eigenvalue bounds Now we move our focus completely toward automorphic representations rather than on their L-functions. In this section and the next we examine the discrete spectrum of the laplacian 1 on L 2 (SLn (Z)\ SLn (R)/ SOn (R)).∗ We normalize our laplacian so that its continuous spectrum on L 2 (SLn (R)/ SOn (R)) spans the interval from n3 − n λ1 SLn (R)/ SOn (R) = 24 to ∞. Because the ring of invariant differential operators R on SLn (R)/ SOn (R) is commutative, we may take a basis of Laplace eigenfunctions that are also common eigenfunctions of the operators in R . Thus, to each discrete eigenfunction φ ∈ L 2 (SLn (Z)\ SLn (R)/ SOn (R)) we can attach Langlands parameters µ1 , . . . , µn . These describe φ’s eigenvalues under the different operators in R ; in particular, the Laplace eigenvalue satisfies 1φ = λφ,
λ=
µ2 + · · · + µ2n n3 − n − 1 . 24 2
By the Jacquet-Shalika “trivial” bound (see [JS]), | Re µ j | < Thus,
1 , 2
j = 1, . . . , n.
n
λ>
1X n 3 − 4n (Im µ j )2 + . 2 24
(10)
j=1
We use (10) to bound λ from below. Positivity functions Recall the function πx 1 π|x| |x| cos + sin g1, p = 1 − / cosh(x/2), p p π p
0 < p ≤ log 2.
Define s(r ) = max Re l1,1/2 (ir + σ ), where the maximum is taken over σ ∈ [−(1/2), 3/2]. ∗ Of course, our methods carry over to some congruence covers, but we restrict our attention to the fulllevel congruence group SLn (Z) here.
THE HIGHEST LOWEST ZERO
107
1.25 1 0.75 0.5 0.25 r −100
−50
50
100
−0.25 Figure 14. The graph of s(r )
Criteria 6.1 Pn If the sum j=1 s(r j ) is negative, then there is no cuspidal eigenfunction in L 2 (SLn (Z)\ SLn (R)/ SOn (R)) whose Langlands parameters µ1 , . . . , µn have Im µ j = r j . If φ is a cusp form, then the archimedean Ramanujan-Selberg conjectures assert that π∞ is tempered; that is, Re µ j = 0. A consequence is that λcusp ≥ (n 3 − n)/24. This was proven unconditionally in [M] using a similar positivity argument. Here we can derive some stronger results and different applications. 6.2 (A trivial bound) If φ is a cusp form in L 2 (SLn (Z)\ SLn (R)/ SOn (R)), then with the above notation, PROPOSITION
n X j=1
r 2j > 51.84 1 +
1 . n−1
Proof P The plot in Figure 14 shows s(r ) < 0 for |r | < 7.2. Thus, nj=1 s(r j ) ≥ 0 only if at least one |r j | ≥ 7.2. Since the r j are constrained to have r 1 + · · · + rn = 0, this means that n 7.2 2 X r 2j ≥ 7.22 + (n − 1) . n−1 j=1
108
STEPHEN D. MILLER
Theorem 1.4 follows immediately from Proposition 6.2 and (10). 6.1. Extreme values P 2 P Given d and the constraints r j = d, r j = 0, if the largest value obtained by P s(r j ) is negative, then Criteria 6.1 implies λ > (d/2) + (n 3 − 4n)/24. 6.3 If (r1 , . . . , rn ) is an extremal point of
PRINCIPLE ∗
n X
s(r j )
j=1
subject to the constraints n X j=1
r j = 0,
n X j=1
r 2j = d,
then the r j assume at most three distinct values. Proof By Lagrange multipliers, there are real constants c1 , c2 ∈ R such that s 0 (r1 ), . . . , s 0 (rn ) = c1 (r1 , . . . , rn ) + c2 (1, . . . , 1);
that is, the points (r j , s 0 (r j )) all lie on the intersection of some line and the graph of y = s 0 (x). But no line crosses this graph in more than three places. Even though Figure 15 shows only the range |x| ≤ 100, it is legal to use this principle in this paper. Another crossing would give a value of r j so large that it would not enter into our subsequent bounds. THEOREM 6.4 We have the following bounds on the Laplace eigenvalue of a cuspidal eigenfunction of 1 in L 2 (SLn (Z)\ SLn (R)/ SOn (R)):
n λ≥ ∗ Some
3 87.625
4 108
5 140.875
6 167
7 201.125
8 232
may not consider the justification to be a proof, but, as we indicate, it can be verified in the applications we use it for.
THE HIGHEST LOWEST ZERO
109
0.4 0.2
−100
−50
50
100
r
−0.02 −0.04
Figure 15. The graph of s 0 (r )
Proof By the last proposition, we need only consider the case where there are A r 1 ’s, B r2 ’s, and C r3 ’s, with r1 , r2 , r3 ∈ R, Ar1 + Br2 + Cr3 = 0,
Ar12 + Br22 + Cr32 = d,
and then try to find a large value of d such that As(r1 ) + Bs(r2 ) + Cs(r3 ) is always negative. Since A, B, and C are all positive integers that sum to n, this is a finite calculation. We take A, B, C > 0 by allowing some of the values of r 1 , r2 , and r3 to coincide. Then in terms of the parameter r 3 , either q ACr3 + AB −C 2r32 + A(D − C) + B(Dr 32 − Cr32 ) r1 = − A (A + B) and r2 =
− (BCr3 ) +
q
or instead r1 =
−ACr3 +
q
AB −C 2r32 + A(D − Cr32 ) + B(D − Cr32 ) B (A + B)
AB −C 2r32 + A(D − C) + B(Dr 32 − Cr32 ) A (A + B)
110
STEPHEN D. MILLER
and r2 = −
(BCr3 ) +
q
AB −C 2r32 + A(D − Cr32 ) + B(D − Cr32 )
. B (A + B) Actually, the second set of solutions and the first are interchanged upon r 3 ↔ −r3 , so they take the same values. For a given n, we need only enumerate the integer triples of A, B, C with A ≥ B ≥ C > 0 and plot q ACr3 + AB −C 2r32 + A(D − C) + B(Dr 32 − Cr32 ) As − A (A + B) q − (BCr3 ) + AB −C 2r32 + A(D − Cr32 ) + B(D − Cr32 ) + Cs(r 3) + Bs B (A + B) over the range √ √ √ A+B D A+B D −√ √ ≤ r3 ≤ √ √ . C A+ B +C C A+ B +C √
One finds that the following values of d work: n d
3 174
4 212
5 273
6 318
7 376
8 424
Remark 6.5 Theorem 1.3 shows that there exists a positive constant c > 0 such that λcusp −
n3 − n > cn, 24
n = 1, 2, . . . .
The argument of Theorem 6.4 gives a much better constant.
cusp
6.2. Some open problems about λ1
(SLn (Z)\ SLn (R)/ SOn (R))
CONJECTURE 6.6 Fix a positive integer k = 1, 2, . . . , and denote the kth cuspidal eigenvalue of 1 cusp on L 2 (SLn (Z)\ SLn (R)/ SOn (R)) as λk (SLn (Z)\ SLn (R)/ SOn (R)). Then the sequence
n λcusp (SLn (Z)\ SLn (R)/ SOn (R)) − k
n
n 3 −n 24
| n = 1, 2, . . .
o
(11)
THE HIGHEST LOWEST ZERO
111
has a limiting distribution. Question 6.7 Is the sequence in (11) also bounded from above as well as from below? 7. Bounds on noncuspidal eigenvalues A. Lubotzky asked if the bound λ≥
n3 − n 24
might additionally hold for the entire nonzero discrete spectrum of 1 on L 2 (SLn (Z)\ SLn (R)/ SOn (R)), that is, not just for cusp forms alone. Although from the point of view of automorphic forms the cusp forms are most essential, the entire discrete spectrum enters into considerations in differential geometry. In fact, there are nonconstant, noncuspidal, square-integrable residues of Eisenstein series on L 2 (SLn (Z)\ SLn (R)/ SOn (R)) which are discrete Laplace eigenfunctions, and they are never tempered (that is, they violate Re µ j = 0). The first example of one on SLn (Z)\ SLn (R)/ SOn (R) violating λ ≥ (n 3 − n)/24 occurs for n = 68. THEOREM 1.5 There exists a discrete Laplace eigenfunction
φ ∈ L 2 SL68 (Z)\ SL68 (R)/ SO68 (R) such that 1φ = λφ φ,
λφ ≈ 12916.6 <
683 − 68 = 13098.5. 24
Yet for n ≤ 67, the bound
n3 − n 24 is valid for every nonzero discrete eigenvalue of 1 on L 2 (SLn (Z)\ SLn (R)/ SOn (R)). λ≥
Of course, the failure of λ ≥ (n 3 − n)/24 at n = 68 is the typical case for large n, as the argument demonstrates. The key idea here is the classification of the discrete spectrum in terms of cusp forms. It was first conjectured by H. Jacquet [J] and later proven by C. Moeglin and J.-L. Waldspurger [MW]. Let us now describe how discrete eigenfunctions can be constructed. Factor n = ra, and let φ be a cusp form on SLa (Z)\ SLa (R)/ SOa (R). The group SLn (R) has a rank r − 1 parabolic subgroup P of type (a, a, . . . , a) whose Levi component is L = GLa (R)r ∩ SLn (R).
112
STEPHEN D. MILLER
The cusp form φ extends as a product to the r copies of SLa (R) in L in the obvious way. Given h = (h 1 , . . . , h r ) ∈ Cr such that h 1 + · · · + h r = 0, we can form a character of the split Levi component A of P, and the Eisenstein series E(P, g, φ, h). If φ has Langlands parameters µ1 , . . . , µa , then E(P, g, φ, h) has Langlands parameters a
a
z }| { z }| { (µ1 + h 1 , µ2 + h 1 , . . . , µa + h 1 , µ1 + h 2 , . . . , µ1 + h r , . . . , µa + h r ). | {z } ra=n
Furthermore, E(P, g, φ, h) has a pole of order r − 1 at h = (r − 1)/2, (r − 3)/2, . . . , −(r − 1)/2 , and its (r − 1)th iterated residue there is a discrete L 2 eigenfunction of 1. Moreover, all discrete L 2 -eigenfunctions of 1 arise this way. For example, if r = 1, these are just cusp forms, and if a = 1, they are constant functions. We compute that the residue’s Laplace eigenvalue is a
2λ −
r
2 XX n3 − n r −1 =− −k µj + 12 2 =
j=1 k=1 a X −r µ2j j=1
−a
r3 − r . 12
Incidentally, Maass forms with Laplace eigenvalue 1/4 are known to exist on congruence quotients of SL2 (R)/ SO2 (R). Using this procedure, one may already construct a discrete residue on a congruence quotient of SL4 (R)/ SO4 (R) which violates the λ ≥ (43 − 4)/24 bound. Proof of Theorem 1.5 First, Hejhal [H] has computed that cusp
λ1
(SL2 (Z)\ SL2 (R)/ SO2 (R)) = 91.1413 · · ·,
corresponding to µ1 = −µ2 ≈ 9.534i. Thus, the Laplace eigenvalue of a residue formed from the Hejhal Maass form has 2λ −
r3 − r n3 − n ≈ r (181.8) − 2 , 12 12
and this difference is positive for √ r < 6 · 181.8 + 1 ≈ 33.04. For r = 34, we have λ ≈ 12916.6 <
683 − 68 = 13098.5. 24
THE HIGHEST LOWEST ZERO
113
If in fact there were an example of a residue for n = ra < 68, with −r
a X j=1
µ2j − a
r3 − r < 0, 12
we would necessarily have v u a u 12 X t µ2j + 1. r> − a
(12)
j=1
We already know that r > 1 since cusp forms obey the λ ≥ (n 3 − n)/24 bound. Thus, we can restrict to the cases r ≥ 2, a = 1, . . . , 34. Using our pre-existing bounds P for µ2j , we conclude that ra ≥ 68 (see Table 1 for details). 8. Cuspidal cohomology The positivity inequality can be applied to products of L-functions that have poles, for example, Rankin-Selberg L-functions L(s, π ⊗ π) ˜ of cuspidal automorphic forms m π on GLn . If {µ jk } j=1,k=1 are the archimedean 0R parameters, the inequality reads Z
R
m X m X g(x) e x/2 + e−x/2 d x + 2 Re l(µ jk ) + g(0) log D ≥ 0.
(13)
j=1 k=1
The new term in (13) as compared to (4) comes from the poles of L(s, π ⊗ π). ˜ Also, here we have simply dropped the coefficients entirely because (L 0 /L)(s, φ) has a Dirichlet series with nonpositive coefficients (see [RS] for a verification of this) and so there is no restriction on the support of g. N If π = p≤∞ π p comes from a constant-coefficients cohomological cusp form on GLn (AQ ), then π∞ is of either the form n π∞ = IndGL P(2,2,...,2) (D2 , D4 , . . . , Dn ),
n even,
or n π∞ = IndGL P(1,2,2,...,2) sgn(·) , D3 , D5 , . . . , Dn ,
n odd.
(sgn is the sign character, = 0 or 1, and Dk denotes the kth discrete series on GL2 , corresponding to weight k holomorphic forms). Thus, if n is written as 2m + t, t = 0 or 1, the archimedean µ jk can be computed via the recipe summarized in [RS] and are the following multisets: {µ j,k } = {t + j + k, t − 1 + j + k, |k − j|, 1 + |k − j| | 1 ≤ j, k ≤ m} ∪
{0, j, j, j + 1, j + 1 | 1 ≤ j ≤ m} . {z } | omit if t = 0
(14)
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STEPHEN D. MILLER
Table 1. This table completes the proof of Theorem 1.5. Suppose a residue of a cusp form on GLa occurred on some GLn , n = ra < 68 with Laplace eigenvalue less than or equal to (n 3 − n)/24. The second column gives the upper bound P r ≤ [68/a]. The third column gives a lower bound for − aj=1 µ2j (from Proposition 6.2), but the fourth gives an upper bound on P − aj=1 µ2j which would be satisfied by such a residue with low eigenvalue (as derived in (12) in the proof of Theorem 1.5). The inconsistency of these two inequalities is a contradiction showing that the discrete Laplace spectrum on SLn (Z)\ SLn (R)/ SOn (R) is contained in {0} ∪ [(n 3 − n)/24, ∞) for n < 68. a
r ≤ [ 68 a ]
Lower bound for −
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
22 17 13 11 9 8 7 6 6 5 5 4 4 4 4 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2
171.25 211 271.75 316.5 374.25 422 56.07 55.10 54.27 53.55 52.91 52.32 51.79 51.29 50.83 50.38 49.96 49.56 49.18 48.80 48.44 48.09 47.75 47.41 47.08 46.76 46.44 46.12 45.81 45.51 45.20 44.91
Pa
j=1
µ2j
Upper bound for − 120.75 96.00 70 60 46.66 42.00 36 29.16 32.08 24 26 17.5 18.75 20 21.25 12 12.66 13.33 14 14.66 5.75 6 6.25 6.5 6.75 7 7.25 7.5 7.75 8 8.25 8.5
Pa
j=1
µ2j
THE HIGHEST LOWEST ZERO
115
Table 2. The (numerical) proof of the cohomology theorem. The R left-hand side of (13), R g(x) e x/2 + e−x/2 d x Pm Pm +2 Re j=1 k=1 l(µ jk ) + g(0) log D, must be positive if the cusp form exists, and this table shows it is negative for n < 27. n 2 3 4 5 6 7
t 3 6 6 6 6 6
LHS(13) −2.821 −8.113 −17.02 −28.30 −38.51 −50.30
n 15 16 17 18 19 20
t 6 6 6 6 6 6
LHS(13) −111.4 −112.1 −112.4 −109.2 −105.4 −97.87
n 9 10 11 12 13 14
t 6 6 6 6 6 6
LHS(13) −71.43 −80.27 −89.68 −96.45 −103.4 −107.5
n 22 23 24 25 26
t 6 6 6 6 6
LHS(13) −77.30 −64.06 −46.70 −28.18 −5.388
THEOREM 1.6 For 1 < n < 27, the constant-coefficients cuspidal cohomology of SL n (Z) vanishes: · SLn (Z); C = 0. Hcusp
Proof Let g p (x) =
1−
πx 1 π|x| |x| cos + sin / cosh(x/2). p p π p
Then h 1, p (r ) = gˆ 1, p (r ) is positive in the critical strip | Im r | < 1/2. For our cohomological forms D = 1 at full level, and with the µ jk ’s as above, we arrive at a contradiction to the positivity inequality (see Table 2).
Remarks We proved this for n < 23 in [M] with the Rankin-Selberg L-functions but without the Weil formula (instead using the Mittag-Leffler expansion). Fermigier [F] proved a weaker result using the Weil formula but with the standard L-function. Theorem 8.1 surpasses both. Nothing is known about these cuspidal Betti numbers for n ≥ 27, let alone if they are ever nonzero. Acknowledgments. We wish to thank Don Blasius, William Duke, Benedict Gross, Alexander Lubotzky, Andrew Odlyzko, Ilya Piatetski-Shapiro, Vladimir Rokhlin, Peter Sarnak, Jean-Pierre Serre, Gunther Steil, Andrew Wiles, and Gregg Zuckerman for their discussions. We wish to thank Princeton University and the University of California at San Diego. All numerical computations were made with Mathematica v.3
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on an Intel Pentium II 300 MHz system running Windows NT 4.0 and Slackware Linux 2.0.30. References [BDHI] D. BUMP, W. DUKE, J. HOFFSTEIN, and H. IWANIEC, An estimate for the Hecke eigenvalues of Maass forms, Internat. Math. Res. Notices 1992, 75–81. MR 93d:11047 104 [F] S. FERMIGIER, Annulation de la cohomologie cuspidale de sous-groupes de congruence de GLn (Z), Math. Ann. 306 (1996), 247–256. MR 97h:11056 87, 115 [H] D. A. HEJHAL, Eigenvalues of the Laplacian for Hecke triangle groups, Mem. Amer. Math. Soc. 97 (1992), no. 469. MR 93f:11043 102, 112 [J] H. JACQUET, “On the residual spectrum of GL(n)” in Lie Group Representations, II, Lecture Notes in Math. 1041, Springer, Berlin, 1984, 185–208. MR 85k:22045 111 [JS] H. JACQUET and J. A. SHALIKA, “Rankin-Selberg convolutions: Archimedean theory” in Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of His Sixtieth Birthday (Ramat Aviv, Israel, 1989), I, Israel Math. Conf. Proc. 2, Weizmann, Jerusalem, 1990, 125–207. MR 93d:22022 87, 106 [LRS] W. LUO, Z. RUDNICK, and P. SARNAK, On Selberg’s eigenvalue conjecture, Geom. Funct. Anal. 5 (1995), 387–401. MR 96h:11045 [M] S. D. MILLER, Spectral and cohomological applications of the Rankin-Selberg method, Internat. Math. Res. Notices 1996, 15–26. MR 97a:11079 85, 86, 87, 107, 115 ´ [MW] C. MOEGLIN and J.-L. WALDSPURGER, Le spectre r´esiduel de GL(n), Ann. Sci. Ecole Norm. Sup. (4) 22 (1989), 605–674. MR 91b:22028 111 [O] A. M. ODLYZKO, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: A survey of recent results, S´em. Th´eor. Nombres Bordeaux (2) 2 (1990), 119–141. MR 91i:11154 84, 85 ¨ [R] W. ROELCKE, Uber die Wellengleichung bei Grenzkreisgruppen erster Art, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1953/1955, 159–267. MR 18:476d [RS] Z. RUDNICK and P. SARNAK, Zeros of principal L-functions and random matrix theory, Duke Math J. 81 (1996), 269–322. MR 97f:11074 88, 113
Department of Mathematics, Yale University, Box 208283, New Haven, Connecticut 06520-8283, USA; current, Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1,
ON LEVEL-ZERO REPRESENTATIONS OF QUANTIZED AFFINE ALGEBRAS MASAKI KASHIWARA
Abstract We study the properties of level-zero modules over quantized affine algebras. The proof of the conjecture on the cyclicity of tensor products by T. Akasaka and the author is given. Several properties of modules generated by extremal vectors are proved. The weights of a module generated by an extremal vector are contained in the convex hull of the Weyl group orbit of the extremal weight. The universal extremal weight module with level-zero fundamental weight as an extremal weight is irreducible, and it is isomorphic to the affinization of an irreducible finite-dimensional module. Contents 1. Introduction . . . . . . . . . . . . . . . . . 2. Review on crystal bases . . . . . . . . . . . 2.1. Quantized universal enveloping algebras 2.2. Crystals . . . . . . . . . . . . . . . . 2.3. Schubert decomposition of crystal bases 2.4. Global bases . . . . . . . . . . . . . . 3. Extremal weight modules . . . . . . . . . . 3.1. Extremal vectors . . . . . . . . . . . . 3.2. Dominant weights . . . . . . . . . . . 4. Affine quantum algebras . . . . . . . . . . . 4.1. Affine root systems . . . . . . . . . . 4.2. Affinization . . . . . . . . . . . . . . 4.3. Simple crystals . . . . . . . . . . . . 5. Affine extremal weight modules . . . . . . . 5.1. Extremal vectors—affine case . . . . . 5.2. Fundamental representations . . . . . . 6. Existence of global bases . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117 119 119 120 121 122 123 123 124 125 126 131 132 133 133 135 142
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1, Received 30 October 2000. Revision received 11 December 2000. 2000 Mathematics Subject Classification. Primary 20G05. This work benefits from a “Chaire Internationale de Recherche Blaise Pascal de l’Etat et de la R´egion d’Ile-deFrance, g´er´ee par la Fondation de l’Ecole Normale Sup´erieure.” 117
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MASAKI KASHIWARA
6.1. Regularized modified operators . 6.2. Existence theorem . . . . . . . . 7. Universal R-matrix . . . . . . . . . . . 8. Good modules . . . . . . . . . . . . . 9. Main theorem . . . . . . . . . . . . . 10. Combinatorial R-matrices . . . . . . . 11. Energy function . . . . . . . . . . . . 12. Fock space . . . . . . . . . . . . . . . 12.1. Some properties of good modules 12.2. Wedge spaces . . . . . . . . . . 12.3. Global basis of the Fock space . . 13. Conjectural structure of V (λ) . . . . . . Appendices . . . . . . . . . . . . . . . . . A. Proof of the formula (6.2) . . . . . . B. Formulas for the crystal B(U˜ q (g)) . . References . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
142 143 146 148 153 155 158 160 160 164 165 169 171 171 173 174
1. Introduction In this paper, we study the level-zero representations of quantum affine algebras. This paper is divided into three parts: extremal weight modules, the conjecture in [1] on the cyclicity of the tensor products of fundamental representations, and the global basis of the Fock space. In [11], as a generalization of highest weight vectors, the notion of extremal weight vectors is introduced, and it is shown that the universal module generated by an extremal weight vector has favorable properties: this has a crystal base, a global basis, and so on. The main purpose of the first part (§2–§5) is to study such modules in the affine case and to prove the following two properties. (a) If a module is generated by an extremal vector with weight λ, then all the weights of this module are contained in the convex hull of the Weyl group orbit of λ. (b) Any module generated by an extremal vector with a level-zero fundamental weight $i is irreducible, and it is isomorphic to the affinization of an irreducible finite-dimensional module W ($i ) (see Th. 5.17 and Prop. 5.16 for an exact statement). In the second part, we prove the following theorem∗ , which is conjectured in [1] (1) (1) and proved in the case of An and Cn . ∗ M.
Varagnolo and E. Vasserot [22] prove the same conjecture in the simply laced case by a different method.
QUANTIZED AFFINE ALGEBRAS
119
THEOREM
If aν /aν+1 has no pole at q = 0 (ν = 1, . . . , m −1), then W ($i1 )a1 ⊗· · ·⊗W ($im )am is generated by the tensor product of the extremal vectors. In the course of the proof, one uses the global basis on the tensor products of the affinizations of W ($iν ), especially the fact that the transformation matrix between the global basis of the tensor products and the tensor products of global bases is triangular. Among the consequences of this theorem (see §9), we mention here the following one. Under the conditions of the theorem above, there is a unique homomorphism up to a constant multiple W ($i1 )a1 ⊗ · · · ⊗ W ($im )am −→ W ($im )am ⊗ · · · ⊗ W ($i1 )a1 , and its image is an irreducible Uq0 (g)-module. This phenomenon is analogous to the morphism from the Verma module to the dual Verma module. Conversely, combining with a result of V. Drinfeld [4], any irreducible integrable Uq0 (g)-module is isomorphic to the image for some {(i 1 , a1 ), . . . , (i m , am )}. Moreover, {(i 1 , a1 ), . . . , (i m , am )} is unique up to a permutation. In the third part (§12), we prove the existence of the global basis on the Fock space. The plan of the paper is as follows. In §2–§4, we review some of the known results of crystal bases. Then, in §5, we give a proof of (a) and (b). In §6, we prove a sufficient condition for a module to admit a global basis: very roughly speaking, it is enough to have a global basis in the extremal weight spaces. In §7, we review the universal R-matrix and the universal conjugation operator. After introducing the notion of good modules (roughly speaking, a module with a global basis), we prove in §9 the above theorem in the framework of good modules. After preparations in §10–§11 on the combinatorial R-matrix and the energy function, we prove in §12 the properties of good modules which are postulated for the existence of the wedge products and the Fock space in [13]. Finally, we show that the Fock space admits a global basis. In the case of the vector representation of (1) g = An , the global basis of the corresponding Fock space has already been constructed by B. Leclerc and J.-Y. Thibon [14] (see also [15], [21]). In the last section, we present conjectures on the structure of V (λ). 2. Review on crystal bases In this section, we review very briefly the quantized universal enveloping algebra and crystal bases. We refer the reader to [12], [8], and [11].
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MASAKI KASHIWARA
2.1. Quantized universal enveloping algebras We define the quantized universal enveloping algebra Uq (g). Assume that we are given the following data: P : a free Z-module (called a weight lattice), I : an index set (for simple roots), αi ∈ P for i ∈ I (called a simple root), h i ∈ P ∗ = HomZ (P, Z) (called a simple coroot), ( · , · ) : P × P → Q a bilinear symmetric form. We denote by h · , · i : P ∗ × P → Z the canonical pairing. The data above are assumed to satisfy the following axioms: (αi , αi ) > 0 for any i ∈ I , 2(αi , λ) for any i ∈ I and λ ∈ P, hh i , λi = (αi , αi ) (αi , α j ) ≤ 0 for any i, j ∈ I with i 6= j.
(2.1) (2.2) (2.3)
Let us choose a positive integer d such that (αi , αi )/2 ∈ Z d −1 for any i ∈ I . Now let q be an indeterminate, and set K = Q(qs ),
where qs = q 1/d .
(2.4)
Definition 2.1 The quantized universal enveloping algebra Uq (g) is the algebra over K generated by the symbols ei , f i (i ∈ I ), and q(h) (h ∈ d −1 P ∗ ) with the following defining relations: (1) q(h) = 1 for h = 0; (2) q(h 1 )q(h 2 ) = q(h 1 + h 2 ) for h 1 , h 2 ∈ d −1 P ∗ ; (3) q(h)ei q(h)−1 = q hh,αi i ei and q(h) f i q(h)−1 = q −hh,αi i f i for any i ∈ I and h ∈ d −1 P ∗ ; (4) [ei , f j ] = δi j (ti − ti−1 )/(qi − qi−1 ) for i, j ∈ I ; here qi = q (αi ,αi )/2 and ti = q ((αi , αi )/2)h i ; (5) (Serre relation) for i 6 = j, b X
(k)
(b−k)
(−1)k ei e j ei
=
k=0
b X
(k)
(−1)k f i
(b−k)
f j fi
= 0;
k=0
here b = 1 − hh i , α j i and (k)
ei
= eik /[k]i !,
(k)
fi
[k]i = (qik − qi−k )/(qi − qi−1 ),
= f ik /[k]i !, [k]i ! = [1]i · · · [k]i .
QUANTIZED AFFINE ALGEBRAS
121
Sometimes we need an algebraically closed field containing K , for example, [ b= K C (q 1/n ) ,
(2.5)
n
b. and we consider Uq (g) as an algebra over K We denote by Uq (g)Q the subalgebra of Uq (g) over Q[qs±1 ] generated by the (n)
(n)
ei ’s, the f i ’s (i ∈ I ), and q h (h ∈ d −1 P ∗ ). Let us denote by W the Weyl group, the subgroup of GL(P) generated by the simple reflections si : si (λ) = λ − hh i , λiαi . P ± Let 1 ⊂ Q = i Zαi be the set of roots. Let 1P = 1 ∩ Q ± be the set of positive and negative roots, respectively. Here Q ± = ± i Z≥0 αi . Let 1re be the set re of real roots: 1re ± = 1± ∩ 1 . 2.2. Crystals We do not review the notion of crystals, but we refer the reader to [12], [8], and [11]. We say that a crystal B over Uq (g) is a regular crystal if, for any J ⊂ I such that {αi ; i ∈ J } is of finite-dimensional type, B is, as a crystal over Uq (g J ), isomorphic to the crystal bases associated with an integrable Uq (g J )-module. Here Uq (g J ) is the subalgebra of Uq (g) generated by e j , f j ( j ∈ J ), and q h (h ∈ d −1 P ∗ ). By [11], the Weyl group W acts on any regular crystal. This action S is given by ( hh ,wt(b)i f˜i i b if hh i , wt(b)i ≥ 0, Ssi b = −hh i ,wt(b)i e˜i b if hh i , wt(b)i ≤ 0. Let us denote by Uq− (g) (resp., Uq+ (g)) the subalgebra of Uq (g) generated by the f i ’s (resp., by the ei ’s). Then Uq− (g) has a crystal base denoted by B(∞) (see [8]). The unique weight vector of B(∞) with weight zero is denoted by u ∞ . Similarly, Uq+ (g) has a crystal base denoted by B(−∞), and the unique weight vector of B(−∞) with weight zero is denoted by u −∞ . Let ψ be the ring automorphism of Uq (g) that sends qs , ei , f i , and q(h) to qs , f i , ei , and q(−h). It gives a bijection B(∞) ' B(−∞) by which u ∞ , e˜i , f˜i , εi , ϕi , wt correspond to u −∞ , f˜i , e˜i , ϕi , εi , − wt. Let us denote by U˜ q (g) the modified quantized universal enveloping algebra L ˜ ˜ λ∈P Uq (g)aλ (see [11]). Then Uq (g) has a crystal base B(Uq (g)). As a crystal, ˜ B(Uq (g)) is regular and isomorphic to G B(∞) ⊗ Tλ ⊗ B(−∞). λ∈P
Here Tλ is the crystal consisting of a single element tλ with εi (tλ ) = ϕi (tλ ) = −∞ and wt(tλ ) = λ.
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MASAKI KASHIWARA
Let ∗ be the anti-involution of Uq (g) that sends q(h) to q(−h), and qs , ei , f i to themselves. The involution ∗ of Uq (g) induces an involution ∗ on B(∞), B(−∞), B(U˜ q (g)). Then e˜i∗ = ∗ ◦ e˜i ◦ ∗, and so on, give another crystal structure on B(∞), B(−∞), B(U˜ q (g)). We call it the star crystal structure. In the case of B(U˜ q (g)), these two crystal structures are compatible, and B(U˜ q (g)) may be considered as a crystal over g ⊕ g. Hence, for example, Sw∗ , the Weyl group action on B(U˜ q (g)) with respect to the star crystal structure, is a crystal automorphism of B(U˜ q (g)) with respect to the original crystal structure. In particular, the two Weyl group actions Sw and Sw∗ 0 commute with each other. The formulas concerning B(U˜ q (g)) are given in Appendix B. Note that we always have εi (b) + ϕi∗ (b) = εi∗ (b) + ϕi (b) ≥ 0
for any b ∈ B(∞).
(2.6)
2.3. Schubert decomposition of crystal bases For w ∈ W with a reduced expression si1 · · · si` , we define the subset Bw (∞) of B(∞) by Bw (∞) = { f˜ia11 · · · f˜ia` ` u ∞ ; a1 , . . . , a` ∈ Z≥0 }. (2.7) Then Bw (∞) does not depend on the choice of a reduced expression. We refer the reader to [9] on the details of Bw (∞) and its relationship with the Demazure module. We have (see [9]) (i) Bw (∞)∗ = Bw−1 (∞); (ii) if w0 ≤ w, then Bw0 ( ∞) ⊂ Bw (∞); (iii) if si w < w, then f˜i Bw (∞) ⊂ Bw (∞); (iv) e˜i Bw (∞) ⊂ Bw (∞) t {0}; (v) if both b and f˜i b belong to Bw (∞), then all f˜ik b (k ≥ 0) belong to Bw (∞). Here ≤ is the Bruhat order. Set [ B w (∞) = Bw (∞) \ Bw0 (∞) . w0 <w
P. Littelmann [16] showed that B(∞) =
G
B w (∞).
w∈W
We have B w (∞)∗ = B w−1 (∞).
(2.8)
If si w < w, then e˜imax B w (∞) ⊂ B siw (∞),
f˜i B w (∞) ⊂ B w (∞).
(2.9)
In particular, εi (b) > 0 for any b ∈ B w (∞). Here we use the notation e˜imax b = ε (b) e˜i i b.
QUANTIZED AFFINE ALGEBRAS
123
2.4. Global bases Let A ⊂ K be the subring of K consisting of rational functions in qs without pole at qs = 0. Let − be the automorphism of K sending qs to qs−1 . Set K Q := Q[qs , qs−1 ]. Let V be a vector space over K , let L 0 be an A-submodule of V , let L ∞ be an Asubmodule, and let VQ be a K Q -submodule. Set E := L 0 ∩ L ∞ ∩ VQ . Definition 2.2 ([8]) We say that (L 0 , L ∞ , VQ ) is balanced if each of L 0 , L ∞ , and VQ generates V as a K vector space, and if the following equivalent conditions are satisfied. (i) E → L 0 /qs L 0 is an isomorphism. (ii) E → L ∞ /qs−1 L ∞ is an isomorphism. (iii) (L 0 ∩ VQ ) ⊕ (qs−1 L ∞ ∩ VQ ) → VQ is an isomorphism. (iv) A ⊗Q E → L 0 , A ⊗Q E → L ∞ , K Q ⊗Q E → VQ , and K ⊗Q E → V are isomorphisms. Let − be the ring automorphism of Uq (g) sending qs , q h , ei , f i to qs−1 , q −h , ei , f i . h (n) (n) Let Uq (g)Q be the K Q -subalgebra of Uq (g) generated by ei , f i , and qn (h ∈ P ∗ ). Let M be a Uq (g)-module. Let − be an involution of M satisfying (au)− = a¯ u¯ for any a ∈ Uq (g) and u ∈ M. In this paper we call such an involution a bar involution. Let (L , B) be a crystal base of an integrable Uq (g)-module M. Let MQ be a Uq (g)Q -submodule of M such that (MQ )− = MQ
and
(u − u) ∈ (qs − 1)MQ
for every u ∈ MQ .
(2.10)
Definition 2.3 If (L , L, MQ ) is balanced, we say that M has a global basis. ∼ E := L ∩ L ∩ MQ be the inverse of E −→ ∼ L/qs L. In such a case, let G : L/qs L −→ Then {G(b); b ∈ B} forms a basis of M. We call this basis a (lower) global basis. The global basis enjoys the following properties (see [8], [10]). (i) G(b) = G(b) for any b ∈ B. (ii) For any n ∈ Z≥0 , {G(b); εi (b) ≥ n} is a basis of the K Q -submodule P (m) MQ . m≥n f i (iii) For any i ∈ I and b ∈ B, we have X i 0 f i G(b) = 1 + εi (b) i G( f˜i b) + Fb,b 0 G(b ). b0
Here the sum ranges over b0 ∈ B such that εi (b0 ) > 1 + εi (b). The coefficient 1−εi (b0 ) i Fb,b Q[qs ]. 0 belongs to qs qi
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MASAKI KASHIWARA
A similar statement holds for ei G(b).
3. Extremal weight modules 3.1. Extremal vectors Let M be an integrable Uq (g)-module. A vector u ∈ M of weight λ ∈ P is called extremal (see [1], [11]) if we can find vectors {u w }w∈W satisfying the following properties: uw = u
for w = e;
if hh i , wλi ≥ 0, then ei u w = 0 and if hh i , wλi ≤ 0, then f i u w = 0 and
(3.1) (hh ,wλi) fi i u w = u si w ; (−hh ,wλi) ei i = u si w .
(3.2) (3.3)
Hence if such {u w } exists, then it is unique and u w has weight wλ. We denote u w by Sw u. Similarly, for a vector b of a regular crystal B with weight λ, we say that b is an extremal vector if it satisfies the following similar conditions: we can find vectors {bw }w∈W such that bw = b
for w = e;
(3.4)
hh ,wλi if hh i , wλi ≥ 0, then e˜i bw = 0 and f˜i i bw = bsi w ;
if hh i , wλi ≤ 0, then f˜i vw = 0 and
−hh ,wλi e˜i i bw
= bsi w .
(3.5) (3.6)
Then bw must be Sw b. For λ ∈ P, let us denote by V (λ) the Uq (g)-module generated by u λ with the defining relation that u λ is an extremal vector of weight λ. There are, in fact, infinitely many linear relations on u λ . We proved in [11]∗ that V (λ) has a global crystal base (L(λ), B(λ)). Moreover, the crystal B(λ) is isomorphic to the subcrystal of B(∞) ⊗ tλ ⊗ B(−∞) consisting of vectors b such that b∗ is an extremal vector of weight −λ. We denote by the same letter u λ the element of B(λ) corresponding to u λ ∈ V (λ). Then u λ ∈ B(λ) corresponds to u ∞ ⊗ tλ ⊗ u −∞ . Note that, for b1 ⊗ tλ ⊗ b2 ∈ B(∞) ⊗ tλ ⊗ B(−∞) belonging to B(λ), one has εi∗ (b1 ) ≤ max hh i , λi, 0 and ϕi∗ (b2 ) ≤ max − hh i , λi, 0 for any i ∈ I. (3.7) For any w ∈ W , u λ 7→ Sw−1 u wλ gives an isomorphism of Uq (g)-modules: ∼ V (wλ). V (λ) −→ [11], it is denoted by V max (λ) because I thought there would be a natural Uq (g)-module whose crystal base is the connected component of B(λ).
∗ In
QUANTIZED AFFINE ALGEBRAS
125
Similarly, letting Sw∗ be the Weyl group action on B(U˜ q (g)) with respect to the star ∼ crystal structure and regarding B(λ) as a subcrystal of B(U˜ q (g)), Sw∗ : B(U˜ q (g)) −→ B(U˜ q (g)) induces an isomorphism of crystals ∼ B(wλ). Sw∗ : B(λ) −→ For a dominant weight λ, V (λ) is an irreducible highest weight module of highest weight λ, and V (−λ) is an irreducible lowest weight module of lowest weight −λ.
3.2. Dominant weights Definition 3.1 For a weight λ ∈ P and w ∈ W , we say that λ is w-dominant (resp., w-regular) if −1 re hβ, λi ≥ 0 (resp., hβ, λi 6= 0) for any β ∈ 1re + ∩ w 1− . If λ is w-dominant and w-regular, we say that λ is regularly w-dominant. If w = si` · · · si1 is a reduced expression, then we have −1 re 1re + ∩ w 1− = {si 1 · · · si k−1 αi k ; 1 ≤ k ≤ `}.
Hence λ is w-dominant (resp., w-regular) if and only if hh ik , sik−1 · · · si1 λi ≥ 0
resp., hh k , sik−1 · · · si1 λi 6= 0 .
(3.8)
Conversely, one has the following lemma. 3.2 For i 1 , . . . , il ∈ I and a weight λ, assume that LEMMA
hh ik , sik−1 sik−1 · · · si1 λi > 0
for k = 1, . . . , l.
Then w = sil · · · si1 is a reduced expression. Proof By the induction on l, we may assume that sil−1 · · · si1 is a reduced expression. If l(w) < l, then there exists k with 1 ≤ k ≤ l − 1 such that sil−1 · · · sik+1 (h ik ) = −h il . Hence hh il , sil−1 · · · si1 λi = −hh ik , sik−1 sik−1 · · · si1 λi < 0, which is a contradiction. This lemma implies the following lemma.
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LEMMA 3.3 Let w1 , w2 ∈ W , and let λ be an integral weight. If λ is regularly w2 -dominant and w2 λ is regularly w1 -dominant, then `(w1 w2 ) = `(w1 ) + `(w2 ) and λ is regularly w1 w2 -dominant. Here ` : W → Z is the length function.
3.4 Let λ ∈ P, and let b1 ∈ B w1 (∞), b2 ∈ B w2 (−∞). If b := b1 ⊗ tλ ⊗ b2 belongs to B(λ), then (i) λ is regularly w1 -dominant and −λ is regularly w2 -dominant, (ii) `(w1 w2−1 ) = `(w1 ) + `(w2 ), (iii) and we have PROPOSITION
Sw∗ 2 (b1 ⊗ tλ ⊗ b2 ) ∈ Bw
−1 1 w2
(∞) ⊗ tw2 λ ⊗ u −∞ ,
Sw∗ 1 (b1 ⊗ tλ ⊗ b2 ) ∈ u ∞ ⊗ tw1 λ ⊗ Bw
−1 2 w1
(−∞).
More generally, if w1 = w0 w00 with `(w1 ) = `(w0 ) + `(w00 ), then Sw∗ 00 (b1 ⊗ tλ ⊗ b2 ) ∈ B w0 (∞) ⊗ tw00 λ ⊗ Bw2 w00 −1 (−∞). Proof Assume that w1 si < w1 . Then c := εi∗ (b1 ) > 0 by (2.9). Hence hh i , λi ≥ c > 0 by (3.7). We have e˜i∗ max b1 ∈ B¯ w1 si (∞) and b0 = Si∗ (b1 ⊗ tλ ⊗ b2 ) = (e˜i∗ max b1 ) ⊗ tsi λ ⊗ (e˜i∗ hh i ,λi−c b2 ).
(3.9)
Hence λ is regularly w1 -dominant by the induction on the length of w1 . The other statement in (i) is similarly proved, and (ii) follows from (i) and Lemma 3.3. In (3.9), e˜i∗ hh i ,λi−c b2 belongs to Bw2 si (−∞) since (ii) implies w2 si > w2 . Repeating this, we obtain (iii). 4. Affine quantum algebras In the sequel we assume that g is affine. 4.1. Affine root systems Although the materials in this subsection are more or less classical, we review the affine algebras in order to fix the notation. Let g be an affine Lie algebra, and let t be its Cartan subalgebra (assuming that they are defined over Q). Let I be the index set of simple roots, let αi ∈ t∗ be the simple roots, and let h i ∈ t be the simple coroots (i ∈ I ). We choose a Cartan subalgebra t such that {αi }i∈I and {h i }i∈I are linearly independent and dim t = rank g + 1. Let
QUANTIZED AFFINE ALGEBRAS
127
us set the root lattice and coroot lattice by M Q= Zαi ⊂ t∗ and
Q∨ =
i
M
Zh i ⊂ t.
i
P P Set Q ± = ± i Z≥0 αi and Q ∨ ± = ± i Z≥0 h i . Let δ ∈ Q + be a unique element satisfying {λ ∈ Q; hh i , λi = 0 for every i} = Zδ. Similarly, we define c ∈ Q ∨ + by {h ∈ Q ∨ ; hh, αi i = 0 for every i} = Zc. We write X X δ= ai αi and c= ai∨ h i . (4.1) i
i
We take a W -invariant nondegenerate symmetric bilinear form (·, ·) on t∗ normalized by (δ, λ) = hc, λi for any λ ∈ t∗ . (4.2) Then this symmetric form has the signature (dim t − 1, 1). We sometimes identify t and t∗ by this symmetric form. By this identification, δ and c correspond to each other. We have (αi , αi ) ai∨ = ai . (4.3) 2 Note that (αi , αi )/2 takes the values 1, 2, 3, 1/2, 1/3. Hence we have for each i, (αi , αi ) ∈Z 2
or
2 ∈ Z. (αi , αi )
(4.4)
If g is untwisted, then 2/(αi , αi ) is an integer. Let us set t∗cl = t∗ /Q δ, and let cl : t∗ → t∗cl be the canonical projection. We have M t∗cl ' (Q h i )∗ . i∈I ∗0 ∗ ∗0 Set t∗0 = {λ ∈ t∗ ; hc, λi = 0} and t∗0 cl = cl(t ) ⊂ tcl . Then tcl has a positive-definite symmetric form induced by the one of t∗ .
4.1 For any a ∈ Q, LEMMA
cl : λ ∈ t∗ ; (λ, λ) = a and (λ, δ) 6 = 0 → t∗cl \ t∗0 cl is bijective. Proof Let λ ∈ t∗ such that (λ, δ) 6= 0. Setting µ = λ + xδ for x ∈ Q, we have (µ, µ) = (λ + xδ, λ + xδ) = (λ, λ) + 2x(λ, δ). Hence λ+ xδ has square length a if and only if x = (a −(λ, λ))/2(λ, δ).
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As a corollary we have the following proposition. PROPOSITION 4.2 The space t∗ endowed
with an invariant symmetric form as above, with simple roots and coroots, is unique up to a canonical isomorphism. Proof For example, take ρ ∈ t∗ such that hh i , ρi = 1 for any i and (ρ, ρ) = 0. Lemma 4.1 guarantees its existence and its uniqueness. The αi ’s and ρ form a basis of t∗ . In particular, for any Dynkin diagram isomorphism ι (i.e., a bijection ι : I → I such that hh ι(i) , αι( j) i = hh i , α j i), there exists a unique isomorphism of t∗ that sends αi to αι(i) and leaves the symmetric form invariant. Let 1 ⊂ t∗ be the root system of g, and let 1re be the set of real roots: 1re = 1 \ Z δ. For β ∈ t∗ with (β, β) 6 = 0, we set β ∨ = 2β/(β, β). Then 1∨ := {β ∨ ; β ∈ 1re } ∪ (Z c \ {0}) ⊂ t is the root system for the dual Lie algebra of g. We set 1± = 1 ∩ Q±. Let us denote by 1cl the image of 1e by cl. Then 1cl is a finite subset of t∗0 cl , and (1cl , t∗0 ) is a (not necessarily reduced) root system. We call an element of 1 a cl cl classical root. Let O(t∗ ) be the orthogonal group of t∗ with respect to the invariant symmetric form. Let O(t∗ )δ be the isotropy subgroup of δ; that is, O(t∗ )δ = {g ∈ O(t∗ ); gδ = δ}. Then there are canonical group homomorphisms cl : O(t∗ )δ → GL(t∗cl )
and
cl0 : O(t∗ )δ → O(t∗0 cl ).
The homomorphism cl : O(t∗ )δ → GL(t∗cl ) is injective. For β ∈ 1e , let sβ be the corresponding reflection λ 7→ λ − hβ ∨ , λiβ. Let W be the Weyl group, that is, the subgroup of GL(t∗ ) generated by the sβ ’s. Since W ⊂ O(t∗ )δ , there are group homomorphisms W → GL(t∗cl ) and W → O(t∗0 cl ). Let us denote by Wcl the image of W → O(t∗0 ). Then W is the Weyl group of cl cl ∗0 the root system (1cl , tcl ). For ξ ∈ t∗0 , we set T (λ) = λ + (δ, λ)ξ − (ξ, λ)δ −
(ξ, ξ ) (δ, λ)δ. 2
(4.5)
Then T belongs to O(t∗ )δ , and T depends only on cl(ξ ). For ξ0 ∈ t∗0 cl , let us define −1 ∗ t (ξ0 ) ∈ O(t )δ as the right-hand side of (4.5) with ξ ∈ cl (ξ0 ). Then ∗ ∗0 t : t∗0 (4.6) cl → Ker cl0 : O(t )δ → GL(tcl ) is a group isomorphism.
QUANTIZED AFFINE ALGEBRAS
129
We have g ◦ t (ξ ) ◦ g −1 = t cl0 (g)(ξ ) for g ∈ O(t∗ )δ and ξ ∈ t∗0 cl .
(4.7)
For β ∈ t∗ such that (β, β) 6 = 0, let us denote by sβ the reflection sβ (λ) = λ − (β ∨ , λ)β . Then we have for β ∈ t∗0 such that (β, β) 6 = 0, sβ−aδ sβ = t (aβ ∨ ).
(4.8)
Wcl is generated by {si ; i 6 = i 0 }.
(4.9)
There exists i 0 such that
(2)
If g is not isomorphic to A2n , such an i 0 is unique up to a Dynkin diagram automor(2) phism and (αi0 , αi0 ) = 2, ai0 = ai∨0 = 1. In the case of A2n , there are two choices of i 0 , two extremal nodes, and (αi0 , αi0 ) = 1 or 4, and accordingly ai0 = 2 or 1, ai∨0 = 1 or 2. For α ∈ 1re or α ∈ 1cl , we set (α, α) cα = max 1, 2 and ci = cαi . Then we have, for any α ∈ 1re , {n ∈ Z; α + nδ ∈ 1} = Z cα .
(4.10)
We set ∨ Q cl = cl(Q), Q∨ cl = cl(Q ), P Here Q ∨ = α∈1re Zα ∨ . We have an exact sequence
e = Q cl ∩ Q ∨ Q cl .
cl0 t e− 1 −→ Q → W −−→ Wcl −→ 1.
(4.11)
(4.12)
e ∩ Q>0 cl(α) with the smallest length. For any α ∈ 1re , let α˜ be the element in Q We set ˜ = {α; 1 ˜ α ∈ 1re }. e is the root lattice of 1. ˜ is a reduced root system, and Q ˜ Then 1 Remark 4.3 Any affine Lie algebra is either untwisted or the dual of an untwisted affine algebra (2) or A2n .
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MASAKI KASHIWARA
(i) (ii)
e = Q ∨ ⊂ Q cl , 1 ˜ = cl(1∨ re ), α˜ = α ∨ . If g is untwisted, then Q cl e = Q cl ⊂ Q ∨ , 1 ˜ = cl(1re ), If g is the dual of an untwisted algebra, then Q cl α˜ = α. (2) e = Q cl = Q ∨ , 1 ˜ = cl(1re ) = cl(1∨ re ). For any α ∈ 1re , If g = A2n , then Q cl one has ( cl(α) if (α, α) 6 = 4, α˜ = cl(α)/2 if (α, α) = 4.
(iii)
Note that (α − δ)/2 ∈ 1re if (α, α) = 4. (2) If g 6 = A2n , then α˜ = cα α ∨ . PROPOSITION
4.4
e For ξ ∈ Q, X X 1 X (β, ξ ) /cβ = (β ∨ , ξ )+ . l t (ξ ) = (β, ξ )+ /cβ = 2 β∈1cl
β∈1cl
˜ β∈1
Here a+ = max(a, 0). Proof For β ∈ 1cl , let us denote by β 0 the unique element of 1+ such that cl(β 0 ) = β and β 0 − nδ 6∈ 1+ for any n > 0. Note that (β, ξ ) ∈ cβ Z. We have t (ξ )−1 1− ∩ 1+ = γ ∈ 1+ ; γ − (γ , ξ )δ ∈ 1− , and l(t (ξ )) is the number of elements in this set. By setting γ = β 0 + ncβ δ, it is isomorphic to (β, n) ∈ 1cl × Z; n ≥ 0 and β 0 + ncβ − (β, ξ ) δ ∈ 1− = (β, n) ∈ 1cl × Z; 0 ≤ n < (β, ξ )/cβ . Since (β, ξ )/cβ is an integer, we have X l t (ξ ) = (β, ξ )/cβ + . β∈1cl
The other equalities easily follow. 4.5 e For ξ ∈ Q and w ∈ Wcl , COROLLARY
l t (wξ ) = l t (ξ ) .
QUANTIZED AFFINE ALGEBRAS
131
We choose a weight lattice P ⊂ t∗ satisfying αi ∈ P and h i ∈ P ∗ for any i ∈ I ; for every i ∈ I, there exists 3i ∈ P such that hh j , 3i i = δ ji .
(4.13)
We set P 0 = λ ∈ P; hc, λi = 0 ,
Pcl = cl(P) ⊂ t∗cl ,
and
Pcl0 = cl(P 0 ). (4.14)
We have Pcl =
M (Zh i )∗ . i∈I
LEMMA 4.6 e the following two conditions are equivalent: For λ ∈ P 0 and µ ∈ Q, (i) λ and µ are in the same Weyl chamber (i.e., for any α ∈ 1re , (cl(α), µ) > 0 implies (α, λ) ≥ 0); (ii) λ is t (µ)-dominant.
Proof For α ∈ 1re , let us take α 0 ∈ (α+Zδ)∩1+ such that cl(α 0 ) = cl(α) and α 0 −nδ 6∈ 1+ for any n ∈ Z>0 . Then for α = α 0 + nδ ∈ 1+ , α ∈ 1+ ∩ t (µ)−1 1− ⇔ t (µ)α = α − (α, µ)δ = α 0 + n − (α, µ) δ ∈ 1− ⇔ 0 ≤ n < (α, µ). (i) ⇒ (ii) Now assume that α = α 0 + nδ ∈ 1+ ∩ t (µ)−1 1− . Then 0 ≤ n < (α, µ), and (i) implies (α, λ) ≥ 0 (ii) ⇒ (i) Assume that (α, µ) > 0. Then taking n = 0, α 0 ∈ 1+ ∩ t (µ)−1 1− , and hence (α, λ) = (α 0 , λ) ≥ 0. The following lemma is similarly proved. LEMMA 4.7 e the following two conditions are equivalent: For λ ∈ P 0 and µ ∈ Q, (i) for any α ∈ 1cl , (α, µ) > 0 implies (α, λ) > 0; (ii) λ is regularly t (µ)-dominant.
Let us choose i 0 ∈ I as in (4.9), and let W0 be the subgroup of W generated by e {si ; i ∈ I \ {i 0 }}. Then W is a semidirect product of W0 and Q.
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LEMMA 4.8 e and w ∈ W0 . If ξ is regularly w-dominant, then Let ξ ∈ Q l t (ξ ) = l t (ξ )w−1 + l(w).
Proof We prove the assertion by the induction on l(w). Write w = si w0 with w > w 0 and i 6 = i 0 . Then l(t (ξ )) = l(t (ξ )w0−1 ) + l(w0 ). Hence it is enough to show t (ξ )w0−1 > t (ξ )w0−1 si or, equivalently, t (ξ )w0−1 αi ∈ 1− . We have t (ξ )w0−1 αi = w0
−1
αi − (w0 ξ, αi )δ.
Since (w0 ξ, αi ) > 0, the coefficient of αi0 in t (ξ )w 0−1 αi is negative, and hence t (ξ )w0−1 αi is a negative root. 4.2. Affinization Let P and Pcl be as in (4.13). We denote by Uq (g) the quantized universal enveloping algebra with P as a weight lattice. We denote by Uq0 (g) the quantized universal enveloping algebra with Pcl as a weight lattice. Hence Uq0 (g) is a subalgebra of Uq (g) generated by the ei ’s, the f i ’s, and q h (h ∈ d −1 (Pcl )∗ ). When we talk about an integrable Uq (g)-module (resp., Uq0 (g)-module), the weight of its element belongs to P (resp., Pcl ). L Let M be a Uq0 (g)-module with the weight decomposition M = λ∈Pcl Mλ . We L define a Uq (g)-module Maff with a weight decomposition Maff = λ∈P (Maff )λ by (Maff )λ = Mcl(λ) . The actions of ei and f i are defined in an obvious way, so that the canonical homomorphism cl : Maff → M is Uq0 (g)-linear. We define the Uq0 (g)-linear automorphism ∼ Mcl(λ) = Mcl(λ+δ) −→ ∼ (Maff )λ+δ . z of Maff with weight δ by (Maff )λ −→ Let us choose 0 ∈ I satisfying Wcl is generated by {si ; i 6 = 0},
and
a0 = 1.
(4.15)
P (2) Recall that δ = i ai αi . When g = A2n , α0 is the longest simple root. Choose a section s : Pcl → P of cl : P → Pcl such that s(cl(αi )) = αi for any i ∈ I \ {0}. Then M is embedded into Maff by s as a vector space. We have an isomorphism of Uq0 (g)-modules Maff ' K [z, z −1 ] ⊗ M.
(4.16)
Here ei ∈ Uq0 (g) and f i ∈ Uq0 (g) act on the right-hand side by z δi0 ⊗ ei and z −δi0 ⊗ f i .
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133
Similarly, for a crystal with weights in Pcl , we can define its affinization Baff by G Baff = Bcl(λ) . (4.17) λ∈P
Uq0 (g)-module
If an integrable M has a crystal base (L , B), then its affinization Maff has a crystal base (L aff , Baff ). For a ∈ K , we define the Uq0 (g)-module Ma by Ma = Maff /(z − a)Maff .
(4.18)
4.3. Simple crystals In [1], we defined the notion of simple crystals and studied their properties. Definition 4.9 We say that a finite regular crystal B (with weights in Pcl0 ) is a simple crystal if B satisfies the following: (1) there exists λ ∈ Pcl0 such that the weight of any extremal vector of B is contained in Wcl λ; (2) ](Bλ ) = 1. Simple crystals have the following properties (see [1]). LEMMA 4.10 A simple crystal B is connected. LEMMA 4.11 The tensor product of simple crystals is also simple. PROPOSITION 4.12 A finite-dimensional integrable Uq0 (g)-module with a simple crystal base is irreducible.
5. Affine extremal weight modules 5.1. Extremal vectors—affine case We now prove one of the main results of this paper. In the sequel we employ the notation ε (b)
e˜imax b = e˜i i
b,
ϕ (b) f˜imax b = f˜i i b,
and similarly for e˜i∗ max and f˜i∗ max .
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MASAKI KASHIWARA
THEOREM 5.1 Let λ ∈ P 0 be a level zero weight. Then the weight of any extremal vector of B(λ) is contained in cl−1 cl(W λ).
Proof We regard B(λ) as a subcrystal of B(∞) ⊗ tλ ⊗ B(−∞) ⊂ B(U˜ q (g)). We show that cl(wt(b)) and − cl(wt(b∗ )) are in the same Wcl -orbit whenever b and b∗ are extremal vectors. For any b1 ⊗ tλ ⊗ b2 , we have f˜imax (b1 ⊗ tλ ⊗ b2 ) = b10 ⊗ tλ ⊗ f˜imax b2
for some b10 .
(For the action of f˜imax , etc., on B(U˜ q (g)), see Appendix B.) Hence any extremal vector b ∈ B(λ) has the form b1 ⊗ tλ ⊗ u −∞ after applying the f˜imax ’s. Hence we may further assume the following conditions on b: b has the form b1 ⊗ tλ ⊗ u −∞ ;
(5.1)
for any vector of the form b10 ⊗ tµ ⊗ u −∞ in {Sw Sw∗ 0 b; w, w0 ∈ W }, the length of wt(b10 ) is greater than or equal to the length of wt(b1 ).
(5.2)
P P Here the length of i m i αi is by definition i |m i |. Take i ∈ I . We write λi = hh i , λi and wti (b1 ) = hh i , wt(b1 )i for brevity. Note that we have εi∗ (b1 ) ≤ max(λi , 0). We show wti (b1 ) ≥ 0 for every i in several steps. (1) The case λi ≤ 0 and λi + wti (b1 ) ≤ 0. Since b1 ⊗ tλ ⊗ u −∞ is a lowest weight vector in the i-string, one has ϕi (b) = max(ϕi (b1 ) + λi , 0) = 0, and hence ϕi (b1 ) + λi ≤ 0. Similarly, εi∗ (b) = 0 because b∗ is a highest weight vector in the i-string. Therefore one has −ϕ (b )−λ Si∗ Si (b1 ⊗ tλ ⊗ u −∞ ) = f˜i∗ −λi (e˜i max b1 ⊗ tλ ⊗ e˜i i 1 i u −∞ )
= ( f˜i∗ ϕi (b1 ) e˜i max b1 ) ⊗ tsi λ ⊗ u −∞ . The last equality follows from Si∗ Si (b) = ( f˜i∗ k e˜i max b1 ) ⊗ tsi λ ⊗ u −∞ for some k. Hence the minimality of b1 gives 0 ≤ ϕi (b1 ) − εi (b1 ) = wti (b1 ). (2) The case λi > 0 and λi + wti (b1 ) ≤ 0. We show that this case cannot occur. In this case, as in (i), ϕi (b1 ) + λi ≤ 0.
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135
On the other hand, ϕi∗ (b1 ⊗ tλ ⊗ u −∞ ) = max(εi∗ (b1 ) − λi , 0) = 0 implies εi∗ (b1 ) ≤ λi . Hence we obtain (the first inequality by (2.6)) 0 ≤ εi∗ (b1 ) + ϕi (b1 ) = εi∗ (b1 ) − λi + ϕi (b1 ) + λi ≤ 0, which implies εi∗ (b1 ) = λi and ϕi (b1 ) = −λi . Then we have e˜i∗ max (b1 ⊗ tλ ⊗ u −∞ ) = (e˜i∗ max b1 ) ⊗ tsi λ ⊗ u −∞ . Hence the minimality of wt(b1 ) implies εi∗ (b1 ) = 0, and this contradicts εi∗ (b1 ) = λi > 0. (3) The case λi ≥ 0 and λi + wti (b1 ) ≥ 0. In this case, one has εi (b) = ϕi∗ (b) = 0, and hence ϕi (b) = λi +wti (b1 ), which implies ϕi (b)−(λi −εi∗ (b1 )) = ϕi∗ (b1 ) ≥ 0. Hence we have λi −ε (b1 ) ϕ (b) Si Si∗ (b1 ⊗ tλ ⊗ u −∞ ) = f˜i i (e˜i∗ max b1 ⊗ tsi λ ⊗ e˜i i u −∞ ) ∗
ϕi∗ (b) ∗ max e˜i b1 ) ⊗ tsi λ
= ( f˜i
⊗ u −∞ .
Hence we have ϕi∗ (b1 ) ≥ εi∗ (b1 ) or, equivalently, wti (b1 ) ≥ 0. (4) The case λi ≤ 0 and λi + wti (b1 ) ≥ 0. We have immediately wti (b1 ) ≥ 0. In all the cases we have wti (b1 ) ≥ 0. Since wt(b1 ) is of level zero, one has P 0 = hc, wt(b1 )i = i ai∨ wti (b1 ), which implies that wti (b1 ) = 0 for every i or, equivalently, cl(wt(b1 )) = 0. COROLLARY 5.2 For any λ ∈ P, the weight of any vector in B(λ) is contained in the convex hull of W λ.
Proof In the positive level case (i.e., hc, λi > 0), λ is conjugate to a dominant weight and B(λ) is isomorphic to the crystal base of an irreducible highest weight module. In this case, the assertion is well known. We can argue similarly in the negative level case. Assume that the level of λ is zero. Note that all vectors in B(λ) can be reached at an extremal vector after applying e˜imax and f˜imax by [11]. Hence the assertion follows from Theorem 5.1. Note that cl−1 cl(W λ) is contained in the convex hull of W λ, provided that cl(λ) 6 = 0. The following theorem is an immediate consequence of Corollary 5.2.
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THEOREM 5.3 Let M be an integrable Uq0 (g)-module, and let u be a vector in M of weight λ ∈ Pcl . Then the following conditions are equivalent: (i) u is an extremal vector; (ii) the weights of Uq0 (g)u are contained in the convex hull of Wcl λ; (iii) Uq0 (g)β u = 0 for any β ∈ 1cl such that (β, λ) ≥ 0.
In particular, for any λ ∈ P, V (λ) is isomorphic to the Uq (g)-module generated by a weight vector u of weight λ with (iii) in Corollary 5.2 and the following integrability condition as defining relations: 1+hh i ,λi
fi
u=0
if hh i , λi ≥ 0
and
1−hh i ,λi
ei
u=0
if hh i , λi ≤ 0.
5.2. Fundamental representations Let us take 0∨ ∈ I such that Wcl is generated by{si ; i 6 = 0∨ }
and
a0∨∨ = 1.
(5.3)
P ∨ (2) ∨ Recall that c = i ai h i . When g = A2n , 0 is the shortest simple root. We set ∨ I0∨ = I \ {0 }. For i ∈ I0∨ , we set $i = 3i − ai∨ 30∨ ∈ P 0 . L Hence we have Pcl0 = i∈I ∨ Z cl($i ). We say that λ ∈ P is a basic weight if cl(λ) is 0 Wcl -conjugate to some cl($i ) (i ∈ I0∨ ). Note that this notion does not depend on the choice of 0∨ . 5.4 P Assume that λ = i∈J $i for some subset J of I0∨ . Then one has the following: (i) any extremal vector of B(λ) is in the W -orbit of u λ ; (ii) B(λ) is connected. PROPOSITION
Proof Part (ii) follows from (i) because any vector is connected with an extremal vector. Let us prove (i). We use arguments similar to the proof of Theorem 5.1. Let us take an extremal vector b ∈ B(λ). Among the vectors in Sw Sw∗ 0 b with the form b1 ⊗ tµ ⊗ u −∞ , we take one such that wt(b1 ) has the smallest length. Then the proof in Theorem 5.1 shows that cl(wt(b1 )) = 0. Hence one has ( ∗ f˜i εi (b1 ) e˜i∗ max (b1 ) ⊗ tsi µ ⊗ u −∞ if µi ≥ 0, Si Si∗ (b1 ⊗ tµ ⊗ u −∞ ) = f˜∗ εi (b1 ) e˜i max (b1 ) ⊗ tsi µ ⊗ u −∞ if µi ≤ 0. i
QUANTIZED AFFINE ALGEBRAS
137
In both cases, the length of b1 remains unchanged after applying Si Si∗ . Therefore, applying Sw0−1 Sw∗ 0−1 , we can assume that w0 = 1 and µ = λ. For i ∈ I \ J , we have λi ≤ 0, which implies εi∗ (b1 ) = 0. If i ∈ J , then λi = 1, and hence εi∗ (b1 ) (≤ λi ) must be 0 or 1. On the other hand, we have λi −εi∗ (b1 )
Si∗ (b1 ⊗ tλ ⊗ u −∞ ) = e˜i∗ max b1 ⊗ tλ ⊗ e˜i
u −∞ .
If εi∗ (b1 ) = 1, then this contradicts the minimality of wt(b1 ). Hence εi∗ (b1 ) = 0 for every i ∈ J . Thus we have εi∗ (b1 ) = 0 for every i ∈ I , and hence b1 = u ∞ . Thus we obtain u λ = Sw b. The following theorem is a particular case of Proposition 5.4. THEOREM 5.5 If λ ∈ P is a basic weight, then any extremal vector of B(λ) is in the W -orbit of u λ .
We now study further properties of B(λ) for a basic weight λ. LEMMA 5.6 Let λ be a basic weight. Then {w ∈ W ; wλ = λ} is generated by {sβ ; β ∈ 1re + , (β, λ) = 0}.
Proof ∨ We may assume that λ = 3 j − a ∨j 3∨ 0 for some j ∈ I0 . Since a similar statement holds for (Wcl , t∗0 cl ), it is enough to show that t (ξ ) is contained in the subgroup G e generated by {sβ ; β ∈ 1re + , (β, λ) = 0}, provided that ξ ∈ Q and (ξ, λ) = 0. ∨ We have saδ−β sβ = t (aβ ) by (4.8). In particular, one has t (cβ β ∨ ) ∈ G whenever β ∈ 1re satisfies (β, λ) = 0. (2) e (ξ, λ) = 0} is (1) The case where g 6= A2n . It is enough to show that {ξ ∈ Q; ∨ e generated by {cβ β ; β ∈ 1cl , (β, λ) = 0}. In this case, Q has a basis {ci αi∨ ; i ∈ I0∨ }. e (ξ, λ) = 0} is generated by {ci α ∨ ; i ∈ I ∨ \ { j}}. Hence {ξ ∈ Q; i 0 L (2) e = Q = i∈I ∨ Zα˜ i . Hence {ξ ∈ (2) The case where g = A2n . In this case, Q 0 e (ξ, λ) = 0} has a basis {α˜ i ; i ∈ I ∨ \ { j}}. Hence the result follows from Q; 0
t (α˜ i ) =
( sδ−αi sαi s(δ−αi )/2 sαi
if (αi , αi ) = 2, if (αi , αi ) = 4.
Note that (δ − αi )/2 is a real root in the last case.
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LEMMA 5.7 For any β ∈ 1re and any λ ∈ P such that sβ λ = λ, we have Ssβ (u ∞ ⊗ tλ ⊗ u −∞ ) = u ∞ ⊗ tλ ⊗ u −∞ .
Proof Set aλ = u ∞ ⊗ tλ ⊗ u −∞ . We assume that β ∈ 1re + . We prove the assertion by the induction on the length of β. If β is a simple root, it is obvious. Otherwise, we can write β = si γ for a positive real root γ whose length is less than that of β. We have Ssβ = Si Ssγ Si . Set µ = si λ. Then sγ µ = µ, and hence we have Ssγ aµ = aµ by the induction hypothesis. Since Si Si∗ aλ = aµ or, equivalently, Si aλ = Si∗ aµ , we have Ssβ aλ = Si Ssγ Si aλ = Si Ssγ Si∗ aµ = Si Si∗ aµ = aλ . Lemmas 5.6 and 5.7 imply the following proposition. PROPOSITION 5.8 Let λ be a basic weight. (i) If w ∈ W satisfies wλ = λ, then Sw u λ = u λ and Sw∗ u λ = u λ . ∼ B(µ) does not depend on w ∈ (ii) For µ ∈ W λ, the isomorphism Sw∗ : B(λ) −→ W such that µ = wλ. Here we regard B(λ) and B(µ) as subcrystals of B(U˜ q (g)).
Remark 5.9 For a general λ ∈ P 0 , it is not true that the extremal weights of B(λ) belong to W λ. (1) For example, in λ = 2(31 − 30 ) in the A1 -case, f 0 f 1 u λ is an extremal vector with weight λ − δ. Remark 5.10 It is not true in general that wλ = λ implies Sw u λ = u λ . For example, in the case of (1) g = A2 , and λ = 31 + 32 − 230 , set w1 = t (α1 ) = s1 s0 s2 s1 and w2 = t (α2 ) = s1 s0 s1 s2 . Then w1 λ = w2 λ = λ − δ, but Sw1 u λ 6= Sw2 u λ . 5.11 For any λ ∈ P, Sw u λ = u λ if and only if w ∈ W is in the subgroup generated by {sβ ; β is a real root such that (β, λ) = 0}. CONJECTURE
Theorem 5.5 and Proposition 5.8 immediately imply the following result. 5.12 Assume that λ is a basic weight. PROPOSITION
QUANTIZED AFFINE ALGEBRAS
(i) (ii)
139
We have B(λ)λ = {u λ }. B(λ) is connected.
Proof Let b ∈ B(λ)λ . Then Theorem 5.5 implies b = Sw u λ for some w ∈ W with wλ = λ, and Proposition 5.8 implies Sw u λ = u λ . In order to show the finite multiplicity theorem for B($i ), we need the following result. 5.13 Assume that λ = cl(3i1 − ai∨1 30 ) for some i 1 ∈ I0∨ and µ ∈ Wcl λ. If w ∈ W satisfies l(w) ≥ ]Wcl and if µ is regularly w-dominant, then there exist w0 , w00 ∈ W such that w = w0 w00 , l(w) = l(w0 ) + l(w00 ), and λ = w00 µ. LEMMA
Proof Let w = si1 · · · sil be a reduced expression of w. Since l ≥ ]Wcl , there exists 0 ≤ j < k ≤ l such that cl(si1 · · · si j ) = cl(si1 · · · sik ). Hence si j+1 · · · sik = t (ξ ) for some e \ {0}. Replacing µ with sik+1 · · · si` µ, we reduce the lemma to the following ξ ∈ Q sublemma. 5.14 e If ξ ∈ Q \ {0} and if µ ∈ Wcl λ is regularly t (ξ )-dominant, then there exists w1 ∈ W such that λ = w1 µ and l(t (ξ )) = l(t (ξ )w1−1 ) + l(w1 ). SUBLEMMA
Proof Let us take w ∈ W0∨ := hsi ; i ∈ I0∨ i such that µ = wλ and λ is regularly wdominant. By Lemma 4.7, for β ∈ 1cl , (β, ξ ) > 0 implies (β, µ) > 0. Hence (β, w−1 ξ ) > 0 implies (β, λ) > 0. In particular, (β, λ) = 0 (resp., (β, λ) > 0) implies (β, w−1 ξ ) = 0 (resp., (β, w−1 ξ ) ≥ 0). For i ∈ I0∨ \ {i 1 }, (αi , w−1 ξ ) = 0 because (αi , λ) = 0. Moreover, (αi1 , w−1 ξ ) ≥ 0 because (αi1 , λ) > 0. Hence we have w−1 ξ = cλ for c > 0. Hence w−1 ξ is regularly w-dominant. Corollary 4.5 and Lemma 4.8 imply that l t (ξ ) = l t (w−1 ξ ) = l t (w−1 ξ )w −1 + l(w) = l(w) + l w−1 t (ξ ) . Then the sublemma follows by setting w1 = w−1 t (ξ ). PROPOSITION 5.15 Let λ ∈ P be a basic weight. Then for every ξ ∈ P, B(λ)ξ is a finite set.
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MASAKI KASHIWARA
Proof For w ∈ W and µ ∈ W λ, we define a subset Aw (µ) of B(U˜ q (g)) by Aw (µ) = b ⊗ tµ ⊗ u −∞ ∈ B(µ); b ∈ B w (∞) , and then we set Aw =
F
µ∈W λ
Aw (µ). Note that Aw (µ) is a finite set. One has
B(λ) ⊂
[ w,w1 ∈W
Sw∗ 1 Aw (w1−1 λ) .
We first show B(λ) ⊂
[ w1 ∈W, w∈W with `(w)≤N
Here N := ]Wcl . For b := b1 ⊗ tµ ⊗ u −∞ in Aw , we show [ b∈ w1 ∈W, w0 ∈W with `(w0 )≤N
Sw∗ 1 (Aw ).
(5.4)
Sw∗ 1 (Aw0 )
by the induction on `(w). Proposition 3.4 implies that µ is regularly w-dominant. We may assume that `(w) > N . By Lemma 5.13, there exists w1 = w0 w00 such that l(w) = l(w0 ) + l(w00 ), w0 6= 1, and λ0 := w00 µ satisfies cl(λ0 ) = cl(λ). By Proposition 3.4, one has Sw∗ 00 (b1 ⊗ tµ ⊗ u −∞ ) = b10 ⊗ tλ0 ⊗ b20 with b10 ∈ B w0 (∞) and b20 ∈ Bw00 −1 (−∞). Take i ∈ I such that w0 si < w 0 . Then λi0 > 0 implies i = i 1 . Hence c := εi∗ (b10 ) ≤ λi0 = 1. One has Si∗ (b10 ⊗ tλ0 ⊗ b20 ) = (e˜i∗ max b10 ) ⊗ tsi λ0 ⊗ e˜i∗ λi −c b20 . 0
(5.5)
If c = 1, then λi0 − c = 0. Take x ∈ W such that b20 ∈ B x (−∞). Then x ≤ w00 −1 since b20 ∈ Bw00 −1 (−∞). Since e˜i∗ max b10 ∈ B w0 si (∞), Proposition 3.4 implies Sx∗ (e˜i∗ max b10 ) ⊗ tsi λ ⊗ b20 ∈ Bw0 si x −1 (∞) ⊗ txsi λ ⊗ u −∞ . Since `(w0 si x −1 ) < `(w), the induction proceeds. Next assume that c = 0. Then λ j ≤ 0 for j ∈ I \ {i 1 } implies ε∗j (b10 ) = 0 for every j ∈ I . Hence b10 = u ∞ . This contradicts w0 6= 1 and b10 ∈ B w0 (∞). Thus we have proved (5.4).
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141
For µ ∈ W λ, set
[
C(µ) =
Aw (µ).
w∈W with `(w)≤N
Taking w ∈ W such that µ = wλ, we set ˜ C(µ) := Sw∗ −1 C(µ) ⊂ B(λ). ˜ By Proposition 5.8, C(µ) does not depend on the choice of w. We have the following: ˜ (i) C(µ) is a finite set; ˜ (ii) there is a finite subset F of Q independent of µ such that Wt(C(µ)) ⊂ µ + F. Hence, for any ξ ∈ P, [ [ ˜ ˜ B(λ)ξ ⊂ C(µ) C(µ) ξ = ξ µ∈W λ
µ∈W λ∩(ξ −F)
is a finite set. We have thus obtained the following properties of V (λ). PROPOSITION 5.16 Let λ ∈ P 0 be a basic
weight. Wt(V (λ)) is contained in the intersection of λ + Q and the convex hull of W λ. We have dim V (λ)µ = 1 for any µ ∈ W λ. We have dim V (λ)µ < ∞ for any µ ∈ P. We have Wt(V (λ)) ∩ (λ + Z δ) ⊂ W λ. V (λ) is an irreducible Uq (g)-module. Any nonzero integrable Uq (g)-module generated by an extremal weight vector of weight λ is isomorphic to V (λ). Moreover, V (λ) has a global base. (i) (ii) (iii) (iv) (v) (vi)
For any µ ∈ W λ, let us denote by u µ the unique global basis in V (λ)µ . Since u µ is an extremal vector with weight µ, we have the Uq (g)-linear homomorphism V (µ) → V (λ) that sends u µ ∈ V (µ) to u µ ∈ V (λ). This homomorphism is in fact an isomorphism. Set λ = $i . One has {n ∈ Z; $i + nδ ∈ W $i } = Zdi ,
(5.6)
where di = ($i , α˜ i ). Note that di = max(1, (αi , αi )/2) ∈ Z except in the case di = 1 (2) when g = A2n and αi is the longest root. Hence one has M M V ($i )µ = V ($i )λ+ndi . µ∈cl−1 cl($i )
n∈Z
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MASAKI KASHIWARA
∼ V ($i ). Since there is a U 0 (g)We have a Uq (g)-linear isomorphism V ($i +di δ) −→ q ∼ linear isomorphism V ($i ) −→ V ($i + di δ) that sends u $i to u $i +di δ , we obtain a Uq0 (g)-linear automorphism z i of V ($i ) of weight di δ, which sends u $i to u $i +di δ . Let us define the Uq0 (g)-module W ($i ) by W ($i ) = V ($i )/(z i − 1)V ($i ).
(5.7)
The following result is now obvious. 5.17 W ($i ) is a finite-dimensional irreducible integrable Uq0 (g)-module. W ($i ) has a global basis with a simple crystal. For any µ ∈ Wt(V ($i )),
THEOREM
(i) (ii) (iii)
W ($i )cl(µ) ' V ($i )µ . We have dim W ($i )cl($i ) = 1. The weight of any extremal vector of W ($i ) belongs to W cl($i ). Wt(W ($i )) is the intersection of cl($i )+Q cl and the convex hull of W cl($i ). 1/d 1/d We have K [z i i ] ⊗ K [zi ] V ($i ) ' W ($i )aff . Here the action of z i i on the left-hand side corresponds to the action of z on the right-hand side defined in §4.2. (viii) V ($i ) is isomorphic to the submodule K [z di , z −di ] ⊗ W ($i ) of W ($i )aff as a Uq (g)-module. Here we identify W ($i )aff with K [z, z −1 ] ⊗ W ($i ) as in (4.16). (ix) Any irreducible finite-dimensional integrable Uq0 (g)-module with cl($i ) as an extremal weight is isomorphic to W ($i )a for some a ∈ K \ {0}. (iv) (v) (vi) (vii)
Proof The irreducibility of W ($i ) follows, for example, by Proposition 4.12, and the other assertions are now obvious. We call W ($i ) a fundamental representation (of level zero).
6. Existence of global bases 6.1. Regularized modified operators ei (n) , For n ∈ Z and i ∈ I , let us define the operator F X (n+k) (k) ei (n) = F fi ei ak (ti ). k≥0,−n
(6.1)
QUANTIZED AFFINE ALGEBRAS
143
Here ak (ti ) =
k(1−n) k (−1)k qi ti
k−1 Y
(1 − qin+2ν ).
ν=0
Then it acts on any integrable Uq (g)-module M. Moreover, it acts also on any Uq (g)Q ei (n) has no pole except q = 0, ∞. Let (L , B) be a submodule MQ . In this sense, F ei (n) has no crystal base of M. Then we have the following result, which says that F n pole at q = 0 and coincides with f˜i at q = 0. PROPOSITION 6.1 ei (n) L ⊂ L, and the action of F ei (n) on L/qs L coincides with f˜n . We have F i
Proof In order to prove this, it is sufficient to prove the following statement. For any weight vector u ∈ M with ei u = 0 and m ∈ Z≥0 , we have ei (n) f (m) u = c f (m+n) u F i i for some c ∈ K := Q(qs ) regular at qs = 0 and c(0) = 1. Set ti u = qil u. Then we can assume that l ≥ n + m. We have
(m)
ak (ti ) f i
(m)
u = ak (qil−2m ) f i
u.
Hence (m)
fi
u=
X k≥0 m X
(n+k) (k) (m) ei f i u
ak (qil−2m ) f i
l −m+k (m−k) = fi u k i k=0 m X l −m+k (m+n) l−2m n + m = ak (qi ) fi u. m−k i k i (n+k) ak (qil−2m ) f i
k=0
Here
n [n]i ! = m i [m]i ![n − m]i !
is the q-binomial coefficient. Hence it is enough to show that A :=
m X k=0
n+m l −m+k ∈ 1 + qi Z[qi ]. m−k i k i
ak (qil−2m )
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MASAKI KASHIWARA
This follows immediately from the following formula, whose proof due to Anne Schilling is given in Appendix A. A=
m X
k(2l−2m−n+2)
qi
k=0
n+2( j−1) m−k k Y Y 1 − q n+2 j 1 − qi i 2j
j=1
1 − qi
2j
j=1
1 − qi
.
(6.2)
6.2. Existence theorem We use the notation and terminologies in §2.4. Let M be an integrable Uq (g)-module, let − be a bar involution of M, and let (L , B) be a crystal base of M. Let MQ be a Uq (g)Q -submodule of M such that (MQ )− = MQ . Set E := L ∩ L ∩ MQ . THEOREM 6.2 Let S be a subset of P. We assume the following conditions: (i) {(ξ, ξ ); ξ ∈ Wt(M)} is bounded from above; (ii) u − u¯ ∈ (qs − 1)MQ for any u ∈ MQ ; (iii) MQ generates M as a vector space over K ; (iv) for any ξ ∈ P \ S, (L ξ , L ξ , (MQ )ξ ) is balanced; (v) any extremal weight (i.e., the weight of an extremal vector) of B is in P \ S; (vi) qs L ∩ L ∩ MQ = 0. Then we have the following: (a) (L , L, MQ ) is balanced; (b) for any n, we have M M f in M = Q(qs )G(b) and ein M = Q(qs )G(b); εi (b)≥n
(c)
MQ =
P
ξ ∈P\S
Uq (g)Q (MQ )ξ and M =
ϕi (b)≥n
P
ξ ∈P\S
Uq (g)Mξ .
Proof The rest of this section is devoted to the proof of this theorem. LEMMA 6.3 The action of − on E is the identity.
Proof For u ∈ E, we have (u − u)/(1 ¯ − qs−1 ) ∈ qs L ∩ L ∩ MQ = 0. By (vi), the homomorphism E → L/qs L is injective. Let us denote by B 0 the intersection of B and the image of this homomorphism. To see (a), it is enough to show that B = B 0 . For b ∈ B 0 , let us denote by G(b) the element E such that
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145
b ≡ G(b) mod qs L. Note that G(b) = G(b) by Lemma 6.3. We prove the following statements by the descending induction on (ξ, ξ ): Bξ = Bξ0 or, equivalently, L ξ , L ξ , (MQ )ξ is balanced, (6.3) X (ε (b)) G(b) − f i i G(e˜imax b) ∈ Q[qs , qs−1 ]G(b0 ) for any b ∈ Bξ , (6.4) εi (b0 )>εi (b)
X
Q[qs , qs−1 ]G(b) =
b∈Bξ , εi (b)≥n
X
(m)
fi
(MQ )ξ +mαi
m≥n
for any n ≥ max 0, −hh i , ξ i ,
(6.5)
as well as the similar statements replacing f i with ei . If (ξ, ξ ) is big enough, those statements are trivially satisfied by (i). Now assuming (6.3) – (6.5) for ξ such that (ξ, ξ ) > a, let us prove them for ξ with (ξ, ξ ) = a. LEMMA 6.4 Let i ∈ I . Set k = max(0, −hh i , ξ i). (a) If e˜imax b ∈ B 0 , then b ∈ B 0 and (εi (b))
G(b) − f i
G(e˜imax b) ∈
X
Q[qs , qs−1 ]G(b0 ).
b0 ∈Bξ0 εi (b0 )>εi (b)
(b)
In particular, any b ∈ Bξ with εi (b) > k is contained in B 0 . P P (m) −1 We have (MQ )ξ +mαi for any n > k. b∈Bξ0 Q[qs , qs ]G(b) = m≥n f i εi (b)≥n
Similar statements hold after exchanging ei and f i . Proof Let us prove the lemma by the descending induction on n (in the case (a), n means εi (b), and hence n ≥ k). If n is big enough, they are true by hypothesis (i) in ei (n) G(b1 ) satisfies Theorem 6.2. Let us prove (a). Set b1 = e˜imax b. Then u = F b ≡ u mod qs L and X X (m) (n) (m+n) (m) Z[qs , qs−1 ] f i u − f i G(b1 ) ∈ ei G(b1 ) ⊂ f i (MQ )ξ +mαi . m>n
m>0
Induction hypothesis (b) implies that the last space is contained in X Q[qs , qs−1 ]G(b0 ). b0 ∈Bξ0 εi (b0 )>n
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MASAKI KASHIWARA
P (n) Hence we can write u − f i G(b1 ) = b0 cb0 G(b0 ), where b0 ranges over b0 ∈ B 0 with εi (b0 ) > n and cb0 ∈ Q[qs , qs−1 ]. Hence we can write cb0 − cb0 = cb0 0 − cb0 0 with P P (n) cb0 0 ∈ qs Q[qs ]. Then v := u − b0 cb0 0 G(b0 ) = f i G(b1 ) + b0 (cb0 − cb0 0 )G(b0 ) satisfies v = v, and hence it belongs to E. Moreover, one has b ≡ v mod qs L. Hence b belongs to B 0 , and G(b) = v. To complete the proof of (a), it is enough to remark that e˜imax b ∈ B 0 when εi (b) > k, because wt(e˜imax (b)), wt(e˜imax (b)) > (wt(b), wt(b)). Let us prove (b). The left-hand side is contained in the right-hand side by (a) and the induction hypothesis on n. Let us show the opposite inclusion. Set η = ξ + nαi with n > k. Then we have (η, η) > (ξ, ξ ), and (6.5) holds for η. Hence we have X X (m) (MQ )η ⊂ Q[qs , qs−1 ]G(b) + f i (MQ )η+mαi , εi (b)=0, b∈Bη0
m>0
which implies (n)
f i (MQ )η ⊂
X
(n)
Q[qs , qs−1 ] f i
G(b) +
εi (b)=0, b∈Bη0
⊂
X
(m)
fi
MQ
m>n (n)
Q[qs , qs−1 ] f i
εi (b)=0 b∈Bη0
X
G(b) +
X
Q[qs , qs−1 ]G(b).
εi (b)>n b∈Bξ0
The desired inclusion follows from (a). LEMMA 6.5 We have Bξ ⊂ B 0 .
Proof Let b ∈ Bξ . By hypothesis (v), there exists X l · · · X 1 b whose weight is outside S, where X ν is e˜imax or f˜imax . Hence, by the induction on l, we may assume that e˜imax b or f˜imax b is contained in B 0 . Then Lemma 6.4 implies b ∈ B 0 . Properties (6.3) and (6.4) are now obvious, and (6.5) easily follows from Lemmas 6.4 and 6.5. Thus the induction proceeds, and we complete the proof of (a), (b) in Theorem 6.2. Finally, let us prove (c). Set X X 0 M0 = Uq (g)Mξ and MQ = Uq (g)Q (MQ )ξ . ξ ∈P\S
ξ ∈P\S
Set L 0 = L ∩ M 0 . Then L 0 is invariant by e˜i and f˜i . By hypothesis (v), any vector in B is connected with a vector whose weight is outside S. Hence B is contained in
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147
L 0 /qs L 0 ⊂ L/qs L. This shows that (L 0 , B) is a crystal base of M 0 , and L 0 /qs L 0 = 0 = L/qs L. Thus we can apply Theorem 6.2 to M 0 . Hence we obtain L 0 ∩ L 0 ∩ MQ 0 = K ⊗ (L 0 ∩ L 0 ∩ M 0 ) = M . This completes the proof of L ∩ L ∩ MQ , and MQ Q Q Q Theorem 6.2. 7. Universal R-matrix In this section, we review the universal R-matrix introduced by Drinfeld and the universal bar involution introduced by G. Lusztig. Although we mainly use the coproduct 1 in this article, 1(q h ) = q h ⊗ q h , 1(ei ) = ei ⊗ ti−1 + 1 ⊗ ei , 1( f i ) = f i ⊗ 1 + ti ⊗ f i ,
(7.1)
we introduce another coproduct 1 = (− ⊗ −) ◦ 1 ◦ −, 1(q h ) = q h ⊗ q h , 1(ei ) = ei ⊗ ti + 1 ⊗ ei , 1( f i ) = f i ⊗ 1 + ti−1 ⊗ f i .
(7.2)
Let Mν (ν = 1, 2) be Uq (g)-modules with weight decomposition. Let us denote by M1 ⊗ M2 the tensor product of M1 and M2 with the Uq (g)-module structure induced by 1, and by M1 × M2 the Uq (g)-module induced by 1. Then there is an isomorphism q −( · , · ) : M1 × M2 → M2 ⊗ M1 given by
q −( · , · ) (x × y) = q −(wt(x),wt(y)) y ⊗ x.
b Uq− (g) by Let us define the ring Uq+ (g) ⊗ M Y b Uq− (g) = Uq+ (g) ⊗ Uq+ (g)λ ⊗ Uq− (g)µ .
(7.3)
ξ ∈Q ξ =λ+µ
Uq+ (g)
Uq− (g)
b Uq− (g) → K . ModThe counits → K and → K induce ε : Uq+ (g) ⊗ ifying Drinfeld’s construction in [3] of a universal R-matrix, Lusztig has shown that b Uq− (g) satisfying the following propthere exists a unique intertwiner 4 ∈ Uq+ (g) ⊗ erties: 4 ◦ 1(a) = 1(a) ◦ 4 for any a ∈ Uq (g), normalized by ε(4) = 1. Then it satisfies 4 ◦ 4 = 4 ◦ 4 = 1.
(7.4)
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MASAKI KASHIWARA
We introduce the completion of the tensor products as follows. We set Y b M2 ) = F(λ,µ) (M1 ⊗ (M1 )λ+γ ⊗ (M2 )µ−γ , γ ∈Q +
Y
b M2 ) = F>(λ,µ) (M1 ⊗
(M1 )λ+γ ⊗ (M2 )µ−γ ,
γ ∈Q + \{0}
and then b M2 = M1 ⊗
X
b M2 ) ⊂ F(λ,µ) (M1 ⊗
λ,µ∈P
Y
(M1 )λ ⊗ (M2 )µ .
λ,µ∈P
e M2 in the opposite direction, Sometimes we use another completion M1 ⊗ Y e M2 ) = F(λ,µ) (M1 ⊗ (M1 )λ−γ ⊗ (M2 )µ+γ , γ ∈Q +
and then e M2 = M1 ⊗
X
e M2 ) ⊂ F(λ,µ) (M1 ⊗
λ,µ∈P
Y
(M1 )λ ⊗ (M2 )µ .
λ,µ∈P
They have the structure of a Uq (g)-module by 1 and contain M1 ⊗ M2 as a Uq (g)-submodule. b M the same vector space M ⊗ We denote by M1 ⊗ 2 1 b M2 with the action of Uq (g) b M contains M × M as a Uq (g)-submodule. induced by 1. Then M1 ⊗ 2 1 2 We have an isomorphism b M −→ ∼ M2 ⊗ e M1 . q −( · , · ) : M1 ⊗ 2 The operator 4 induces an isomorphism bM . ∼ M1 ⊗ b M2 −→ M1 ⊗ 2 b M ), and the homomorphism inb M2 ) to F(λ,µ) (M1 ⊗ Then 4 sends F(λ,µ) (M1 ⊗ 2 duced by 4, b M2 )/F>(λ,µ) (M1 ⊗ b M2 ) M1λ ⊗ M2µ ' F(λ,µ) (M1 ⊗ b M )/F b −→ F(λ,µ) (M1 ⊗ 2 >(λ,µ) (M1 ⊗ M2 ) ' M1λ ⊗ M2µ ,
(7.5)
is equal to the identity. b M2 → M2 ⊗ e M1 , called the universal R-matrix, is The intertwiner R univ : M1 ⊗ given by q −( · , · ) 4 bM − b M2 −− e M1 . R univ : M1 ⊗ → M1 ⊗ → M2 ⊗ (7.6) 2 −−−
QUANTIZED AFFINE ALGEBRAS
149
It is an isomorphism. Assume that M1 and M2 have a bar involution. Then (7.4) implies that 4 −⊗− bM − b M2 −− b M2 cuniv : M1 ⊗ → M1 ⊗ → M1 ⊗ 2 −−
b M2 as observed by Lusztig [17]. We call it the universal is a bar involution on M1 ⊗ bar involution. 8. Good modules Let us take a finite-dimensional integrable Uq0 (g)-module M. We consider the following conditions on M: M has a bar involution,
(8.1)
M has a crystal base L(M), B(M) ,
(8.2)
M has a global base,
(8.3)
B(M) is a simple crystal.
(8.4)
In this paper, we say that a Uq0 (g)-module M is a good Uq0 (g)-module if M satisfies the above conditions. The level-zero fundamental representations W (ϕi ) are good Uq0 (g)-modules. A good Uq0 (g)-module is always irreducible (see Prop. 4.12). Let M1 and M2 be good Uq0 (g)-modules. Then we have b (M2 )aff = K [z 1 /z 2 ] ⊗ K [z 1 /z 2 ] (M1 )aff ⊗ (M2 )aff , (M1 )aff ⊗ e (M1 )aff = K [z 1 /z 2 ] ⊗ K [z 1 /z 2 ] (M2 )aff ⊗ (M1 )aff . (M2 )aff ⊗ Here z ν is the Uq0 (g)-linear automorphism of weight δ on (Mν )aff introduced in §4.2. 8.1 K (z 1 /z 2 ) ⊗ K [z 1 /z 2 ] ((M1 )aff ⊗ (M2 )aff ) K (z 1 /z 2 ) ⊗ K [z 1 /z 2 ] Uq (g)[z 1±1 , z 2±1 ].
LEMMA
is
an
irreducible
module
over
Proof Since M1 ⊗ M2 has a simple crystal base by Lemma 4.11, it is irreducible by Proposition 4.12. Then the lemma follows from the fact that the specialization of (M1 )aff ⊗ (M2 )aff at the special point z 1 /z 2 = 1 is irreducible. By the result of Section 7, we have the bar involution b (M2 )aff → (M1 )aff ⊗ b (M2 )aff . cuniv : (M1 )aff ⊗ It commutes with z 1 and z 2 . Let u ν be the extremal vector with dominant weight λν b (M2 )aff of Mν (ν = 1, 2), and set u = u 1 ⊗ u 2 . Then we have (M1 )aff ⊗ = λ +λ 1
2
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MASAKI KASHIWARA
K ((z 1 /z 2 ))u. Hence, by (7.5), we have cuniv (u) = ϕ(z 1 /z 2 )u
or, equivalently,
4(u) = ϕ(z 1 /z 2 )u
(8.5)
for some ϕ(z 1 /z 2 ) ∈ K [[z 1 /z 2 ]] with ϕ(0) = 1. We define b (M2 )aff → (M1 )aff ⊗ b (M2 )aff cnorm : (M1 )aff ⊗ by cnorm = cuniv ◦ ϕ(z 1 /z 2 )−1 . Then it satisfies cnorm (u) = u. LEMMA 8.2 b (M2 )aff satisfying The operator cnorm is a unique endomorphism of (M1 )aff ⊗ cnorm (u 1 ⊗ u 2 ) = u 1 ⊗ u 2 and cnorm (av) = acnorm (v) for any a ∈ b (M2 )aff . Uq (g)((z 1 /z 2 ))[z 2±1 ], v ∈ (M1 )aff ⊗
Proof It is enough to show that a Uq (g)[z 1±1 , z 2±1 ]-linear homomorphism b (M2 )aff f : (M1 )aff ⊗ (M2 )aff → (M1 )aff ⊗ vanishes if f (u 1 ⊗ u 2 ) = 0. By Lemma 8.1, K (z 1 /z 2 ) ⊗ K [z 1 /z 2 ] (M1 )aff ⊗ (M2 )aff is an irreducible module over K (z 1 /z 2 )[z 2±1 ] ⊗ Uq (g). Hence the assertion follows. b (M2 )aff , which we call the normalHence cnorm defines a bar involution on (M1 )aff ⊗ ized bar involution. In particular, we have ϕ(z)ϕ(z) = 1. In the sequel, we use the normalized bar involution to define a global basis. The universal R-matrix b (M2 )aff → (M2 )aff ⊗ e (M1 )aff R univ : (M1 )aff ⊗ sends u 1 ⊗ u 2 to q −(λ1 ,λ2 ) ϕ(z 1 /z 2 )u 2 ⊗ u 1 with the same function ϕ given in (8.5). Hence, setting R norm = q (λ1 ,λ2 ) ϕ(z 1 /z 2 )−1 R univ , we have an intertwiner b (M2 )aff → (M2 )aff ⊗ e (M1 )aff R norm : (M1 )aff ⊗ that sends u 1 ⊗u 2 to u 2 ⊗u 1 . We call R norm the normalized R-matrix. Both R-matrices commute with z 1 and z 2 .
QUANTIZED AFFINE ALGEBRAS
151
By (7.6) and (7.5), we have, for any vν ∈ (Mν )aff , R norm (v1 ⊗ v2 ) ≡ q hλ1 ,λ2 i−hwt(v1 ),wt(v2 )i v2 ⊗ v1 Y (M2 )aff wt(v )−ξ ⊗ (M1 )aff wt(v mod
1 )+ξ
2
ξ ∈Q + \{0}
.
(8.6)
We also have R norm : (M1 )aff ⊗ (M2 )aff → K (z 1 /z 2 ) ⊗ K [z 1 /z 2 ] (M2 )aff ⊗ (M1 )aff e (M1 )aff . ,→ (M2 )aff ⊗
We generalize these observations to the case of tensor products of several modules. Let Mν (ν = 1, . . . , m) be good Uq0 (g)-modules with a crystal base (L ν , Bν ). Let (Mν )aff be its affinization. Then (Mν )aff has a crystal base (L ν )aff , (Bν )aff . Let λν ∈ P be a dominant extremal weight of (Mν )aff , and let u ν be the extremal global basis with weight λν . We denote the canonical automorphism (Mν )aff of weight δ by zν . Then m O M := (Mν )aff = (M1 )aff ⊗ · · · ⊗ (Mm )aff ν=1
has a structure of a
±1 ]-module. K [z 1±1 , . . . , z m
Set
M = (M1 )aff ⊗ · · · ⊗ (Mm )aff ,
(8.7)
MQ = (M1Q )aff ⊗ · · · ⊗ (Mm Q )aff ,
(8.8)
and let (L(M), B(M)) be the tensor product of the crystal bases of the (Mν )aff ’s. We set m O b ⊗ M = K [z 1 /z 2 , . . . , z m−1 /z m ] (Mν )aff . K [z 1 /z 2 ,...,z m−1 /z m ]
ν=1
We also set
b = A [z 1 /z 2 , . . . , z m−1 /z m ] L( M) bQ = Q [z 1 /z 2 , . . . , z m−1 /z m ] M
⊗
A[z 1 /z 2 ,...,z m−1 /z m ]
⊗
Q[z 1 /z 2 ,...,z m−1 /z m ]
m O ν=1 m O
(L ν )aff , (Mν )affQ .
ν=1
Similarly to the case of the tensor product of two modules, we can define the b by universal bar involution of M Y cuniv = (− ⊗ · · · ⊗ −) ◦ 4i j , 1≤i< j≤m
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MASAKI KASHIWARA
where 4i j is the operator 4 acting on the ith and jth components of the tensor prodb It satisuct. Normalizing cuniv , we obtain the normalized bar involution cnorm on M. fies, by setting u = u 1 ⊗ · · · ⊗ u m , cnorm (u) = u. Moreover, it satisfies for vν ∈ (Mν )aff , cnorm (v1 ⊗ · · · ⊗ vm ) ≡ v1 ⊗ · · · ⊗ vm
mod
Y
(M1 )aff
ξ1 ,...,ξm
wt(v1 )+ξ1
⊗ · · · ⊗ (Mm )aff
wt(vm )+ξm
.
(8.9) P Pµ Here the product ranges over ξ1 , . . . , ξm ∈ Q with m ν=1 ξν = 0 and ν=1 ξν ∈ Q + \ {0} (µ = 1, . . . , m − 1). Since cnorm is expressed by a triangular matrix, the well-known argument of triangular matrices implies the following result. 8.3 b ∩ (cnorm L( M)) b ∩M bQ = 0. We have qs L( M) b such For any b = b1 ⊗ · · · ⊗ bm ∈ B(M), there exists a unique G(b) ∈ L( M) norm b that c (G(b)) = G(b) and b ≡ G(b) mod qs L( M). Moreover, G(b) has the form X 0 G(b) = G(b1 ) ⊗ · · · ⊗ G(bm ) + cb0 ,··· ,bm0 G(b10 ) ⊗ · · · ⊗ G(bm ).
LEMMA
(i) (ii) (iii)
1
0 Here the infinite sum ranges over b10 ⊗ · · · ⊗ bm ∈ B(M) such that Pm Pm Pµ 0 0 wt(b ) = wt(b ) and (wt(b ) − wt(b ν ν )) ∈ Q + \ {0} ν ν ν=1 ν=1 ν=1 (µ = 1, . . . , m − 1). Moreover, cb0 ,...,bm0 ∈ qs Q[qs ]. 1
Later we see that this infinite sum is in fact a finite sum. Set ±1 ]u. N = Uq (g)[z 1±1 , . . . , z m b stable by the bar involution cnorm . Set λ = Then N is a submodule of M Then we have Nλ+Zδ := =
M n∈Z m O ν=1
Nλ+nδ =
m O (Mν )aff ν=1
(Mν )aff
λν +Zδ
Pm
λ+Zδ
±1 = K [z 1±1 , . . . , z m ](u 1 ⊗ · · · ⊗ u m ).
ν=1 λν .
QUANTIZED AFFINE ALGEBRAS
153
Hence one has N µ = Mµ
for any µ ∈ W λ + Zδ.
(8.10)
Define L(N ) = L(M) ∩ N , N Q = MQ ∩ N , B(N ) = B(M). Then L(N )/qs L(N ) ⊂ L(M)/qs L(M). 8.4 B(N ) is a basis of L(N )/qs L(N ), and (L(N ), B(N )) is a crystal base of N .
LEMMA
Proof Since B(N ) is a basis of L(M)/qs L(M), it is enough to show that B(N ) is contained in L(N )/qs L(N ). Since every vector in B(N ) is connected with an extremal vector with weight in λ + Zδ and since extremal vectors with such a weight are u up to the ±1 , we obtain the desired result. action of z 1±1 , . . . , z m Setting S = Wt(M) \ (W λ + Zδ), we can apply Theorem 6.2 to N . The hypotheses in the theorem are satisfied by Lemmas 8.3 and 8.4, and we obtain the following theorem. 8.5 (L(N ), cnorm L(N ), NQ ) is balanced. Hence N has a global base. We have NQ = Uq (g)Q [z 1 , . . . , z m ]u.
THEOREM
(i) (ii)
Furthermore, Lemma 8.3 implies the following proposition. 8.6 For any bν ∈ B((Mν )aff ) (ν = 1, . . . , m), we have X 0 G(b1 ⊗ · · · ⊗ bm ) = G(b1 ) ⊗ · · · ⊗ G(bm ) + cb0 ,...,bm0 G(b10 ) ⊗ · · · ⊗ G(bm ). PROPOSITION
1
Qm
Pm 0 )∈ 0 Here the sum ranges over (b10 , . . . , bm ν=1 B((Mν )aff ) such that ν=1 wt(bν ) = Pµ Pm 0 ν=1 wt(bν ) and ν=1 (wt(bν )−wt(bν )) ∈ Q + \{0} (µ = 1, . . . , m −1). Moreover, 0 ). cb0 ,...,bm0 ∈ qs Q[qs ], and cb0 ,...,bm0 vanishes except for finitely many (b10 , . . . , bm 1
1
By specializing at z ν = 1, we obtain the following proposition.
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MASAKI KASHIWARA
PROPOSITION 8.7 The tensor product of good Uq0 (g)-modules is also a good Uq0 (g)-module.
9. Main theorem The following theorem is conjectured in [1] in the special case when all the Mν are fundamental representations. Note that, as seen by the proof, the theorem holds even if we consider Uq (g) as an algebra over the algebraically closed P b := b, and replace A with the subring field K C((q 1/n )), aν as elements of K n>0 P b := b. A C[[q 1/n ]] of K n>0
9.1 Let Mν (ν = 1, . . . , m) be good Uq0 (g)-modules. Let aν ∈ K . Assume that aν /aν+1 ∈ A for ν = 1, . . . , m − 1. Then (M1 )a1 ⊗ (M2 )a2 ⊗ · · · ⊗ (Mm )am is generated by u 1 ⊗ · · · ⊗ u m . Assume that (Mν )∗ (ν = 1, . . . , m) are good Uq0 (g)-modules, and aν+1 /aν ∈ A for ν = 1, . . . , m − 1. Then any nonzero submodule of (M1 )a1 ⊗ (M2 )a2 ⊗ · · · ⊗ (Mm )am contains u 1 ⊗ · · · ⊗ u m .
THEOREM
(i)
(ii)
Proof Since (ii) is the dual statement of (i), it is enough to prove (i). Let us embed the crystal Bν of Mν into (Bν )aff as in (4.16). Let ψ : (M1 )aff ⊗· · ·⊗ (Mm )aff → (M1 )a1 ⊗ · · · ⊗ (Mm )am be the canonical projection. Then ψ(G(b1 ) ⊗ · · · ⊗ G(bm )) (bν ∈ Bν ) forms a basis of (M1 )a1 ⊗ (M2 )a2 ⊗ · · · ⊗ (Mm )am . Since ψ(G(b1 ⊗ · · · ⊗ bm )) are in Uq0 (g)(u 1 ⊗ · · · ⊗ u m ), it is enough to show that they also generate (M1 )a1 ⊗ (M2 )a2 ⊗ · · · ⊗ (Mm )am as a vector space. By Proposition 8.6, we can write G(b1 ⊗ · · · ⊗ bm ) = G(b1 ) ⊗ · · · ⊗ G(bm ) +
X
0 m cbk10 ,...,k G(z k1 b10 ) ⊗ · · · ⊗ G(z km bm ). ,...,b0 1
m
Qm 0 ) ∈ Here the summation ranges over the set of (b10 , . . . , bm ν=1 Bν and P m m (k1 , . . . , km ) ∈ Z such that ν=1 kν = 0 and k1 + · · · + kν ≥ 0 (ν = 1, . . . , m). m Moreover, we have cbk10 ,...,k 0 ∈ qs Q[qs ]. 1 ,...,bm On the other hand, we have 0 ψ G(z k1 b10 ) ⊗ · · · ⊗ G(z km bm ) 0 km = (a1k1 · · · am )ψ G(b10 ) ⊗ · · · ⊗ G(bm ) 0 = (a1 /a2 )k1 (a2 /a3 )k1 +k2 · · · ψ G(b10 ) ⊗ · · · ⊗ G(bm ) ∈ L,
QUANTIZED AFFINE ALGEBRAS
155
L where L = bν ∈B(Mν ) A G(b1 ) ⊗ · · · ⊗ G(bm ) . Hence we have ψ G(b1 ⊗ · · · ⊗ bm ) ≡ ψ G(b1 ) ⊗ · · · ⊗ G(bm ) mod qs L . Then Nakayama’s lemma implies that ψ(G(b1 ⊗· · ·⊗bm )); bν ∈ B(Mν ) generates (M1 )a1 ⊗ · · · ⊗ (Mm )am . We can apply Theorem 9.1 to the fundamental representations. THEOREM 9.2 Let aν ∈ K , and let i ν ∈ I0∨ (ν = 1, . . . , m). (i) Assume that aν /aν+1 ∈ A for ν = 1, . . . , m −1. Then W ($i1 )a1 ⊗W ($i2 )a2 ⊗ · · · ⊗ W ($im )am is generated by u $i1 ⊗ · · · ⊗ u $im . (ii) Assume that aν+1 /aν ∈ A for ν = 1, . . . , m − 1. Then any nonzero submodule of W ($i1 )a1 ⊗ W ($i2 )a2 ⊗ · · · ⊗ W ($im )am contains u $i1 ⊗ · · · ⊗ u $im .
b and A, b they imply the Since these theorems hold even if we replace K and A with K following consequences, as shown in [1]. PROPOSITION 9.3 Assume that M j is a good Uq0 (g)-module with dominant extremal vector u j . The normalized R-matrix
Ri,norm j (x, y) : (Mi )x ⊗ (M j ) y → (M j ) y ⊗ (Mi )x b does not have a pole at x/y = a ∈ A. Here Ri,norm j (x, y) is the intertwiner (Mi )x ⊗ (M j ) y → (M j ) y ⊗ (Mi )x so normalized that it sends u i ⊗ u j to u j ⊗ u i . Let ψi j (x, y) be the denominator of Ri,norm j (x, y). Then one has ψi j (x, y) ∈ 1 + A[x/y]qs x/y.
(9.1)
For the sake of simplicity, we assume that M j as well as its dual M ∗j are good Uq0 (g)-modules, and we let u j be a dominant extremal vector of M j . 9.4 The extremal vector u 1 ⊗ · · · ⊗ u m generates (M1 )a1 ⊗ · · · ⊗ (Mm )am if and only if Ri,norm j (x, y) has no pole at x/y = ai /a j for any 1 ≤ j < i ≤ m. Any nonzero submodule (M1 )a1 ⊗ · · · ⊗ (Mm )am contains u 1 ⊗ · · · ⊗ u m if and only if Ri,norm j (x, y) has no pole at x/y = ai /a j for any 1 ≤ i < j ≤ m.
PROPOSITION
(i) (ii)
156
(iii)
MASAKI KASHIWARA
(M1 )a1 ⊗ · · · ⊗ (Mm )am is irreducible if and only if Ri,norm j (x, y) does not have a pole at x/y = ai /a j for any 1 ≤ i, j ≤ m (i 6 = j).
PROPOSITION 9.5 If M and M 0 are irreducible finite-dimensional integrable Uq0 (g)-modules, then M ⊗ Mz0 is an irreducible Uq0 (g)-module except for finitely many z.
10. Combinatorial R-matrices Let M1 and M2 be two good Uq0 (g)-modules. Let u ν be the extremal vector of Mν with dominant weight (ν = 1, 2). Let ψ(z 1 /z 2 ) be the denominator of the normalized R-matrix, normalized by ψ ∈ K [z 1 /z 2 ] with ψ(0) = 1. Then, by (9.1), we have ψ(z) ∈ 1 + qs z A[z].
(10.1)
We have an intertwiner ψ(z 1 /z 2 )R norm : (M1 )aff ⊗ (M2 )aff → (M2 )aff ⊗ (M1 )aff . We first prove the following proposition. PROPOSITION
10.1
We have ψ(z 1 /z 2 )R norm L(M1 )aff ⊗ L(M2 )aff ⊂ L(M2 )aff ⊗ L(M1 )aff . Proof Set M = (M1 )aff ⊗ (M2 )aff , and let L be the smallest crystal lattice of M containing A[z 1±1 , z 2±1 ](u 1 ⊗ u 2 ). Then L is contained in L(M). Since every vector in B(M) is connected with some z 1m u 1 ⊗ z 2n u 2 , L/qs L → L(M)/qs L(M) is surjective. Hence, by the following well-known lemma, there exists g such that g ∈ 1 + qs A[z 1±1 , z 2±1 ]
and
gL(M) ⊂ L .
10.2 Let R be a commutative ring, let a ∈ R, and let F be a finitely generated R-module. If F = a F, then there exists b ∈ 1 + a R such that bF = 0. LEMMA
Let us define M 0 and L 0 in a way similar to M and L by exchanging M1 and M2 . The operator T = ψ(z 1 /z 2 )R norm : M → M 0 commutes with e˜i , f˜i , z 1 , z 2 , and it satisfies T (u 1 ⊗ u 2 ) ∈ L(M 0 ) by (10.1). Hence we have T L ⊂ L(M 0 ).
QUANTIZED AFFINE ALGEBRAS
157
Taking g as above, we obtain gT L(M) ⊂ T L ⊂ L(M 0 ). Since L(M 0 ) is a free A[z 1±1 , z 2±1 ]-module of finite rank, the proposition follows from the following lemma. 10.3 Let F be a free A[z 1±1 , z 2±1 ]-module, and let g be an element in 1 + qs A[z 1±1 , z 2±1 ]. If u ∈ K ⊗ F satisfies gu ∈ F, then u belongs to F. LEMMA
A
Since the proof is elementary, we do not give it. As a corollary of Proposition 10.1 and (10.1), we obtain the following. COROLLARY
10.4
We have b (M2 )aff R norm L (M1 )aff ⊗ CONJECTURE
e L(M1 )aff . ⊂ L (M2 )aff ⊗
10.5
We have ψ(z 1 /z 2 ) (M1 )aff ⊗ (M2 )aff ⊂ Uq (g)[z 1±1 , z 2±1 ](u 1 ⊗ u 2 ). Set N = Uq (g)[z 1±1 , z 2±1 ](u 1 ⊗ u 2 ) ⊂ (M1 )aff ⊗ (M2 )aff , N 0 = Uq (g)[z 1±1 , z 2±1 ](u 2 ⊗ u 1 ) ⊂ (M2 )aff ⊗ (M1 )aff . Then R norm gives an isomorphism ∼ N 0. R norm : N −→ In §8, we saw that N (resp., N 0 ) has a crystal base (L(N ), B(M1 )aff ⊗ B(M2 )aff ) resp., (L(N 0 ), B(M2 )aff ⊗ B(M1 )aff ) . Hence R norm induces an isomorphism ∼ B(M2 )aff ⊗ B(M1 )aff . R comb : B(M1 )aff ⊗ B(M2 )aff −→ We have R comb (zb1 ⊗ b2 ) = (1 ⊗ z)R comb (b1 ⊗ b2 ), R comb (b1 ⊗ zb2 ) = (z ⊗ 1)R comb (b1 ⊗ b2 ).
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MASAKI KASHIWARA
Hence we have a commutative diagram: R comb
B(M1 )aff ⊗ B(M2 )aff y
−−−→
B(M1 ) ⊗ B(M2 )
−−→
∼
B(M2 )aff ⊗ B(M1 )aff y
(10.2)
B(M2 ) ⊗ B(M1 )
Hence one obtains the following proposition. PROPOSITION 10.6 If B1 and B2 are crystal bases of good Uq0 (g)-modules, then B1 ⊗ B2 ' B2 ⊗ B1 .
By Corollary 10.4, we have R norm G(b1 ⊗ b2 ) = G R comb (b1 ⊗ b2 ) .
(10.3)
Setting R comb (b1 ⊗ b2 ) = b20 ⊗ b10 with bν , bν0 ∈ B(Mν )aff , we define S(b1 ⊗ b2 ) = wt(b10 ) − wt(b1 ) = wt(b2 ) − wt(b20 ) ∈ Q. P By (8.6), we have S(b1 ⊗ b2 ) ∈ Q + := i∈I Z≥0 αi . On the other hand, we have S(z 1 b1 ⊗ b2 ) = S(b1 ⊗ z 2 b2 ) = S(b1 ⊗ b2 ), and hence it induces a map S : B(M1 ) ⊗ B(M2 ) → Q + .
(10.4)
This map S is characterized by the following properties (note that B(M1 ) ⊗ B(M2 ) is connected): S(u 1 ⊗ u 2 ) = 0, S(b1 ⊗ b2 ) + αi S f˜i (b1 ⊗ b2 ) = S(b1 ⊗ b2 ) − αi S(b ⊗ b ) 1
2
(10.5) if f˜i (b1 ⊗ b2 ) = ( f˜i b1 ) ⊗ b2 and f˜i (b0 ⊗ b0 ) = ( f˜i b0 ) ⊗ b0 , 2
1
2
1
if f˜i (b1 ⊗ b2 ) = b1 ⊗ ( f˜i b2 ) and f˜i (b0 ⊗ b0 ) = b0 ⊗ ( f˜i b0 ), 2
1
2
1
otherwise. (10.6)
11. Energy function In this section, we assume that M is good, and we investigate the properties of Maff ⊗2 . In this case, we have e Maff . R norm = ι¯ ◦ cnorm : Maff ⊗ Maff → Maff ⊗
(11.1)
QUANTIZED AFFINE ALGEBRAS
159
b Maff → Maff ⊗ e Maff is given by Here ι¯ : Maff ⊗ ι¯(v ⊗ v 0 ) = q (λ,λ)−(wt(v),wt(v )) v 0 ⊗ v. 0
Indeed, R norm and ι¯ ◦ cnorm are Uq (g)-linear homomorphisms sending u ⊗ u to itself, z ⊗ 1 to 1 ⊗ z, and 1 ⊗ z to z ⊗ 1. Such a homomorphism is unique. Similarly, the identity being a unique automorphism of B(M)⊗2 , (10.2) implies that there exists a unique map H : B(M)aff ⊗2 → Z such that R comb (b1 ⊗ b2 ) = (z −H (b1 ⊗b2 ) b1 ) ⊗ (z H (b1 ⊗b2 ) b2 ). Hence one has S(b1 ⊗ b2 ) = wt(b2 ) − wt(b1 ) + H (b1 ⊗ b2 )δ. We call H the energy function. We have H ((zb1 ) ⊗ b2 ) = H (b1 ⊗ (z −1 b2 )) = H (b1 ⊗ b2 ) + 1. It is easy to see that G B(M)aff ⊗2 = H −1 (n) n∈Z
is the decomposition of B(M)⊗2 aff into the minimal regular subcrystals invariant by z ⊗ z (cf. [6]). Embedding B(M) into B(M)aff as in (4.16), the energy function restricted on B(M)⊗2 is also characterized by the following two properties: H (v ⊗ v) = 0 for any extremal vector v of B(M), if i 6= 0 and f˜i (b1 ⊗ b2 ) 6= 0, H (b1 ⊗ b2 ) H (b1 ⊗ b2 ) + 1 if i = 0 and H f˜i (b1 ⊗ b2 ) = f˜i (b1 ⊗ b2 ) = ( f˜i b1 ) ⊗ b2 6= 0, H (b ⊗ b ) − 1 if i = 0 and 1 2 f˜ (b ⊗ b ) = b ⊗ ( f˜ b ) 6= 0. i
1
2
1
i 2
Set ⊗2 N := Uq (g) (z ⊗ z)±1 , z ⊗ 1 + 1 ⊗ z (u ⊗ u) ⊂ Maff , ⊗2 ⊗2 N 0 := Ker R norm − 1 : Maff → K (z ⊗ z −1 ) ⊗ K [z⊗z −1 ] Maff . ⊗2 Then we have N ⊂ N 0 ⊂ Maff . Define L(N ) = L(Maff )⊗2 ∩ N , and define L(N 0 ) ⊗2 ⊗2 similarly. Then one has L(N )/qs L(N ) ⊂ L(N 0 )/qs L(N 0 ) ⊂ L(Maff )/qs L(Maff ). Set ⊗2 B0 (Maff ) := b1 ⊗ b2 ∈ B(Maff )⊗2 ; H (b1 ⊗ b2 ) = 0 .
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Then n ⊗2 ⊗2 (z ⊗ 1)b; n ∈ Z≥0 , b ∈ B0 (Maff ) t (1 ⊗ z n )b; n ∈ Z>0 , b ∈ B0 (Maff ) ⊗2 ⊗2 is a basis of L(Maff )/qs L(Maff ). ⊗2 ⊗2 Let us define the subset B 0 of L(Maff )/qs L(Maff ) by ⊗2 B 0 := z n ⊗ 1 + δ(n 6 = 0)(1 ⊗ z n ) b; n ∈ Z≥0 , b ∈ B0 (Maff ) .
Here, for a statement P, we define δ(P) by ( 1 if P is true, δ(P) = 0 if P is false. Then B 0 is linearly independent. LEMMA 11.1 We have B 0 ⊂ L(N )/qs L(N ). Moreover, (L(N ), B 0 ) and (L(N 0 ), B 0 ) are crystal bases of N and N 0 , respectively.
Proof It is enough to show that B 0 ⊂ L(N )/qs L(N ) and that B 0 is a basis of ⊗2 L(N 0 )/qs L(N 0 ). Since B0 (Maff ) is a minimal subcrystal invariant by z ⊗ z, we have ⊗2 B0 (Maff ) ⊂ L(N )/qs L(N ). Since L(N )/qs L(N ) is invariant by z n ⊗ 1 + 1 ⊗ z n , we have B 0 ⊂ L(N )/qs L(N ). It remains to prove that B 0 generates L(N 0 )/qs L(N 0 ). Since R norm = 1 on N 0 , we have R norm = 1 on L(N 0 )/qs L(N 0 ), and hence L(N 0 )/qs L(N 0 ) ⊂ F := {v ∈ (L(Maff )/qs L(Maff ))⊗2 ; R norm (v) = v}. ⊗2 Since the action of R norm on (L(Maff )/qs L(Maff ))⊗2 = Q⊕B(Maff ) is given by R comb , we can easily see that B 0 is a basis of F. PROPOSITION 11.2 N = N 0 , and it has
a global basis {G(b); b ∈ B 0 }.
Proof We apply Theorem 6.2 for N and N 0 . Set ⊗2 NQ := Uq (g)Q (z ⊗ z)±1 , z ⊗ 1 + 1 ⊗ z (u ⊗ u) ⊂ Maff , ⊗2 and define NQ0 similarly. Set S = Wt(Maff ) \ (W (2λ) + Zδ). For ξ = 2wλ + nδ L (w ∈ W , n ∈ Z), setting H = ν+µ=n K (z ν ⊗ z µ + z µ ⊗ z ν )u ⊗2 wλ , we have H ⊂ Nξ ⊂ Nξ0 ⊂ H . Hence Nξ = Nξ0 = H , and condition (iv) in Theorem 6.2 is satisfied for N and N 0 . Condition (v) follows from the fact that the weight of any extremal
QUANTIZED AFFINE ALGEBRAS
161
vector of B(Maff )⊗2 is in W (2λ) + Zδ. Hence all the conditions in Theorem 6.2 are satisfied for N and N 0 , and both N and N 0 have a global basis. These two global bases coincide, and hence N = N 0 . COROLLARY 11.3 If b1 , b2 ∈ B(M)aff satisfy H (b1 ⊗ b2 ) = 0, then G(b1 ⊗ b2 ) ∈ Uq (g)Q (z ⊗ z)±1 , z ⊗ 1 + 1 ⊗ z (u ⊗ u).
Moreover, denoting by N0 the vector subspace generated by {G(b1 ⊗ b2 ); H (b1 ⊗ b2 ) = 0}, one has Uq (g)Q (z ⊗ z)±1 , z ⊗ 1 + 1 ⊗ z (u ⊗ u) = Q[z ⊗ 1 + 1 ⊗ z] ⊗Q N0 , Maff ⊗2 = Q[z ±1 ⊗ 1, 1 ⊗ z ±1 ] ⊗Q[(z⊗z)±1 ] N0 = Q[z ±1 ⊗ 1] ⊗Q N0 .
12. Fock space 12.1. Some properties of good modules In [13], we defined the wedge spaces and the Fock spaces for a finite-dimensional integrable Uq0 (g)-module V . In that paper, we assumed several conditions on V . In this section, we show that all those conditions are satisfied whenever V is a good module with a perfect crystal base. In [13], we employed the reversed coproduct. Adapting the notation to ours, those conditions read as follows. We set N := Uq (g)[(z ⊗ z)±1 , z ⊗ 1 + 1 ⊗ z](u ⊗ u) ⊂ (Vaff )⊗2 with an extremal vector u of V of weight λ. (G) V is good. Let (L , B) be the crystal base of V . (P) B is a perfect crystal. (L) Let s : Q → Z be the additive function such that s(αi ) = 1, and let ` : Baff → Z be the function defined by `(b) = s(wt(b) − wt(u)). Then one has H (b1 ⊗ b2 ) ≤ 0 ⇒ `(b1 ) ≤ `(b2 ). (D) (R)
We have ψ ∈ 1 + qs z A[z]. Here ψ(x/y) is the denominator of the normalized R-matrix R norm : Vx ⊗ Vy → Vy ⊗ Vx . For every pair (b1 , b2 ) in Baff with H (b1 ⊗ b2 ) = 0, there exists Cb1 ,b2 ∈ N of the form X Cb1 ,b2 = G(b1 ) ⊗ G(b2 ) − ab0 ,b0 G(b10 ) ⊗ G(b20 ). b10 ,b20
1
2
162
MASAKI KASHIWARA 2 such that Here the sum ranges over (b10 , b20 ) ∈ Baff
H (b10 ⊗ b20 ) > 0, `(b1 ) < `(b10 ) ≤ `(b2 ), `(b1 ) ≤ `(b20 ) < `(b2 ), and the coefficients ab0 ,b0 belong to Q[qs , qs−1 ]. 1
2
12.1 ([13]) V We assume that (G), (L), (D), and (R). Then the wedge space m Vaff has a basis {G(b1 ) ∧ · · · ∧ G(bm )}, where (b1 , . . . , bm ) ranges over (Baff )m with H (b j ⊗b j+1 ) > 0 ( j = 1, . . . , m − 1). THEOREM
For the other consequences and the Fock space, see §12.2, §12.3, and [13]. In this section we prove the following theorem. THEOREM 12.2 Assume that V is a good Uq0 (g)-module. Then all the properties above except (P) are satisfied.
In fact, we prove here a little bit stronger results. In the sequel, we assume that V is a good Uq0 (g)-module. The property (D) has already been proved in (10.1). The following lemma immediately implies (L). LEMMA 12.3 If H (b1 ⊗ b2 ) ≤ 0, then wt(b2 ) − wt(b1 ) ∈ Q + .
Proof By (10.4), we have S(b1 ⊗ b2 ) = wt(b2 ) − wt(b1 ) + H (b1 ⊗ b2 )δ ∈ Q + . Hence if H (b1 ⊗ b2 ) ≤ 0, then wt(b2 ) − wt(b1 ) ∈ Q + . In order to prove the remaining property, we prove the following result on global bases. PROPOSITION 12.4 Assume that H (b1 ⊗ b2 ) = 0. Write
G(b1 ⊗ b2 ) =
X b10 , b20 ∈B(M)aff
ab0 ,b0 G(b10 ) ⊗ G(b20 ). 1
2
(12.1)
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163
Then we have ab1 ,b2 = 1, ab0 ,b0 = q (λ,λ)−(wt b1 ,wt b2 ) ab0 ,b0 . 0
2
0
1
1
2
If ab0 ,b0 6 = 0, then 1
2
wt(b10 ) ∈ wt(b1 ) + Q + ∩ wt(b2 ) − Q + , wt(b20 ) ∈ wt(b1 ) + Q + ∩ wt(b2 ) − Q + . Moreover, wt(b10 ) = wt(b1 ) implies (b10 , b20 ) = (b1 , b2 ), and wt(b10 ) = wt(b2 ) implies (b10 , b20 ) = (b2 , b1 ). Proof We have seen that wt(b10 ) ∈ wt(b1 ) + Q + , wt(b20 ) ∈ wt(b2 ) − Q + , and wt(b10 ) = wt(b1 ) implies (b10 , b20 ) = (b1 , b2 ). Since R norm G(b1 ⊗ b2 ) = G(b1 ⊗ b2 ) and since cnorm G(b1 ⊗ b2 ) = G(b1 ⊗ b2 ), we have ι¯G(b1 ⊗ b2 ) = G(b1 ⊗ b2 ) by (11.1). Hence we have X 0 0 G(b1 ⊗ b2 ) = q (λ,λ)−(wt b1 ,wt b2 ) ab0 ,b0 G(b20 ) ⊗ G(b10 ), 1
b10 ,b20 ∈B(M)aff
2
which gives ab0 ,b0 = q (λ,λ)−(wt b1 ,wt b2 ) ab0 ,b0 . Hence we obtain the remaining asser1 2 2 1 tions. 0
0
CONJECTURE 12.5 Conjecturally, we have H (b10 ⊗ b20 ) ≥ 0 if ab0 ,b0 6 = 0. 1
2
Let us set I+ (b) = b0 ∈ Baff ; wt(b0 ) − wt(b) ∈ Q + \ {0} t {b}, I− (b) = b0 ∈ Baff ; wt(b) − wt(b0 ) ∈ Q + \ {0} t {b}. The following lemma immediately implies (R). 12.6 For every pair (b1 , b2 ) in Baff , there exists Cb1 ,b2 ∈ N of the form X Cb1 ,b2 = G(b1 ) ⊗ G(b2 ) − ab0 ,b0 G(b10 ) ⊗ G(b20 ). LEMMA
b10 ,b20
1
2
2 such that H (b0 ⊗ b0 ) > 0 and b0 , b0 ∈ Here the sum ranges over (b10 , b20 ) ∈ Baff 1 2 1 2 I+ (b1 ) ∩ I− (b2 ), and the coefficients ab0 ,b0 belong to Q[qs , qs−1 ]. 1
2
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MASAKI KASHIWARA
Proof We prove this by the induction on `(b2 ) − `(b1 ). Note that the assertion is trivial when H (b1 ⊗ b2 ) > 0. We may assume that H (b1 ⊗ b2 ) ≤ 0. Then (L) implies `(b2 ) − `(b1 ) ≥ 0. Set n := −H (b1 ⊗ b2 ). Then H (z n b1 ⊗ b2 ) = 0. Hence G(z n b1 ⊗ b2 ) ∈ N . By Proposition 12.4, we can write X G(z n b1 ⊗ b2 ) = z n G(b1 ) ⊗ G(b2 ) + ab0 ,b0 G(b10 ) ⊗ G(b20 ), b10 ,b20
1
2
where the sum ranges over (b10 , b20 ) with b10 , b20 ∈ I+ (z n b1 ) ∩ I− (b2 ) and b10 6 = z n b1 . In particular, one has `(z −n b10 ) > `(b1 ). Then z −n ⊗ 1 + δ(n > 0)1 ⊗ z −n G(z n b1 ⊗ b2 ) = G(b1 ) ⊗ G(b2 ) + δ(n > 0)G(z n b1 ) ⊗ G(z −n b2 ) X + ab0 ,b0 G(z −n b10 ) ⊗ G(b20 ) + δ(n > 0)G(b10 ) ⊗ G(z −n b20 ) (b10 ,b20 )∈I
1
2
⊗2 belongs to NQ = N ∩ (Maff )Q . Hence, modulo NQ , G(b1 ) ⊗ G(b2 ) is a linear combin −n nation of G(z b1 ) ⊗ G(z b2 ) (n > 0), G(z −n b10 ) ⊗ G(b20 ), and G(b10 ) ⊗ G(z −n b20 ). When n > 0, we have `(z −n b2 ) − `(z n b1 ) < `(b2 ) − `(b1 ), and the induction hypothesis implies that G(z n b1 ) ⊗ G(z −n b2 ) is, modulo NQ , a linear combination of G(b100 ) ⊗ G(b200 ) with H (b100 ⊗ b200 ) > 0 and b100 , b200 ∈ I+ (z n b1 ) ∩ I− (z −n b2 ) ⊂ I+ (b1 ) ∩ I− (b2 ). Similarly, we have `(b20 ) − `(z −n b10 ) < `(b2 ) − `(b1 ). Hence, modulo NQ , −n G(z b10 ) ⊗ G(b20 ) is a linear combination of G(b100 ) ⊗ G(b200 ) with H (b100 ⊗ b200 ) > 0 and b100 , b200 ∈ I+ (z −n b10 ) ∩ I− (b20 ) ⊂ I+ (b1 ) ∩ I− (b2 ). Finally, since `(z −n b20 )−`(b10 ) ≤ `(b20 )−`(z −n b10 ) < `(b2 )−`(b1 ), the induction hypothesis implies that G(b10 ) ⊗ G(z −n b20 ) modulo NQ is a linear combination of G(b100 ) ⊗ G(b200 ) with H (b100 ⊗ b200 ) > 0 and b100 , b200 ∈ I+ (b10 ) ∩ I− (z −n b20 ) ⊂ I+ (b1 ) ∩ I− (b2 ).
12.2. Wedge spaces Let us recall the construction of the wedge space in [13]. Let V be a good Uq0 (g)module with an extremal global basis u. Let us set ⊗2 N = Uq (g) (z ⊗ z)±1 , z ⊗ 1 + 1 ⊗ z (u ⊗ u) ⊂ Vaff , (12.2) Nm =
m−2 X j=0
⊗m Vaff ⊗ j ⊗ N ⊗ V ⊗(m−2− j) ⊂ Vaff .
(12.3)
QUANTIZED AFFINE ALGEBRAS
The wedge space
Vm
165
Vaff is defined by m ^
⊗m /Nm . Vaff = Vaff
For v1 , . . . , vm ∈ Vaff , let v1 ∧ · · · ∧ vm denote the image of v1 ⊗ · · · ⊗ vm by the V V V ⊗m ⊗m projection Vaff → m Vaff . Let L( m Vaff ) ⊂ m Vaff be the image of L(Vaff ). ⊗m For b = b1 ⊗ · · · ⊗ bm ∈ B(Vaff ), we set G pure (b) = G(b1 ) ∧ · · · ∧ G(bm ). ⊗m ⊗m For b ∈ B(Vaff ), let G(b) be the global basis of Vaff , and let G ∧ (b) be its image Vm in Vaff . We set
B
m ^
⊗m Vaff = b1 ⊗ · · · ⊗ bm ∈ B(Vaff ); H (bν ⊗ bν+1 ) > 0
for ν = 1, . . . , m − 1 .
Then in [13], the following properties are proved: V V (i) {G pure (b); b ∈ B( m Vaff )} is a basis of L( m Vaff ); Vm Vm V (ii) identifying B( Vaff ) with a subset of L( Vaff )/qs L( m Vaff ) by G pure , V V V (L( m Vaff ), B( m Vaff )) is a crystal base of m Vaff . On the other hand, the following proposition follows from Proposition 8.6. PROPOSITION 12.7 ⊗(m +m ) ⊗m 1 ⊗m 2 For b1 ∈ B(Vaff ) and b2 ∈ B(Vaff ), one has the following equality in Vaff 1 2 :
G(b1 ⊗ b2 ) = G(b1 ) ⊗ G(b2 ) +
X b10 ,b20
cb0 ,b0 G(b10 ) ⊗ G(b20 ). 1
2
⊗m 1 ⊗m 2 Here the sum ranges over (b10 , b20 ) ∈ B(Vaff ) × B(Vaff ) such that wt(b10 ) − 0 wt(b1 ) = wt(b2 ) − wt(b2 ) ∈ Q + \ {0}, and the coefficients satisfy cb0 ,b0 ∈ qs Q[qs ]. 1
2
Set ⊗m ⊗m B0 (Vaff ) = b1 ⊗ · · · ⊗ bm ∈ B(Vaff ); H (bν ⊗ bν+1 ) = 0 or ν = 1, . . . , m − 1 , M K G(b). Nm0 = ⊗m b∈B0 (Vaff )
Arguments similar to those in Proposition 11.2 and Corollary 11.3 show the following proposition.
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MASAKI KASHIWARA
PROPOSITION
12.8
We have ±1 sym ⊗m ±1 sym Uq (g) ⊗ Q[z 1±1 , . . . , z m ] u = Q[z 1±1 , . . . , z m ]
⊗ Q[(z 1 ···z m )±1 ]
Nm0 .
⊗m Here z ν is the automorphism of Vaff induced by the action of z on the νth factor, and ±1 ±1 sym Q[z 1 , . . . , z m ] is the ring of symmetric Laurent polynomials. In particular, for any Laurent polynomial f (z 1 , . . . , z m ) symmetric in (z ν , z ν+1 ) for some ν, f (z 1 , . . . , z m )Nm0 ⊂ Nm .
Vm ±1 ] sym -module, Since Nm is a Q[z 1±1 , . . . , z m Vaff has the structure of a Vm ±1 ] sym -module. We denote by B the operator on Q[z 1±1 , . . . , z m Vaff given by n Pm n z . Then the B ’s commute with one another. n ν=1 ν LEMMA 12.9 ⊗m For any b ∈ B(Vaff ), one has either G ∧ (b) = 0 or G ∧ (b) = ±G ∧ (b0 ) for some V m 0 b ∈ B( Vaff ).
Proof Set b = z a1 b1 ⊗ · · · ⊗ z am bm with H (bν ⊗ bν+1 ) = 0. Then we have by Proposition 12.8, G(b) ≡ ±G(z aσ (1) b1 ⊗ · · · ⊗ z aσ (m) bm ) for any permutation σ . Hence we may assume that (a1 , . . . , am ) is a decreasing sequence. If there is ν such that aν = aν+1 , then G(b) ∈ Nm by Proposition 12.8. V Otherwise, b belongs to B( m Vaff ). 12.3. Global basis of the Fock space The purpose of this subsection is to define the global basis of the Fock space. Let us now assume that V is a good Uq0 (g)-module with perfect crystal base (L , B) of level `. Let us recall that a simple crystal B is called perfect of level ` if it satisfies the following conditions. P (P1) Any b ∈ B satisfies hc, ε(b)i = hc, ϕ(b)i ≥ `. Here ε(b) = i εi (b) cl(3i ) ∈ P Pcl and ϕ(b) = i ϕi (b) cl(3i ) ∈ Pcl . (`) (P2) Set Pcl = {λ ∈ Pcl ; hc, λi = ` and hh i , λi ≥ 0 for every i}, the set of dominant weights of level `, and Bmin = {b ∈ B; hc, ε(b)i = `}. Then the two maps (`) (`) ε : Bmin −→ Pcl and ϕ : Bmin −→ Pcl are bijective.
QUANTIZED AFFINE ALGEBRAS
167 (1)
For example, the vector representation of An is a good Uq0 (g)-module with a perfect crystal base of level 1. Let (Baff )min be the inverse image of Bmin by the map Baff → B. Let us take a sequence {bn }n∈Z in (Baff )min such that ϕ(bn ◦ ) = ε(bn−1 ◦ )
H (bn ◦ ⊗ bn−1 ◦ ) = 1.
and
Such a sequence is called a ground state. Take a sequence {λn }n∈Z in P such that λn = λn−1 + wt(bn ◦ )
cl(λn ) = ϕ(bn ◦ ) = ε(bn−1 ◦ ).
and
In [13], the Fock spaces Fr (r ∈ Z) are constructed, and they satisfy the following properties. (F1) Fr is an integrable Uq (g)-module. (F2) We have Wt(Fr ) ⊂ λr + Q − . (F3) There exist Uq0 (g)-linear endomorphisms Bn (n ∈ Z \ {0}) of Fr with weight nδ satisfying the boson commutation relations [Bn , Bm ] = δ−n,m an for some an ∈ K \ {0}. V (F4) There exists a Uq (g)-linear map · ∧ · : Fr ⊗ m Vaff → Fr −m such that V V 0 (u ∧ v) ∧ v 0 = u ∧ (v ∧ v 0 ) for u ∈ Fr , v ∈ m Vaff , and v 0 ∈ m Vaff . (F5) Bn (u ∧ v) = (Bn u) ∧ v + u ∧ (z n v) for n ∈ Z \ {0}, u ∈ Fr , and v ∈ Vaff . (F6) There is a nonzero vector vacr ∈ Fr of weight λr , (Fr )λr = K vacr . Moreover, one has vacr +1 ∧ G(br ◦ ) = vacr . (F7) We have {u ∈ Fr ; Bn u = 0 for any n > 0 and ei u = 0 for any i} = K varr . P (F8) Let K [B−1 , B−2 , . . .] = K [Bn ; n 6= 0]/ m>0 K [Bn ; n 6= 0]Bm be the ∼ Fr as Fock space of the boson algebra. Then K [B−1 , B−2 , . . .] ⊗ V (λr ) −→ a K [Bn ; n 6= 0] ⊗ Uq (g)-module. Here 1 ⊗ u λr corresponds to vacr . (F9) Let B(Fr ) be the set of sequences {bn }n≥r satisfying H (bn+1 ⊗ bn ) > 0 bn = bn
for any n ≥ r, ◦
for n r.
For b = {bn }n≥r ∈ B(Fr ), set G pure (b) = vacn ∧ G(bn−1 ) ∧ · · · ∧ G(br ) for n r . Then {G pure (b); b ∈ B(Fr )} is a basis of Fr . S V (F10) Set L(Fr ) = n≥r vacn ∧ L( n−r Vaff ). Then (L(Fr ), B(Fr )) is a crystal base of Fr . Here B(Fr ) is identified with a subset of L(Fr )/qs L(Fr ) by G pure . (k) (F11) We have f i vacr = vacr +1 ∧ G( f˜ik br ◦ ). Now we show that the Fock space Fr has a global basis. First let us define a bar involution c on Fr such that c(vacr ) = vacr ,
(12.4)
[Bn , c] = 0
(12.5)
for any n > 0.
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MASAKI KASHIWARA
By (F8), there exists a unique bar involution on Fr satisfying the conditions above. Note that c ◦ B−n ◦ c = an an −1 B−n for n > 0, since [Bn , an −1 B−n ] = 1 implies that an −1 B−n is c-invariant. We set m−r X ^ (Fr )Q = vacm ∧ (Vaff )Q . m≥r
12.10 ⊗m Let b := b1 ⊗ · · · ⊗ bm be an element of Baff . ◦ ∧ (a) If H (br ⊗ b1 ) ≤ 0, then vacr ∧ G (b) = vacr ∧ G pure (b) = 0 hold in Fr −m . (b) We have vacr +1 ∧ G ∧ (br ◦ ⊗ b) = vacr ∧ G ∧ (b). LEMMA
Proof (a) We have G(b) =
X
cb0 ,b0 G(b10 ) ⊗ G(b0 ), 1
where the sum ranges over b10 ∈ Baff and b0 ∈ Baff ⊗(m−1) such that wt(b10 ) − wt(b1 ) ∈ Q + . Since H (br ◦ ⊗ b1 ) ≤ 0, we have `(br −1 ◦ ) < `(b1 ) ≤ `(b10 ) by [13, Lem. 4.2.2]. Since Wt(Fr −1 ) ⊂ λr −1 + Q − by (F2), one has vacr ∧ G(b10 ) = 0. Hence we obtain vacr ∧ G ∧ (b) = 0. The proof of vacr ∧ G pure (b) = 0 is similar. (b) The proof is similar to that of (a). One has X G(br ◦ ⊗ b) = G(br ◦ ) ⊗ G(b) + cb0 ,b0 G(b00 ) ⊗ G(b0 ), 0
where the sum ranges over b00 ∈ Baff and b0 ∈ Baff ⊗m such that wt(b00 ) − wt(br ◦ ) ∈ Q + \ {0}. Then by the same reasoning on the weight of Wt(Fr ), we have vacr +1 ∧ G(b00 ) = 0. By Lemma 12.10, for b = {bn }n≥r ∈ B(Fr ), G(b) := vacm ∧ G ∧ (bm−1 ⊗ · · · ⊗ br ) does not depend on m such that b j = b j ◦ for j ≥ m. LEMMA 12.11 {G(b); b ∈ B(Fr )} is a basis of the A-module L(Fr ).
Proof Since b ≡ G(b) mod qs L(Fr ), {G(b); b ∈ B(Fr )} is linearly independent. Hence it is enough to show that it generates L(Fr ).
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169
V Let b = (b1 , . . . , bm ) ∈ B( m Vaff ). For any integer N , we can write X X G pure (b) = ab0 G ∧ (b0 ) + cb00 G pure (b00 ). b0
b00
⊗m 00 ranges over B(V ⊗m ) with Here b0 ranges over B(Vaff ) and b00 = b100 ⊗ · · · ⊗ bm aff 00 ◦ `(b1 ) > N . Taking `(bm+r −1 ) as N , one has vacm+r ∧ G pure (b00 ) = 0. Hence one has X ab0 vacm+r ∧ G ∧ (b0 ). vacm+r ∧ G pure (b) = ⊗m ) b0 ∈B(Vaff
Now it is enough to apply Lemmas 12.9 and 12.10. THEOREM 12.12 {G(b); b ∈ B(Fr )} is a global basis of Fr .
Proof It remains to prove that the G(b)’s are invariant by the bar involution c. Let E be the vector space over Q generated by {G(b); b ∈ B(Fr )}. Then vacr +m ∧ G ∧ (b) ⊗m is contained in E for any b ∈ B(Vaff ) by Lemmas 12.9 and 12.10. We define the 0 involution c of Fr by c 0 (v) = v
for any v ∈ E
and c 0 (av) = a c 0 (v)
for any v ∈ Fr and a ∈ K .
We show that c 0 = c. In order to see this, it is enough to show the following properties: c 0 (vacr ) = vacr , 0
c commutes with Bn
(12.6) if n > 0,
c (av) = ac (v) for any v ∈ Fr and a ∈ Uq (g). 0
0
(12.7) (12.8)
The property (12.6) is obvious. Let us first show that c 0 commutes with Bn (n > 0). This follows from the fact V that Bn (vacr +m ∧ G ∧ (b)) = vacr +m ∧ Bn G ∧ (b) holds for b ∈ B( m Vaff ) and from the fact that Bn G ∧ (b) belongs to E. Let us show (12.8). We evidently have q h ◦ c 0 = c 0 ◦ q −h for every h ∈ P ∗ . The conjugation c 0 commutes with ei because, for b ∈ B(Fr ), ei G(b) belongs to Q[qs + qs−1 ] ⊗ E.
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Finally, let us show that c 0 commutes with f i . To see this, we prove f i c 0 (v) = i v) for any weight vector v ∈ Fr by the induction on wt(v). For any j ∈ I , one has, by using the commutativity of c 0 and ei ,
c0( f
ti − ti−1 0 e j f i c 0 (v) − c 0 ( f i v) = f i e j + δi j c (v) qi − qi−1 ti − ti−1 v − c 0 f i e j + δi j qi − qi−1 = f i c 0 (e j v) − c 0 ( f i e j v). Since this vanishes by the induction hypothesis, f i c 0 (v) − c 0 ( f i v) is a highest weight vector. Similarly, it is annihilated by all the Bn ’s (n > 0). Since the weight of f i c 0 (v)− c 0 ( f i v) is not λr , it must vanish by (F7). Thus we obtain (12.8). Remark 12.13 (1) In the case when g = An and V is the vector representation, the global basis of the Fock space was introduced by Leclerc and Thibon [14], [15]. D. Uglov [20] general(1) (1) ized this to the case when g = An ⊕ Am and when V is the tensor product of the vector representations. The connection of global bases of Fock space and KazhdanLusztig polynomials has also been studied by Varagnolo and Vasserot [21] and O. Schiffmann [19]. 13. Conjectural structure of V (λ) In this section, we present conjectures that clarify the structure of V (λ) and its crystal base B(λ) for λ ∈ P 0 . The paper by J. Beck, V. Chari, and A. Pressley [2] should help to solve them. These conjectures are closely related to those of Lusztig [18]. P Let λ be a dominant integral weight of level zero. We write λ = i∈I ∨ m i $i . 0 N N Then the module i∈I ∨ V (m i $i ) contains the extremal vector i∈I ∨ u m i $i whose 0 0 weight is λ. Here we can take any ordering of I0∨ to define the tensor product. Hence we have a Uq (g)-linear morphism O 8λ : V (λ) → V (m i $i ) i∈I0∨
sending u λ to
N
i∈I0∨
u m i $i .
13.1 8λ is a monomorphism. We have 8−1 λ (⊗i∈I ∨ L(m i $i )) = L(λ).
CONJECTURE
(i) (ii)
0
QUANTIZED AFFINE ALGEBRAS
171
By 8λ , we have an isomorphism of crystals O ∼ B(λ) −→ B(m i $i ).
(iii)
i∈I0∨
Next we consider the case when λ is a multiple of a fundamental weight. There is a morphism of Uq (g)-modules 9m,i : V (m$i ) → V ($i )⊗m 0 sending u m$i to u ⊗m $i . Let z i be the Uq (g)-linear automorphism of V ($i ) of weight di δ introduced in §5.2, and let z ν (ν = 1, . . . , m) be the operator of V ($i )⊗m obtained by the action of z i on the νth factor. It is again a Uq0 (g)-linear automorphism of V ($i )⊗m of weight di δ. Let B0 (m$i ) be the connected component of B(m$i ) containing u m$i , and let B0 (V ($i )⊗m ) be the connected component of B($i )⊗m containing u ⊗m $i .
13.2 9m,i is a monomorphism. −1 We have 9m,i L(V ($i )⊗m ) = L(m$i ). ∼ B0 (V ($i )⊗m ) by 9m,i . Moreover, the global basis G(b) with B0 (m$i ) −→ b ∈ B0 (m$i ) is sent to the corresponding global basis of Uq (g)u ⊗m $i ⊂ W ($i )⊗m constructed in Theorem 8.5. aff Let S be the set of Schur Laurent polynomials in z 1 , . . . , z m , that is, the set of characters of GL(m) ((z 1 , . . . , z m ) that are the components of the diagonal matrices). Then {G(b); b ∈ B(m$i )} is sent by 9m,i to {aG(b); b ∈ B0 (V ($i )⊗m ), a ∈ S}.
CONJECTURE
(i) (ii) (iii)
(iv)
Note that, for a, a 0 ∈ S and b, b0 ∈ B0 (V ($i )⊗m ), a G(b) = a 0 G(b0 ) holds if and only if a 0 = (z 1 · · · z m )r a and b = (z 1 · · · z m )r b0 for some r ∈ Z. These conjectures imply the following conjecture on U˜ q (g) analogous to the Peter-Weyl theorem. For λ ∈ P, let B0 (λ) be the connected component of B(λ) conF taining u λ . Note that if hc, λi 6 = 0, then B0 (λ) = B(λ). We consider λ∈P B0 (λ) × F B(−λ) as a crystal over g ⊕ g. The Weyl group W acts on λ∈P B0 (λ) × B(−λ) by Sw∗ × Sw∗ : B0 (λ) × B(−λ) → B0 (wλ) × B(−wλ). 13.3 ∼ B(U˜ q (g)) as a crystal over g × g. B0 (λ) × B(−λ) /W −→
CONJECTURE
F
λ∈P
Here the usual crystal structure on B(U˜ q (g)) corresponds to that of B0 (λ), and the star crystal structure on B(U˜ q (g)) corresponds to that of B(−λ). The isomorphism
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MASAKI KASHIWARA
sends u λ ⊗ b ∈ B0 (λ) × B(−λ) to b∗ ∈ B(U˜ q (g)).
Appendices A. Proof of the formula (6.2) In this appendix, we give a proof of (6.2) due to Anne Schilling. Let us define (a)n = (a; q)n =
n−1 Y
(1 − aq i ).
i=0
Then in terms of (a; q)n , the q-binomial in this paper is given as m (q 2 ; q 2 )m = q n(n−m) 2 2 . n i (q ; q )n (q 2 ; q 2 )m−n Hence, replacing n → 2n and q → q 1/2 in (6.2), it reads as follows. LEMMA
A.1
We have m X
(−1)k q (1/2)k(k+1−2m)−nm
k=0
(q n )k (q)2n+m (q)`−m+k (q)m−k (q)2n+k (q)k (q)`−m =
m X
q k(`−m−n+1)
k=0
(q n )k (q n+1 )m−k . (A.1) (q)k (q)m−k
Proof Using (see [5, (I.10) in Appendix]) (a)m−k =
(a)m q k (k )−mk q 2 , − a (q 1−m /a)k
equation (A.1) may be rewritten in hypergeometric notation as q
−nm (q
2n+1 ) m
(q)m
3 82
q −m , q n , q `−m+1 ;q q 2n+1 , 0
−m n (q n+1 )m q ,q `−m−2n+1 = ;q . 2 81 (q)m q −m−n However, this formula readily follows from [5, (III.7) in Appendix] with the replacements n → m,
b → qn,
c → q −n−m ,
z → q `−m−2n+1 .
QUANTIZED AFFINE ALGEBRAS
173
B. Formulas for the crystal B(U˜ q (g)) In this table, b1 ∈ B(∞), b2 ∈ B(−∞), λ ∈ P, b = b1 ⊗ tλ ⊗ b2 , λi = hh i , λi, and wti (b1 ) = hh i , wt(b1 )i: b∗ = b1∗ ⊗ t−λ−wt(b1 )−wt(b2 ) ⊗ b2∗ ; εi (b) = max εi (b1 ), εi (b2 ) − λi − wti (b1 ) ; ϕi (b) = max ϕi (b1 ) + λi + wti (b2 ), ϕi (b2 ) ; wt∗ (b) = wt(b∗ ) = −λi ; εi∗ (b) = max εi∗ (b1 ), ϕi∗ (b2 ) + λi ; ϕi∗ (b) = max εi∗ (b1 ) − λi , ϕi∗ (b2 ) ; e˜i b1 ⊗ tλ ⊗ b2 if ϕi (b1 ) ≥ εi (b2 ) − λi , e˜i b = b1 ⊗ tλ ⊗ e˜i b2 if ϕi (b1 ) < εi (b2 ) − λi ; ( f˜i b1 ⊗ tλ ⊗ b2 if ϕi (b1 ) > εi (b2 ) − λi , f˜i b = b1 ⊗ tλ ⊗ f˜i b2 if ϕi (b1 ) ≤ εi (b2 ) − λi ; ( e˜i∗ b1 ⊗ tλ−αi ⊗ b2 if εi∗ (b1 ) ≥ ϕi∗ (b2 ) + λi , ∗ e˜i b = b1 ⊗ tλ−αi ⊗ e˜i∗ b2 if εi∗ (b1 ) < ϕi∗ (b2 ) + λi ; ( f˜i∗ b1 ⊗ tλ+αi ⊗ b2 if εi∗ (b1 ) > ϕi∗ (b2 ) + λi , ∗ f˜i b = b1 ⊗ tλ+αi ⊗ f˜i∗ b2 if εi∗ (b1 ) ≤ ϕi∗ (b2 ) + λi ; e˜imax b = e˜imax b1 ⊗ tλ ⊗ e˜i c b2 , where c = max εi (b2 ) − ϕi (b1 ) − λi , 0 ; f˜imax b = f˜i c b1 ⊗ tλ ⊗ f˜imax b2 , where c = max ϕi (b1 ) − εi (b2 ) + λi , 0 ; ∗ max ∗ ∗ e˜i b1 ⊗ tλ−(ϕi∗ (b2 )+λi )αi ⊗ e˜i∗ ϕi (b2 )−εi (b1 )+λi b2 if εi∗ (b1 ) − ϕi∗ (b2 ) − λi ≤ 0, ∗ max e˜i b= e˜i∗ max b1 ⊗ tλ−εi∗ (b1 )αi ⊗ b2 if εi∗ (b1 ) − ϕi∗ (b2 ) − λi ≥ 0; ∗ ε∗ (b )−ϕ ∗ (b )−λ f˜i i 1 i 2 i b1 ⊗ tλ+(εi∗ (b1 )−λi )αi ⊗ f˜i∗ max b2 if εi∗ (b1 ) − ϕi∗ (b2 ) − λi ≥ 0 , f˜i∗ max b = b1 ⊗ tλ+ϕi∗ (b2 )αi ⊗ f˜i∗ max b2 if εi∗ (b1 ) − ϕi∗ (b2 ) − λi ≤ 0 .
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Now assume that b = b1 ⊗ tλ ⊗ u −∞ . If b is extremal, wt (b )+λ if εi (b) = 0, f˜i i 1 i b1 ⊗ tλ ⊗ u −∞ Si b = max −ϕ (b )−λ i i 1 e˜i b1 ⊗ tλ ⊗ e˜i u −∞ if ϕi (b) = 0. If b∗ is extremal, ( Si∗ b
=
if εi∗ (b) = 0,
f˜i∗ −λi b1 ⊗ tsi λ ⊗ u −∞ ∗ λi −εi∗ (b1 )
e˜i∗ max b1 ⊗ tsi λ ⊗ e˜i
u −∞
if ϕi∗ (b) = 0.
Acknowledgment. The author would like to thank Anne Schilling who kindly provided a proof of the formula (6.2). References [1]
[2] [3]
[4] [5]
[6]
[7] [8] [9] [10] [11] [12]
T. AKASAKA and M. KASHIWARA, Finite-dimensional representations of quantum
affine algebras, Publ. Res. Inst. Math. Sci. 33 (1997), 839–867. MR 99d:17017 117, 118, 123, 132, 153, 154 J. BECK, V. CHARI, and A. PRESSLEY, An algebraic characterization of the affine canonical basis, Duke Math. J. 99 (1999), 455–487. MR 2000g:17013 169 V. G. DRINFELD, “Quantum groups” in Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986), Vol. 1, Amer. Math. Soc., Providence, 1987, 798–820. MR 89f:17017 147 , A new realization of Yangians and the quantum affine algebras, Soviet Math. Dokl. 36, no. 2 (1988), 212–216. MR 88j:17020 118 DZH. GASPER and M. RAKHMAN, Basic Hypergeometric Series (in Russian), “Mir,” Moscow, 1993, MR 95d:33017; English translation in Encyclopedia Math. Appl. 35, Cambridge Univ. Press, Cambridge, 1990. MR 91d:33034 171, 172 S.-J. KANG, M. KASHIWARA, K. MISRA, T. MIWA, T. NAKASHIMA, and A. NAKAYASHIKI, “Affine crystals and Vertex models” in Infinite Analysis (Kyoto, 1991), Part A, Adv. Ser. Math. Phys. 16, World Sci., River Edge, N.J., 1992, 449–484. MR 94a:17008 158 , Perfect crystals of quantum affine Lie algebras, Duke Math. J. 68 (1992), 499–607. MR 94j:17013 M. KASHIWARA, On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465–516. MR 93b:17045 119, 120, 122 , The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J. 71 (1993), 839–858. MR 95b:17019 121 , Global crystal bases of quantum groups, Duke Math. J. 69 (1993), 455–485. MR 94b:17024 122 , Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73 (1994), 383–413. MR 95c:17024 117, 119, 120, 123, 135 , “On crystal bases” in Representations of Groups (Banff, Alberta, 1994), CMS Conf. Proc. 16, Amer. Math. Soc., Providence, 1995, 155–197. MR 97a:17016 119, 120
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M. KASHIWARA, T. MIWA, J.-U. H. PETERSEN, and C. M. YUNG, Perfect crystals and
q-deformed Fock spaces, Selecta Math. (N.S.) 2 (1996), 415–499. MR 98f:17012 118, 160, 161, 164, 166, 167 B. LECLERC and J.-Y. THIBON, Canonical bases of q-deformed Fock spaces, Internat. Math. Res. Notices 1996, 447–456. MR 97h:17023 118, 169 , “Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials” in Combinatorial Methods in Representation Theory (Kyoto, 1998), Adv. Stud. Pure Math. 28, Kinokuniya, Tokyo, 2000, 155–200. MR 1 864 481 118, 169 P. LITTELMANN, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), 499–525. MR 96m:17011 121 G. LUSZTIG, Introduction to Quantum Groups, Progr. Math. 110 Birkh¨auser, Boston, 1993. MR 94m:17016 148 , “Quantum groups at v = ∞” in Functional analysis on the Eve of the 21st century (New Brunswick, N.J., 1993), Vol. 1, Progr. Math. 131 Birkh¨auser, Boston, 1995, 199–221. MR 97g:17014 169 O. SCHIFFMANN, The Hall algebra of a cyclic quiver and canonical bases of Fock spaces, Internat. Math. Res. Notices 2000, 413–440. MR 2001g:16032 169 D. UGLOV, “Canonical bases of higher-level q-deformed Fock spaces and Kazhdan-Lusztig polynomials” in Physical Combinatorics (Kyoto, 1999), ed. M. Kashiwara and T. Miwa, Progr. Math. 191, Birkh¨auser, Boston, 2000, 249–299. MR 2001k:17030 169 M. VARAGNOLO and E. VASSEROT, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), 267–297. MR 2001c:17029 118, 169 , Standard modules of quantum affine algebras, Duke Math. J. 111 (2002), 509–533. 118
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606–8502, Japan.
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1,
CUSPIDALITY OF SYMMETRIC POWERS WITH APPLICATIONS HENRY H. KIM and FREYDOON SHAHIDI
Abstract The purpose of this paper is to prove that the symmetric fourth power of a cusp form on GL(2), whose existence was proved earlier by the first author, is cuspidal unless the corresponding automorphic representation is of dihedral, tetrahedral, or octahedral type. As a consequence, we prove a number of results toward the Ramanujan1/9 Petersson and Sato-Tate conjectures. In particular, we establish the bound qv for unramified Hecke eigenvalues of cusp forms on GL(2). Over an arbitrary number field, this is the best bound available at present. 1. Introduction In this paper we prove a criterion for cuspidality of the fourth symmetric powers of cusp forms on GL(2), whose existence was established earlier by the first author. As a consequence, we show that a cuspidal representation has a noncuspidal symmetric fourth power if and only if it is of either dihedral, tetrahedral, or octahedral type. We then prove a number of corollaries toward both the Ramanujan-Petersson and Sato-Tate conjectures for cusp forms on GL(2) by establishing analytic properties of several new symmetric power L-functions attached to them. N More precisely, let A be the ring of adeles of a number field F. Let π = v πv be a cuspidal automorphic representation of GL2 (A) with central character ωπ . Fix a positive integer m, and let Symm : GL2 (C) −→ GLm+1 (C) be the mth symmetric power representation of GL2 (C) on symmetric tensors of rank m (cf. [28], [30]). By the local Langlands correspondence (see [4], [5], [17]), Symm (πv ) is well defined for every v. Then Langlands functoriality in this case is equivalent to the fact N m that Symm (π) = v Sym (πv ) is an automorphic representation of GLm+1 (A). It is convenient to introduce Am (π) = Symm (π ) ⊗ ωπ−1 (denoted by Adm (π) in [28]). DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 1, Received 30 March 2001 2000 Mathematics Subject Classification. Primary 11F30, 11F70, 11R42 Kim’s work partially supported by National Science Foundation grant numbers DMS9988672 and DMS9729992 (at the Institute for Advanced Study) and by the Clay Mathematics Institute, as a Prize Fellow. Shahidi’s work partially supported by National Science Foundation grant number DMS9970156 and by the Clay Mathematics Institute, as a Prize Fellow. 177
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KIM and SHAHIDI
If m = 2, A2 (π) = Ad(π) and it is the well-known Gelbart-Jacquet lift in [2]. If m = 3, we proved in [13] and [12] that Sym3 (π) is an automorphic representation of GL4 (A) and gave a criterion for when it is cuspidal. In [10], the first author proved that Sym4 (π) is an automorphic representation of GL5 (A). If Sym3 (π) is cuspidal, Sym4 (π)is either cuspidal or unitarily induced from cuspidal representations of GL2 (A) and GL3 (A). In this paper, we give a criterion for when Sym4 (π ) is cuspidal. More precisely, we have the following. 3.3.7 Sym4 (π) ⊗ ωπ−1 is a cuspidal representation of GL5 (A), except in the following three cases: (1) π is monomial; (2) π is not monomial and A3 (π) is not cuspidal; this is the case when there exists a nontrivial gr¨ossencharacter µ such that Ad(π) ' Ad(π) ⊗ µ; (3) A3 (π) is cuspidal, but there exists a nontrivial quadratic character η such that A3 (π) ' A3 (π ) ⊗ η, or, equivalently, there exists a nontrivial gr¨ossencharacter χ of E such that Ad(π E ) ' Ad(π E ) ⊗ χ , where E/F is the quadratic extension determined by η and π E is the base change of π. In this case, A4 (π) = σ1 σ2 , where σ1 = π(χ −1 ) ⊗ ωπ and σ2 = Ad(π) ⊗ (ωπ η). Cases (1), (2), and (3) are equivalent to π being of dihedral, tetrahedral, and octahedral type, respectively. THEOREM A4 (π ) =
We give several applications of the cuspidality of third and fourth symmetric powers. First, following Ramakrishnan [23], we prove that given a cuspidal representation of GL2 (A), the set of tempered places has lower Dirichlet density of at least 34/35. Next, we prove the meromorphic continuation and a functional equation for each of the sixth, seventh, eighth, and ninth symmetric power L-functions for cuspidal representations of GL2 (A). An immediate corollary (cf. [27, Lemma 5.8]) is that if N πv is an unramified local component of a cuspidal representation π = v πv , then −1/9 1/9 qv < |αv |, |βv | < qv , where diag(αv , βv ) is the Satake parameter for πv . The archimedean analogue of 1/9, using this approach, is proved in [11]. When F = Q, using the ideas in [19], the bound 1/9 can be improved to 7/64 + ε, ∀ε > 0. This is the subject of an appendix in [10] by Kim and Sarnak. For an arbitrary number field, 1/9 remains the best bound available at present. Finally, we prove that partial fifth, sixth, seventh, and eighth symmetric power L-functions attached to a cuspidal representation π of GL2 (A) with trivial central character such that Sym4 (π) is cuspidal are all invertible at s = 1, and we apply this fact to the Sato-Tate conjecture (see [25]), following Serre’s method (see [30, Appendix]). Namely, we show that for every > 0 there are sets T + and T − of
SYMMETRIC POWERS
179
positive lower (Dirichlet) densities such that av > 2 cos(2π/11) − for all v ∈ T + , and av < −2 cos(2π/11) + for all v ∈ T − , where av = αv + βv . Note that 2 cos(2π/11) = 1.68 . . . . 2. Cuspidality of the symmetric cube Suppose π is a cuspidal representation of GL2 (A). We review the properties of the symmetric cube Sym3 (π) (see [13]). Recall that A3 (π) = Sym3 (π) ⊗ ωπ−1 . 2.1. π a monomial cuspidal representation That is, π ⊗ η ' π for a nontrivial gr¨ossencharacter η. Then η2 = 1 and η determines a quadratic extension E/F. According to [14], there is a gr¨ossencharacter χ of E such that π = π(χ), where π(χ) is the automorphic representation whose local factor at v is the one attached to the representation of the local Weil group induced from χv . Let χ 0 be the conjugate of χ by the action of the nontrivial element of the Galois group. Then the Gelbart-Jacquet lift (adjoint) of π is given by Ad(π ) = π(χχ 0
−1
) η.
There are two cases. Case 1: χ χ 0 −1 factors through the norm. That is, χχ 0 −1 = µ ◦ N E/F for a gr¨ossencharacter µ of F. Then π(χχ 0 −1 ) is not cuspidal. In fact, π(χχ 0 −1 ) = µ µη. In this case, −1 A3 (π) = π(χχ 0 ) π = (µ ⊗ π ) (µη ⊗ π ). Case 2: χ χ 0 −1 does not factor through the norm. In this case, π(χχ 0 −1 ) is a cuspidal representation. Then A3 (π) = π(χχ 0
−1
) π = π(χ 2 χ 0
−1
) π.
Here we use the fact that π(χ) E = χ χ 0 (see [24, Proposition 2.3.1]) and that π 0 π = I FE (π E0 ⊗ χ ) if π = π(χ) (see [24, §3.1]). 2.2. π not monomial In this case, Ad(π) is a cuspidal representation of GL3 (A). We recall from [13] the following. THEOREM 2.2.1 Let σ be a cuspidal representation of GL2 (A). Then the triple L-function L S (s, Ad(π) × π × σ ) has a pole at s = 1 if and only if σ ' π ⊗ χ and Ad(π ) ' Ad(π ) ⊗ (ωπ χ) for some gr¨ossencharacter χ .
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KIM and SHAHIDI
By Theorem 2.2.1, we have the following. 2.2.2 Let π be a nonmonomial cuspidal representation of GL2 (A). Then A3 (π ) is not cuspidal if and only if there exists a nontrivial gr¨ossencharacter µ such that Ad(π ) ' Ad(π ) ⊗ µ. In that case, THEOREM
A3 (π) = (π ⊗ µ) (π ⊗ µ2 ). 3. Cuspidality of the symmetric fourth Suppose π is a cuspidal representation of GL2 (A F ). Let A4 (π ) = Sym4 (π ) ⊗ ωπ−1 . We review the properties of Sym4 (π) (see [10]). 3.1. π monomial Suppose π is a monomial cuspidal representation given by π = π(χ). Then the Gelbart-Jacquet lift of π is given by Ad(π ) = π(χχ 0 −1 ) η. Case 1: χ χ 0 −1 factors through the norm. Then (see Section 2.1) since ∧2 (A3 (π)) = A4 (π) ⊕ ωπ , A4 (π) = (π π ) ωπ = ωπ ωπ µωπ ηωπ µηωπ . We used the fact that η and µ are quadratic gr¨ossencharacters. Case 2: χ χ 0 −1 does not factor through the norm. Then (see Section 2.1) A4 (π) = π(χ 2 χ 0
−1
−1 ) π ωπ = π(χ 3 χ 0 ) π(χ 2 ) ωπ .
3.2. π a nonmonomial representation such that Sym3 (π) is not cuspidal This is the case when there exists a nontrivial gr¨ossencharacter µ such that Ad(π ) ' Ad(π) ⊗ µ. Note that µ3 = 1. Then A3 (π) = (π ⊗ µ) (π ⊗ µ2 ). Hence ∧2 (A3 π) = Sym2 (π ) ωπ ωπ µ ωπ µ2 . So A4 (π) = Sym2 (π ) ωπ µ ωπ µ2 . PROPOSITION 3.2.1 Suppose π is a nonmonomial representation such that A3 (π) is not cuspidal; that is, Ad(π) ' Ad(π) ⊗ µ. Then L(s, π, Sym4 ⊗ ωπ−1 ) has a pole at s = 1 if and only if ωπ = µ or µ2 . In particular, if ωπ = 1, L(s, π, Sym4 ) is holomorphic at s = 1.
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3.3. Both Ad(π) and Sym3 (π) cuspidal Throughout this paper S is always a set of places of F such that for v 6∈ S every representation is unramified. The first author showed in [10] that ∧2 (A3 (π )) is an automorphic representation of GL6 (A) and that ∧2 (A3 (π )) = A4 (π) ωπ . Hence A4 (π) is an automorphic representation of GL5 (A), either cuspidal or induced from cuspidal representations of GL2 (A) and GL3 (A). We want to give a criterion for when A4 (π) is cuspidal. First we note that (1) L S s, σ ⊗ A3 (π ), ρ2 ⊗ ∧2 ρ4 = L S s, σ × A4 (π) L S (s, σ ⊗ ωπ ). LEMMA 3.3.1 A4 (π ) is not cuspidal if and only if
L S (s, σ ⊗ A3 (π), ρ2 ⊗ ∧2 ρ4 ) has a pole at s = 1 for some cuspidal representation σ of GL2 (A). Proof Since A4 (π ) is either cuspidal or induced from cuspidal representations of GL2 (A) and GL3 (A), A4 (π) is not cuspidal if and only if L S (s, σ × A4 (π )) has a pole at s = 1 for some cuspidal representation σ of GL2 (A). Our assertion follows from (1) since L S (s, σ ⊗ ωπ ) is invertible at s = 1. In order to find a criterion for the pole of L(s, σ ⊗ A3 (π), ρ2 ⊗ ∧2 ρ4 ), we need the following unpublished result of H. Jacquet, I. Piatetski-Shapiro, and J. Shalika (cf. [22]). THEOREM 3.3.2 (Jacquet, Piatetski-Shapiro, and Shalika) Let π be a cuspidal automorphic representation of GL4 (A) such that there exist a gr¨ossencharacter χ and a finite set S of places as above for which L S (s, π, ∧2 ⊗ χ −1 ) has a pole at s = 1. Then there exists a globally generic cuspidal automorphic representation τ of GSp4 (A) with central character χ such that π is the Langlands functorial lift of τ under the natural embedding L GSp4 = GSp4 (C) ,→ GL4 (C).
The following paragraph is a brief sketch of the steps of the proof of the theorem, which we are including at a referee’s suggestion. We thank Dinakar Ramakrishnan for helping us with its preparation. Theorem 3.3.2 is proved using the dual reductive pair (GO6 (A), GSp4 (A)). More precisely, one considers the low-rank isogeny of SL4 and SO6 to lift a cuspidal repreN sentation π = v πv of GL4 (A) to one, still denoted by π, on GO6 (A), provided that the central character of πv is trivial on ±I for each v. One can then compute the theta lift of π to an automorphic representation of GSp4 (A) by integrating functions in the
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space of π in the usual way against the θ -function, a function on the two-fold cover of GSp12 (A). It is this integral that, in view of [6], is in fact equal to the residue of L S (s, π, ∧2 ⊗ χ −1 ) at s = 1, where χ is the central character of the theta lift. Hence the nonvanishing of the theta lift of π to GSp4 (A) is equivalent to the existence of a pole for L S (s, π, ∧2 ⊗ χ −1 ) at s = 1. Here S is a finite set of places for which v ∈ /S implies that πv is unramified. The lift is irreducible and globally generic. We now look at how the Satake parameter behaves under the map L GSp4 = GSp GSp4 (C) ,→ GL4 (C). Suppose τv is an unramified representation given by Ind B 4 µ ⊗ ν ⊗ λ, where µ, ν, λ are unramified quasicharacters of Fv× and µ ⊗ ν ⊗ λ is the character of the torus which assigns to diag(x, y, t y −1 , t x −1 ) the value µ(x)ν(y)λ(t). Note that the central character is χv = µνλ2 . Then the Satake parameter corresponding to τv is (see, e.g., [31, p. 95]) diag(µνλ, µλ, λ, νλ). Here we identify µ with µ($ ); we do the same with ν and λ. The Satake parameter for ∧2 (πv ) is diag(µ2 νλ2 , µνλ2 , µν 2 λ, µλ2 , νλ2 , µνλ2 ) = diag(χv µ, χv ν, χv , χv µ−1 , χv ν −1 , χv ). Hence, if σv is an unramified representation of GL2 (Fv ) given by π(η1 , η2 ), then L(s,σv ⊗ πv , ρ2 ⊗ ∧2 ρ4 )−1 =
2 Y (1 − χv ηi µ±1 qv−s )(1 − χv ηi ν ±1 qv−s )(1 − χv ηi qv−s ) i=1 2 Y × (1 − χv ηi qv−s ) i=1
= L s, (σv ⊗ χv ) × τv
−1
L(s, σv ⊗ χv )−1 .
(2)
Here L(s, (σv ⊗χv )×τv ) is the degree 10 Rankin-Selberg L-function for GL2 × GSp4 . Note that if τ 0 is any irreducible constituent of τ |Sp4 (A) , then L s, (σv ⊗ χv ) × τv = L s, (σv ⊗ χv ) × τv0 . We now apply the above observation to A3 (π ), where π is a cuspidal representation of GL2 (A). Since ∧2 (A3 (π)) = A4 (π) ωπ , L S (s, A3 (π ), ∧2 ⊗ ωπ−1 ) has a pole at s = 1. So there exists a generic cuspidal representation τ of GSp4 (A) with central character ωπ . Let τ 0 be any irreducible constituent of τ |Sp4 (A) . Then by (1) and (2), we have L S s, σ × A4 (π) = L S s, (σ ⊗ ωπ ) × τ 0 .
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Recall that if we consider the maximal Levi subgroup GL2 × Sp4 ⊂ Sp8 with the cuspidal representation (σ ⊗ ωπ ) ⊗ τ 0 (see [27]), we obtain the following L-function as a normalizing factor in the constant term of the Eisenstein series: L S s, (σ ⊗ ωπ ) × τ 0 L S (2s, ωσ ωπ2 ). Note that if (ωσ ωπ2 )2 6= 1, (σ ⊗ ωπ ) ⊗ τ 0 is not self-contragredient. Hence we have the following. PROPOSITION 3.3.3 If (ωσ ωπ2 )2 6= 1, then L S (s, (σ ⊗ ωπ ) × τ 0 ) is holomorphic at s = 1.
Proof By following the proof of [8, Theorem 3.2], we can show that the completed Lfunction L(s, (σ ⊗ ωπ ) × τ 0 ) is entire. Since (σ ⊗ ωπ ) ⊗ τ 0 is not self-contragredient, by [7, Proposition 2.1] the global intertwining operator M(s, (σ ⊗ ωπ ) ⊗ τ 0 , w0 ) is holomorphic for Re s > 0. Using [8, Proposition 3.4], the local normalized intertwining operators N (s, (σv ⊗ωπv )⊗τv0 , w0 ) are holomorphic and nonzero for Re s ≥ 1/2. Hence we see that the L-function L(s, (σ ⊗ωπ )×τ 0 )L(2s, ωσ ωπ2 ) is holomorphic for Re s ≥ 1/2. Since L(2s, ωσ ωπ2 ) has no zeros for Re s ≥ 1/2, L(s, (σ ⊗ ωπ ) × τ 0 ) is holomorphic for Re s ≥ 1/2. Our assertion follows from the functional equation. Using the integral representation, we have the following. THEOREM 3.3.4 (Ginzburg-Rallis-Soudry [3]) If ωσ ωπ2 = 1, then L S (s, (σ ⊗ ωπ ) × τ 0 ) is holomorphic at s = 1.
Proof This follows immediately from the integral representation for L S (s, (σ ⊗ ωπ ) × τ 0 ). The possible poles come from the poles of the Eisenstein series attached to (GL2 , σ ⊗ ωπ ) for the split group SO5 , which in turn come from the poles of symmetric square L-function L S (s, σ ⊗ ωπ , Sym2 ). The last L-function is entire if ωσ ωπ2 = 1. COROLLARY 3.3.5 If ωσ ωπ2 = 1, or if (ωσ ωπ2 )2 6= 1, then L S (s, σ × A4 (π)) is holomorphic at s = 1. PROPOSITION 3.3.6 Let π be a cuspidal representation of GL2 (A) such that both 5 = Ad(π ) and A3 (π ) are cuspidal. Then the following are equivalent. (1) A4 (π) is not cuspidal.
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(2)
There exists a quadratic extension E/F such that A3 (π) ' A3 (π ) ⊗ η, where η is the quadratic gr¨ossencharacter attached to E/F via class field theory. This means that the base change (A3 (π)) E is not cuspidal. (Note that A3 (π E ) ' (A3 (π )) E .) There exists a quadratic extension E/F such that 5 E ' 5 E ⊗ χ for a nontrivial gr¨ossencharacter χ of E, where 5 E is the base change of 5 to E.
(3)
Proof Consider L S s, (σ ⊗ ωπ ) × 5 × 5 = L S s, (σ ⊗ ωπ ) × 5 L S s, σ ⊗ A3 (π ), ρ2 ⊗ ∧2 ρ4 . Since L S (s, (σ ⊗ ωπ ) × 5) does not have a pole or a zero at s = 1, L S (s, (σ ⊗ ωπ ) × 5 × 5) has a pole at s = 1 if and only if L S (s, σ ⊗ A3 (π), ρ2 ⊗ ∧2 ρ4 ) has a pole at s = 1. Statement (1) implies statement (2). Since A4 (π) is not cuspidal, L S (s, σ ⊗ 3 A (π ), ρ2 ⊗ ∧2 ρ4 ) has a pole at s = 1 for some σ by Lemma 3.3.1. Hence, by Corollary 3.3.5, ω2 = 1, ω 6= 1, where ω = ωσ ωπ2 . Let E/F be the quadratic extension attached to ω via class field theory. Let (A3 (π)) E be the base change of A3 (π ). Consider the equality L S s,σ E ⊗ A3 (π) E , ρ2 ⊗ ∧2 ρ4 = L S s, σ ⊗ A3 (π ), ρ2 ⊗ ∧2 ρ4 L S s, (σ ⊗ ω) ⊗ A3 (π ), ρ2 ⊗ ∧2 ρ4 . Note that L S s, σ E ⊗ (A3 (π)) E , ρ2 ⊗ ∧2 ρ4 has a pole at s = 1 if and only if L S (s, (σ E ⊗ ωπ E ) × 5 E × 5 E ) has a pole at s = 1. Suppose that (A3 (π)) E is cuspidal. Then 5 E 6 ' 5 E ⊗ χ for any nontrivial character by Theorem 2.2.2. If σ is not monomial, then σ E is cuspidal, and hence L S (s, σ E ⊗(A3 (π)) E , ρ2 ⊗∧2 ρ4 ) is holomorphic at s = 1. If σ is monomial, then σ E is an automorphic representation induced from two gr¨ossencharacters. Hence again L S (s, (σ E ⊗ ωπ E ) × 5 E × 5 E ) is holomorphic at s = 1. Therefore L S (s, σ ⊗ A3 (π ), ρ2 ⊗ ∧2 ρ4 ) is holomorphic at s = 1 for any σ . This is a contradiction. Statement (2) is equivalent to statement (3). Suppose (A3 (π )) E is not cuspidal. Since (A3 (π)) E is equivalent to A3 (π E ), A3 (π E ) is not cuspidal. So Ad(π E ) ' Ad(π E )⊗χ for a nontrivial gr¨ossencharacter of E. Since 5 E is equivalent to Ad(π E ), we have 5 E ' 5 E ⊗ χ for a nontrivial gr¨ossencharacter χ of E. Statement (3) implies statement (1). Let σ 0 = σ ⊗ ωπ = π(χ ) be the monomial representation of GL2 (A F ) attached to χ . Let η be the quadratic character attached to E/F. Then σ E0 = χ χ 0 . Consider the equality L S (s, σ E0 × 5 E × 5 E ) = L S (s, σ 0 × 5 × 5)L S s, (σ 0 ⊗ η) × 5 × 5 .
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Then L S (s, σ E0 × 5 E × 5 E ) = L S (s, (χ ⊗ 5 E ) × 5 E )L S (s, (χ 0 ⊗ 5 E ) × 5 E ). Since χ ⊗ 5 E ' 5 E , L S (s, (χ ⊗ 5 E ) × 5 E ) has a pole at s = 1. Hence either L(s, σ 0 × 5 × 5) or L S (s, (σ 0 ⊗ η) × 5 × 5) has a pole at s = 1. This implies that either L S (s, σ ⊗ τ, ρ2 ⊗ ∧2 ρ4 ) or L S (s, (σ ⊗ η) ⊗ τ, ρ2 ⊗ ∧2 ρ4 ) has a pole at s = 1. Hence A4 (π) is not cuspidal. By the above proposition, we see the following. THEOREM 3.3.7 A4 (π ) = Sym4 (π)
⊗ ωπ−1 is a cuspidal representation of GL5 (A) except in the fol-
lowing three cases: (1) π is monomial; (2) π is not monomial and A3 (π) is not cuspidal; this is the case when there exists a nontrivial gr¨ossencharacter µ such that Ad(π) ' Ad(π) ⊗ µ; (3) A3 (π) is cuspidal and there exists a nontrivial quadratic character η such that A3 (π) ' A3 (π ) ⊗ η, or, equivalently, there exists a nontrivial gr¨ossencharacter χ of E such that Ad(π E ) ' Ad(π E ) ⊗ χ , where E/F is the quadratic extension determined by η. In this case, A4 (π) = σ1 σ2 , where σ1 = π(χ −1 ) ⊗ ωπ and σ2 = Ad(π ) ⊗ (ωπ η). Proof We only need to prove the last assertion. By the proof of Proposition 3.3.6, L S (s, σ˜ 1 × ^ 3 (π) × (A3 (π) ⊗ η)). It has a pole at A4 (π )) has a pole at s = 1. Consider L(s, A s = 1 since A3 (π) ' A3 (π) ⊗ η. By formal calculation, ^ 3 (π) × A3 (π) ⊗ η L S s, A = L S s, π, Sym6 ⊗ (ωπ−3 η) L S s, A4 (π) ⊗ (ωπ−1 η) × L S s, Ad(π) ⊗ η L S (s, η) = L S s, Ad(π) ⊗ (ωπ−1 η) × A4 (π) L S (s, η). The L-function L(s, η) has no zeros at s = 1, and therefore L S s, Ad(π) ⊗ (ωπ−1 η) × A4 (π) has a pole at s = 1. Hence σ2 = Ad(π) ⊗ (ωπ η). We note that since χ is a nontrivial cubic character, χ|A∗F = 1, and the central character of π(χ) is η · χ|A∗F = η. (of the proof ) Let π be a cuspidal representation of GL2 (A F ) such that A3 (π) is cuspidal. If A4 (π ) COROLLARY
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is not cuspidal, then L(s, π, Sym6 ⊗ (ωπ−3 η)) has a pole at s = 1. Here η is as in Theorem 3.3.7(3). Finally, based on Langlands’s calculations in [18], it is reasonable to claim the following. CONJECTURE
Let π be a nonmonomial cuspidal representation of GL2 (A). Then (1) Sym3 (π) is not cuspidal if and only if π is of tetrahedral type; (2) Sym3 (π) is cuspidal, but Sym4 (π) is not, if and only if π is of octahedral type; (3) Sym4 (π) and Sym5 (π ) are cuspidal, but Sym6 (π) is not, if and only if π is of icosahedral type. Remark. The final form of part (3) of the conjecture is an outcome of a number of communications with J.-P. Serre as well as calculations done in [9]. The purpose of our next proposition is to demonstrate the first two parts of the conjecture. PROPOSITION 3.3.8 Let π be a nonmonomial cuspidal representation of GL2 (A). Then (1) Sym3 (π) is not cuspidal if and only if π is of tetrahedral type; (2) Sym3 (π) is cuspidal, but Sym4 (π) is not, if and only if π is of octahedral type.
Proof Part (1) is [13, Lemma 6.5]. Part (2) is proved the same way. In fact, observe first that by Proposition 3.3.6(3) there exists a quadratic extension E/F such that 5 E ∼ = 5E ⊗ χ for a nontrivial gr¨ossencharacter χ . Notice that 5 E = Ad(π E ) and that therefore Ad(π E ) ∼ = Ad(π E ) ⊗ χ . By Theorem 2.2.2, Sym3 (π E ) is not cuspidal, and therefore, by part (1), π E is of tetrahedral type. Consequently, there exists a two-dimensional tetrahedral representation σ E of W E such that π E = π(σ E ). Since σ E is invariant under Gal(E/F), it can be extended to a two-dimensional continuous representation σ of W F , which is now octahedral. Let π 0 = π(σ ), which is of octahedral type. Clearly, π E0 ∼ = π E , and therefore π 0 ∼ = π ⊗ ηa for some a = 0, 1, where η is the gr¨ossencharacter attached to E/F. But σ is unique only up to twisting by a power of η, and therefore by changing the choice of η if necessary, we have π ∼ = π 0 = π(σ ). We are done.
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4. Applications We give several applications of cuspidality of third and fourth symmetric powers. In N this section we let π = v πv be a cuspidal representation of GL2 (A) unless otherwise specified. Let πv be an unramified local component with the Satake parameter diag(αv , βv ). Set av (π) = αv + βv . THEOREM 4.1 Let S(π ) be the set of places where πv is tempered. Then
34 δ S(π ) ≥ . 35 Proof We follow [23]. Let ω be the central character of π . Let av = av (π ). Then by direct computation we see that av (Sym2 (π)) = av2 −ωv and av (A4 (π)) = ωv−1 av4 −3av2 +ωv . For m, k, l nonnegative integers to be chosen below, let η = m[ω] k Sym2 (π) l A4 (π). Then av (η) = mωv + k(av2 − ωv ) + l(ωv−1 av4 − 3av2 + ωv ) = (m − k + l)ωv + (k − 3l)av2 + lωv−1 av4 . Let T (π, 2) = {v| |av | ≥ 2}. Then note that for v ∈ T (π, 2), αv = αq r , βv = for |α| = 1 and r ≥ 0 (see [23, Claim 4.6]). So except for finitely many places, S(π) = {v| |av | ≤ 2}. If v ∈ T (π, 2), av = α(q r + q −r ). So av (η) ≥ m + 3k + 5l.
αq −r ,
Hence v ∈ T (η, m + 3k + 5l). Thus, by [23, (4.4)], m2 + k2 + l2 δ T (η, m + 3k + 5l) ≤ . (m + 3k + 5l)2 This holds for every choice of (nonnegative) triples (m, k, l). It can be verified that the minimum of the right-hand side occurs when k = 3m, l = 5m, yielding 1 δ T (π, 2) ≤ . 35 Higher symmetric power L-functions and the Sato-Tate conjecture N In the following, let π = v πv be a cuspidal representation of GL2 (A), and let S be a finite set of places, including the archimedean ones, such that πv is unramified for
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v ∈ / S. Let diag(αv , βv ) be the Satake parameter for πv for v ∈ / S. Then the partial mth symmetric power L-function is defined to be L(s, πv , Sym )
m −1
m Y = (1 − αvm−i βvi qv−s ), i=0
L S (s, π, Sym ) = m
Y
L(s, πv , Symm ).
v ∈S /
PROPOSITION 4.2 Let π be a cuspidal representation of GL2 (A) such that Sym3 (π ) is cuspidal. Then L S (s, π, Sym5 ) is invertible for Re s ≥ 1; that is, it is holomorphic and nonzero for Re s ≥ 1.
Proof Consider L S s, Sym2 (π ) × Sym3 (π) = L S (s, π, Sym5 )L S s, Sym3 (π) ⊗ ωπ L S (s, π ⊗ ωπ2 ). Note that L S (s, Sym3 (π) ⊗ ωπ )L S (s, π ⊗ ωπ2 ) is invertible for Re s ≥ 1. Note also that the left-hand side is invertible for Re s ≥ 1. Hence our result follows. PROPOSITION 4.3 Let π be a cuspidal representation of GL2 (A) such that Sym3 (π ) is cuspidal. Then every partial sixth symmetric power L-function has a meromorphic continuation and satisfies a standard functional equation. Moreover, if ωπ3 = 1, they are all invertible for Re s ≥ 1.
Proof By standard calculations, L S s, Sym3 (π) × Sym3 (π) = L S (s, π, Sym6 )L S s, Sym4 (π) ⊗ ωπ L S s, Sym2 (π) ⊗ ωπ2 L S (s, ωπ3 ). Meromorphic continuation and a functional equation follow. If ωπ3 = 1, then Sym3 (π ) is self-contragredient. This implies that the lefthand side has a pole at s = 1, while in the right-hand side L(s, ωπ3 ) has a pole at s = 1. By assumption, Sym4 (π) is either cuspidal or of the form σ1 σ2 , where σ1 and σ2 are cuspidal representations of GL2 (A) and GL3 (A), respectively. Hence L S (s, Sym4 (π) ⊗ ωπ )L S (s, Sym2 (π) ⊗ ωπ2 ) is invertible for Re s ≥ 1. This implies our last claim.
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PROPOSITION 4.4 If Sym4 (π ) is cuspidal, then partial sixth symmetric power L-functions are all invertible for Re s ≥ 1.
Proof Consider the equality L S s, Sym2 (π) × Sym4 (π) = L S (s, π, Sym6 )L S s, Sym4 (π) ⊗ ωπ L S s, Sym2 (π ) ⊗ ωπ2 . Since the left-hand side and L S (s, Sym4 (π) ⊗ ωπ )L S (s, Sym2 (π ) ⊗ ωπ2 ) are invertible for Re s ≥ 1, the same holds for L S (s, π, Sym6 ). COROLLARY
Let π be a cuspidal representation of GL2 such that Sym3 (π) is cuspidal. If ωπ3 6= 1 and Sym3 (π) is self-contragredient, then L(s, π, Sym6 ) has a pole at s = 1. PROPOSITION 4.5 Let π be a cuspidal representation of GL2 (A) such that Sym3 (π ) is cuspidal. Then every partial seventh symmetric power L-function has a meromorphic continuation and satisfies a standard functional equation. Moreover, they are all invertible for Re s ≥ 1.
Proof By standard calculations, L S s, Sym3 (π) × Sym4 (π ) = L S (s, π, Sym7 )L S (s, π, Sym5 ⊗ ωπ )L S s, Sym3 (π) ⊗ ωπ2 L S (s, π ⊗ ωπ3 ). Meromorphic continuation and functional equations then follow. By assumption, Sym4 (π ) is either cuspidal or of the form σ1 σ2 , where σ1 and σ2 are cuspidal representations of GL2 (A) and GL3 (A), respectively. Hence the left-hand side is invertible for Re s ≥ 1. In the proof of Proposition 4.2, using L S (s, (Sym2 (π) ⊗ ωπ ) × Sym3 (π )), one can see that L S (s, π, Sym5 ⊗ ωπ ) is invertible for Re s ≥ 1. From this, we obtain our assertion. 4.6 Let π be a cuspidal representation of GL2 such that Sym3 (π) is cuspidal. Then every partial eighth symmetric power L-function has a meromorphic continuation and satisfies a standard functional equation. If Sym4 (π) is cuspidal and ωπ4 = 1, then they are all invertible for Re s ≥ 1. PROPOSITION
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Proof By standard calculations, L S s, Sym4 (π) × Sym4 (π) = L S (s, π, Sym8 )L S (s, π, Sym6 ⊗ ωπ )L S s, Sym4 (π ) ⊗ ωπ2 × L S s, Sym2 (π) ⊗ ωπ3 L S (s, ωπ4 ).
This proves the meromorphic continuation and functional equation. If Sym4 (π ) is cuspidal and ωπ4 = 1, then Sym4 (π) is self-contragredient. Hence the left-hand side has a simple pole at s = 1, while in the right-hand side L(s, ωπ4 ) has a simple pole at s = 1. By Proposition 4.4, and by considering L S (s, Sym2 (π) × (Sym4 (π ) ⊗ ωπ )), we see that L S (s, π, Sym6 ⊗ ωπ )L S s, Sym4 (π) ⊗ ωπ2 L S s, Sym2 (π ) ⊗ ωπ3 is invertible for Re s ≥ 1. This completes our claim. 4.7 Let π be a cuspidal representation of GL2 (A) such that Sym3 (π ) is cuspidal. Then every partial ninth symmetric power L-function has a meromorphic continuation and satisfies a standard functional equation. If Sym4 (π) is cuspidal, then L S (s, π, Sym9 ) has at most a simple pole or a simple zero at s = 1. If Sym4 (π ) is not cuspidal, then L S (s, π, Sym9 ) is invertible for Re s ≥ 1. PROPOSITION
Proof Suppose first that Sym4 (π) is cuspidal. Consider the case E 8 − 2 of [27]. Let M be a maximal Levi subgroup, and denote by A the connected component of its center. Since E 8 is simply connected, the derived group M D of M is simply connected as well, and hence M D = SL4 × SL5 . Thus M = (GL1 × SL4 × SL5 )/(A ∩ M D ), where A ∩ M D ' Z/20Z. Let πi , i = 1, 2, be cuspidal representations of GL4 (A) and GL5 (A) with central characters ωi , i = 1, 2, respectively. Let πi0 , i = 1, 2, be irreducible constituents of π1 |SL4 (A) and π2 |SL5 (A) , respectively. Then 6 = ω15 ω28 ⊗ π10 ⊗ π20 can be considered a cuspidal representation of M(A). We then get the L-function L(s, π1 ⊗ π2 , ρ4 ⊗ ∧2 ρ5 ) as our first L-function. In fact, there are five
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L-functions in the constant term of the Eisenstein series, namely, L S (s, 6, r1 ) = L S (s, π1 ⊗ π2 , ρ4 ⊗ ∧2 ρ5 ), L S (s, 6, r2 ) = L S s, π1 ⊗ (π˜ 2 ⊗ ω2 ), ∧2 ρ4 ⊗ ρ5 , L S (s, 6, r3 ) = L S s, π˜ 1 × (π2 ⊗ ω1 ω2 ) , L S (s, 6, r4 ) = L S (s, π˜ 2 , ∧2 ρ5 ⊗ ω1 ω22 ), L S (s, 6, r5 ) = L S (s, π1 ⊗ ω1 ω22 ). (See [27] for the trivial central character case and [11] for the general case for detailed calculations.) Each of the L-functions, especially L S (s, 6, r1 ), has a meromorphic continuation and satisfies a standard functional equation (see [27]). We apply the above to π1 = A3 (π ) and π2 = Sym4 (π ). By standard calculations, we have L S (s,π1 ⊗ π2 , ρ4 ⊗ ∧2 ρ5 ) = L S (s, π, Sym9 )L S (s, π, Sym7 ⊗ ωπ )L S (s, π, Sym5 ⊗ ωπ2 )2 2 × L S s, Sym3 (π) ⊗ ωπ3 L S (s, π ⊗ ωπ4 ). The meromorphic continuation and functional equation of L S (s, π, Sym9 ) now follow from those of L S (s, π1 ⊗ π2 , ρ4 ⊗ ∧2 ρ5 ). Moreover, 2 L S (s, π, Sym7 ⊗ ωπ )L S (s, π, Sym5 ⊗ ωπ2 )2 L S s, Sym3 (π) ⊗ ωπ3 L S (s, π ⊗ ωπ4 ) is invertible at s = 1 by Propositions 4.2 and 4.5. So it is enough to prove that L S (s, 6, r1 ) has at most a simple pole or simple zero at s = 1. Q5 By [26], the product i=1 L S (1 + is, 6, ri ) does not have a zero at s = 0. But none of the L-functions L S (s, 6, ri ), i = 3, 4, 5, has a pole at s = 1. In fact, they are all entire and have no zeros for Re s ≥ 1. (See [7] for the case r4 and ω1 ω22 = 1. The general case can be seen to be the same by observing that the twisted exterior square L-function appears as the normalizing factor of a certain Eisenstein series if we consider Spin2n (cf. [11]). Alternatively, by direct calculation, we have L S (s, 6, r4 ) = L S (s, π, Sym6 ⊗ ωπ15 )L S (s, Sym2 (π) ⊗ ωπ17 ). The necessary properties of L S (s, π, Sym6 ⊗ ωπ15 ), and in particular its invertibility at s = 1, are now obtained by considering L S (s, Sym2 (π ) × (Sym4 (π ) ⊗ ωπ15 )) in the proof of Proposition 4.4.) The second L-function, L S (s, 6, r2 ), appears as the first L-function in the case D8 − 3. Hence it has at most a simple pole at s = 1. Alternatively, by direct calculation, we see that, since ∧2 (A3 (π)) = A4 (π) ⊕ ωπ , L S s,π1 ⊗ (π˜ 2 ⊗ ω2 ), ∧2 ρ4 ⊗ ρ5 = L S s, Sym4 (π ) × Sym4 (π) ⊗ ωπ5 L S s, Sym4 (π) ⊗ ωπ7 .
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Hence L S (s, π1 ⊗ (π˜ 2 ⊗ ω2 ), ∧2 ρ4 ⊗ ρ5 ) has at most a simple pole at s = 1 and is invertible for Re s > 1. Therefore L S (s, 6, r1 ) has at most a simple zero at s = 1. Q5 By [27], the product i=1 L S (is, 6, ri ) has at most a simple pole at s = 1. Q5 However, i=2 L S (is, 6, ri ) has no zeros at s = 1. Therefore L S (s, 6, r1 ) has at most a simple pole at s = 1. Next, suppose Sym4 (π) is not cuspidal. Then Sym4 (π) = σ1 σ2 , where σ1 and σ2 are cuspidal representations of GL2 (A) and GL3 (A), respectively. In this case, by standard calculations, L S (s,π1 ⊗ π2 , ρ4 ⊗ ∧2 ρ5 ) = L S (s, π1 × σ1 × σ2 )L S (s, π1 ⊗ ωσ1 )L S s, π1 × (σ˜ 2 ⊗ ωσ2 ) . Here L S (s, π1 ⊗ ωσ1 )L S (s, π1 × (σ˜ 2 ⊗ ωσ2 )) is invertible for Re s ≥ 1. By [13], L S (s, π1 × σ1 × σ2 ) = L S s, π1 × (σ1 σ2 ) , where σ1 σ2 is the functorial product that is an automorphic representation of GL6 (A). Since σ1 is monomial (see Theorem 3.3.7), by the main theorem of [13], σ1 σ2 is either cuspidal or unitarily induced from two cuspidal representations of GL3 (A). Hence L S (s, π1 × σ1 × σ2 ) is invertible for Re s ≥ 1, and therefore the same conclusion holds for L S (s, π, Sym9 ). PROPOSITION 4.8 Let π be a cuspidal representation of GL2 (A) such that Sym3 (π ) is cuspidal. Let diag(αv , βv ) be the Satake parameter for an unramified local component. Then 1/9 |αv |, |βv | < qv . If Sym4 (π) is not cuspidal, then the full Ramanujan conjecture is valid.
Proof If Sym4 (π ) is cuspidal, use Proposition 4.7 and [27, Lemma 5.8]. If Sym4 (π) is not cuspidal, then by Proposition 3.3.8, π is of Galois type for which |αv | = |βv | = 1. The following result coincides with Langlands’s calculations in [18]. PROPOSITION 4.9 Let π be a nonmonomial cuspidal representation of GL2 (A) with a trivial central character. Suppose m ≤ 9. (1) Suppose Sym3 (π) is not cuspidal. Then L S (s, π, Symm ) is invertible at s = 1, except for m = 6, 8; the L-functions L S (s, π, Sym6 ) and L S (s, π, Sym8 ) each have a simple pole at s = 1.
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Suppose Sym3 (π) is cuspidal, but Sym4 (π) is not. Then L S (s, π, Symm ) is invertible at s = 1 for m = 1, . . . , 7 and m = 9; the L-function L S (s, π, Sym8 ) has a simple pole at s = 1.
Proof (1) By Theorem 2.2.2 and Section 3.2, Sym3 (π) = (π ⊗ µ) (π ⊗ µ2 ),
Sym4 (π) = Sym2 (π ) µ µ2 ,
where µ is a nontrivial gr¨ossencharacter such that Ad(π) ' Ad(π )⊗µ. We explicitly calculate L S (s, π, Symm ). Let 5 = Sym2 (π). Then L S (s, π, Sym3 ) = L S (s, π ⊗ µ)L S (s, π ⊗ µ2 ), L S (s, π, Sym4 ) = L S (s, 5)L S (s, µ)L S (s, µ2 ). They are both invertible for Re s ≥ 1. From the equality in Proposition 4.2, we have L S (s, π, Sym5 ) = L S (s, π)L S (s, π ⊗ µ)L S (s, π ⊗ µ2 ), which is clearly invertible for Re s ≥ 1. Using the equality in Proposition 4.3, we have L S (s, π, Sym6 ) = L S (s, 5)2 ·L S (s, 1). Since L(s, 5) is invertible at s = 1, L S (s, π, Sym6 ) has a simple pole at s = 1. From the equality in Proposition 4.5, we have L S (s, π, Sym7 ) = L S (s, π)2 L S (s, π ⊗ µ)L S (s, π ⊗ µ2 ). Hence it is invertible for Re s ≥ 1. For L S (s, π, Sym8 ), consider the equality in Proposition 4.4 with ωπ = 1; we have L S (s, 5 × 5) = L S (s, π, Sym6 )L S (s, µ)L S (s, µ2 ). Hence L S (s, 5 × 5) = L S (s, 5)2 L S (s, 1)L S (s, µ)L S (s, µ2 ). Then from the equality in Proposition 4.6, L S (s, π, Sym8 ) = L S (s, 5)2 L S (s, µ)L S (s, µ2 )L S (s, 1), and therefore L S (s, π, Sym8 ) has a simple pole at s = 1. For L S (s, π, Sym9 ), consider the equality in Proposition 4.7 with ωπ = 1; by standard calculations, we see that L S (s, π, Sym9 ) = L S (s, π ⊗ η)2 L S (s, π ⊗ η2 )2 L S (s, π ). We therefore conclude that L S (s, π, Sym9 ) is invertible for Re s ≥ 1. (2) In this case, Sym4 (π ) = σ1 σ2 , where σ1 and σ2 are cuspidal representations of GL2 (A) and GL3 (A), respectively. If ωπ = 1, then σ1 and σ2 are self-contragredient. We only have to discuss the case m = 8. Consider the equality in Proposition 4.6 with ωπ = 1 in which the left-hand side has a double pole at s = 1. But L S (s, π, Sym6 )L S (s, π, Sym4 )L S (s, π, Sym2 ) is invertible at s = 1. Thus L S (s, π, Sym8 ) has a simple pole at s = 1.
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Remark. The above proposition is no longer true if the central character is not trivial (see Proposition 3.2.1, Corollaries to Theorem 3.3.7 and Proposition 4.4). We now give an application of these properties of symmetric power L-functions to the Sato-Tate conjecture (see [25], [30]). This we do by following Serre’s method N (see [30, Appendix]). In what follows, let π = v πv be a cuspidal representation of GL2 (A) with a trivial central character such that Sym4 (π) is cuspidal. We also assume that π satisfies the Ramanujan-Petersson conjecture. Recall that we let av = αv + βv , where πv is an unramified local component with the Satake parameter diag(αv , βv ). We use exactly the same notation as in [30, Appendix]. First let us calculate Tn (x), the polynomial that gives the trace of the nth symmetric power of an element of SL2 (C) whose trace is x: T0 = 1,
T2 = x 2 − 1,
T1 = x,
T4 = x 4 − 3x 2 + 1,
T3 = x 3 − 2x,
T5 = x 5 − 4x 3 + 3x,
T7 = x 7 − 6x 5 + 10x 3 − 4x,
T6 = x 6 − 5x 4 + 6x 2 − 1,
T8 = x 8 − 7x 6 + 15x 4 − 10x 2 + 1,
T9 = x 9 − 8x 7 + 21x 5 − 20x 3 + 5x. Next recall the quantity P I (Tn ) = lim
N →∞
qv ≤N
Tn (av )
π(N )
= kn ,
where the nth symmetric power L-function has an order −kn at s = 1 and π(N ) is the number of places such that qv ≤ N . Hence I (T0 ) = 1 and I (Tn ) = 0 for n = 1, . . . , 8 by Propositions 4.2–4.6. Let I (T9 ) = k. By Proposition 4.7, we know that k ∈ {−1, 0, 1}. We now calculate I (x n ): I (x) = 0, I (x 5 ) = 0,
I (x 2 ) = 1,
I (x 6 ) = 5,
I (x 3 ) = 0,
I (x 7 ) = 0,
I (x 4 ) = 2,
I (x 8 ) = 14,
I (x 9 ) = k.
With notation as in [30, Appendix], d = 9 and m = 4. Using this, we can calculate the orthogonal polynomials, P0 , P1 , . . . , P4 to get P0 = 1,
P1 = T1 = x,
P2 = T2 = x 2 − 1,
P4 = T4 = x 4 − 3x 2 + 1, (1) (2)
P3 = T3 = x 3 − 2x,
P5 = T5 − k(x 4 − 3x 2 + 1).
There are three possibilities: √ √ k = 0: P5 = T5 = x 5 − 4x 3 + 3x; the roots are − 3, −1, 0, 1, 3; k = 1: P5 = T5 − T4 = x 5 − x 4 − 4x 3 + 3x 2 + 3x − 1; the roots are −2 cos(2kπ/11), k = 1, 2, 3, 4, 5;
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k = −1: P5 = T5 + T4 = x 5 + x 4 − 4x 3 − 3x 2 + 3x + 1; the roots are 2 cos(2kπ/11), k = 1, 2, 3, 4, 5. Note that 2 cos(2π/11) ∼ = 1.68 . . . .
THEOREM 4.10 For every > 0, there are sets T + and T − of positive lower (Dirichlet) density such that av > 1.68 . . . − for all v ∈ T + and av < −1.68 . . . + for all v ∈ T − .
Remark. In [30, Appendix] and [23], where the √ third, fourth, and fifth symmetric power L-functions were used, the result is av > 2 − . In the next proposition, we do not assume the Ramanujan-Petersson conjecture for π. PROPOSITION 4.11 Let π be a cuspidal representation of GL2 (A) with a trivial central character. Then for every > 0, there exists a set T of positive lower (Dirichlet) density such that |av | > 1.68 . . . − for all v ∈ T .
Remark. Let σ be a two-dimensional continuous representation of W F , the Weil group of F/F. Assume that there exists an automorphic cuspidal representation π(σ ) of GL2 (A) preserving root numbers and L-functions for pairs (cf. [16], [13]). This is possible except perhaps when σ is icosahedral of special kind (cf. [13, §10]). When σ is of icosahedral type and π(σ ) exists, L(s, Sym4 σ ) is a five-dimensional irreducible, and therefore entire, Artin L-function. But it is not primitive since Sym4 σ is monomial (cf. [9]). We refer to [9] for an interesting application of this to automorphic induction. Acknowledgments. We would like to thank Joseph Shalika for his suggestion that we use an unpublished result of his with Herv´e Jacquet and Ilya Piatetski-Shapiro in the proof of Corollary 3.3.5. Thanks are also due to Dinakar Ramakrishnan and JeanPierre Serre for many useful discussions and communications. Finally, the authors would like to thank the Institute for Advanced Study and particularly the organizers of the Special Year in the Theory of Automorphic Forms and L-functions, E. Bombieri, H. Iwaniec, R. P. Langlands, and P. Sarnak, for inviting them to participate. References [1]
J. ARTHUR and L. CLOZEL, Simple Algebras, Base Change, and the Advanced Theory
of the Trace Formula, Ann. of Math. Stud. 120, Princeton Univ. Press, Princeton, 1989. MR 90m:22041
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S. GELBART and H. JACQUET, A relation between automorphic representations of
´ GL(2) and GL(3), Ann. Sci. Ecole Norm. Sup. (4) 11 (1978), 471–552. MR 81e:10025 178 D. GINZBURG, S. RALLIS, and D. SOUDRY, L-functions for symplectic groups, Bull. Soc. Math. France 126 (1998), 181–244. MR 2000b:22017 183 M. HARRIS and R. TAYLOR, The Geometry and Cohomology of Some Simple Shimura Varieties, Ann. of Math. Stud. 151, Princeton Univ. Press, Princeton, 2001. 177 G. HENNIART, Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique, Invent. Math. 139 (2000), 439–455. MR 2001e:11052 177 H. JACQUET and J. A. SHALIKA, “Exterior square L-functions” in Automorphic Forms, Shimura Varieties, and L-functions, II (Ann Arbor, Mich., 1988), ed. L. Clozel and J. Milne, Academic Press, Boston, 1990, 143–226. MR 90j:11003 182 H. KIM, Langlands-Shahidi method and poles of automorphic L-functions: Application to exterior square L-functions, Canad. J. Math. 51 (1999), 835–849. MR 2000f:11058 183, 191 , Langlands-Shahidi method and poles of automorphic L-functions, II, Israel J. Math. 117 (2000), 261–284. MR 2001i:11059a 183 , An example of a non-normal quintic automorphic induction, preprint, 2000. 186, 195 , Functoriality for the exterior square of GL4 and symmetric fourth of GL2 , preprint, 2000. 178, 180, 181 , On local L-functions and normalized intertwining operators, preprint, 2000. 178, 191 H. KIM AND F. SHAHIDI, Functorial products for GL2 × GL3 and functorial symmetric cube for GL2 , C. R. Acad. Sci. Paris S´er. I Math. 331 (2000), 599–604. MR 1 799 096 178 , Functorial products for GL2 × GL3 and the symmetric cube for GL2 , to appear in Ann. of Math. (2). 178, 179, 186, 192, 195 J.-P. LABESSE AND R. P. LANGLANDS, L-indistinguishability for SL(2), Canad. J. Math. 31 (1979), 726–785. MR 81b:22017 179 R. P. LANGLANDS, “Problems in the theory of automorphic forms” in Lectures in Modern Analysis and Applications, III, Lecture Notes in Math. 170, Springer, Berlin, 1970, 18–61. MR 46:1758 , Base change for GL(2), Ann. of Math. Stud. 96, Princeton Univ. Press, Princeton, 1980. MR 82a:10032 195 , “On the classification of irreducible representations of real algebraic groups” in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, ed. P. J. Sally Jr. and D. A. Vogan, Math. Surveys Monogr. 31, Amer. Math. Soc., Providence, 1989, 101–170. MR 91e:22017 177 , “Beyond endoscopy” to appear in Volume in Honor of Joseph Shalika’s Sixtieth Birthday, ed. D. Ramakrishnan and F. Shahidi, Johns Hopkins Univ. Press, Baltimore. 186, 192 W. LUO, Z. RUDNICK, and P. SARNAK, “On the generalized Ramanujan conjecture for GL(n)” in Automorphic Forms, Automorphic Representations, and Arithmetic
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(Fort Worth, Tex., 1996), Proc. Sympos. Pure Math. 66, Part 2, Amer. Math. Soc., Providence, 1999, 301–310. MR 2000e:11072 178 ´ C. MOEGLIN and J.-L. WALDSPURGER, Le spectre r´esiduel de GL(n), Ann. Sci. Ecole Norm. Sup. (4) 22 (1989), 605–674. MR 91b:22028 , Spectral decomposition and Eisenstein series: Une paraphrase de l’Ecriture, Cambridge Tracts in Math. 113, Cambridge Univ. Press, Cambridge, 1995. MR 97d:11083 D. PRASAD and D. RAMAKRISHNAN, “On the global root numbers of GL(n) × GL(m)” in Automorphic Forms, Automorphic Representations, and Arithmetic (Fort Worth, Tex., 1996), Proc. Sympos. Pure Math. 66, Part 2, Amer. Math. Soc., Providence, 1999, 311–330. MR 2000f:11060 181 D. RAMAKRISHNAN, On the coefficients of cusp forms, Math. Res. Lett. 4 (1997), 295–307. MR 98e:11064 178, 187, 195 , Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Ann. of Math. (2) 152 (2000), 45–111. MR 2001g:11077 179 J.-P. SERRE, Abelian `-adic Representations and Elliptic Curves, W. A. Benjamin, New York, 1968. MR 41:8422 178, 194 F. SHAHIDI, On certain L-functions, Amer. J. Math. 103 (1981), 297–355. MR 82i:10030 191 , On the Ramanujan conjecture and finiteness of poles for certain L-functions, Ann. of Math. (2) 127 (1988), 547–584. MR 89h:11021 178, 183, 190, 191, 192 , Third symmetric power L-functions for GL(2), Compositio Math. 70 (1989), 245–273. MR 90m:11081 177 , A proof of Langlands’ conjecture on Plancherel measures: Complementary series for p-adic groups, Ann. of Math. (2) 132 (1990), 273–330. MR 91m:11095 , “Symmetric power L-functions for GL(2)” in Elliptic Curves and Related Topics, ed. H. Kisilevsky and M. R. Murty, CRM Proc. Lecture Notes 4, Amer. Math. Soc., Providence, 1994, 159–182. MR 95c:11066 177, 178, 194, 195 D. SOUDRY, The CAP representations of GSp(4, A), J. Reine Angew. Math. 383 (1988), 87–108. MR 89e:11032 182
Kim Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada;
[email protected] Shahidi Department of Mathematics, Purdue University, West Lafayette, Indiana 47906, USA;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2,
GLOBAL ASYMPTOTICS FOR MULTIPLE INTEGRALS WITH BOUNDARIES E. DELABAERE and C. J. HOWLS
Dedicated to Fr´ed´eric Pham on the occasion of his 65th birthday. Abstract Under convenient geometric assumptions, the saddle-point method for multidimensional Laplace integrals is extended to the case where the contours of integration have boundaries. The asymptotics are studied in the case of nondegenerate and of degenerate isolated critical points. The incidence of the Stokes phenomenon is related to the monodromy of the homology via generalized Picard-Lefschetz formulae and is quantified in terms of geometric indices of intersection. Exact remainder terms and the hyperasymptotics are then derived. A direct consequence is a numerical algorithm to determine the Stokes constants and indices of intersections. Examples are provided. 1. Introduction The asymptotic behaviour as k → ∞ in the complex plane C of complex oscillatory integrals Z I0 (k) =
0
e−k f (z) g(z) dz (1) ∧ · · · ∧ dz (n) ,
(1)
with f, g : Cn → C analytic functions of the variable z = (z (1) , . . . , z (n) ) and 0 a chain of real dimension n, has been the study of much work, both theoretical and practical. A discussion of the history of the problem can be found in V. Arnold, A. Varchenko, and S. Gussein-Zad`e [3] and D. Kaminski and R. Paris [40]. Applications of these integrals in optics are detailed in [53] and the references therein. Much work has focused on obtaining the asymptotic expansions themselves. Here we focus on deriving “global asymptotics” in all sectors of the complex k-plane for these integrals when the contours of integration are finitely bounded. Interest in this area has been renewed recently following the ideas of R. Balian and C. Bloch [6] and F. Pham [63], DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2, Received 21 March 2000. Revision received 31 January 2001. 2000 Mathematics Subject Classification. Primary 30E15, 30E20, 34M40, 34M60, 41A80; Secondary 32C30, 32S05, 34M35 44A10. ´ Authors’ work supported by the French Minist`ere des Affaires Etrang` eres and the British Council under Alliance Partnership Programme grant number 98018. 199
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[65], the development of the resurgence (see [23]) and hyperasymptotic theories (see [7]), and the work of Kaminski and Paris [40], [41], and C. Howls [36]. The main approach used in deriving the (global) asymptotics of such Laplace integrals is a generalisation of the Riemann-Debye saddle-point method, which can be reduced to the following algorithm (see [25]): (1) the identification of all possible critical points; (2) the topological operation of pushing the integration contour in Cn toward the directions of steepest descent, forming a chain of integration hypersurfaces; (3) the local study near the critical points of the phase function and near the boundary of the hypersurfaces of integration 0, and the computation of the relevant asymptotic expansions; (4) the derivation of the exact remainder terms, their reexpansions in terms of distant critical points, and the calculation of the associated Stokes constants, thereby explicitly linking the contributions from all relevant critical points. The first and third parts have been extensively discussed at leading order (see, e.g., Arnold, Varchenko, and Gussein-Zad`e [3], V. Vassiliev [71], and B. Gaveau [26]) and for real variables (see R. Wong [76]). The second topic has been studied practically for real variables in terms of flows by Kaminski [39]. The third topic has been studied in great detail by Varchenko from the theoretical viewpoint, relating the characteristic exponents of the asymptotic expansions to mixed Hodge structure in vanishing cohomologies (see [69], [70]). Substantial practical progress in the derivation of asymptotic expansions with exact remainder terms for polynomial exponents using Mellin integral representations has been made by Kaminski and Paris [40], [41] and by G. Liakhovetski and Paris [43] using a Newton polygon to identify the appropriate contributions. The fourth point was studied for unbounded integration contours by Howls [36]. Our goal here is to combine the four points above so as to produce for the first time exact remainder terms and a self-consistent numerical algorithm to determine the Stokes constants when 0 is a bounded domain. The asymptotic expansions are well known when f is a polynomial function (and g is “well behaved at infinity”) and the contour of integration 0 is an unbounded nchain of hypersurfaces of integration, satisfying a convergence criterion at infinity <(k f ) → +∞.
(2)
Under convenient geometric assumptions (effectively, that all critical points are isolated and that no critical points at infinity occur), Pham showed geometrically in [63] that the chain 0 (resp., integral I0 (k)) can be decomposed as a finite sum µα µα XX XX 0= Nα j (0)0α j resp., I0 (k) = Nα j (0)I0α j (k) . (3) α
j=1
α
j=1
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201
j
Here Nα (0) are integers and the first sum runs over the finite set of critical points z α of j the phase function f , while the 0α denote a basis of µα (Milnor number) independent steepest-descent n-folds associated with the critical point z α . Such a steepest-descent n-fold is easy to describe locally when the critical point is a Morse singularity, yielding the notion of the Lefschetz thimble of geometers. In general, the local analysis near the (isolated) critical points z λ of the phase function f reduces to a description of the topology of the generic fiber f = t, t ∈ C near z λ , whose (reduced) homology is the vanishing homology of geometers. From the above decomposition, the asymptotics follow from a local analysis near the critical points. Moreover, this geometrical viewpoint also yields any algebraic Stokes phenomena, that is, discontinuities of the decomposition (3) for special values of arg(k). These discontinuities are given by the Picard-Lefschetz formulae, and their use in deriving exact remainder terms, Stokes constants, and Riemann sheet structure was discussed in the hyperasymptotic study by Howls [36]. The purpose of this paper is to extend the results of [36] to consider integrals of type (1) when one adds a set {S1 , S2 , . . . , Sm } of m ≤ n smooth irreducible affine hypersurfaces as possible boundaries for the contour of integration. Assuming that f is a polynomial function, we show (under some convenient geometrical assumptions) a hyperasymptotic extension of known results in real asymptotic analysis (see [76]): the chain 0 can be decomposed again as a finite sum 0=
µα XX α
Nα j (0)0α j
(4)
j=1
with integers Nα (0), where the sum runs over the (finite) set of critical values f α of (1) the critical points of the phase function f and (2) the critical points of the restricted function f | on the boundary. The above description follows from a decomposition of the space of (relative) homology classes of n-chains satisfying the descent condition. Some discontinuities in decomposition (4) usually appear under variations of the phase of k. For a geometer, this phenomenon can be understood in terms of indices of intersection, described by generalized Picard-Lefschetz formulae. For the analyst, this is the Stokes phenomenon, and the previous indices of intersection are now seen as Stokes multipliers. The first viewpoint helps in understanding the quantized nature of these Stokes multipliers but, as a rule, fails at the stage of concrete computation. Understanding these indices of intersection as Stokes multipliers gives a numerical method of computation, using the tools and ideas of hyperasymptotic theory. The structure of the paper is as follows. In Section 2 we briefly describe the space of contours of integration that will be considered and the decomposition property. This is done in an intuitive manner, saving the proofs and technicalities for Ap-
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pendix A. In Section 3 we apply this decomposition to integrals of type (1), deriving the asymptotics. In Section 4 we discuss the Stokes phenomenon, describing the socalled resurgence properties. Here again, the technicalities of the proofs are left to Appendix B. In Section 5 we demonstrate how to derive the exact remainder term for a truncated asymptotic expansion, together with the calculation of the associated Stokes constants and thence the hyperasymptotic reexpansions. In Section 6 we provide an example. We conclude in Section 7 with a discussion of the hypotheses and some related topics; among other subjects, the problem of the “confluent” case is briefly described, and some links between oscillating integrals with boundaries and differential equations are suggested. We also mention some open problems. By their very nature, the complex oscillatory integrals are crossroads for different scientific communities, from pure mathematicians to physicists and chemists (cf. [15] and the references therein). This paper has been written in such a way as to be readable by as large an audience as possible; consequently, although some comments in the text will appear trivial to geometers, they are intended to be helpful to the nonspecialist. Conversely, the procedure for deriving the new hyperasymptotic formulae associated with these integrals may be familiar to an (exponential) asymptotician, but this is explained in Section 5 for the convenience of others. 2. Contours of integration and assumptions This section contains the assumptions we make about the types of integrals to be treated, together with the definitions of the technical geometrical terms that we use. For the benefit of the geometer, the technicalities of the explanations and proofs can be found in Appendix A. We treat only the case when f is a polynomial. As in the (n = 1)-dimensional case, the C-space of the values of f plays a central role and is called the Borel plane or the t-plane. In what follows, we assume that k has a given phase −θ: k = |k| exp(−iθ) ∈ C\{0}.
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2.1. Contours of integration with no boundary In Howls [36] and Pham [63], integrals of type (1) are considered for unbounded (nreal)-dimensional contours of integration 0, traveling between asymptotic valleys at infinity where <(k f ) → +∞. This condition ensures the convergence of the integral and the validity of the Stokes theorem (at least when the growth of the function g at infinity remains small in comparison to the decay of the exponential involving f ). It is assumed from now on that g is a polynomial function, although we believe the results to be more widely applicable.
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As in [63], the set of these unbounded integration contours is denoted by Hn9 (Cn ). The properties of Hn9 (Cn ) are essentially governed by the behavior of the function f near its critical points z α , where the gradient ∇ f vanishes. In principle, an integral I0 over one such contour can be reduced to the sum of integrals over a sequence, called a chain, of “steepest-descent” contours 0α , each of which encounters a single critical point z α of f and follows the flow of the vector field ∇(< k f ). This is true at least when the phase of k is “generic” in the sense that each 0α encounters no more than one critical point, so that no Stokes phenomenon occurs. This contour decomposition follows from the work of Pham [63], under two main geometrical assumptions. (1) All critical points are isolated. This excludes those f that contain a ridge of critical points (see, for instance, [76]). (2) There are no critical points at infinity. Since these two requirements are central to the paper, it is worth pausing to give an explanation. To explore the space of integration contours, it is necessary to study the topology of the level hypersurfaces f (−1) (t) for t ∈ C, and particularly for t near a critical value f α for which the fiber f (−1) ( f α ) becomes singular. This amounts to studying the geometry near the corresponding critical values, which is completely understood (since J. Milnor [50]) only when isolated critical points are concerned. Hence we have the first assumption. The definition of critical points at infinity is subtle but can be illustrated with an example (from [10]). A straightforward computation shows that the polynomial 2
f (z (1) , z (2) ) = z (1) z (2) + 2z (1)
(6)
has no critical point, but, nevertheless, the fiber f −1 (0) differs from the other level hypersurfaces f (−1) (t). This can be seen by deforming this polynomial into 2
f k (z (1) , z (2) ) = z (1) z (2) + 2z (1) + k 2 z (2) .
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For k ∈ C\{0}, the polynomials f k have two nondegenerate critical points (resp., values) at (z (1) , z (2) ) = (ik, i/k) and (−ik, −i/k) (resp., 2ik and −2ik). However, when k → 0, the two Morse singularities evaporate to infinity while the two critical values converge to zero. This gives rise to a critical point at infinity, with zero for its corresponding bifurcation value. We see later in Section 7.1 how this bifurcation value coming from a critical point at infinity indeed affects the (hyper)asymptotics. Nevertheless, the general topology in the locality of such critical points at infinity is complicated and not yet well understood (apart from the dimension n = 2; see [32] and [22] for a survey of recent results), and this is the reason for the second assumption.
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2.2. Contours of integration with boundaries Returning to the general case of integrals of type (1), suppose that the contour 0 now encounters a sequence of boundaries {S1 , S2 , . . . , Sm }, these being m ≤ n smooth complex (n −1)-dimensional hypersurfaces. For instance, the contour may run from a boundary into an asymptotic valley at infinity where <(k f ) → +∞ for convergence, or it may run between two finitely placed boundaries. We assume that each boundary Si is defined by a polynomial equation Pi (z) = 0 with Pi ∈ C[z]. This prevents the boundaries from behaving too wildly at infinity, thus ensuring (1) the convergence of the integrals and (2) the validity of the Stokes theorem. Notation. The set of these contours 0, which we also refer to as cycles, is denoted by Hn9 (Cn , S) with S = S1 ∪ · · · ∪ Sm . The affine hypersurfaces {S1 , S2 , . . . , Sm } are assumed to satisfy the following hypotheses. HYPOTHESIS H1 The m hypersurfaces S1 , S2 , . . . , Sm are in general position; that is, they may cross, but they are not tangential at the crossing points.
Notation. For brevity, the intersections of boundary surfaces Si1 ∩ · · · ∩ Si p (1 ≤ i 1 < · · · < i p ≤ m) are denoted by S(i1 ,...,i p ) . These are smooth submanifolds of complex codimension p by Hypothesis H1. From the polynomial nature of the phase function f and the Bertini-Sard theorem, it follows that the set 3() of critical values of f is finite. Moreover, from the algebraic assumptions on the Si , the restriction f (i1 ,...,i p ) := f | S(i1 ,...,i p ) of f to the intersection of the boundaries S(i1 ,...,i p ) also has a finite set 3(i1 ,...,i p ) of critical values (see [50]). S We denote by 3 = 3(i1 ,...,i p ) the set of all these critical values. Notice here that if m = n, then the (finite) set of points S1 ∩ · · · ∩ Sn are considered critical points. In one-dimensional integrals, this corresponds exactly to linear endpoints. To avoid complications at infinity, we assume the following hypothesis. HYPOTHESIS H2 We have that f , as well as each f (i1 ,...,i p ) (1 ≤ i 1 < · · · < i p ≤ m), has no singularity at infinity.
We avoid nonisolated critical points.
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HYPOTHESIS H3 All critical points of f , as well as those of the restricted functions f (i1 ,...,i p ) from S(i1 ,...,i p ) to C, are isolated.
It is useful also to assume a one-to-one correspondence between critical points z α in n dimensions and their critical values f α = f (z α ) in one dimension. This is the subject of the following hypothesis. H4 Each f α ∈ 3 is the image of a single critical point z α of f or one of the f | S(i1 ,...,i p ) . HYPOTHESIS
At this point it is helpful to introduce some definitions. Definition 2.1 The depth of a (restricted) critical point z α is the maximal number of boundaries p on which it lies. Following the one-to-one equivalence between critical value and critical point (see Hypothesis H4), we can obviously extend the definition of depth to critical values. This definition allows us to classify the critical points according to their depth in the following way.∗ Definition 2.2 Critical points of the first type are critical points of depth p = 0. These are critical points of the unrestricted f and do not lie on a boundary. Definition 2.3 Critical points of the second type are those of depth p > 0 and are critical only for f restricted to exactly p boundaries. For such a critical point, and if S1 , . . . , S p are the p boundaries (up to a reordering), it follows from Hypothesis H1 and Hadamard’s lemma (see [2]) that there exist new local coordinates (s (1) , . . . , s ( p) , s ( p+1) , . . . , s (n) ) such that we can write f as f = f α + s (1) + · · · + s ( p) + F(s ( p+1) , . . . , s (n) ) with s (i) = 0 as a local equation for the boundary Si . ∗
Our nomenclature differs slightly from Wong’s [76].
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Definition 2.4 Critical points of the third type are all other cases (including when an actual critical point of the unrestricted f accidently lies on a boundary). We discuss only critical points of the first two types due to the extra complications that can arise in third-type cases. HYPOTHESIS H5 No critical value of the third type occurs.
2.3. The decomposition theorem In the presence of the boundaries S1 , S2 , . . . , Sm , it is by no means obvious that it is still possible to decompose the original integration contour into a sequence of steepest-descent contours, each passing over either a single critical point of f or its restriction f (i1 ,...,i p ) . While this may be obvious for single-dimensional integrals, and while local results in real dimensions are obtainable by using “neutralisers” (in terms of Laplace integrals; see Wong [76]), until now it has not been demonstrated explicitly in higher complex dimensions. In Section A.3, by extending ideas of Pham [63] and taking advantage of Hypotheses H1 and H2, it is shown that a given cycle of integration can be decomposed through a continuous deformation (isotopy) into a chain of cycles 0α such that their projections by f are straight half-lines L α := ( f α , ∞ exp(iθ )), where f α belongs to the set of critical values 3. This is true at least when the finite set of these L α are two-by-two disjoint, and this is guaranteed for a “generic” phase of k. Note that on each of these 0α the exponential factor exp(−k f (z)) has the greatest exponential decay; that is, 0α satisfies the steepest-descent condition. Under Hypotheses H3 and H4, we show in Section A.4 that the investigation of each cycle 0α essentially reduces to a local analysis near its relevant critical point z α . Assuming these results, Sections 2.3.1 and 2.3.2 help us to understand the local geometry near a critical point, introducing objects such as “Lefschetz thimbles” and “vanishing cycles.” These objects are most easily defined initially for the unbounded contour case. We do this in Section 2.3.1 before demonstrating how they are modified in the presence of boundaries. 2.3.1. The geometry near a first-type critical point In the Borel plane, we consider restrictions to (sufficiently) small closed discs Dα centred on the critical values f α , of radius r0 . This corresponds to local truncations of the steepest-descent contours 0α in neighbourhoods of the critical point z α . We first consider the case of a nondegenerate critical point z α . The function f
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has a Morse singularity there, and the Morse lemma ensures the existence of local coordinates (s (1) , . . . , s (n) ) such that 2
2
2
2
f − f α = s (1) + · · · + s (n)
2
2
= x (1) + · · · + x (n) − (y (1) + · · · + y (n) ) + 2i(x (1) y (1) + · · · + x (n) y (n) ),
(9)
where x (i) = <(s (i) ) is the real part of s (i) , and y (i) = =(s (i) ) is the imaginary part. Assuming for simplicity that θ = 0, and under the truncation condition | f (s) − f α | ≤ r0 , we see that the best realisation of the steepest-descent conditions is given by the so-called Lefschtez thimble,∗ defined by 2 2 0αt0 = (s (1) , . . . , s (n) ) ∈ Cn y (1) = · · · = y (n) = 0 , x (1) + · · · + x (n) ≤ r0 . (10) t The Lefschtez thimble 0α0 is unique up to the choice of orientation. The standard one is that defined by the coordinates (x (1) , . . . , x (n) ) (see [63]). The case θ 6 = 0 is easily deduced from the case k > 0 by the linear mapping (s (1) , . . . , s (n) ) 7 → (eiθ/2 s (1) , . . . , eiθ/2 s (n) ).
(11)
t
Note that, from its very definition, a Lefschetz thimble 0α0 is mapped by f onto t the closed segment L α0 := ( f α , t0 ), where t0 = f α + r0 exp(iθ).
Z1 γα t Z2 0αt t
Figure 1. Lefschetz thimble 0α0 for n = 2 and its boundary t t γα0 = ∂[0α0 ] t
An associated geometrical object is the vanishing cycle, γα0 (see Figure 1) being the (n − 1)-real dimensional oriented boundary of the Lefschetz thimble, where f = ∗ In
what follows, the term “absolute” is added to these Lefschtez thimbles to differentiate them from the “relative” Lefschtez thimbles introduced in Section 2.3.2.
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t0 ; when θ = 0, 2 2 γαt0 = (s (1) , . . . , s (n) ) ∈ Cn y (1) = · · · = y (n) = 0 , x (1) + · · · + x (n) = r0 . (12) The orientation is determined by that of the Lefschetz thimble. It is convenient to refer to the vanishing cycle in terms of a boundary operator ∂, γαt0 = ∂ [0αt0 ].
(13)
A Lefschetz thimble encountering a first-type nondegenerate critical point can thus be S geometrically parametrised in that locality as the union of all vanishing cycles t γαt t as t runs along L α0 . We note briefly that it is possible to extend the concepts of a Lefschetz thimble and a vanishing cycle to the case of an isolated degenerate critical point of the first type, that is, when the Hessian determinant of f vanishes. The idea is as follows. Under a small (generic) deformation, the degenerate critical point z α splits into a finite number µα of nondegenerate critical points, µα being the multiplicity or Milnor t number of the critical point.∗ We thus have a basis of µα Lefschetz thimbles 0α0j , j = 1, . . . , µα , and their corresponding vanishing cycles. Returning to the original unperturbed f , the basis of Lefschetz thimbles is deformed into a basis of µα “folded” Lefschetz thimbles. t Note that, starting with this local description of Lefschetz thimbles 0α0 , we can extend them globally by following the flow of the vector field ∇(< k f ) with the vanishing cycles as initial data. This defines what are called the (absolute) steepestdescent n-folds 0α . This is true at least when no Stokes phenomenon occurs, and this is guaranteed if the half-line L α , onto which 0α is mapped by f , does not encounter any other critical value f β ∈ 3() of f . 2.3.2. The geometry near a second-type critical point When boundaries are present, the definition of a Lefschetz thimble and a vanishing cycle needs clarification. In the Borel plane, we again consider a restriction to a small enough closed disc Dα of radius r0 centered on a critical value f α of depth p. Localising near the corresponding critical point z α , we use the local coordinates of (8). If we assume that F has a nondegenerate critical point, then it may be written as 2
F = s ( p+1) + · · · + s (n) ∗
2
(14)
There are many equivalent definitions for the Milnor number. The nonspecialist may be referred to [58] a b or [2]. For instance, the Milnor number for the singularity z (1) + z (2) (a, b ∈ N\{0}) is (a − 1)(b − 1). Of course µα = 1 for a nondegenerate critical point.
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by changing the n − p local coordinates (s ( p+1) , . . . , s (n) ) if necessary. This is a consequence of the Morse lemma applied on restriction to the S1 ∩ · · · ∩ S p , which of course does not affect the local equations s (i) = 0 for the boundary Si , i = 1, . . . , p. Assuming for a moment that θ = 0, we easily obtain 0αt0 = (s (1) , . . . , s (n) ) ∈ Cn y (1) = · · · = y (n) = 0 , x (1) ≥ 0, . . . , x ( p) ≥ 0, 2 2 (15) x (1) + · · · + x ( p) + x ( p+1) + · · · + x (n) ≤ r0 , with x (i) = <(s (i) ) and y (i) = =(s (i) ) for the best realization of the steepest-descent conditions, under the truncation condition | f (s) − f α | ≤ r0 . t We refer to 0α0 as a relative Lefschetz thimble. Again, the relative Lefschetz thimble is unique, up to the choice of the orientation. The orientation defined by the coordinates (x (1) , . . . , x (n) ) is called the standard orientation. The case of general θ is deduced from the case k > 0 by the simple linear mapping (s (1) , . . . , s ( p) , s ( p+1) , s (n) ) 7 → (eiθ s (1) , . . . , eiθ s ( p) , eiθ/2 s ( p+1) eiθ/2 s (n) ). t
(16) t
The relative Lefschetz thimble 0α0 is mapped by f onto the closed segment L α0 := ( f α , t0 ), where t0 = f α + r0 exp(iθ). The concept of a (relative) Lefschetz thimble now being clear, its companion vanishing cycle is defined in terms of a reduction process. This algorithm is important in the derivation of the asymptotics of Laplace integrals of type (1), and so we now explain it. This is done on the basis of Figure 2, where n = 3 and p = 2. We assume t θ = 0 for simplicity. We first introduce ∂ [0α0 ], where ∂ is the boundary operator that t selects the part of the boundary of 0α0 where f = t0 : 2
∂ [0αt0 ] = {x (1) ≥ 0, x (2) ≥ 0, x (1) + x (2) + x (3) = r0 }.
(17)
t
The relative Lefschetz thimble 0α0 can thus be locally parametrised as the union t0 t t ∂ [0α ] with t running on L α .
S
2
Reading the equality in (17) as x (2) + x (3) = r − x (1) = r1 with 0 ≤ x (1) ≤ r , S we see that each ∂ [0αt ] can itself be parametrised by the union t1 ∂1 ◦ ∂ [0αt1 ] for t1 running on L tα (and x1 = r −r1 ). Here ∂1 is the boundary operator that selects the part of the boundary of ∂ [0αt1 ] lying on S1 , where f |S1 = t1 and t1 = f α + r1 exp(iθ); in other words, 2 ∂1 ◦ ∂ [0αt1 ] = {x (2) ≥ 0, x (2) + x (3) = r1 }. (18) Similarly introducing the boundary operator ∂2 , we get a parametrisation of each ∂1 ◦ ∂ [0αt1 ] in terms of the sequence over t2 ∈ L tα1 of the boundaries 2
∂2 ◦ ∂1 ◦ ∂ [0αt2 ] = {x (3) = r2 }
(19)
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S1
0011 a
00000000000 11111111111 11111111111 00000000000 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111
b
S2
0011
Figure 2. Relative Lefschetz thimble 0αt for n = 3 and p = 2. The vanishing cycle is ∂2 ◦ ∂1 ◦ ∂[0αt ] = [b] − [a].
(and x2 = r1 − r2 ) lying on S2 ∩ S1 , where f |S2 ∩ S1 = t2 and t2 = f α + r2 exp(iθ ). t The boundary ∂2 ◦ ∂1 ◦ ∂ [0α0 ] is precisely the vanishing cycle γαt2 associated t with the relative Lefschetz thimble 0αt2 . More generally, the vanishing cycle γα0 for a critical point of the second type of depth p is the boundary on the simultaneous intersection of (15) with the p + 1 constraining boundaries, including the boundary condition f = t0 . In boundary operator notation, we have γαt0 = ∂i p ◦ · · · ◦ ∂i1 ◦ ∂[0αt0 ],
(20)
where the ∂i p is the operator that selects the part of the boundary lying on the boundary t Si p . Its orientation is deduced from that prescribed to 0α0 . Note that the reduction process (20) is not canonical, in the sense that it is defined up to a permutation of the p +1 boundary operators. However, the resulting vanishing cycle does not depend on the order. When F has a degeneracy of multiplicity µα , we obtain a basis of µα folded t relative Lefschetz thimbles 0α0j , j = 1, . . . , µα , and the reduction process (20) to associate a (unique) vanishing cycle to a given (folded) relative fold Lefschetz thimble is unchanged. t To extend the local description of a relative Lefschetz thimble 0α0 into a global notion of a relative steepest-descent n-fold 0α is less obvious than in the “absolute case,” where no boundary interferes (see [48]). This can be performed (cf. App. A), at least when no Stokes phenomenon occurs.
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2.3.3. The corner case In the above description of the behaviour of a second-type critical point, it is implicitly assumed that the depth p is less than n. The case where the depth is n, which corresponds to a corner, is actually simpler. As illustrated in Figure 3, the reduction process stops at the ( p = n)th iteration, the boundary ∂ p−1 ◦ · · · ◦ ∂1 ◦ ∂[0]
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being just the end critical corner point, and no vanishing cycle needs to be defined. S3 S1
S2 Figure 3. Relative Lefschetz thimble for n = p = 3. No vanishing cycle occurs.
2.4. Conclusion Assuming that θ is generic, so that the set of half-lines L α = f α + eiθ R+ ( f α ∈ 3) are two-by-two disjoint, we saw in Section 2.3 how each critical point of multiplicity µα may be associated with µα independent (absolute or relative) steepest-descent nfolds. The set of all these steepest-descent n-folds then defines a basis for the contours of integration in the following sense. THEOREM 2.1 In the presence of the boundaries S1 , S2 , . . . , Sm , every contour of integration of integrals of type (1) can be decomposed uniquely as a chain (cf. (4)) of steepest-descent n-folds associated to the critical points, restricted or actual, of f .
This result is just a na¨ıve formulation of Theorem A.2 proved in Appendix A. Note, moreover, that by adding the Milnor numbers of each critical point (with the convention µ = 1 for a corner), we have a simple way of keeping an account of the number of possible independent steepest-descent contours.
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Example By way of an example, consider the function 4
2
f (z (1) , z (2) ) = z (1) + z (1) z (2) + z (2)
3
(22)
in the presence of a boundary S1 : z (1) − z (2) = 1. The phase function f has no critical point at infinity. By dimensionality, the question of critical points at infinity is not relevant for the restricted functions on the boundary (there are no critical points at infinity in dimension 1). The phase function f has two isolated critical points of the first type. One is nondegenerate; hence µ1 = 1 and corresponds to the critical value f 1 = −1/19683. The other critical point is 0 ∈ C2 ; it is degenerate, with f 2 = 0 for its critical value. Theoretically, the computation of its Milnor number µ2 follows from a result of V. Palamodov and H. Grauert (see, e.g., [58]) via an algorithm (see, e.g., [19], [2]). Here we can take advantage of the semi-quasi homogeneity of the polynomial f (see [2]); in the neighbourhood of the critical point, the singular locus f = 0 is isomorphic to X (Y 2 + X 3 ) = 0, that is, the union of a cusp and a smooth curve. In the language of catastrophe theory, this singularity is classified as the parabolic umbilic, D5 (cf. [2]), and one finds that µ2 = 5. The restricted function f (1) on S1 has three nondegenerate critical points, and the critical values f (1)1 , f (1)2 , f (1)3 are of the second type. To conclude, we have a basis set of 9 (1+5+3×1) possible independent steepest surfaces into which a general integration contour can be decomposed.∗ Variation with θ . The above analysis has been carried out for fixed generic phase 9(θ ) θ0 = −arg(k). When θ ranges from θ0 to θ0 + 2π, a given cycle 0 θ0 ∈ Hn 0 (Cn , S) 9(θ ) is deformed continuously into a cycle 0 θ ∈ Hn (Cn , S), according to the continuous variation of the convergence criterion <(k f ) → +∞ at infinity.† Note, however, the following. • In general, the deformed cycle 0 θ0 +2π differs from 0 θ0 ; therefore, as a rule, the integrals (1) are ramified at k = 0. (Just think, for instance, of the Airy function, where f (z) = z − z 3 /3.) • When nongeneric phases θ are crossed, the decomposition of a given cycle with respect to a basis of steepest-descent n-folds may encounter discontinui-
other words, H29 (Cn , S1 ) is a free Z-module of rank 9. 9(θ) translates in terms of sheaves; the family of groups of homology Hn (Cn , S) makes up a local system of free Z-modules of finite rank on the circle S of directions. We denote this local system by Hn9 (Cn , S)S .
∗ In
† This
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ties. This is the Stokes phenomenon. We return to this point later (see Section 4). 3. Integral representations and asymptotics We are now ready to study the analytical and asymptotic properties of the Laplacetype integrals (1). We recall that g = g(z) is a complex polynomial function, and we denote by ω = g(z) dz (1) ∧ · · · ∧ dz (n) the corresponding holomorphic differential n-form. It follows from the above analysis (see Section 2) that the mapping Z 0 7 → I0 (k) = e−k f (z) ω (23) 0
translates the geometrical properties of the family of spaces of integration contours 9(θ ) (Hn (Cn , S))θ ∈S into analytic properties of integral functions defined on the universal covering of C\{0}.∗ Moreover, to give a complete description of this representation, it is enough (1) to consider the action of (23) on each cycle of a basis of steepest-descent nfolds for generic θ (see Theorems 2.1 or A.2) and (2) to analyse the possible discontinuities (Stokes phenomena) when nongeneric θ are crossed. The second point is discussed in Section 4. Here we concentrate on the first point, the asymptotics with |k| → +∞. We thus consider a steepest-descent n-fold 0 = 0α (θ) for a given generic θ. One can make two preliminary remarks. (1) It follows from its very definition that f maps the steepest-descent n-fold 0 onto the half-line L α . This ensures the exponential decay of e−k f (z) at infinity along 0 for k in all closed subsectors of o n π and |k| > 0 . (24) 6θ = arg(k) + θ < 2 It thus follows that the Laplace integral Iα (k) =
Z 0
e−k f (z) ω
(25)
defines an analytic function in 6θ . (2) Using the notation of Sections 2.3.1 and 2.3.2, let us “truncate” our chain R 0 as 0 t0 with t0 ∈ L α and close to f α .† Then the difference integral 0−0 t0 e−k f (z) ω other words, (23) gives a representation of the local system of homology Hn9 (Cn , S)S in terms of the sheaf of holomorphic functions on C\{0}. † With the notation of Sections A.4.1 and A.4.2, 0 t0 is a relative cycle in H (X , X t0 ) if depth( f ) = 0 n α α α t and in Hn (X α , X α0 ∪ Y1 ∪ · · · ∪ Y p ) if depth( f α ) > 0. ∗ In
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defines an analytic function in k which is exponentially decreasing at infinity inside all closed subsectors of 6θ and is therefore “flat” (i.e., asymptotic to the zero function) at infinity. This means that Iα (k) and Z Iαt0 (k) = e−k f (z) ω (26) 0 t0
have the same Poincar´e asymptotics at infinity inside all subsectors of 6θ . We can now compute these asymptotics by the saddle-point method. The result of course depends on the type of critical point we consider, and this is the purpose of Sections 3.1 and 3.2. 3.1. Absolute steepest-descent contours (no boundaries) As an introduction to the bounded case, we first study the asymptotics associated with a critical value f α = f (z α ) ∈ 3( ) of the first type. This case is well known: the asymptotics are governed by the local behaviour of the integrand near the isolated critical point z α . We briefly recall the ideas of B. Malgrange [45] (see also [63], [25], and [3, Section 11]), which relate the asymptotics directly to the geometry of the phase function. Furthermore, we demonstrate the Borel summability of these asymptotics, which can therefore be considered as an exact encoding of the function Iα (k) within a Stokes sector. Writing Z I0 (k) = e−k fα
0
e−k( f (z)− fα ) ω,
(27)
one can assume that f α = 0 without loss of generality, and we furthermore assume that θ = 0 for simplicity. We now consider the truncated integral (26) and use the boundary operator ∂ (cf. (13) or (A.12)) to reduce the dimensionality. By the Stokes theorem (see [45]) and Leray residue theory (see [42]), Z Z t0 Z ω t0 −k f (z) −kt b b Iα (k) = e ω= dte Jα (t) with Jα (t) = , (28) t t d 00 0 ∂[0 ] f t where ω/d f t denotes the Leray residue differential (n − 1)-form of ω,∗ ∂[0 t ] being the vanishing cycle.† It is known from [9] and [45] that Jbα (t) defines an analytic function on the univerbα (t) is a solution sal covering D^ α \{0} of the punctured disc Dα \{0}. More precisely, J is, ω = d f ∧ ω/d f . This (n − 1)-quotient form ω/d f is holomorphic along each nonsingular fibre X αt = {z ∈ f −1 (t), z near z α and t near f α }. † That is, ∂[0 t ] ∈ H t n−1 (X α ). ∗ That
GLOBAL ASYMPTOTICS FOR BOUNDED MULTIPLE INTEGRALS
215
of a so-called Picard-Fuchs equation (or Gauss-Manin connection; see [61], for instance) d l Jbα d l−1 Jbα + q (t) + · · · + ql (t) Jbα = 0 (29) 1 dt l dt l−1 with l ≤ µα (the multiplicity of the singularity), which has (at most) a regular singular point at the origin. It follows from the theory of Fuchs (see [74]) that Jbα (t) admits in each sector a < arg(t) < b a convergent series expansion of the form X Jbα (t) = ar,s (t)t r (ln t)s , (30) r,s
where •
•
•
the r ∈ Q, r > −1, belong to the finite set of distinct monodromy exponents of the classical monodromy operator in homology∗ (cf. [45]); to each r is associated a set of s ∈ N with s ≤ inf{µα − 1, n − 1} (see [3] for more details); of course, s = 0 for nondegenerate critical points; P ar,s (t) = j≥0 ar,s, j t j are convergent Taylor series. It remains to use the standard integral Z +∞ d l 0(λ + 1) (31) e−kt t λ (ln t)l dt = dλ k (λ+1) 0 t
to conclude from Watson’s lemma that Iα0 (k); hence Iα (k) has Jα (k) =
X r,s
Tr,s (k)
(ln k)s k r +1
(32)
for its asymptotics when |k| → ∞ in all closed subsectors of 6θ , where Tr,s (k) are formal Gevrey-1 series expansions.† ∗ Define
a base point t0 near (but different from) the critical value, and consider (for t0 ) a basis of vanishing cycles generating the vanishing homology. Consider the deformation of these cycles when t goes around the critical value, starting at and coming back to t0 . The resulting cycles still define a basis of the homology: comparing the two bases, one gets an invertible matrix with integer coefficients describing the monodromy of the vanishing homology. In the case of a nondegenerate critical point (only one vanishing cycle), the possible monodromy arises from the possible self-intersection of the vanishing cycle. In the case of a self-intersection, we get a square root singularity at the origin for Jbα (t); otherwise, Jbα (t) is analytic near t0 . In the case of a degenerate critical point, each of the deformed cycles is described in general as a linear combination (with integer coefficients) in terms of the preliminary basis. Write the eigenvalues of this matrix under the form exp(−2iπλ). Then these λ are rationals (see [45]), and, moreover, they are (up to an addition of an integer) the exponents r in the series expansion (30). It may happen that the monodromy matrix has multiple eigenvalues, resulting in possible logarithmic terms in (30). In Kaminski and Paris’s scheme (see [40], [41]), this corresponds to multiple poles for the integrand for the associated Mellin-Barnes integral representation. P j † That is, T (k) = j r,s j≥0 Tr,s; j /k , and there exist Cr,s > 0 and Ar,s > 0 such that |Tr,l; j | ≤ Cr,s Ar,s 0( j) (see Malgrange [47], J. P. Ramis [66], or M. Loday-Richaud [44]).
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Now we can use our freedom to change θ slightly so that L α still does not meet other critical values. The resulting integral is just the analytic continuation of the previous one, the asymptotics at infinity being preserved. This proves that the asymptotics (32) are valid inside a wider sector of aperture greater than π. Therefore formal expansion (32) is Borel resummable with Iα (k) as its Borel sum (Watson’s theorem;∗ see [47]), and we thus obtain the following equivalent, one-dimensional, integral representation: Z ∞eiθ
Iα (k) =
dte−kt Jbα (t).
0
(33)
In summary, for general value f α , we have the following theorem. THEOREM 3.1 Let f α be a first-type critical value. The integral Iα (k) admits e−k fα Jα (k) as its asymptotic series expansion for k → ∞ in 6θ = {|k| > 0, | arg(k) + θ| < π/2}; in other words, Iα (k) ∼ e−k fα Jα (k) in 6θ , with
Jα (k) =
X r,s
Tr,s (k)
(ln k)s , k r +1
(34)
where Tr,s (k) =
X
Tr,s; j /k j
(35)
j≥0
belongs to C[[k −1 ]]1 , the differential algebra of Gevrey-1 series expansions (see [66]). The r ∈ Q run over a finite spectral set, and to each r is associated a set of s ∈ N satisfying s ≤ inf{µα − 1, n − 1}. Conversely, Iα (k) (resp., Jbα (t)) can be considered as the Borel sum (resp., the mi-
nor in the resurgence theory of J. Ecalle† ) of e−k fα Jα (k) in the direction of argument θ.
Remark. Note that we are here treating the Tr,s and Jα as formal series expansions, although they can be interpreted also as (Poincar´e) asymptotic expansions. The notational approach used here may be unfamiliar to some readers, but it is used to maintain consistency with the Borel and resurgence viewpoint (see [47], [23], [12]). Note also that in Theorem 3.1 the direction of summation should be considered ^ the universal covering of C\{0}. as a direction in C\{0}, ∗ This † See
is not to be confused with his lemma for standard integral expansions. Ecalle [23], or [12], [18], or [16] for a short introduction to resurgence theory.
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217
COROLLARY 3.1 If z α is a nondegenerate quadratic critical point of f , then Iα (k) ∼ e−k fα Jα (k) in 6θ , with
Jα (k) =
1
k
T (k) n/2 α
and
X g(0)(2π)n/2 Tα (k) = √ + T α /k j ∈ C[[k −1 ]], Hess( f )(0) j≥1 j
(36) where Hess( f ) is the hessian determinant. The choice of the root depends on the orientation of the vanishing cycle. This corollary is well known (cf. [25], [76]). One method for the practical computation of the T jα is described in [17]. 3.2. Relative steepest-descent contours (bounded case) This case corresponds to a critical value of the second type and is the main focus of this paper. Let f α ∈ 3(1,..., p) be a critical value of the restricted function f (1,..., p) with p > 0 the depth. We apply the reduction process developed in Section 2.3.2 (resp., Section A.4.2) to analyse the asymptotics of our bounded integral. Again without loss of generality, we assume that f α = 0 and θ = 0. To compute the asymptotics, it is enough to consider the truncated integral. Since f = 0 is not a singular level, by following formula (17) and its comments, we can apply Fubini’s theorem to write Z t0 Z ω t0 −kt Iα (k) = dte , (37) 0 ∂[0 t ] d f t where ω/d f t is the (holomorphic) restriction of the differential quotient (n − 1)form ω/d f to the level f = t, and ∂[0 t ] is that part of the boundary of 0 t lying on the fibre f −1 (t).∗ The same reduction can be repeated step by step. We use the local coordinates (s (1) , . . . , s ( p) , s ( p+1) , . . . , s (n) ) so that f is given by formula (8), and we define the set of functions f 1,2,...,i : (s (1) , . . . , s (n) ) 7 → f (0, . . . , 0, s (i+1) , . . . , s (n) ). At the second step (if p ≥ 2), from Fubini’s theorem we have Z Z t Z ω ω b Jα (t) = = dt1 | f =t, f1 =t1 , (38) t ∂[0 t ] d f 0 ∂1 ◦∂[0 1 ] d f ∧ d f 1 where ω/d f ∧ d f 1 | f =t, f1 =t1 denotes the (holomorphic) restriction of the differential quotient (n − 2)-form ω/d f ∧ d f 1 along the nonsingular level f = t, f 1 = t1 : here again, (38) is a simple translation of formula (18) with its comments. At the pth step the class of homology of [0 t ] belongs to Hn (X α , X αt ∪ Y1 ∪ · · · ∪ Y p ), and we select that part of its boundary lying on X αt . ∗ Precisely,
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DELABAERE and HOWLS
we obtain Iαt0 (k)
t0
Z
dte
=
−kt
0
t
Z
t p−2
Z dt1 · · ·
0
Z dt p−1
0
∂( p−1,...,1,0) [0
t p−1
]
ω( p−1,...,1,0) ,
(39)
where ∂( p−1,...,1,0) = ∂ p−1 ◦ · · · ◦ ∂1 ◦ ∂,
(40)
while ω( p−1,...,1,0) =
ω | f =t, f1 =t1 ,..., f1,...,( p−1) =t p−1 d f ∧ d f 1 ∧ · · · ∧ d f 1,...,( p−1)
(41)
is the corresponding Leray quotient (n − p)–differential form. If p = n (corner critical point), then ω( p−1,...,1,0) is just a holomorphic function of (t, t1 , . . . , t p−1 ) and the reduction process stops here. If p ≤ n − 1, it then remains to use the same argument as in Section 3.1 to obtain Z t0 Z t Z t p−1 Z t0 −kt Iα (k) = dte dt1 · · · dt p ω( p,...,1,0) , (42) 0
0
0
∂( p,...,1,0) [0 t p ]
where ∂ p ◦ · · · ∂1 ◦ ∂[0 t p ] is the vanishing cycle of the critical point. In the case p = n, the integral Z t Z t p−2 Z Jbα (t) = dt1 · · · dt p−1 ω( p−1,...,1,0) 0
0
∂( p−1,...,1,0) [0
t p−1
(43)
]
P j+ p−1 as its convergent Taylor defines an analytic function on Dα , with j≥0 a j t expansion. In the case p ≤ n − 1, by using the results from Section 3.2, the function Z h α (t, t1 , . . . , t p ) = ω( p,...,1,0) , (44) ∂( p,...,1,0) [0 t p ]
considered as a function of t p with (t, t1 , . . . , t p−1 ) as a parameter, is defined as an analytic function on the universal covering D^ α \{ f α } and admits in each sector a < arg(t) < b a convergent series expansion of the form X ar,s (t, t1 , . . . , t p−1 , t p )t rp (ln t p )s ,
(45)
r,s
where •
•
the r belong to a finite set of distinct monodromy exponents of the classical monodromy operator in homology, and the s ∈ N satisfy s ≤ inf{µα − 1, n − p − 1}; P j ar,s (t, t1 , . . . , t p−1 , t p ) = j≥0 ar,s, j (t, t1 , . . . , t p−1 )t p are convergent Taylor series with analytic dependence with respect to (t, t1 , . . . , t p−1 ).
GLOBAL ASYMPTOTICS FOR BOUNDED MULTIPLE INTEGRALS
It thus follows that the function Z t Z Jbα (t) = dt1 · · · 0
t p−1
Z dt p
0
∂( p,...,1,0) [0 t p ]
219
ω( p,...,1,0)
(46)
has the same properties, with Jbα (t) =
X
br,s (t)t r + p (ln t)s
(47)
r,s
P j for its convergent series expansion, where br,s (t) = j≥0 br,s, j t p are convergent Taylor series. As in Section 3.1, we finally recast Iα (k) as a one-dimensional integral representation Z ∞eiθ
Iα (k) =
0
dte−kt Jbα (t),
(48)
and we obtain the following theorem. THEOREM 3.2 If f α is a second-type critical value of depth p, the integral Iα (k) admits e−k fα Jα (k) as its asymptotic series expansion for k → ∞ in 6θ (i.e., Iα (k) ∼ e−k fα Jα (k) in 6θ ), where 1 X T j /k j ∈ C[[k −1 ]]1 , (49) for p = n, Jα (k) = p k j≥0
for p ≤ n − 1,
Jα (k) =
X r,s
Tr,s (k)
(ln k)s , k r + p+1
(50)
P j −1 where Tr,s (k) = j≥0 Tr,s; j /k belongs to C[[k ]]1 . The rational r ’s run over a finite spectral set associated with the critical point, and to each r is associated a set of s ∈ N satisfying s ≤ inf{µα − 1, n − p − 1}. Conversely, Iα (k) (resp., Jbα (t)) is the Borel sum (resp., the minor) of e−k fα Jα (k) in the direction of argument θ. The asymptotics are simpler in the case of a nondegenerate critical point (see, for instance, [17], [76], [3]), arising from p boundaries. COROLLARY 3.2 When the singular point is quadratic, Iα (k) ∼ e−k fα Jα (k) in 6θ , with
Jα (k) =
1
k
T (k) (n+ p)/2 α
for the asymptotic series expansion.
and
Tα (k) =
∞ Tα X j j=0
kj
(51)
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DELABAERE and HOWLS
4. Stokes phenomenon We described in the previous section the asymptotics of the multiple Laplace integrals of type (1), as well as their representation as Borel sums for generic summation directions (θ ). To obtain the global asymptotics, we must analyse the Stokes phenomena. This amounts to analysing the singularities in the Borel plane of the analytic continuations of (the minors) Jbα (t). How this can be obtained from generalised PicardLefschetz formulae (detailed in Section B.1) is discussed in Section 4.1. Against this Borel viewpoint, we demonstrate in Section 4.2 how the Stokes phenomenon can be understood directly in terms of steepest-descent contours, thus keeping the geometric ideas of the saddle-point method as in the one-dimensional case. In what follows, it is convenient to assume the following hypothesis. HYPOTHESIS H6 All singular points are nondegenerate.
4.1. Ramifications We return to the Jbα (t) of Theorems 3.1 and 3.2, together with their corollaries. From (36) and (51), if f α has depth pα , then locally near the origin, Jbα (t) = t (n+ pα −2)/2 Hα (t),
(52)
where Hα is holomorphic near zero. Thus when n + pα is even, Jbα (t) is an analytic function at the origin, but when n + pα is odd, Jbα (t) has a square-root singularity there.∗ Note that the Jbα (t) have been defined as (germs of ramified) analytic functions at the origin. If f α 6= 0, it is necessary to take into account translations by the complex numbers f αβ = f β − f α . (53) The generalised Picard-Lefschetz formulae described in Section B.1 allow us to identify the type of singularity in the analytic continuations of each Jbα (t). Denoting by Var = ρ − I the “variation” operator, where I is the identity operator and ρ is the analytic continuation around f αβ anticlockwise, it follows from formula (B.3) that for t near f αβ , Var Jbα (t) = καβ Jbβ (t − f αβ ), (54) where καβ is a (positive or negative) integer. This is consistent with the geometric argument (B.5). The vanishing homology Hn− pα −1 (S p ∩ · · · ∩ S1 ∩ X αt ) is a trivial covering on the circle of directions S when n + pα is even, but it is a two-fold covering when n + pα is odd.
∗
GLOBAL ASYMPTOTICS FOR BOUNDED MULTIPLE INTEGRALS
221
PROPOSITION 4.1 If pβ is the depth of the critical value f β , then • when n − pβ is even (so that Jbβ (t) is not ramified around t = 0),
Jbα (t) = καβ Jbβ (t − f αβ )
•
ln(t − f αβ ) + Hol(t − f αβ ) 2iπ
(55)
with Hol a holomorphic function near zero,∗ and when n − pβ is odd, Jbα (t) = −
καβ Jbβ (t − f αβ ) + Hol(t − f αβ ). 2
(56)
Arguments developed in Malgrange [45] show that Jbα (t) remains bounded (when n ≥ 2, the analysis being obvious when n = 1) near f αβ . Proposition 4.1 now results from the variation formulae (54) and Riemann’s removable singularity theorem (see [28]). We show in Section 4.2 how the above properties can be derived independently. 4.2. Stokes phenomenon We consider a singular direction (θ ) and assume that the closed half-line L α = L α (θ) meets a singular value f β ; this (possibly) gives rise to a Stokes phenomenon, which can be described as follows. Assume for simplicity that L α meets no singular point − − + + other than f β . We define the half-lines L − α = L α (θ ) and L β (resp., L α = L α (θ ) + and L β ) by slightly rotating L α and L β clockwise (resp., anticlockwise; see Figure 4). We perform the same rotations for the other half-lines L λ . For fixed θ − (resp., θ + ), define a basis of steepest-descent n-folds (0α − , 0β − , . . . , 0λ− , . . .) (resp., (0α + , 0β + , . . . , 0λ+ , . . .)). From our hypothesis, we can assume that (0β + , . . . , 0λ+ , . . .) is deduced from (0β − , . . . , 0λ− , . . .) by an isotopy (continuous deformation) when the argument runs from θ − to θ + , so that we can remove the upper scripts ± in the notation. Concerning the 0α ± ’s, we can assume only that the one is deduced from the other by a local deformation near the critical point z α . This may not work globally due to the critical value f β . We thus get the decomposition 0α − = 0α + + 0α ± ,
(57)
0α ± = 0α − − 0α + = καβ 0β .
(58)
where necessarily
∗ In other words, J bα (t) and καβ Jbβ (t − f αβ ) ln(t − f αβ )/2iπ are equal when one considers them as microfunctions.
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DELABAERE and HOWLS
L+ α L+ β fα
fβ fβ
fα
L− β L− α
Figure 4. Generic Stokes phenomenon: Below, before the Stokes phenomenon; above, after the Stokes phenomenon
In this formula, καβ can be understood geometrically as an index of intersection,∗ and therefore it is a positive or negative integer, the sign depending on the orientations of 0α and 0β . We now apply (58) to our integral representation. We start with the integral Iα (k) =
Z 0α−
e
−k f (z)
ω=e
−k f α
∞eiθ
Z
−
e−kt Jbα (t) dt,
0
(59)
which defines an analytic function within the sector 6θ − = {|k| > 0, | arg(k)+θ − | < π/2}. We know that Iα (k) extends analytically over C\{0} as a multivalued function, and from (58) we see that Z Z Iα (k) = e−k f (z) ω + καβ e−k f (z) ω (60) 0α+
0β+
for k ∈ 6θ + , that is, that Iα (k) = e
−k f α
∞eiθ
Z
+
e 0
−kt
Jbα (t) dt + καβ e−k fβ
∞eiθ
Z 0
+
e−kt Jbβ (t) dt.
(61)
Formula (61) provides a complete description of the Stokes phenomenon. We can recast these results in the framework of resurgence theory, using the notation of Section 3. From (59), Iα (k) is the left (lateral) Borel sum in the direction of argument θ of the ∗ Compare
h0β? , 0α ± i.
with Section B.2. With the convention of formula (B.8), καβ is given by the equality καβ =
GLOBAL ASYMPTOTICS FOR BOUNDED MULTIPLE INTEGRALS
223
resurgent symbol e−k fα Jα (k), Iα (k) = e−k fα S(θ − ) Jα (k),
(62)
and from (61) the action on Jα (k) of the Stokes automorphism S(θ) is given by S(θ) Jα (k) = Jα (k) + καβ e−k fαβ Jβ (k),
(63)
1 fαβ Jα (k) = καβ Jβ (k),
(64)
which finally yields where 1 fαβ is the “alien derivative operator” (see [23], [12], [18], [16]) at f αβ . It is important to notice that the alien derivatives (as well as the directions of summations) have to be indexed over the two-fold covering of C\{0} (i.e., the Riemann surface of the square root) when n + pα is odd. The sign of καβ depends on which sheet of this covering is under consideration. 4.3. Conclusion In (64), the index of intersection καβ now appears as a so-called Stokes multiplier. Note that (64) can be understood in terms of singularity in the Borel plane. The analytic continuation of the minor Jbα (t) of Jα (k) along a straight half-line (0, f αβ ∞) encounters a singularity at t = f αβ if καβ 6= 0, and its variation there (as defined in Section 4.1) is καβ Jbβ (t − f αβ ). This is nothing but (54). To compute the constants καβ and thus to obtain the complete resurgence structure, it is helpful to keep in mind their two interpretations. The geometric description in terms of indices of intersection has already demonstrated their quantised nature. This helps again to show that some of them necessarily vanish. This stems from the S fact that the set of critical values 3 = 3(i1 ,...,i p ) has a natural hierarchy that follows directly from the stratification of S. This means that a relative steepest-descent n-fold 0α having S1 , . . . , S p (say) as its boundary can be eventually affected only by critical values f β belonging to the subset 3α ⊂ 3 of critical values of f or of the restricted f | to one of the Si , i = 1, . . . , p, or their different intersections. Otherwise, καβ = 0 necessarily.∗ This is almost all that we can learn from the geometry. To get quantitative information about the remaining indices of intersections καβ , we have to turn to the hyperasymptotic analysis, interpreting this time the καβ as Stokes multipliers. This is the aim of the next section. It is convenient for this purpose to represent the results of (64) in terms of the series expansions Tα introduced in formulae (36) – (51). Hereafter, formula (65) is simply derived from formula (64) by the Leibniz rule (see [23], [16]). ∗ See
also the second remark of Section B.1.
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DELABAERE and HOWLS
THEOREM 4.1 The series expansions Tα (k) ∈ C[[k −1 ]]1 are Gevrey-1 resurgent resummable. If f α is a critical value of depth pα , the adjacent singularities∗ of the minor Tbα (t) of Tα (k) are (at most) the f αβ = f β − f α , with f β ∈ 3α . Moreover, in the (generic) case where these relevant f αβ have distinct phases,
1 fαβ Tα (k) = καβ k ( pα − pβ )/2 Tβ (k),
(65)
where pβ (≤ pα ) is the depth of f β , whereas the καβ are integers. Remark. In (65), the alien derivatives can now be indexed over C\{0}. This does not make (65) ambiguous. One has to keep in mind that the sign of the index of intersection καβ depends on the orientations of the steepest-descent n-folds 0α and 0β , on which the determination of the square root k ( pα − pβ )/2 also depends. 5. Hyperasymptotic analysis: Calculation of Stokes constants We assume here for simplicity that the values f αβ have distinct phases. Theorem 4.1 thus applies, allowing us to identify the singularity types. The results from the previous section can now be used to deduce the exact remainder term for a truncated asymptotic expansion about any of the singularities. As we have converted integral (1) into a one-dimensional Laplace integral (Borel sum), the procedure follows closely that of Howls [36] and Olde Daalhuis [55], allowing us to be brief. It is important to stress that at no point in the hyperasymptotic procedure detailed in this section are the full infinite asymptotic/formal power series used. All the expansions are finite and exactly terminated by the appropriate (hyperasymptotic) remainder term. We now describe the hyperasymptotic analysis for the (slowly varying) related analytic function Tα (k) = k (n+ pα )/2 ek f α Iα (k) = S(θ ) Tα (k) = T0α +
∞eiθ
Z 0
dt e−kt Tbα (t).
(66)
Here S(θ ) Tα (k) is the Borel sum of the formal series expansions Tα (k) in the nonsingular direction of argument θ , and therefore Tα (k) ∼ Tα (k) for k → ∞ in 6θ . We represent the local behaviour of the function Tbα (t) in terms of a Cauchy integral representation I Tbα (u) 1 b du . (67) Tα (t) = 2iπ u=t u−t ∗
The adjacent singularities of a germ of analytic functions at the origin are the singularities of the analytic continutions of this germ along half-lines emanating from the origin.
GLOBAL ASYMPTOTICS FOR BOUNDED MULTIPLE INTEGRALS
225
γαβ1 f αβ1 γα
f αβ2
γαβ2
Figure 5. The path γα on the left. Its deformation is on the right with the contributions γαβ from the adjacent singularities.
After a binomial expansion to the order of truncation N required, we have exactly Tα (k) =
N −1 X r =0
Trα + Rα (k, N ), kr
(68)
where Rα (k, N ) =
1 2iπk N
Z 0
+∞
dwe−w w N −1
Z γα
du
Tbα (u) . (1 − w/ku)u N
(69)
The contour γα encircles the positive real axis as in Figure 5. We then deform γα (see Figure 5) to encounter the other singularities f αβ that are adjacent (see [36]). By a suitable restriction of the type of integrand functions we consider, or by going to sufficiently high truncation N , the arcs at infinity (see Figure 5) make no contribution (see [36]).∗ We thus have Z X Z +∞ 1 Tbα (u) −w N −1 Rα (k, N ) = dwe w du . (70) N 2iπk (1 − w/ku)u N 0 γαβ adjacent f αβ
∗ The general case can be treated as in Olde Daalhuis [55]: instead of working with full Borel sums, one uses truncated Laplace integrals (essentially changing summation to presummation in resurgence-speak; cf. [12], [18]). Pushing the circular arcs far enough away, their contributions can be bounded away to an exponential level smaller than the one to which the hyperasymptotics are eventually taken, that is, less than exp(−M|k|) for any chosen M > 0.
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DELABAERE and HOWLS
At each of the singularities β, we make a change of variables and collapse the contour γαβ onto the associated cut. The results of the Picard-Lefschetz analysis, embodied in (65), then guarantee that the discontinuities generate self-similar integrands, with contours now over the critical point β. The final result is Tα (k) =
N −1 X r =0
Trα + Rα (k, N ) kr
(71)
with Rα (k, N ) = −
Z [−θαβ ] N −1+( pα − pβ )/2 1 X καβ −t0 f αβ t0 dt e Tβ (t0 ), 0 2iπ k − t0 k N −1 0
(72)
καβ 6 =0
R [η] R ∞eiη (see [55]), with θαβ denoting the phase where we have used the notation 0 = 0 ∗ of f αβ . The indices pα and pβ are the depths of (number of boundaries associated with) α and β, respectively. The minus sign comes from the fact that the orientation of the contours γαβ is in the opposite sense to the convention used to define the index of intersection καβ (compare the γαβ in Figure 5 with the path L + β in Figure 4, for instance). The (quantised and, as yet, unknown) Stokes constants associated with α and β are nonzero integers if the singularity is adjacent, defined up to a sign depending ( p − p )/2 on the branch of t0 α β (see the remark following Theorem 4.1). We determine the Stokes constants by resorting to a resurgence formula for the coefficients in the expansions themselves. Using the fact that TNα = k N Rα (k, N ) − Rα (k, N + 1) Z +∞ v καβ 1 X −v N −1+( pα − pβ )/2 =− dv e v T , (73) β N +( pα − pβ )/2 2iπ f αβ 0 f καβ 6=0
αβ
we substitute the corresponding exactly terminated asymptotic expansions of all the adjacent Tβ of type (71) with v/ f αβ playing the role of k. If the next-nearest singularity is some distance further away from α than the nearest, we obtain the usual (Dingle type; see [21]) leading-order approximation to the late terms: N −N καβ 0(N + ( pα − pβmin )/2) βmin TNα = − min T 1 + O 0 N +( pα − pβmin )/2) 2iπ |k f αβmin | f αβmin as N → ∞.
(74)
analytic function Tβ can be thought of as the Borel sum of the series expansion Tβ in the direction of arg( f αβ ). ∗ The
GLOBAL ASYMPTOTICS FOR BOUNDED MULTIPLE INTEGRALS
227
Here f αβmin is the distance in the Borel plane to the nearest potential singularity (which may or may not be adjacent). As the Stokes constants are quantised integers, we only need to determine καβ to an accuracy of within 1/2 to infer its value. The large parameter in (74) is now N . An appropriately high value of N will give that accuracy if we know the N th term, f αβmin , and the number of boundaries that α, β sit on ( pα , pβ ). We can then move to determine whether the next-nearest neighbour is adjacent using the same level of approximation. The calculations may be checked by including further terms in the series expansion taking larger values of N and/or higher |k|. Note that this procedure is a simplification of the method in [36] and [55] and reduces the work required to determine the full set of Stokes constants. However, in this form it only works for integrals due to the quantised nature of the Stokes constants. Even this new method cannot determine all the καβ at this stage. Stokes constants from next-nearest neighbours β1 can be determined, provided that | f βmin β1 | < | f αβmin |. By summing the nearest neighbour contributions to the least term and including a single term of the next-nearest contribution, we may again shorten the method of [36] and [55] and use the following to determine καβ1 : N1 −1 καβmin X 0(N0 + ( pα − pβmin )/2 − r1 ) βmin Tr1 N0 +( pα − pβmin )/2−r1 2πi f αβmin r1 =0 N −N0 καβ1 0(N0 + ( pα − pβ1 )/2) β1 0 =− , T0 1 + O N +( p − p )/2 α β 0 2πi |k f αβ1 | 1 f αβ1 N0 = |k| | f αβmin | + | f βmin β1 | , and N1 = N0 − |k f αβ1 |.
TNα0 +
(75)
This can be demonstrated by an intricate use of Stirling’s approximation. If this is not sufficient to determine all the καβ , one must resort to hyperasymptotic approximations and successively reexpand the remainders for each β. The result of this incestuous iterative reexpansion and substitution of the exact remainder term into itself is the treelike hyperasymptotic expansion (see Howls [36]) TNαα 0
X καβ 1 = 2iπ καβ1 6 =0
−
X
αβ1
N1
X−1
Trβ1 K (1) (0; α, β1 , N0α + 1, r )
r =0
X καβ κβ β 1 1 2 (2iπ)2
καβ1 6 =0 κβ1 β2 6 =0
αβ1 β2
N2
X−1 r =0
αβ
Trβ2 K (2) (0; α, β1 , β2 , N0α + 1, N1 1 , r )
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DELABAERE and HOWLS
+ · · · + (−1)m−1
X καβ1 6 =0
···
X κβm−1 βm 6 =0
καβ1 · · · κβm−1 βm (2iπ )m
αβ1 ···βm −1
Nm
X
Trβm
r =0
α···β
αβ
× K (m) (0; α, β1 , . . . , βm , N0α + 1, N1 1 , . . . , Nm−1 m−1 , r ) X X καβ1 · · · κβm−1 βm αβ rαβ1 ···βm (N0α , N1 1 , . . . , Nmαβ1 ···βm ), + ··· (2iπ )m καβ1 6 =0
κβm−1 βm 6 =0
(76) where K (1) k; α, β1 , N0α , r N0α + ( pα − pβ1 )/2 − r (1) =F k; , f αβ1 αβ
K (2) (k; α, β1 , β2 , N0α , N1 1 , r ) =F
(2)
αβ1
k;
N0α − N1
+ 1 + ( pα − pβ1 )/2, f αβ1 ,
αβ1
N1
+ ( pβ1 − pβ2 )/2 − r f β1 β2
! , (77)
and, more generally, α...β αβ K (m) k; α, β1 , . . . , βm , N0α , N1 1 , . . . , Nm−1 m−1 , r αβ N α − N1 1 + 1 + ( pα − pβ1 )/2, . . . , (m) =F k; 0 f αβ1 , α···βm−1 α···βm−2 + ( pβm−2 − pβm−1 )/2, − Nm−1 Nm−2 f βm−2 βm−1 ,
...,
α···β
Nm−1 m−1 + ( pβm−1 − pβm )/2 − r f βm−1 βm
.
(78) αβ ···β
The Fr 1 m are the canonical hyperterminants (see [7], [35], [36], [55]) (0) F (z) = 1, M −1 R [−θ ] t 0 (α) z; M0 F = 0 0 e−t0 σ0 0z−t0 dt0 , σ0 (l+1) z; M0 , . . . , Ml F σ0 , . . . , σl M −1 M −1 R [−θ0 ] R [−θ ] t0 0 ···tl l = 0 · · · 0 l e−(t0 σ0 +···+tl σl ) (z−t0 )(t dt0 · · · dtl , 0 −t1 )···(tl−1 −tl )
(79)
where θi is the phase of σi ∈ C\{0}, <Mi > 1. When ph σ j = ph σ j+1 (mod 2π), the t j -path of integration is deformed to the left or the right as in [55], yielding
GLOBAL ASYMPTOTICS FOR BOUNDED MULTIPLE INTEGRALS
229
lateral canonical hyperterminants, intrinsically linked to the lateral summations of the resurgence theory (see [12], [16], [18]). These multiple integrals converge when −θ0 < ph z < 2π − θ0 and can be evaluated easily by the methods of Olde Daalhuis [54], [56] with the convention that µαβ |Olde
Daalhuis
=
( pα − pβ ) . 2
(80)
Thus we can employ his truncations and error estimates directly to minimise the overall remainder terms. After m stages of hyperasymptotics, the optimal truncations that globally minimise the remainder term are X N0α = |k| × min | f βl βl+1 | , | f αβ1 | + καβ1 6=0 κβl βl+1 6 =0
αβ1
N1
αβ1 β2
N2
l=1···m
= max 0, N0α − |k f αβ1 | , αβ1
= max 0, N1 .. .
− |k f β1 β2 | ,
αβ ···βm−1
1 Nmαβ1 ···βm = max 0, Nm−1
− |k f βm−1 βm | .
(81)
At each level of hyperasymptotics, if a καβ is not yet known, then it is initially included, regardless of whether it subsequently turns out to be zero (at which point all branches containing this constant may be immediately pruned). Knowing only the relevant f αβ , pα , pβ along each of the branches and using the truncations (81) allows each καβ to be determined from an algebraic system of equations as outlined in [36] and [55]. It might appear that hyperasymptotic expansions are an unnecessary technical numerical detail. However, we know of no other general and systematic analytical or numerical method that is a practical tool for calculating the Stokes constants of these types of integrals. Following Theorem 4.1, there is one qualitative difference between the unbounded and bounded integral case. In the unbounded integrals, every distant quadratic critical point β that could be seen by the initial one α could, in turn, see α itself. In the bounded case, some of the α (and even β) arise only because of the presence of the boundaries. If α lies on a boundary S1 ∩ S2 , then α can see β only if β lies on one of the strata S1 ∩ S2 , S1 , or S2 , or if it arises from the phase function f itself. If β is a quadratic critical point arising from the phase function f , since this is a fundamental property of the integrand, β is a likely candidate for adjacency to all the boundary α. There is thus a hierarchy that can be inferred and can be used to simplify the hyperasymptotic analysis and deduction of Stokes constants. If pβ > pα , we may
230
DELABAERE and HOWLS
deduce that καβ = 0 immediately since β exists only because of the presence of an extra boundary that α knows nothing about. Note that the opposite inference cannot be made; unlike the unbounded case, κβα may differ from καβ , and so care must be taken in calculations. Once the Stokes constants have been determined, it is possible to obtain a (hyper)exponential accurate approximation to (1) via the full expansion αβ1
N0α −1
N1 −1 X X X Tα καβ1 r − Trβ1 K (1) (k; α, β1 , N0α , r ) Tα (k) = α N0 −1 kr 2iπk κ 6 =0 r =0 r =0 αβ1
+
X
X
καβ1 6=0 κβ1 β2
αβ1 β2
N2
καβ1 κβ1 β2 α
(2iπ )2 k N0 −1 6=0
+ · · · + (−1)
m
X καβ1 6=0
···
X−1
αβ
Trβ2 K (2) (k; α, β1 , β2 , N0α , N1 1 , r )
r =0
καβ1 · · · κβm−1 βm
X κβm−1 βm 6 =0 αβ
αβ1 ···βm −1
Nm
α (2iπ )m k N0 −1
X
Trβm
r =0
α···β
× K (m) (k; α, β1 , . . . , βm , N0α , N1 1 , . . . , Nm−1 m−1 , r ) X X καβ1 · · · κβm−1 βm + ··· Rαβ1 ···βm α m k N0 −1 (2iπ ) κ 6 =0 κ 6 =0 αβ1
× (k,
βm−1 βm
N0α ,
αβ N1 1 , . . . ,
Nmαβ1 ···βm ).
(82)
Using truncations (81), this yields an accuracy of O(e−M|k| ) at the Mth iterαβ ···β ation (see [55]), leaving an unevaluated remainder Rαβ1 ···βm (k, N0α , . . . , Nm 1 m ). This expression also widens the domain of validity of the original asymptotics and automatically and exactly incorporates any Stokes phenomenon through the hyperterminants, as explained in [36] and [55]. 6. Example We illustrate the theory with the following example, for which explicit Borel transforms can be deduced as benchmarks against which to test the hyperasymptotic analysis. We take 2
f (z) = z (1) + 2z (2) + 3z (3) + z (1) z (2) + z (2) z (4) + z (3) z (4) + z (2) + z (4)
2
(83)
and g(z) = 1. The boundary is S = S1 ∪ S2 ∪ S3 , where S1 , S2 , S3 are hypersurfaces defined by the equations S1 : z (1) = 2i, S2 : z (2) = 1 + 2i, and S3 : z (3) = 2 − 3i. The set of isolated critical values on the different strata is thus • 3() = { f 0 = 11} and z 0 = (3, −1, 7, −3),
GLOBAL ASYMPTOTICS FOR BOUNDED MULTIPLE INTEGRALS
231
3(1) = { f (1) = 39/4 + 3i} and z (1) = (2i, 1/2 − i, 11/2 + i, −3), 3(3) = { f (3) = 7 − 15i/2} and z (3) = (1/2 − 3i/2, −1, 2 − 3i, −1/2 + 3i/2), • 3(1,2) = { f (1,2) = 1 + 6i} and z (12) = (2i, 1 + 2i, 5 − 2i, −3), • 3(1,3) = { f (1,3) = 11 − 19i/3} and z (13) = (2i, −2/3 − 7i/3, 2 − 3i, −2/3 + 8i/3), • 3(1,2,3) = { f (1,2,3) = −1 + 9i/2} and z (123) = (2i, 1 + 2i, 2 − 3i, −3/2 + i/2). We choose θ = 0 for the (generic) direction of summation. To the family of closed half-lines drawn on Figure 6 corresponds a basis of six steepest-descent contours ([00 ], . . . , [0(1,2,3) ]) generating the space H49 (C4 , S) of allowed cycles of integration (cf. Theorem 2.1). • •
[1, 2] 6
[1, 2, 3] 4
[1] 2 2
−2
4
6
8
10
[]
12
0
−2
−4
−6
[1, 3] [3]
−8
Figure 6. The set of singularities and the family of half-lines L α in the complex Borel t-plane
6.1. Asymptotics We now describe the representation of the integration contours space H49 (C4 , S) via Laplace integrals. From Theorem 2.1, it is enough to consider the Laplace integrals over each of the steepest-descent contours [00 ], . . . , [0(1,2,3) ]. We recall that the Lefschetz thimble, in principle, determines completely the homology of the steepest-descent contour, and thus we do not provide a detailed global description of [00 ], . . . , [0(1,2,3) ]. We have just to prescribe the orientations, which we take to be those of the standard Lefschetz thimble (cf. Sections 2.3.1 and 2.3.2).
232
DELABAERE and HOWLS
√ In what follows, the square root k refers to the usual determination (real positive for positive real k). • Straightforward computations give I00 (k) = •
Z 00
e−k f (z) dz (1) ∧ · · · ∧ dz (4) =
4π 2 −11k e . k2
Convenient reductions show that the integral Z I0(1) (k) = e−k f (z) dz (1) ∧ · · · ∧ dz (4) 0(1)
(84)
(85)
can be written as 4iπ 3/2 I0(1) (k) = 3/2 e−(39/4+3i)k k
+∞
Z 0
dt e−kt √ . 5 − 12i − 4t
(86)
It follows that I0(1) (k) is the Borel sum of T(1) (k)k −(4+1)/2 e−(39/4+3i)k , where T(1) (k) = 4iπ
3/2 3 + 2i
13
∞ X 20 + 48i j 0( j + 1/2) − j k ∈ C[[k −1 ]]1 . 169 0(1/2)
(87)
j=0
I0(1) (k) can be analytically continued by rotating the direction of summation, but a Stokes phenomenon occurs due to the adjacent singularity at t = 5/4 − 3i = f 0 − f (1) = f (1),0 . R −k f (z) dz (1) ∧ · · · ∧ dz (4) yields • Similarly, I0 (k) = (3) 0(3) e I0(3) (k) =
4iπ 3/2 −(7−15i/2)k e k 3/2
+∞
Z 0
dt e−kt √ . 16 + 30i − 4t
(88)
This is the Borel sum of T(3) (k)k −5/2 e−(7−15i/2)k , where ∞
T(3) (k) = 2iπ 3/2
5 − 3i X 16 − 30i j 0( j + 1/2) − j k ∈ C[[k −1 ]]1 . 17 289 0(1/2)
(89)
j=0
A Stokes phenomenon occurs due to the adjacent singularity at t = 4 + 15i/2 = f 0 − f (3) = f (3),0 . R −k f (z) dz (1) ∧ · · · ∧ • In the same way as above, the integral I0 (12) (k) = 0(12) e dz (4) is conveniently reduced to Z +∞ dt π I0(12) (k) = 2 e−(1+6i)k e−kt √ . √ k 35/4 − 3i − t(3/2 − i + i 35/4 − 3i − t) 0 (90)
GLOBAL ASYMPTOTICS FOR BOUNDED MULTIPLE INTEGRALS
233
This is the Borel sum of T(12) (k)k −(4+2)/2 e−(1+6i)k , where j ∞ X 5 + 7i j+1 X 14 − 10i l 0( j + l + 1) − j T(12) (k) = −2iπ k 148 37 0(l + 1) j=0
∈ C[[k
−1
l=0
]]1 .
(91)
Stokes phenomena occur due to adjacent singularities at t = 35/4 − 3i = f (1) − f (12) = f (12),(1) , and at t = 10 − 6i = f 0 − f (12) = f (12),0 for the Borel transform Td (12) (t). R −k f (z) dz (1) ∧ · · · ∧ dz (4) , which can be • Continuing, we have I0 (13) (k) = 0(13) e written as √ Z Z t 3 3π −(11−19i/3)k +∞ −kt I0(13) (k) = e dte dt1 2k 0 0 1 p ×√ . √ 15/4 − 28i + 3t1 3(t − t1 ) + (3/2 + 9i/4 + 15/4 − 28i + 3t1 /2)2 (92) This gives I0(13) (k) as the Borel sum of T(13) (k)k −3 e−(11−19i/3)k , with ∞
2π X (−1) j 567 + 369i j+1 T(13) (k) = √ 5650 3 j=0 3 j ×
X j X j−l1 l1 =0 l2 =0
49 + 57i l1 −l2 0( j − l1 + l2 + 1)0( j + l1 − l2 + 1) − j k 113 0(l1 + 1)0(l2 + 1)0( j − l1 − l2 + 1)
∈ C[[k −1 ]]1 .
(93)
From (92), a Stokes phenomenon occurs due to an adjacent singularity at t = −5/4 + 28i/3 = f (1) − f (13) = f (13),(1) (to see this, set t1 = t in the integrand) and also at t = −4 − 7i/6 = f (3) − f (13) = f (13),(3) (t1 = 0). Referring to Section 4.1, a singularity at t = 19i/3 = f 0 − f (13) = f (13),0 is also expected when one continues onto another sheet. R −k f (z) dz (1) ∧ · · · ∧ dz (4) may be written • The last integral I0 (123) (k) = 0(123) e in the form I0(123) (k) =
√ Z Z t 12i π +∞ −kt dte dt1 k 3/2 0 0
1 p ×√ √ 8 + 6i − 4t1 ( 16t1 − 12t + 132 − 54i − (4 + 24i) 8 + 6i − 4t1 ) 1 p × ; √ √ (4 − 6i − 8 + 6i − 4t1 + i 16t1 − 12t + 132 − 54i − (4 + 24i) 8 + 6i − 4t1 )
(94)
234
DELABAERE and HOWLS
I0(123) (k) is then the Borel sum of T(123) (k)k −(4+3)/2 with T(123) (k) ∈ C[[k −1 ]]1 , T(123) (k) √ X j j−l j−l1 −l2 ∞ 5 − 13i l1 +l2 +2l3 +1 π (−1) j X X1 X = (2i)l2 +1 3l3 (1 − i)l1 +1 2 j 8 97 2 j=0 l1 =0 l2 =0 l3 =0 1 + 3i 2 j−2l1 −l2 −2l3 +1 0(l + l + 2l + 1)0(2 j − 2l − l − 2l + 1) 1 2 3 1 2 3 × k− j . 5 0(l2 + 1)0(l3 + 1)0( j − l1 − l2 − l3 + 1) (95) Moreover, one can deduce from the integral representation above that a Stokes phenomenon occurs due to an adjacent singularity at t = 2 + 3i/2 = f (12) − f (123) = f (123),(12) (t1 = t) and also at t = 12 − 65i/6 = f (13) − f (123) = f (123),(13) (t1 = 0). For reasons explained in Section 4.1, singularities can be expected at f (123),(1) , f (123),(3) , and f (123),() for some analytic continuations. 6.2. Resurgence formulae The sheet structure in this example can be derived from knowledge of explicit formulae for the Borel transforms of the multiple integrals. In general, the situation is not so simple. Such a study can nevertheless be done numerically with only the asymptotic expansions T(··· ) as inputs by appealing to hyperasymptotics. Since the values f αβ have distinct phases, formulae (74), (75), (76), (81), (82) can thus be applied. Note that from the “one way” adjacency property and the fact that there is only one critical point on each stratum, the hyperasymptotic expansion (82) (or (76)) terminates at most (when T(123) is concerned) at the m = 3 level. By comparison with Figure 6, part of the first sheet structure can be recovered from the leading order formula (74), which yields information about the indices of intersections corresponding to the nearest adjacent singularities for each critical point in turn. One obtains κ(1,2,3),(1,2) = +1,
κ(1,3),(3) = −1,
κ(3),() = +1,
κ(1,2),(1) = −1,
κ(1),() = +1.
(96)
Now one turns to hyperasymptotics to gain more information. We know that α = (1, 2) has βmin (1) for its nearest adjacent singularity, while its next-nearest singularity is β1 = (). Figure 6 shows that expansion (75) may be applied. With k = 1, (75) yields N0 = 13, N1 = 4, and κ(1,2),() ' −0.924 + 0.04i. From the arguments of Section 5, this is quite enough to conclude that κ(1,2),() = −1. Formula (76) at the first level (m = 1) yields κ(1,2,3),(1) = 0,
κ(1,3),() = 0,
κ(1,3),(1) = −1,
(97)
GLOBAL ASYMPTOTICS FOR BOUNDED MULTIPLE INTEGRALS
235
Table 1. The Stokes multipliers
καβ α= () (1) (3) (1,2) (1,3) (1,2,3)
β=() 0 +1 +1 −1 0 0
(1) 0 0 0 −1 −1 0
(3) 0 0 0 0 −1 −1
(1,2) 0 0 0 0 0 +1
(1,3) 0 0 0 0 0 +1
(1,2,3) 0 0 0 0 0 0
but it may also be used to recover our previous results (cf. [36], [55]). For instance, (1,3) (1,3),(1) (1,3),(3) replacing k = 5 in (81) gives N0 = 63, N1 = 16, N1 = 42, and (1,3),() (1,3) (1,3) N1 = 31 as optimal truncations. On replacing N0α by N0 , N0 − 1, and (1,3) (1,3) N0 − 2 in formula (76), ignoring the error term, and explicitly substituting T (1,3) , T
(1,3) , and (1,3) N0 −1
T
N0 (1,3) , we get a system of three linear equations. Solving these equa(1,3) N0 −2
tions yields κ(1,3),(1) , κ(1,3),(3) , and κ(1,3),() . Using some higher hyperasymptotic levels in the same spirit, one can completely determine the sheet structure. The level-two expansion yields κ(1,2,3),(3) = −1, κ(1,2,3),() = 0, (98) while the level-three expansion gives the last unknown index of intersection, κ(1,2,3),(1,3) = +1 (see Table 1). The results of the adjacency calculations are displayed in Table 2. The sheet structure is now in its complete form.
6.3. Hyperasymptotic computations The Stokes multipliers being known, (82) gives a (hyper)exponential accurate approximation for the integrals, yielding the results in Table 2. Based on the adjacency calculations, we have calculated hyperasymptotic approximations to the integrals T(1) , T(3) , T(12) , T(13) for various values of large parameter k. “Exact” values of the integrals were obtained from numerical integration schemes. The accuracy obtainable for T(123) by this approach was sufficient only for comparison with the first hyper-level, and so we have not included this. Nevertheless, the agreement was as expected and consistent with the adjacency of (3), (12), and (13) to (123).
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DELABAERE and HOWLS
Table 2. Hyperasymptotic levels, truncations, approximations, and achieved accuracies for T(1) and k = 4
Level Lowest
Truncations N (1) = 1
Level 0 (Super) Level 1
N (1) = 13 N (1) = 26, N (1, 0) = 1 k=4
Exact
Approximation −3.42666338266567+ 5.13999507399850i −3.66008976352172+ 5.06575794497829i −3.66007435340317+ 5.06575062056987i −3.66007435340317+ 5.06575062056987i
|1 − approx./exact| 3.9 × 10−2 2.7 × 10−6 0 0
Table 3. Hyperasymptotic levels, truncations, approximations, and achieved accuracies for T(3) and k = 1
Level Lowest
Truncations N (3) = 1
Level 0 (Super) Level 1
N (3) = 9
Exact
N (3) = 17, N (3, 0) = 1 k=1
Approximation 1.96529223417590+ 3.27548705695983i 2.19659657698560+ 3.22186580822527i 2.19566048409404+ 3.22216423713061i 2.19566048409404+ 3.22216423713061i
|1 − approx./exact| 6.1 × 10−2 2.5 × 10−4 0 0
Table 4. Hyperasymptotic levels, truncations, approximations, and achieved accuracies for T(12) and k = 1 Level Lowest
Truncations N (12) = 1
Level 0 (Super) Level 1
N (12) = 9
Level 2 Exact
N (12) = 13, N (12, 1) = 3, N (12, 0) = 1 N (12) = 15, N (12, 1) = 5, N (12, 0) = 1, N (12, 1, 0) = 1 k=1
Approximation 0.29717768344768 −0.21226977389120i 0.34290482693409 −0.20268045850289i 0.34300280953350 −0.20290401468035i 0.34299544259126 −0.20290986661454i 0.34299544259128 −0.20290986661454i
|1 − approx./exact| 1.2 × 10−1 6.2 × 10−4 2.4 × 10−6 < 5.1 × 10−14 0
GLOBAL ASYMPTOTICS FOR BOUNDED MULTIPLE INTEGRALS
237
log10 |Hyperterms|
0 −1 −2
(1)
−3
(1, 0)
−4 0
5
10
15
20
25
Figure 7. The size of the terms in the hyperasymptotic expansion of T(1) , from the critical points (1) (ordinary asymptotics) and (0) (hyperasymptotics along path (1,0)), with k = 4. From the truncations (see Table 2 and (81)) and adjacency of the singularities, only the (0) critical point contributes at the first hyperlevel. As this contributes an exact exponential, only one term is generated, and the hyperasymptotics terminates, generating the exact result.
7. Discussion We first discuss our hypotheses, suggesting how the above results may be generalised. 7.1. Critical points at infinity (Hypothesis H2) Throughout this article, we have assumed that no critical point at infinity occurred (see Hypothesis H2). This assumption was a consequence partly of our own ignorance and partly of a lack of general results about critical points at infinity, although this subject is a matter of an intensive current research (see [59], [31], [33], [67], [13]). However, the following simple example may suggest that Hypothesis H2 can be removed in some instances. We go back to the family of polynomials (7), and we study the behaviour near k = 0 of the integral Z (1) (2) I (k) = e− f (z ,z ;k) dz (1) ∧ dz (2) . (99) 0
For nonzero k, this integral (99) fits into our frame, and we may take for 0 an unbounded chain of integration. By a simple argument of quasi-homogeneity, we first
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log10 |Hyperterms|
0
−1 (3) (3, 0)
−2
−3 0
5
10
15
Figure 8. As for Figure 7, but for T(3) , with k = 1. Again only critical point (0) is adjacent, so the hyperasymptotics terminates at the first reexpansion, generating the exact result.
notice that I (k) =
Z 0
e−k f (z
(1) ,z (2) ;1)
dz (1) ∧ dz (2) .
(100)
(We keep the same notation for the chain 0.) Instead of analysing the behaviour near k = 0, we first compute the asymptotics when k → ∞. We consider the critical value +2i (resp., −2i), choose the nonsingular direction (θ = 0), define the associated Lefschetz thimble 02i (resp., 0−2i ), and consider the asymptotics inside the sector | arg(k)| < π/2. With a convenient orientation of 02i and 0−2i , computations (thanks to the algorithm of [17]) suggest that I2i (k) = (π/k + o(k −N ))e−2ik and I−2i (k) = (π/k + o(k −N ))e2ik for all integers N > 0; hence I2i (k) = πe−2ik /k and I−2i (k) = πe+2ik /k exactly by a Borel-summability argument (Watson theorem; see [47]).∗ This would mean that ? I (k) = h02?i , 0iI2i (k) + h0−2 i , 0iI−2i (k)
(101)
∗ More generally, we suspect (by direct computations) that for every polynomial function g(z (1) , z (2) ), we have the equality
Z 0
e−k f (z
(1) ,z (2) ;1)
g(z (1) , z (2) ) dz (1) ∧ dz (2) =
where the a j , b j are complex coefficients depending on g.
6 6 X a j −2ik X b j 2ik e + e , j k kj j=1
j=1
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0
log10 |Hyperterms|
−1 −2 (12)
−3
(1, 0)
−4 (12, 1)
(12, 1, 0)
−5 0
5
10
15
20
Figure 9. As for Figure 7, but for T(12) , with k = 1 showing the various sequence of critical points that contribute to the hyperasymptotics. Here critical points (1) and (0) are adjacent to (12) and so generate first-level hyperasymptotic corrections. A second level is generated by (0) as this is adjacent to the (first-level) contribution (1).
satisfies the differential equation L (k, ∂k )I = 0
with L (k, ∂k ) =
d2 2 d + +4 2 k dk dk
(102)
with a regular singular point at the origin. This property is satisfied if the differential 2(1) (2) 2 form ω0 = (4k 2 z (2) −6z (2) +4)e− f (z ,z ;k) dz (1) ∧dz (2) is exact (Stokes theorem), and actually, 2
ω0 = d 4z (2) e− f (z
(1) ,z (2) ;k)
dz (1) + (−2 + 2z (1) z (2) )e− f (z
(1) ,z (2) ;k)
dz (2) .
Thus I (k) defined by (100) has a simple pole at k = 0, except when h02?i , 0i = ? , 0i where I (k) can be extended analytically at the origin. This suggests the −h0−2 i existence of convenient valleys at infinity such that the integral (99) still converges for k = 0.∗ ∗ We
formulate a conjecture. The integral Z (1) (2) 8(k) = e−k f (z ,z ;0) g(z (1) , z (2) ) dz (1) ∧ dz (2) 0
might be defined along a (steepest-descent) chain around the connected component z (1) = 0 of the special fiber f (·; 0) = 0, whose image in f would be a path λ starting at infinity along the half-line L 0 , running R ∨ around the origin, and returning to infinity. This integral could be recast as 8(k) = λ e−kt ϕ (t) dt, where
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0
log10 |Hyperterms|
−1 −2 −3 −4 −5 0
5
10
15
20
Figure 10. As for Figure 9, but with the terms ordered according to absolute size
Taking for granted that our oscillating integrals can be defined without Hypothesis H2, can we expect an analogy of Theorem 4.1, at least when the critical points at infinity are isolated (see [59], [67])? Consider the integral with boundaries Z (1) (2) I (k) = e−k f (z ,z ;0) dz (1) ∧ dz (2) , (103) 0
where, for instance, 0 = {(z (1) , z (2) ) ∈ [0, +∞[×[1, +∞[}. For <(k) > 0, integral (103) defines an analytic function in k and can be recast as Z ∞ 2 −2k I (k) = e dt e−kt 1 − . (104) t +2 0 Integral (104) has obvious resurgence properties: t = −2 is the sole singularity in the Borel plane, and, taking into account the translation in the Borel plane induced by the exponential factor e−2k = e−k f (1,0;0) , we see that the location of the singularity arises from the sole bifurcation value corresponding to the critical point at infinity of f (z (1) , z (2) ; 0). Nevertheless, comparing this result with Theorem 4.1, we note a major novelty: the singularity is nonintegrable. Removing Hypothesis H2 thus generates a challenging problem for a geometer—to relate these resurgence properties ∨
O
ϕ (t) is an endlessly continuable major (see [18]), whose singularity ϕ (corresponding microfunction) at the origin could be represented by Z O g(z (1) , z (2) ) dz (1) ∧ dz (2) ϕ (t) ≡ , df γ (t) where γ (t) is a semicycle vanishing at infinity (see [29]).
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Table 5. Hyperasymptotic levels, truncations, approximations, and achieved accuracies for T(13) and k = 1 Level Lowest
Truncations N (13) = 1
Level 0 (Super) Level 1
N (13) = 4
Level 2
Exact
N (13) = 12, N (13, 1) = 2, N (13, 3) = 7 N (13) = 15, N (13, 1) = 5, N (13, 3) = 10, N (13, 1, 0) = 1, N (13, 3, 0) = 1 k=1
Approximation 0.36404397859143+ 0.23691750987697i 0.32105083513787+ 0.19423741497334i 0.32676405233855+ 0.19354284669953i 0.32676199784979+ 0.19354375274768i 0.32676199784980+ 0.19354375274769i
|1 − approx./exact| 1.5 × 10−1 1.5 × 10−2 5.9 × 10−6 < 10−14
0
to the geometry of the phase function f (z (1) , z (2) ; 0) through, for instance, extended Picard-Lefschetz formulae (see [31]). 7.2. Nonisolated critical points (Hypothesis H3) Although the asymptotics of oscillating integrals have been calculated for a class of nonisolated critical points in [76], Hypothesis H3 plays an essential role in all parts of our study, from the proof of Theorem A.2 and the description of the asymptotics (see Theorems 3.1 and 3.2) to the Picard-Lefschetz formulae and the resurgence properties (see Theorem 4.1) and hence the hyperasymptotics. However, we believe it is still possible to generate exact remainder terms by other means. 7.3. “Multiple” critical values (Hypothesis H4) We discuss condition H4 using the following example.∗ For q ∈ C{0}, Z (1) (2) I (q, k, s) = e−k f (z ,z ;q) g(z (1) , z (2) ) dz (1) ∧ dz (2) 0
(105)
with 3
3
f (z (1) , z (2) ; q) = qz (1) z (2) − 4z (1) /3 − z (2) /24 − q 3 /3 and 0 an unbounded chain of integration of real dimension 2. The phase function f has four (isolated) nondegenerate critical points, and a little thought shows that the s
When g = z (1) with s ∈ C, (105) is the solution of the differential equation (∂q2 − k 2 q 4 + 2ksq)I = 0. For s = 0, (105) can be reduced to a Hardy integral (see [34]). Note that, for noninteger s, the ramification of the amplitude function around the complex curve z (1) = 0 enlarges the space of independent contours (see [64] for a similar one-dimensional feature).
∗
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(13)
log10 |Hyperterms|
0 (13, 3)
−2
−4
(13, 1) (13, 1, 0)
−6 0
5
10
15
20
25
30
(13, 3, 0)
Figure 11. As for Figure 9, but for T(13) , with k = 1
space of allowed contours of integration is generated by four independent cycles.∗ Nevertheless, Hypothesis H4 is here violated. While one of the critical points (the origin) corresponds to the critical value −q 3 /3, the three others have the same critical value +q 3 /3 for their image by f . However, we can still define four Lefschetz 1 , 0 2 , 0 3 , and compute the asymptotics of the corthimbles properly, say, 0− and 0+ + + 3 1 2 responding I− and I+ , I+ , I+ . This suggests that Hypothesis H4 is essentially a technicality. It remains to extend the generalised Picard-Lefschetz formulae to understand the Stokes phenomenon. The 1 , 0 2 , and 0 3 simultaneously when a contour 0− may intersect (the duals of) 0+ + + Stokes phenomenon occurs. To compute the indices of intersection, one can introduce a suitable deformation of f so that the “multiple critical value” splits into distinct critical values.† For near but different from zero, the family f = f + z (1) 1 , 0 2 , and 0 3 are deformed into 0 , 0 1 , 0 2 , and 0 3 , and can be used: 0− , 0+ − + + + + + ? l l ? , 0 i is also the indices h0+ , 0− i (l = 1, 2, 3) are now computable; hence h0+ − i ? , 0 j i, computable by continuity. Note in this case that the indices of intersection h0+ + i 6= j, are necessarily zero for topological reasons. (The dependence in is regular in the sense of [18].) 3
3
phase function is governed by the monomials z (1) and z (3) near infinity; therefore each variable has three possible asymptotic valleys. This gives a basis of ((3 − 1) × (3 − 1))-cycles. † Note here that Hypothesis H2 remains true under the deformation. In general, under our hypotheses, this is not guaranteed. For instance, although the set of tame polynomials is a dense (constructible) set in the set of polynomials of a given degree (see [11]), it is not open in dimension 3 (see [13]). ∗ The
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log10 |Hyperterms|
0
−2
−4
−6 0
5
10
15
20
25
30
Figure 12. As for Figure 11, but with the terms ordered according to absolute size. Note that here the final exact result is 0.32676 . . . + i0.19354 . . . , although there are several terms larger than this. Cancellations occur between these larger numbers.
7.4. Confluent cases (Hypothesis H5) We give here a flavour of the difficulty arising from third-type singularities. The main difference between third-type and first- or second-type critical values is primarily topological in nature. When a first- or a second-type critical value is considered, we have seen (see Section 2) that the corresponding Milnor fiber has the homotopy of a bouquet of µ spheres, defining a single group of vanishing homology. The situation is quite different for a third-type critical point, where different groups of vanishing homologies can be defined, depending on which strata are considered, with each of these groups playing a role. Figure 13 exemplifies this situation for n = 3. Here z α is a nondegenerate critical point for f ( f α ∈ 3() ), but the same z α is also a nondegenerate critical point for f |S1 and f |S2 . To such a geometry correspond four vanishing homologies: the vanishing cycle γ (the sphere) is a generator of the vanishing homology group H2 (X αt ), the vanishing cycle γ1 (resp., γ2 ) generates the vanishing homology H1 (X αt ∩ S1 ) (resp., H1 (X αt ∩ S2 )), and γ12 is a basis for the remaining vanishing (reduced) homology group H0 (X αt ∩ S1 ∩ S2 ). These four groups can be understood in terms of our “allowed” cycles of integration. In Figure 13, consider the ball B1 , bounded by the vanishing sphere, as a representation of the Lefschetz thimble. Consider similarly one of the two half-balls 1 (resp., B 2 ), bounded by the vanishing sphere and S (resp., S ), and finally one B1/2 1 2 1/2 of the four quarter-balls B1/4 , bounded by the sphere, S1 , and S2 . Then, obviously, all possible allowed (localized) chains of integration can be described as combinations
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1 , B 2 , and B (with integer coefficients) of B1 , B1/2 1/4 . Hence we have the following 1/2 lemma.
LEMMA 7.1 The local homology is a free Z-module of finite rank, isomorphic to the direct sum of H2 (X αt ), H1 (X αt ∩ S1 ), H1 (X αt ∩ S2 ), and H0 (X αt ∩ S1 ∩ S2 ).
With this kind of decomposition in hand, one can then concentrate on the integral representation. With the notation of Section 3.2 and in convenient local coordinates, the phase function f reads f = s (1) + · · · + s (minα −1) + F(s (minα ) , . . . , s (n) ),
(106)
where F has an (isolated) critical point at the origin, while the (considered) boundary is given by the set of local equations s (1) = 0, . . . , s ( p) = 0, with minα ≤ p. Section 3.2 suggests a way to derive the asymptotics by reducing step by step the multiple integral into a one-dimensional Laplace integral. But a new difficulty arises in the confluent case: the reduction process means considering integrals of differential quotient forms where the quotients are differential forms d f (1,...,q) that vanish at the critical point as soon as q ≥ minα . This makes the analysis of the analytic behaviour of the Borel transform Jbα (t) (analogous to (46)) more complicated. It seems that a result similar to Theorem 3.2 works also in this case. (1) The asymptotics are essentially described by Gevrey-1 resurgent (see [23], [18]) asymptotic expansions. Generically, its leading term is Const/k minα +1/d , where d is the distance to the Newton diagram of F (cf. Figure 14; see [71]). (2) The asymptotics are essentially governed by the geometry of the singularity (monodromy of the vanishing homologies). The first assertion boils down to proving that the integral may be written as a Laplace transform of a solution of a Picard-Fuchs differential equation with (at most) a regular singular point at the origin; similar results have been proved in [52]. To us, to provide a precise statement of the second assertion seems to be much harder. It is worthwhile noting here that the class of integrals considered by Kaminski and Paris [40], [41] (see also [49]) enters into the framework of our third-type critical values (when the phase function is a polynomial function). This class of integrals can thus be used to experiment. Consider, for instance, the function of the example in Section 2.4, 4 2 3 f (z (1) , z (2) ) = z (1) + z (1) z (2) + z (2) . (107) We have seen that the origin in C2 is an isolated critical point with µ = 5 for its Milnor number. We concentrate on Z (1) (2) I (k) = e−k f (z ,z ) dz (1) dz (2) , (108) 0
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S2
γ2
γ
γ1
S1
γ12 Figure 13. Confluent case
with 0 = [0, +∞] × [0, +∞], so that the origin is a critical point of type three. As shown in [40], the asymptotics of I (k) when k → +∞ depend heavily on the Newton diagram of the singularity f . In the terminology of Kaminski and Paris, we are here in the one-internal-point case, and the point (1, 2) lies behind the back face of the Newton diagram. Following the key idea of Kaminski and Paris, we transform our integral representation into a new Mellin-Barnes-type integral representation I (k) =
k −7/12 24iπ
Z
i∞
−i∞
0(t)0
1 − t 1 − 2t 0 k −t/12 dt, 4 3
(109)
where the path of integration avoids the origin to the right. The asymptotics are now simply obtained by taking into account the right-hand poles of the integrand.∗ They occur at the points t = 4 j + 1 and t = (3 j + 1)/2, with a sequence of double poles at t = 12 j + 5, j ∈ N. This yields J = I1 + I2 + I3 for the asymptotic expansion, Mellin-Barnes representation can also be used to compute the behaviour of I (k) near the origin: I (k) is analytic on the twelve-fold covering of C\{0}, that is, the Riemann surface of k 1/12 .
∗ The
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where I1 :=
k −2/3 3
k −5/8 I2 := 8 I3 :=
k −1 24
X j∈N, j−1∈3N /
X j∈N, j−3∈8N /
X j∈N
1 + 8j (−1) j 0(1 + 4 j)0 − k − j/3 , 0( j + 1) 3 (−1) j 1 + 3 j 1 − 3 j − j/8 0( )0 k , 0( j + 1) 2 8
0(5 + 12 j) 0(2 + 3 j)0(4 + 8 j)
× ln(k) − 129(5 + 12 j) + 89(4 + 8 j) + 39(2 + 3 j) k − j
(110)
(9 denotes the digamma function (see [57])). It is easy to check that each of the series expansions in J is Gevrey-1, as expected (see assertion (1) above), but more can be said. While the asymptotics are governed by the local properties of f near the critical point, the reduction to a Mellin-Barnes integral representation proceeds from global information, and, as shown in [40], the asymptotics remain valid in a whole open sectorial neighbourhood at infinity with opening | arg k| < 23π/2. We thus deduce (Watson theorem; see [47]) that J is actually Borel resummable for an argument of summation running over ] − 11π, 11π[. One may guess here that when | arg k| = 11π, then the deformed (steepest-descent) chain of integration 0 encounters the other critical point, so that the minor of J has the corresponding negative critical value −1/19683 as singularity for its analytic continuations, thus inducing a Stokes phenomenon. To illustrate assertion (2) above, it is interesting to compare asymptotics (110) to what we would get in the absence of a boundary. The so-called spectral set of r (see [3]) of Theorem 3.1 can be computed in various ways,∗ but the best references here are certainly the works of A. Varchenko [69], [70]. Taking into account that only 4 2 the quasi-homogeneous part z (1) + z (1) z (2) (with type (1/4, 3/8) and weight 1) of f plays a role, one can directly apply known results from [3, Section 13, Theorem 5], which gives {−3/8, −1/8, 0, 1/8, 3/8} for the spectral set. Moreover, all the s of Theorem 3.1 are less than or equal to n − 1 = 1. We thus get I (k) ∼ J (k) when |k| → +∞ with J (k) =
1 X
X
s=0 r ∈{−3/8,−1/8,0,1/8,3/8}
Tr,s (k)
(ln k)s k r +1
(111)
is known, for instance (cf. [46]), that the eigenvalues of the monodromy are exp(−2iπλ), where the λ are the zeros of the polynomial function e b(λ) related to the Bernstein polynomial b(t) by b(t) = (t +1)e b(t). This polynomial can be computed directly from the Newton diagram (cf. [8]), giving the exponents up to translations. ∗ It
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d Figure 14. Newton diagram of the singularity. The dots correspond to the cosets of the monomials 2
3
1, z (2) , z (1) , z (1) , z (1) after multiplication by z (1) z (2) , which may be chosen as a C-basis for the Milnor algebra O/(∂ f ). The distance to the Newton diagram is d = 8/5.
and Tr,s (k) ∈ C[[k −1 ]]. Comparing now (110) with (111), one can remark that the presence of a boundary has enriched the spectral set; while only the quasihomogeneous part of the phase function plays a role in the unbounded case, in the bounded case the two faces of the Newton diagram must be taken into account. 7.5. Integrals and differential equations When unbounded integration contours are considered, integrals (1) with polynomials f and g belong to a class of functions known as “Bernstein functions”; that is, they satisfy a system of holonomic differential equations in C[k]h∂k i (cf. [63]). It is beyond our scope to study in detail how this property extends to our integrals with boundaries. We discuss our example here (see Section 6) only from the viewpoint of differential equations. This suggests new links between our results and those developed in [55] or [44] and gives a new insight into the hierarchy property. The basic integral (84) obviously satisfies L0 (k, ∂k ) I00 (k) = 0,
with L0 (k, ∂k ) =
d 2 + 11 + , dk k
(112)
while (86) is not only a solution of the second-order differential equation L1 (k, ∂k )L0 (k, ∂k ) I0(1) (k) = 0,
with L1 (k, ∂k ) =
d 5 + 39/4+3i+ (113) dk 2k
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but also, more precisely, a solution of the inhomogeneous equation √ iπ 3/2 5 − 12i −(39/4+3i)k L0 (k, ∂k ) I0(1) (k) = e . k 5/2
(114)
It is important to note that in (114) the homogeneous part is a property of the phase function, while the inhomogeneous part is a consequence of the boundary. This inhomogeneity provides a new insight into the hierarchy property as stated in Theorem 4.1. The characteristic equation associated with the irregular singular point at infinity of (113) shows that 5/4 − 3i is the sole (possible) singularity for the Borel transform of I0(1) . Appealing now to resurgence and using the fact that the dotted alien derivative ˙ ω commutes with the usual differentiation (cf. [23], [12], [16]), (114) yields∗ 1 iπ 3/2 √5 − 12i −(39/4+3i)k ˙ ˙ e = 0. (115) L0 (k, ∂k )(1(5/4−3i) I0(1) ) = 1(5/4−3i) k 5/2 The inhomogeneous part thus disappears, and one obtains ˙ 1 (5/4−3i) I0(1) = κ(1),() I00 ,
(116)
where κ(1),() ∈ C is the Stokes multiplier. More interestingly, (90) satisfies Z +∞ π i 3 1 L0 (k, ∂k ) I0(12) (k) = 2 e−(1+6i)k − + −i e−kt dt √ , k 2 k 35/4 − 3i − t 0 (117) 1 − 25i/2 i L1 (k, ∂k )L0 (k, ∂k ) I0(12) (k) = π e−(1+6i)k + 2 , (118) 3 k 2k L2 (k, ∂k )L1 (k, ∂k )L0 (k, ∂k ) I0(12) (k) = 0, (119) d 3 1 with L2 (k, ∂k ) = + 1 + 6i + + . (120) dk k (1 − (25 + 2i)k)k Starting with (119), a third-order linear differential equation with a singularity of rank one at infinity, we see that (35/4 − 3i) and (10 − 6i) are the two possible adjacent singularities. The one-way adjacency (hierarchy property), which is hidden in (119), clearly appears in the lower-order differential equations (117) and (118). One can analyse each Stokes phenomenon as before. We see, for instance, that the right-hand part of (117) encounters a Stokes phenomenon due to the singularity at t = 35/4 − 3i; replacing the functions by their asymptotic series expansion and applying the alien ˙ derivatives 1 (35/4−3i) to (117), we obtain an equation similar to (114).
∗ With
an abuse of notation, in (115) and (116), I0(1) and I00 stand for their asymptotics.
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Appendices A. The space of homology classes We detail here the proofs for Section 2. We freely appeal to homology and geometric integration theories. We refer readers unfamiliar with these topics to, for instance, [75], [68], [60], and [14] (see also [37] for a physicist’s approach). We recall that S1 , S2 , . . . , Sm are m ≤ n smooth irreducible affine hypersurfaces. A.1. Chains of integration and their homology classes First we describe the space of allowed endless contours of integration running between valleys at infinity where <(k f ) → +∞ in the absence of boundaries. Following Pham [62], [63], we introduce the half-planes: for all c > 0, Sc+ = Sc+ (θ ) = t ∈ C, <(te−iθ ) ≥ c , Sc− = Sc− (θ) = t ∈ C, <(te−iθ ) ≤ c . (A.1) Let 9 = 9(θ ) be the family of closed subsets A ∈ Cn such that, for all c > 0, A ∩ f −1 (Sc− ) is compact. As shown in [63], 9 is actually a “family of supports” in Cn in the sense of homology theory, which allows us to define a complex of chains R C?9 (Cn ) such that, for all nonzero k, the integrals e−k f (z) g(z) dz (1) ∧ · · · ∧ dz (n) along the elements of C?9 (Cn ) are convergent and satisfy the Stokes theorem. We recall that the function g is assumed to be polynomial, so that the growth of g at infinity remains small compared with the exponential’s decay. In order to deal with chains bounded by S = S1 ∪ · · · ∪ Sm , it is natural to introduce the relative chain complex C?9 (Cn , S) := C?9 (Cn )/I? C?9 (S),
(A.2)
where I : S ,→ Cn is the natural mapping. The space of contours of integration H?9 (Cn , S) defined in Section 2.2 is precisely the homology of this complex of chains. The fact that integrals of type (1) are still convergent along the elements of the complex of chains (A.2) and satisfy the Stokes theorem follows from arguments used in the appendix of [63]. From the fact that the Si ’s are the zero level of (irreducible) polynomials Pi , the “semialgebraic” nature of the subchain complex C?[9] introduced by Pham is preserved under the mapping I . Throughout this article, integrals (1) are defined on n-cycles belonging to the homology group Hn9 (Cn , S).
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A.2. Global fibration and homology We recall that S1 , S2 , . . . , Sm are assumed to be in general position (see Hypothesis H1), so that the space Cn has a natural (Whitney) stratification.∗ It is also assumed that f and its restriction on each stratum have no singularity at infinity (see Hypothesis H2). When f is concerned, this could mean in practice that it is tame, for instance; that is, there exists δ > 0 such that the set {z ∈ Cn , k grad f (z) k≤ δ} is compact, or even M -tame (see [51]). Of course, the sole tameness condition on f does not prevent critical points at infinity for the restricted function 2 f | on the strata (just consider f (z (1) , z (2) , z (3) ) = z (1) z (2) − z (1) z (3) + z (2) (z (3) − 1) with the boundary z (3) = 1), and this has to be checked otherwise.† Under Hypothesis H2, f realises a topological trivial stratified fibration outside an open ball B ⊂ Cn of large enough radius (topological triviality at infinity). The following theorem is then a direct consequence of the Thom-Mather first isotopy lemma (see, e.g., [27], [60], [20]). A.1 f| We consider S with its natural stratification; then the mapping Cn \ f −1 (3) −→ C\3 is a topological locally trivial stratified fibration. THEOREM
Thus, for all t0 ∈ C\3, there exists an open neighbourhood U ⊂ C\3 of t0 , a stratified set V , and a homeomorphism ϕ : f −1 (U ) → V × U which maps any stratum of f −1 (U ) onto the product of a stratum of V by U . This result is central to our analysis of the homology through its main consequence, the lifting property. As a first consequence, if c > c0 > 0 are large enough that f −1 (Sc+0 ) does not contain any element of 3, then the natural mapping Cn , f −1 (Sc+ ) ,→ Cn , f −1 (Sc+0 ) (A.3) as well as S, S ∩ f −1 (Sc+ ) ,→ S, S ∩ f −1 (Sc+0 ) can be considered as equivalences of homotopy. Thus the complex of chains (resp., C?9 (S)) can be identified with the projective limit C?9 (Cn ) = lim proj C? Cn , f −1 (Sc+ ) ,
(A.4) C?9 (Cn )
c→+∞
respectively, C?9 (S) = lim proj C? S, S ∩ f −1 (Sc+ ) . c→+∞
S S strata are Cn \S, Si \ j6=i Si ∩ S j , Si ∩ S j \ k6=i, j Si ∩ S j ∩ Sk , and so on (see [20]). We have no general practical way of doing this, the gradient tool not being available, as a rule.
∗ The †
(A.5)
GLOBAL ASYMPTOTICS FOR BOUNDED MULTIPLE INTEGRALS
We therefore get the following projective limit of isomorphisms: Hn9 (Cn , S) = Hn Cn ; S, f −1 (Sc+ ) = Hn Cn , S ∪ f −1 (Sc+ )
251
(A.6)
for all c large enough. Now Cn being contractible (and working with a reduced homology), we finally get the isomorphism ∂ Hn9 Cn , S) → Hn−1 (S ∪ f −1 (t)
(A.7)
for t ∈ Sc+ . In the absence of boundaries and if f has only isolated critical points, it is well known that the (reduced) homology Hn−1 ( f −1 (t)) (hence Hn9 (Cn )) of the generic fiber f −1 (t) is a free Z-module of finite rank, its rank µ f ∗ being the sum of the Milnor numbers at the critical points of f in Cn . In the language of the saddle-point method, this translates to being able to decompose the unbounded chain of integration into a chain of steepest-descent n-folds. Following (A.7) and from an algebraic viewpoint, analysing Hn9 (Cn , S) reduces to computing the total Milnor number of the (generic) fiber P = 0, where P is the Qn polynomial ( f − t) i=1 Pi (each Si being defined by the polynomial Pi ). We, however, follow another way for two reasons: (1) our hypotheses (especially Hypothesis H2) do not translate easily in terms of fibration properties of P; (2) having the saddle-point method in mind, it is useful to understand Hn9 (Cn , S) directly in terms of steepest-descent cycles. We do this in Section A.3. A.3. Localization at the target Following ideas developed in [63], in the complex plane we draw the family (L α ) of closed half-lines L α = f α + eiθ R+ for all f α belonging to 3. Assume also that θ has been chosen generically so that no Stokes phenomenon is currently occurring; that is, all these half-lines are two-by-two disjoint. For all f α ∈ 3, let Tα be a closed neighbourhood of L α , retractable by deformation onto L α . It is assumed that all these Tα are disjoint from one another, as shown in Figure A.1. The reduction process hereafter draws heavily on the work of Pham [63], and so the discussion is brief. We start with (A.6), construct a deformation-retraction of C onto Sc+ ∪ fα Tα , and lift it by f (by virtue of Theorem A.1). This gives [ [ Hn Cn , S ∪ f −1 (Sc+ ) = Hn f −1 (Tα ∪ Sc+ ), ( f |S)−1 (Tα ) ∪ f −1 (Sc+ ) . fα
fα
(A.8) ∗ This
is the total Milnor number, usually defined as µ f = dimC C[z]/(∂ f ).
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Tα fα
Lα
Sc+
Figure A.1. The family of half-lines L α and their closed neighbourhoods Tα for θ = 0
By excision and deformation-retraction (using Theorem A.1), we have [ [ ( f |S)−1 (Tα ) ∪ f −1 (Tα ∩ Sc+ ) Hn Cn , S ∪ f −1 (Sc+ ) = Hn f −1 (Tα ), fα
=
M
Hn f
fα −1
(Tα ), ( f |S)−1 (Tα ) ∪ f −1 (Tα ∩ Sc+ ) .
f α ∈3
(A.9) For each f α ∈ 3, let Dα be an open disc centered at f α with a very small radius r and Dα+ (θ) = Dα ∩ {<((t − f α )e−iθ ) ≥ r/2} (cf. Figure A.2). Then, again by excision and deformation-retraction, M Hn Cn , S ∪ f −1 (Sc+ ) = Hn f −1 (Dα ), ( f |S)−1 (Dα )∪ f −1 (Dα+ (θ)) . (A.10) f α ∈3
This result shows that the homology group of our integration chains can be decomposed into a direct sum of subgroups by localisation near each of the target critical values. In order to categorise all these subgroups, we must perform a detailed analysis at the source in Cn , under suitable hypotheses about the critical points. This is the purpose of Section A.4, which draws heavily on existing work from many authors (e.g., [60], [72]; see also [4], [5]), but the latter half contains the new results necessary to extend the saddle-point method.
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Dα+ fα Dα
Figure A.2. Localisation near f α for θ = 0
A.4. Localisation at the source We have reduced the problem to a local analysis above small open discs centered on the critical values. Up until now, nothing was assumed about the corresponding critical points. It is now time to add Hypotheses H2 and H3. Note. In what follows, it is assumed that one works with a reduced homology. A.4.1. First-type case We localise near an (isolated) first-type singular point z α of depth p = 0; hence f α ∈ 3( ) and z α does not belong to S. It is well known since Milnor (see [50]; see also [58], [20]) that one can choose an open ball Bα centered on z α and with radius small enough such that the level set X αt = {z ∈ Bα , f (z) = t} intersects transversely the boundary of Bα for all t ∈ Dα (for the radius r of Dα such that r > 0). This means that the restricted function f | f −1 (Dα ) ∩ Bα from f −1 (Dα ) ∩ Bα to Dα is a trivial fibration, thus allowing us to analyse the homology group by localisation at the source. We use the notation X α = Bα ∩ f −1 (Dα ), X α+ = Bα ∩ f −1 (Dα+ (θ)). It follows from our hypotheses that Hn f −1 (Dα ), ( f |S)−1 (Dα ) ∪ f −1 (Dα+ (θ)) = Hn f −1 (Dα ), f −1 (Dα+ (θ )) (A.11) by an easy argument of deformation-retraction. Moreover, from [63] or [3, Section 11], one has the following isomorphisms: ∂ Hn f −1 (Dα ), f −1 (Dα+ (θ)) = Hn (X α , X α+ ) → Hn−1 (X αt )
(A.12)
for all t ∈ Dα+ (θ). From Milnor [50] again, we obtain that the fibre X αt has the homotopy type of a bouquet (wedge) of µ spheres, where µ = µα is the Milnor number of the critical point z α ; hence Hn−1 (X αt ) = Zµ , the so-called vanishing homology of the critical
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point. When µ = 1, the relative homology Hn (X α , X α+ ) is generated by the Lefschetz thimble 0αt , as shown in Figure 1. A.4.2. Critical values from the boundary: Second-type case We concentrate now on the case where f α is of the second type, with depth p > 0. Up to a reordering, we can assume that f α belongs to 3(1,..., p) . We introduce as above the sets X α = Bα ∩ f −1 (Dα ), X α+ = Bα ∩ f −1 (Dα+ (θ )), and X αt = Bα ∩ f −1 (t) for t ∈ Dα (θ). We write also Yi = Bα ∩ ( f |Si )−1 (Dα ) = X α ∩ Si . Note that Yi = ∅ for all i ∈ / {1, . . . , p} as a consequence of our hypothesis. −1 The restricted function f | f (Dα ) ∩ Bα now realises a trivial fibration of the stratified set. Analysing the homology thus reduces to a local analysis near the critical point z α , and copying arguments in [63, Section I.3], we obtain Hn f −1 (Dα ), ( f |S)−1 (Dα ) ∪ f −1 (Dα+ (θ)) = Hn (X α , X α+ ∪ Y1 ∪ · · · ∪ Y p ) (A.13) as a preliminary result. Following ideas developed in [60, Section 5.2], we now reduce the homology step by step so as to reach the vanishing homology. We first use the fact that X α and Y1 ∪ · · · ∪Y p are contractible;∗ hence the exact sequence of a triple and a deformationcontraction argument yield the isomorphism ∂ Hn (X α , X α+ ∪ Y1 ∪ · · · ∪ Y p ) → Hn−1 X αt , X αt ∩ (Y1 ∪ · · · ∪ Y p ) , [0] 7→ ∂[0]
(A.14)
for t ∈ Dα+ (θ), where ∂ is the boundary operator that selects the part of the boundary lying on X αt . Second, we observe that both X αt and X αt ∩ (Y2 ∪ · · · ∪ Y p ) are contractible, so using the exact sequence of a triple again gives the isomorphism ∂1 Hn−1 X αt , X αt ∩(Y1 ∪ · · · ∪Y p ) → Hn−2 Y1 ∩X αt , Y1 ∩X αt ∩(Y2 ∪ · · · ∪Y p ) , (A.15) where ∂1 is the boundary operator that takes the part of the boundary lying on Y1 . The same argument can be used p times, yielding the following sequence of isomorphisms: Hn (X α , X α+ ∪ Y1 ∪ · · · ∪ Y p ) ∂
→ Hn−1
(A.16)
X αt , X αt ∩ (Y1 ∪ · · · ∪ Y p )
∂1 → Hn−2 Y1 ∩ X αt , Y1 ∩ X αt ∩ (Y2 ∪ · · · ∪ Y p ) ∂i ∂2 → · · · → Hn−i−1 Yi ∩ · · · ∩ Y1 ∩ X αt , Yi ∩ · · · Y1 ∩ X αt ∩ (Yi+1 ∪ · · · ∪ Y p ) ∂i+1
∂ p−1
→ · · · → Hn− p (Y p−1 ∩ · · · ∩ Y1 ∩ X αt , Y p−1 ∩ · · · Y1 ∩ X αt ∩ Y p ),
∗
Just use the local representation (8).
(A.17)
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255
where ∂i is the boundary operator that takes the part of the boundary lying on Yi . Now assuming that p ≤ n − 1 and taking into account that Y p−1 ∩ · · · ∩ Y1 ∩ X αt is still contractible by virtue of Hypothesis H3, it follows from the exact sequence of a pair that ∂p
Hn− p (Y p−1 ∩ · · · ∩ Y1 ∩ X αt , Y p−1 ∩ · · · Y1 ∩ X αt ∩ Y p ) → Hn− p−1 (Y p ∩ · · · ∩ Y1 ∩ X αt ) (A.18) is again an isomorphism. We finally get the isomorphism Hn (X α , X α+ ∪ Y1 ∪ · · · ∪ Y p )
∂ p ◦···◦∂1 ◦∂
−→
Hn− p−1 (Y p ∩ · · · ∩ Y1 ∩ X αt ).
(A.19)
Of course, this isomorphism depends on the chosen ordering of Y1 , . . . , Y p (cf. [5, Chapter 4, Section 1.14]). The homology group Hn− p−1 (Y p ∩· · ·∩Y1 ∩ X αt ) = Hn− p−1 (S p ∩· · ·∩ S1 ∩ X αt ) is the vanishing homology of the critical point; it is isomorphic to the free Z-module Zµα , where µα is the Milnor number of the critical point z α of f (1,..., p) . When µα = 1, the relative homology Hn (X α , X α+ ∪ Y1 ∪ · · · ∪ Y p ) is generated by the relative Lefschetz thimble 0αt , as described in Figure 2. The case of a corner, where p = n, corresponds to the so-called linear pinching case (see [60]) where ∂ p ◦ · · · ◦ ∂ˇi ◦ · · · ∂1 ◦ ∂[0] is just reduced to a point: no vanishing cycle exists, and the local homology Hn (X α , X α+ ∪ Y1 ∪ · · · ∪ Y p ) is generated by a single relative Lefschetz thimble (see Figure 3). A.4.3. Concluding theorem Putting the pieces together, we have shown how the homology group Hn9 (Cn , S) of the allowed contours of integration in the presence of boundaries can be decomposed into a direct sum (see Section A.3) of free Z-modules of finite rank (see Section A.4), at least under Hypothesis H5 (no critical value of the third type). We have also demonstrated a natural way to define the rank by reduction to the vanishing homology (if defined) through localisation on each stratum. Moreover, when the closed halflines L α = f α + eiθ R+ ( f α ∈ 3) are two-by-two disjoint (θ is “generic in the Stokes sense”), then a basis is given by the set of (relative) steepest-descent n-folds (0α )α∈3 , where each of the 0α projects by f onto L α ∈ C. We have thus obtained the following theorem. THEOREM A.2 The space Hn9 (Cn , S) of relative homology classes is a free Z-module of finite rank. Moreover, if all the half-lines L α are two-by-two disjoint, then every cycle can be decomposed into a chain of (relative) steepest-descent n-folds (0α )α∈3 of Hn9 (Cn , S).
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B. Duality and Stokes phenomenon The aim of this appendix is to provide the geometrical tools to understand the Stokes phenomenon, as discussed in Section 4. We discuss the now more or less classical Picard-Lefschetz formulae, which help in understanding the Stokes phenomena as singularities in the Borel plane. Then we see how a suitable duality provides a direct insight into the Stokes phenomenon, in the spirit of the usual saddle-point method. We here make use of Hypothesis H6, namely, that all singular points are nondegenerate, with the following comment. It is possible to extend the results hereafter by allowing degenerate critical points. The generalised Picard-Lefschetz formulae, as well as the nondegeneracy properties of the Kronecker index (see B.6), follow from a local analysis near the critical points, and this can be studied in the degenerate case by local generic deformations.∗ However, the information we would obtain does not translate easily into the language of resurgence of asymptotic expansions. B.1. Generalised Picard-Lefschetz We assume that θ has been chosen generically so that no Stokes phenomenon is occurring. We denote by (0α ) fα a basis of relative steepest-descent n-cycles (or relative 9(θ ) Lefschetz thimbles) of Hn (Cn , S) considered in Section 2 (with a given orientation). We introduce γα (t) = ∂ 0α (t) ∈ Hn−1 X αt , X αt ∩ (S1 ∪ · · · ∪ Sm ) (B.1) as well as its corresponding vanishing cycle. If f α belongs to 3(i1 ,...,iq ) , then eα (t) = ∂iq ◦ · · · ◦ ∂i1 γα (t) ∈ Hn−q−1 (Siq ∩ · · · ∩ Si1 ∩ X αt ),
(B.2)
which vanishes when t → f α along L α (θ). Let t ? be a fixed regular point in the half-complex plane Sc+ (θ) for c > 0 large enough. We denote by (lα ) a system of paths, where lα starts from t ? and travels to f α along a straight line. This system (lα ) is a so-called distinguished system of paths (see [3]), where the cycle eα (t) vanishes when t → f α along lα . For each lα we associate a closed path `α starting from t ? , following lα , running around f α in the positive sense, and returning to t ? along lα (see Figure B.1). This defines a basis (`α ) fα of the free fundamental group π1 (C\3, t ? ). Hence the variation of the homology when t runs along a loop in C\3 with t ? as base point reduces to a description of what happens for each of the `α . We now apply the generalised Picard-Lefschetz formula described in [60] (see also [72], [5]). Starting with the cycle γα (t), we follow its deformation when t runs ∗ Such
local Morsification does not bring into play any global deformation of f , which may destroy Hypothesis H2 (cf. [13]).
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257
Lα
fα t∗
fβ
`β
Figure B.1. The basis (`α ) f α
along the path lβ . We consider the trace γα? of γα (t) in the Milnor ball Bβ as beF (X t \X t ∩ (S ∪ · · · ∪ S )), the closed homology group dual from longing to Hn−1 m 1 β β Hn−1 (X βt , X βt ∩ (S1 ∪ · · · ∪ Sm )) by the Poincar´e duality (covanishing homology). We can now define the variation operator Var when t goes around f β along `β . Assuming that f β belongs to 3(1,..., p) , [60] now yields (
Var γα? = καβ γβ , καβ = (−1)(n− p)(n− p+1)/2 heβ , α i,
(B.3)
where h , i is the (Kronecker) index of intersection. The cycle α ∈ Hn− p−1 (S1 ∩ · · · ∩ S p ∩ X βt ) is deduced from γα? by γα? = δ1 ◦ · · · ◦ δ p α ,
(B.4)
where δi denotes the Leray coboundary operator (cf. [42]) with respect to Si . Moreover, the index of self-intersection is given by ( 2(−1)(n− p)(n− p−1)/2 if n − p is odd, heβ , eβ i = (B.5) 0 if n − p is even.
Remarks. When m = n (corner critical point) and f β belongs to 3(1,...,n) , then the vanishing cycle eβ does not exist. We can give a meaning to the previous equations with the convention eβ = 0; hence heβ , α i− = 0. One can remark also that if f α belongs to 3(i1 ,...,iq ) , equalities (B.3) and (B.4) show that when {1, . . . , p} is not a subset of the set {i 1 , . . . , i q }, then γα? has a trivial variation around f β (i.e., καβ = 0).
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B.2. Duality from the viewpoint of Laplace integrals The generalised Picard-Lefschetz formulae recalled in Section B.1 follow from the duality between vanishing homology and covanishing homology. Here we realise this duality directly from the viewpoint of Laplace integrals by describing the dual space of the homology group Hn9 (Cn , S) with which we started. 9(θ ) We consider a relative n-cycle 0 ∈ Hn (Cn , S) with support in 9(θ) and 9(θ +π ) an n-cycle 0 0 ∈ Hn (Cn \S) with support in 9(θ + π ). Up to a deformation by isotopy, these two cycles can be assumed to be in a general position. It follows now by definition that the intersection 9(θ) ∩ 9(θ + π ) of the families of supports 9(θ) and 9(θ + π) is the family of compact subsets of Cn . This implies that the intersection in Cn of 0 with 0 0 defines a 0-cycle with compact support and hence an integer because H0 (Cn ) = Z. This allows us to define the index of intersection h0 0 , 0i by the bilinear map (B.6) h , i : Hn9(θ+π ) (Cn \S) ⊗ Hn9(θ) (Cn , S) → Z. This is a direct generalisation of the Kronecker index introduced in [63]. Moreover, arguments based on [63] show the following lemma. LEMMA B.1 The bilinear map (B.6) is nondegenerate. 9(θ)
9(θ+π )
This induces a duality between Hn (Cn , S) and Hn (Cn \S). We recall here that θ has been chosen generically so that the lines L α (θ) ∪ L α (θ + π) are disjoint. We first remark that our bilinear map is diagonal for the decomposition of the homology shown in Appendix A. This allows us to localise the study. Assuming that f α belongs to 3(1,...,q) , we focus on Hn (X α , Sq ∪ · · · ∪ S1 ∪ X α+ ). Following Section A.4.2, this group is isomorphic to Hn−q (Sq ∩· · ·∩ S1 ∩ X α , Sq ∩· · ·∩ S1 ∩ X α+ ) via the isomorphism ∂q ◦ · · · ◦ ∂1 . Returning to the notation of Section A.4.2, we now define Dα− = Dα+ (θ + π) = Dα ∩ < (t − f α )e−i(θ +π) ≥ r/2 (B.7) and, respectively, X α− . From [63, Section I.5] (see also the remark hereafter), we know that the spaces of homology Hn−q (Sq ∩ · · · ∩ S1 ∩ X α , Sq ∩ · · · ∩ S1 ∩ X α+ ) and Hn−q (Sq ∩ · · · ∩ S1 ∩ X α , Sq ∩ · · · ∩ S1 ∩ X α− ) are dual. 9(θ ) As a consequence, each n-cycle 0 ∈ Hn (Cn , S) can be decomposed as X 0= h0α? , 0i0α (B.8) fα
with respect to the basis (0α ) fα of (oriented) relative Lefschetz thimbles, where (0α? ) fα is the dual basis (hence h0α? , 0α i = +1). Figure B.2 describes this duality for the Airy pattern.
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259
Remark. A “concrete dual basis” can be built as follows. We start with a basis of 9(θ ) relative Lefschetz thimbles (0α ) fα of Hn (Cn , S) with the standard orientation. Assuming that f α belongs to 3(1,..., p) , we introduce χα = ∂ p ◦ · · · ◦ ∂1 0α ∈ Hn− p (S p ∩ · · · ∩ S1 ∩ X α , S p ∩ · · · ∩ S1 ∩ X α+ ).
(B.9)
We define χα? ∈ Hn− p (S p ∩ · · · ∩ S1 ∩ X α , S p ∩ · · · ∩ S1 ∩ X α− ) as this cycle deduced from χα by rotating the direction from eiθ to ei(θ+π) on S in the positive sense; then χα (resp., χα? ) can be identified with the ascent (resp., descent) 2 2 2 gradient surfaces of a Morse function F = x ( p+1) + · · · + x (n) − y ( p+1) − 2 · · · − y (n) of index n − p in R2(n− p) = C(n− p) . We compare the orientation (∂/∂ x ( p+1) , . . . , ∂/∂ x (n) , ∂/∂ y ( p+1) , . . . , ∂/∂ y (n) ) with the canonical orientation (∂/∂ x ( p+1) , ∂/∂ y ( p+1) , . . . , ∂/∂ x (n) , ∂/∂ y (n) ) of C(n− p) . This yields hχα? , χα i = (−1)(n− p)(n− p−1)/2 .
(B.10)
Using now the Leray coboundary isomorphisms, we can define 0α? = δ1 ◦ · · · ◦ δ p χα? ,
(B.11)
9(θ+π )
which extends as an element of Hn (Cn \S). Using (B.10), we thus obtain a basis 9(θ+π ) (0α? ) fα of n-cycles of Hn (Cn \S) dual to (0α ) fα . Hill
Valley
Valley
01∗
02
01
0a
a
02∗
Hill
0a∗ Hill
Valley
Figure B.2. The 1-cycles (01 , 0−1 , 0a ) as a basis of H19 (C, a) ? , 0 ? ) for the Airy pattern and its dual basis (01? , 0−1 a
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Acknowledgments. The authors are greatly indebted to F. Pham. They thank Dr. A. B. Olde Daalhuis for his interesting comments and the use of his hyperterminant Maple package. E. Delabaere thanks J. Brianc¸on, M. Merle, and Pham Tien Son for helpful discussions. References [1]
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Delabaere D´epartement de Math´ematiques, Unit´e Mixte de Recherche Centre National de la Recherche Scientifique 6093, Universit´e d’Angers, 2 Boulevard Lavoisier, 49045 Angers CEDEX 01, France;
[email protected] Howls Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2,
THE TOPOLOGICAL IHX RELATION, PURE BRAIDS, AND THE TORELLI GROUP SYLVAIN GERVAIS and NATHAN HABEGGER
Abstract We prove that the filtration on the pure braid group on g strands, induced by the lower central series of the Torelli group of a genus g surface with one boundary component, coincides with its lower central series, shifted by one. In particular, the cubic Jacobi relations in the pure braid group are quadratic relations in the Torelli group. 1. Introduction and statement of results The study of the Torelli subgroup of the mapping class group has progressed in recent years. Beginning with the work of D. Johnson [J1], [J2], [J3] in the early 1980s, which computed its abelianization and gave a finite set of generators, and continuing with the work of S. Morita [Mo1] on the Casson invariant and higher Johnson homomorphisms, the work of S. Garoufalidis and J. Levine [GL] has tied the Torelli group and Johnson homomorphisms to the rich theory of perturbative (or finite-type) 3-manifold invariants (see [O], [Le], [LMO], [G], [Ha]). From another direction, a major leap forward in our understanding is the work of R. Hain [Hn], who gave a presentation of the Malcev Lie algebra of the Torelli group. At the group level, a fundamental open problem is to determine whether the Torelli group is finitely presentable or not. Indeed, to our knowledge, there does not currently exist any presentation of the Torelli group with explicit sets of generators and relators in genus greater than or equal to 3.∗ We still lack an understanding of the topological nature of relations within the Torelli group.† In this paper we exhibit a relation, known as the IHX relation in perturbative theory, that exists in the Torelli group of a surface of genus g ≥ 3. This relation seems to have appeared at least implicitly in calculations of Morita [Mo1]. To see how this relation comes about, recall from [H] that the topological IHX DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2, Received 21 November 2000. Revision received 9 May 2001. 2000 Mathematics Subject Classification. Primary 57M27, 57M99, 57R50, 20F36, 20F38, 20F05. ∗ In
genus 2, the Torelli group is free (see [M]). is the obvious commutativity of diffeomorphisms with disjoint support.
† There
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relation in the theory of homology spheres can be considered as arising from the Jacobi relation in the pure braid group on 4 strands. We denote by 6g,r a surface of genus g with r boundary components, and we denote by Mg,r its mapping class group. Note that M0,g+1 is the framed pure braid group on g strands. There is an embedding of 60,g+1 in 6g,1 (given by considering 6g,0 to be the double of 60,g+1 and removing a disk). Then it turns out that the cubic Jacobi relations in M0,g+1 induce quadratic relations in the Torelli group Tg,1 . Before stating our main result, we recall that Mg,1 comes equipped with a filtration · · · ⊂ Mg,1 [2] ⊂ Mg,1 [1] = Tg,1 ⊂ Mg,1 [0] = Mg,1 , where Mg,1 [n] is the subgroup of Mg,1 acting trivially on F/Fn+1 and where F = π1 (6g,1 ); and for a group G, G n denotes its lower central series.∗ We denote by P(g) the pure braid group on g strands. THEOREM 1.1 Let P(g) → Mg,1 denote the standard inclusion (defined above; see also Section 2). Then for all n ≥ 1,
P(g)n+1 = P(g) ∩ (Tg,1 )n = P(g) ∩ Mg,1 [n]. L Let H denote F/F2 , and let L(H ) = ∞ n=1 L n (H ) denote the free Lie algebra on the Z-module H . Define Dn (H ) to be the kernel of the Lie bracket map [ , ] : H ⊗ L n (H ) → L n+1 (H ). The degree n Johnson homomorphism (see, e.g., [GL]) is a map Jn : Mg,1 [n] → Dn+1 (H ) whose kernel is Mg,1 [n + 1]. One has (Tg,1 )n ⊂ Mg,1 [n], and it is an open problem to study the map jn0 : (Tg,1 )n /(Tg,1 )n+1 → Dn+1 (H ). 1.2 The homomorphism jn0 is injective on the image of P(g)n+1 = P(g) ∩ (Tg,1 )n in (Tg,1 )n /(Tg,1 )n+1 . COROLLARY
Recall that in any group, one has the Jacobi identity c a , [b, c] cb , [a, b] ba , [c, a] = 1, ∗ The
lower central series of a group G is defined by G 1 = G, and inductively, G k+1 = [G, G k ] is the subgroup of G k generated by the commutators of G and G k .
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267
where x y denotes yx y −1 and [x, y] denotes x yx −1 y −1 . 1.3 Every Jacobi relation in the pure braid group is a quadratic relation in the Torelli group. COROLLARY
Remark. Theorem 1.1 (and hence Corollary 1.3) holds also for a surface 6 = 6g,0 without boundary, where Mg [n] is defined to be the image of Mg,1 [n] under the natural map Mg,1 → Mg = Mg,0 . Note first that P(g) ⊂ M0,g+1 → Mg is an inclusion (see, e.g., [PR]). Thus, we have P(g)n+1 = P(g) ∩ (Tg )n = P(g) ∩ Mg [n]. To get an equivalent statement of Corollary 1.2 in the closed case, we just have to replace Dn (H ) by a suitable quotient of Dn0 (H ), the kernel of the Lie bracket map [ , ] : H ⊗ Ln (g) → Ln+1 (g), where Ln (g) = π1 (6g )n /π1 (6g )n+1 (see [Mo3]). 2. The standard embedding of 60,g+1 in 6g,1 There are a number of models available for a surface 6g,1 , but one which we find convenient for drawing pictures is the description as a 2-dimensional handlebody with a single index 0-handle and 2g orientation-preserving index 1-handles grouped together in interlaced pairs. Let B denote a 2-disk (considered as the 0-handle) such that its boundary ∂ B is divided into two segments ∂+ B, ∂− B. We choose a point of ∂− B as the base point. Consider a collection p1 , . . . , p4g of points in ∂+ B ordered using an orientation β of ∂+ B. We attach orientation preserving 1-handles, Hiα (resp., Hi ) along neighborhoods of the 0-spheres { p4i−3 , p4i−1 } (resp., { p4i−2 , p4i }) in ∂+ B. The handlebody Sg β B ∪ ( i=1 (Hiα ∪ Hi )) is a surface 6g,1 of genus g with 1 boundary component. β β The core of the handle Hiα (resp., Hi ) is an arc Aiα (resp., Ai ) whose boundary is the 0-sphere { p4i−3 , p4i−1 } (resp., { p4i−2 , p4i }) which we orient so as to go from p4i−3 to p4i−1 (resp., from p4i to p4i−2 ). β We join the endpoints of the arc Aiα (resp., Ai ) by an arc whose interior lies in β the interior of B. This produces a simple closed curve Ciα (resp., Ci ) which we orient β so as to induce the chosen orientation on Aiα (resp., Ai ). These curves can be chosen β to be mutually disjoint, except that Ciα and Ci meet transversely in a single point (see Figure 1). β We choose paths lying in B to connect the curves Ciα (resp., Ci ) to the base point. This yields well-defined elements of π1 (6g,1 ), which are denoted by αi (resp.,
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C1α
C gα
β
C1
β
Cg
β
Figure 1. The curves Ciα and Ci
b
α1
β1
b
b
αg
βg
Figure 2. The curves αi and βi
βi ; see Figure 2). Then, the set {αi , βi |i = 1, . . . , g} is a basis of the free group F = F(2g) = π1 (6g,1 ). With these conventions, ∂(6g,1 ) is represented by the element [α1 , β1 ] · · · [αg , βg ]. Sg The subsurface of 6g,1 consisting of B ∪( i=1 Hiα ) is a surface 60,g+1 of genus zero with g + 1 boundary components. If we remove a small collar neighborhood of ∂60,g+1 \ ∂− B, the resulting surface is still denoted by 60,g+1 . The closure of its complement in 6g,1 is now a connected surface, which can be made planar. It has g + 1 boundary components and hence is homeomorphic to 60,g+1 . 6g,1 is just the double of 60,g+1 , slit open along ∂− B to obtain a boundary component. We call this embedding of 60,g+1 in 6g,1 the standard embedding. It was studied by T. Oda [Od] and Levine [L] (see also A. Hatcher and W. Thurston [HT] for an embedding of 60,2g in 6g,0 ). The fundamental group π1 (60,g+1 ) is the free group F(g) on the set {αi |i = 1, . . . , g}, π1 (60,g+1 ) = F(g) ⊂ F(2g) = π1 (6g,1 ). We define simple closed curves Ci, j in 60,g+1 by taking the connected sum of Ciα and C αj along an arc lying in B (see Figure 3). Let τi denote the Dehn twist along
PURE BRAIDS AND THE TORELLI GROUP
Hiα
β Hi
269
H jα
β
Hj
Ci, j
Figure 3. The curve Ci, j
Ciα , and let τi, j be the Dehn twist along Ci, j . The elements σi, j = τi, j τi−1 τ −1 j are the generators of P(g), the pure braid group on g strands, considered as a subgroup of M0,g+1 = P(g) × Zg . As the τi are central (generating the Zg summand of M0,g+1 ), when writing commutators in the σi, j , we may dispense with the framing corrections τi−1 , τ −1 j and simply write τi, j . We have that the curves Ci, j (and thus the twists τi, j ) are all conjugate in the following sense: there is a diffeomorphism F˜ of 6g,1 which is the identity on ∂6g,1 and which restricts a diffeomorphism F of 60,g+1 to itself and sends Ci, j to C1,2 .∗ Such an F exists since cutting 60,g+1 open along Ci, j results in 2 surfaces, one of which is 60,3 (and hence the other is 60,g ). F can be extended to 6g,1 simply by taking the double of F, slit open along ∂− B. 3. The main construction Consider a simple closed curve D1,2 lying in 62,1 whose homotopy class is the element α1−1 β1−1 α2 β2 . The curve D1,2 may be taken to intersect C1α and C2α transversely in one point. These two intersection points divide D1,2 into two subarcs, one of which may be taken to lie entirely in B and is called A. The other, A0 , can be taken to miss a third arc A0 connecting A to the base point in B. Note that a small neighborhood of A0 ∪ A ∪ C1α ∪ C2α is a surface 60,3 ⊂ 62,1 , isotopic to the standard embedding. 0 the The connected sum of C1α and C2α along A is the curve C1,2 . Denote by C1,2 α α α 0 0 connected sum of C1 and C2 along A (see Figure 4). By pushing A ∪ C1 ∪ C2α into the complement of A0 ∪ A ∪ C1α ∪ C2α , that is, into 62,1 \ 60,3 ⊂ 6g,1 \ 60,g+1 , we 0 along C 0 commutes with elements of M see that the Dehn twist τ1,2 0,g+1 . 1,2 0 0 −1 Since C1,2 and C1,2 are homologous, τ1,2 (τ1,2 ) lies in Tg,1 . necessarily permutes the components of ∂60,g+1 . It thus corresponds to an element of the braid group and not to the pure braid group. ∗F
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GERVAIS and HABEGGER
C1,2
D1,2
0 Figure 4. The curves D1,2 and C1,2
4. Proof of the inclusion P(g)n+1 ⊂ (Tg,1 )n Recall that we have P(g) ⊂ M0,g+1 ⊂ Mg,1 . Set P = P(g), Pn = P(g)n , T = Tg,1 , Tn = (Tg,1 )n . Since P ∩ Tn is a normal subgroup of P, it suffices to see that a normal set of generators of Pn+1 lies in Tn . For n ≥ 1, Pn+1 is normally generated by [σi, j , f ] = [τi, j , f ], f ∈ Pn . Recall that f τi, j is the Dehn twist along the image of ci, j by the diffeomorphism f (or rather a representative of its class). f Since f (ci, j ) is homologous to ci, j , [τi, j , f ] = τi, j (τi, j )−1 lies in T . This proves that P2 ⊂ T . Suppose inductively that Pn lies in Tn−1 . 0 commutes with M We first suppose that {i, j} = {1, 2}. Since τ1,2 0,g+1 , we have 0 0 −1 −1 [τ1,2 , f ] = [τ1,2 (τ1,2 ) , f ]. But as τ1,2 (τ1,2 ) ∈ T and f ∈ Pn ⊂ Tn−1 , we have [τ1,2 , f ] ∈ Tn . ˜ F) be a diffeomorphism (cf. Section 2) of the Now let {i, j} 6= {1, 2}. Let ( F, pair (6g,1 , 60,g+1 ) sending ci, j to c1,2 . Then [τi, j , f ] F = [τ1,2 , f F ] ∈ Tn since f F lies in Pn , because conjugation by F sends Pn to itself.∗ It follows that [τi, j , f ] ∈ Tn since Tn is normal in Mg,1 . 5. Proof of Theorem 1.1 We have seen that for all n ≥ 1, P(g)n+1 ⊂ (Tg,1 )n . Since (Tg,1 )n ⊂ Mg,1 [n], we have P(g)n+1 ⊂ (Tg,1 )n ∩ P(g) ⊂ Mg,1 [n] ∩ P(g). ∗ Note
well that f 7→ f F is not an inner automorphism of P(g).
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271
To prove equality it is therefore enough to show that M[n] ∩ P ⊂ Pn+1 , where P = P(g) and M[n] = Mg,1 [n]. This was shown by Oda [Od] (see also Levine [L]). For the convenience of the reader, we give the details of the argument, as in [L]. For n = 1, the map P/P2 → M[0]/M[1] = Sp(2g) sends the elements σi, j to j,i the matrix OI Ei, j +E in the basis α1 , . . . , αg , β1 , . . . , βg , (or rather their homology I classes). Since the σi, j are a basis of P/P2 and the E i, j +E j,i are linearly independent, it follows that P/P2 injects into Sp(2g). Thus, we get M[1] ∩ P ⊂ P2 . Let Dn (H ) = ker([ , ] : H ⊗ L n (H ) → L n+1 (H )), where H = H1 (6g,1 ) and L where L(H ) = ∞ n=1 L n (H ) denotes the free Lie algebra on the Z-module H . The degree n Johnson homomorphism (see, e.g., [GL]) is a map Jn : M[n] → Dn+1 (H ) which has kernel M[n +1]. On the other hand, if we set H 0 = H1 (60,g+1 ), the degree n Milnor invariants give a map µn : Pn → Dn (H 0 ) whose kernel is Pn+1 (see, e.g., [HL], [HM]). Thus, the inclusion M[n] ∩ P ⊂ Pn+1 is a result of the following claim by induction. CLAIM
Denote by pn : Dn (H ) → Dn (H 0 ) the morphism induced by the map H → H 0 obtained by forgetting the βi ’s. Then the diagram Pn
⊂
- M[n − 1]
µn ? pn Dn (H 0 )
Jn−1 ? Dn (H )
commutes up to sign. We recall briefly the definitions of the maps µn , Jn . Recall that if F is a free group and H = F/F2 , then Fn /Fn+1 ' L n (H ). Recall that a string link σ induces an automorphism Aσ of F(g)/F(g)n+2 . If we let `i denote the longitude in π = π1 (D 2 × I \ σ ), then Aσ (xi ) = xiλi , where the xi ’s are the free generators of π and λi denotes the image of `i in π/πn+1 ≈ F(g)/F(g)n+1 . Thus, if the λi lie in F(g)n /F(g)n+1 = P 0 0 L n (H 0 ), we define µn (σ ) = i x i ⊗ λi ∈ H ⊗ L n (H ). Actually, µn (σ ) lies in 0 Dn (H ). Similarly, if f is an element of Mg,1 [n], then the induced automorphism f ∗ of β β F(2g)/F(2g)n+2 takes αi to αi ηiα and βi to βi ηi , where ηiα , ηi are elements of P β α F(2g)n+1 /F(2g)n+2 = L n+1 (H ), and we set Jn ( f ) = i αi ⊗ ηi − βi ⊗ ηi . Actually, Jn ( f ) lies in Dn+1 (H ).
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i + (α1 )
i − (αi )
l1
l1
l1
γ2
γi
li
li
li
γi+1
Figure 5. The mapping cylinder of f σ and the curves li , li0 , li00 , γi
For σ ∈ P, let us denote by f σ the corresponding element in Mg,1 . If C( f σ ) is the mapping cylinder of f σ , then C( f σ ) is a handlebody of genus 2g (see Figure 5). + − The boundary of C( f σ ) can be decomposed in two pieces: ∂(C( f σ )) = 6g,1 ∪ 6g,1 . ± + ± We denote by i the inclusions of 6g,1 in C( f σ ), and we identify 6g,1 with 6g,1 , and F(2g) with π1 (C( f σ )) via i ∗+ . Then, by definition, we have ηiα = i ∗+ (αi )−1 i ∗− (αi )
and
β
ηi = i ∗+ (βi )−1 i ∗− (βi ).
Now, if li , li0 , li00 and γi are the curves described in Figure 5, then one has (after connecting these curves to the basepoint) i − (αi ) = li−1 i + (αi )li0
and
i − (βi ) = (li00 )−1 i + (βi )li0 ;
thus, one gets ηiα = [αi−1 , li−1 ]li−1li0
and
β ηi = βi−1 , (li00 )−1 (li00 )−1li0 .
But one has (li00 )−1li0 = i − (αi )γi−1li i − (αi )−1 ; thus, one has β ηi = βi−1 , (li00 )−1 i − (αi ), (γi )−1li (γi )−1li . β
On the other hand, since f σ ∈ M0,g+1 ⊂ Mg,1 , the handles Hi remain fixed via f σ and it follows that we have li = li0 , and li00 = γi+1 . Thus, we get β ηiα = [αi−1 , li−1 ] and ηi = βi−1 , (γi+1 )−1 i − (αi ), (γi )−1li (γi )−1li . β
Now, suppose that σ ∈ Pn . Then f σ ∈ M[n−1] and ηiα , ηi ∈ F(2g)n /F(2g)n+1 . Thus, via the projection H → H 0 , we get ηeiα = xi−1 , (l˜i )−1 ∈ L n (H 0 ) and eβ − ηi = i^ (αi ), (γ˜i )−1l˜i (γ˜i )−1l˜i ∈ L n (H 0 ),
PURE BRAIDS AND THE TORELLI GROUP
273
where z˜ denotes the projection of z. On the other hand, one has Aσ (xi ) = (l˜i )−1 xi l˜i , and we have l˜i ∈ F(g)n since σ ∈ Pn . Furthermore, if δi = li00 (li0 )−1li , we have the following inductive relations: δi = [αi−1 , γi li−1 ]γi and γi+1 = βi li0 (li00 )−1 , δi (li00 )−1 δi , which implies, since li = li0 and li00 = γi+1 , −1 −1 δ˜i = γg γ˜i . i+1 = x i , γ˜i (l˜i ) So, we have γ˜i ∈ F(g)n+1 by induction and we get ηeiα = 0
eβ ηi = l˜i
and
in L n (H 0 ).
This proves that pn Jn−1 ( f σ ) = pn
X
X β αi ⊗ ηi − βi ⊗ ηiα = αi ⊗ l˜i = −µn (σ ).
This concludes the proof of Theorem 1.1. 6. A relation in (Tg,1 )2 We want to give explicitly the relation in (Tg,1 )2 obtained from the Jacobi relation via the map P(g)3 → (Tg,1 )2 . We use Johnson’s notation (see [J1]). Let (a1 , . . . , a2k+1 ) be an odd chain, that is, an ordered collection of oriented simple closed curves in a surface 6 whose intersection graph is the one below (defined by using a vertex for each curve, an edge between two vertices if the corresponding curves intersect transversely in a single point, and no edge if they are disjoint), and such that the algebraic intersection ai · ai+1 is +1. The chain
a1
a2
a3
a2k
a2k+1
defines an element of Tg,1 as follows. A neighborhood of a1 ∪ · · · ∪ a2k+1 in 6 is homeomorphic to 6k,2 . Denote by γ (resp., γ 0 ) the boundary component that is on the left (resp., on the right) of a1 , a3 , . . . , a2k+1 . Then γ and γ 0 are homologous and τγ τγ−1 0 is in the Torelli group of 6. This element is written [a1 ; . . . ; a2k+1 ] and is called the k-chain map induced by (a1 , . . . , a2k+1 ). Now, consider the curves C1α , D1,2 , C2α , C3α introduced in Section 2 and 3, and denote by D2,3 a curve in 6g,1 whose homotopy class is α2−1 β2−1 α3 β3 . Then (C1α , D1,2 , C2α , D2,3 , C3α ) forms a 5-chain.
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GERVAIS and HABEGGER C2α
C1,2
C3α
e1,3
C1,2,3
C1,3
C2,3
C1α D1,2
e1,3
C2,3
D2,3
C1,2
C1,2,3
Figure 6
LEMMA
6.1
One has [τ1,2 , τ2,3 ] = [C1α ; D1,2 + C2α + D2,3 ; C3α ]−1 [C1α ; D1,2 ; C2α ]−1 × [C2α ; D2,3 ; C3α ]−1 [C1α ; D1,2 ; C2α ; D2,3 ; C3α ], where D1,2 + C2α + D2,3 denotes the curve τ D−12,3 τ2−1 (D1,2 ). Remark. It is more convenient here to use the model of 6g,1 given in Figure 6. Proof 0 Let us denote by C1,2,3 and C1,2,3 the two boundary curves of a neighborhood in 6g,1 of C1α ∪ D1,2 ∪ C2α ∪ D2,3 ∪ C3α (see Figure 6). Then one has 0 [C1α ; D1,2 ; C2α ; D2,3 ; C3α ] = τ1,2,3 (τ1,2,3 )−1 , 0 0 ). where τ1,2,3 (resp., τ1,2,3 ) is the Dehn twist along C1,2,3 (resp., C1,2,3 α α α The four curves C1 , C2 , C3 , and C1,2,3 bound a subsurface of 60,g+1 which is homeomorphic to 60,4 . Thus, they yield the following lantern relation (see [J1]):
τ1 τ2 τ3 τ1,2,3 = τ1,3 τ2,3 τ1,2 = τ1,2 τ1,3 (τ1,2 )−1 τ1,2 τ2,3 .
(L1 )
From this we get [τ1,2 , τ2,3 ] = τ1,2 (τ1,3 )−1 (τ1,2 )−1 τ1,3 .
(?)
0 On the other hand, C1α , C2α , C3α , and C1,2,3 also bound a subsurface homeomorphic to 60,4 and so yield the lantern relation 0 0 0 τ1 τ2 τ3 τ1,2,3 = τ1,2 τ2,3 τe 0 . 1,3
(L2 )
PURE BRAIDS AND THE TORELLI GROUP
275
0 are precisely the twists considRemark. The twists τ1 , τ2 , τ3 , τ1,2 , τ1,3 , τ2,3 , and τ1,2 0 are the twists ered in the preceding sections, whereas τ1,2 τ1,3 (τ1,2 )−1 , τe0 , and τ2,3 1,3 0 , and C 0 described in Figure 6. along the curves e1,3 , e1,3 2,3
Multiplying (L1 ) by the inverse of (L2 ), we get 0 0 −1 0 −1 τ1,2,3 (τ1,2,3 )−1 = τ1,3 (τe0 )−1 τ2,3 (τ2,3 ) τ1,2 (τ1,2 ) , 1,3
which yields by (?), [τ1,2 , τ2,3 ] = (τe1,3 )−1 τe0 (τe0 )−1 τ1,3 1,3
1,3
0 0 0 = (τe1,3 )−1 τe0 (τ1,2 )−1 τ1,2 (τ2,3 )−1 τ2,3 τ1,2,3 (τ1,2,3 )−1 . 1,3
Now, noticing that 0 −1 τ1,2 (τ1,2 ) = [C1α ; D1,2 ; C2α ],
0 −1 τ2,3 (τ2,3 ) = [C2α ; D2,3 ; C3α ],
and τe1,3 (τe0 )−1 = [C1α ; D1,2 + C2α + D2,3 ; C3α ], 1,3
we get the required relation. Now, suppose that we are given a collection of seven curves λ, λ1 , λ2 , λ01 , λ02 , λ001 , λ002 whose intersection graph is λ2 λ1 λ2
λ1
λ λ1 λ2
and consider the following elements of Tg,1 : Y1 = [λ2 ; λ1 ; λ; λ01 ; λ02 ], Y2 = [λ02 ; λ01 ; λ; λ001 ; λ002 ], A = [λ; λ1 ; λ2 ],
Y3 = [λ002 ; λ001 ; λ; λ1 ; λ2 ],
A0 = [λ; λ01 ; λ02 ],
B1 = [λ2 ; λ1 + λ + λ01 ; λ02 ],
A00 = [λ; λ001 ; λ002 ],
B2 = [λ02 ; λ01 + λ + λ001 ; λ002 ],
B3 = [λ002 ; λ001 + λ + λ1 ; λ2 ].
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THEOREM 6.2 One has the following relation in (Tg,1 )2 ⊂ Tg,1 :
Y3 B3−1 A00 , Y2 B2−1 A0 (A00 )−1 Y2 B2−1 A0 , Y1 B1−1 A(A0 )−1 × Y1 B1−1 A, Y3 B3−1 A00 (A)−1 = 1.
Proof With the notation of Section 2, consider curves D1,3 and D1,4 in 64,1 whose homotopy classes are, respectively, β1−1 α2 α3 β3 and β1−1 α2 α3 β3 α3−1 β3−1 α4 β4 . Then the intersection graph of C1α , D1,2 , C2α , D1,3 , C3α , D1,4 , C4α is C3α D1,3 C2α
D1,2
C1α D1,4 C4α
and there exists a homeomorphism of 6g,1 which sends, respectively, λ, λ1 , λ2 , λ01 , λ02 , λ001 , and λ002 to C1α , D1,2 , C2α , D1,3 , C3α , D1,4 , and C4α . Thus, it suffices to prove the relation in this case. Consider first the 5-chain (C2α , D1,2 , C1α , D1,3 , C3α ). We claim that B1−1 A(A0 )−1 Y1 = [τ1,2 , τ1,3 ]. Indeed, there is a diffeomorphism h of 6g,1 (e.g., h = (τ1 τ2 τ D1,2 )2 ) which sends, respectively, C2α , D1,2 , C1α , D1,3 , C3α , C1,2 , and C1,3 to C1α , D1,2 , C2α , D2,3 , C3α , C1,2 , and C2,3 . Thus, up to a conjugation, the required relation is the one of Lemma 6.1. In the same way, we have the following two relations: B2−1 A0 (A00 )−1 Y2 = [τ1,3 , τ1,4 ]
and
B3−1 A00 A−1 Y3 = [τ1,4 , τ1,2 ].
From these three relations, we deduce 0 −1 (τ1,2 )τ1,4 (τ1,2 ) = [τ1,4 , τ1,2 ]A = Y3 B3−1 A00 , 0 −1 (τ1,4 )τ1,3 (τ1,4 ) = [τ1,3 , τ1,4 ] A00 = Y2 B2−1 A0 , 0 −1 (τ1,3 )τ1,2 (τ1,3 ) = [τ1,2 , τ1,3 ] A0 = Y1 B1−1 A, 0 )−1 , A0 = τ (τ 0 )−1 , and A00 = τ (τ 0 )−1 . where A = τ1,2 (τ1,2 1,3 1,3 1,4 1,4
PURE BRAIDS AND THE TORELLI GROUP
277
Now, the Jacobi identity in P3 gives (τ1,2 )τ1,4 , [τ1,3 , τ1,4 ] (τ1,4 )τ1,3 , [τ1,2 , τ1,3 ] (τ1,3 )τ1,2 , [τ1,4 , τ1,2 ] = 1. 0 , τ 0 , and τ 0 commute with τ , τ , and τ , we get in (T ) the folSince τ1,2 1,2 1,3 1,4 g,1 2 1,3 1,4 lowing relation: Y3 B3−1 A00 , Y2 B2−1 A0 (A00 )−1 Y2 B2−1 A0 ,Y1 B1−1 A(A0 )−1 × Y1 B1−1 A, Y3 B3−1 A00 (A)−1 = 1.
7. The topological IHX relation L Let G be a group. We set L (G) = n≥0 Ln (G), where Ln (G) = G n /G n+1 . ¯ = [a, b], where a ∈ G n , It is a Lie algebra with bracket defined by [a, ¯ b] b ∈ G m represent a¯ ∈ G n /G n+1 , b¯ ∈ G m /G m+1 . The Jacobi group relation [a c , [b, c]][cb , [a, b]][ba , [c, a]] = 1 implies the Jacobi relation in L (G). We will write LnQ (G) for L (G) ⊗ Q. If G ab = G/G 2 , then one has a surjective mapping L(G ab ) → L (G) given by the isomorphism L 1 (G ab ) = G ab = G/G 2 = L1 (G), where L(G ab ) is the free Lie algebra on G ab . We write L(G ab Q ) for the free Q Lie ab ⊗ Q. Then L(G ab ) → L Q (G) is also surjective. The kernel algebra on G ab = G Q Q R(G) of this map is the ideal of relations. If T = Tg,1 , g ≥ 4, we deduce a relation in R2 (T ), which we call the IHX relation, as it arises from the Jacobi relation in P = P(g). First, recall that the degree 1 Johnson homomorphism is a map J1 : T → ∧3 (H ), H = H1 (6g,1 ). Rationally, this gives an isomorphism L1Q (T ) = TQab → ∧3 (H Q ). Thus, R2 (T ) lies in L 2 (∧3 (H Q )). There is an obvious relation in R2 (T ) (see [Hn]) coming from the commutativity of diffeomorphisms having disjoint support, namely, [a1 ∧ a2 ∧ b2 , b3 ∧ a3 ∧ a4 ], where (ai , bi )1≤i≤g is a symplectic basis of H . Substituting a1 + b3 for b3 also gives a relation. The difference gives the relation [a1 ∧ a2 ∧ b2 , a1 ∧ a3 ∧ a4 ]. The Jacobi relation in P3 /P4 determines a relation in T2 /T3 and hence in Q L2 (T ). We can compute examples as follows. For a = σ1,2 , we have [a, [b, c]] = 0 )−1 , [b, c]]. We claim that J (τ (τ 0 )−1 ) = α ∧ α ∧ β − α ∧ α ∧ β . [τ1,2 (τ1,2 1 1,2 1,2 1 2 2 1 2 1 To see this, note that the curves C1α , D1,2 , C2α of Section 3 define a 3-chain and that J1 ([x, y, z]) = −x ∧ y ∧ z for a 3-chain [x, y, z] (see [J2]). Thus, 0 −1 J1 τ1,2 (τ1,2 ) = −α1 ∧ (−β1 + α2 + β2 ) ∧ α2 = α1 ∧ α2 ∧ β2 − α1 ∧ α2 ∧ β1 . If we take b = σ1,3 , c = σ1,4 , then J1 ([b, c]) = α1 ∧ α3 ∧ α4 . Thus, [σ1,2 , [σ1,3 , σ1,4 ]] defines [α1 ∧ α2 ∧ β2 − α1 ∧ α2 ∧ β1 , α1 ∧ α3 ∧ α4 ] in L2Q (T ). Now note that there is a diffeomorphism f taking M4,1 to itself which on homology takes α1 , β1 to themselves and cyclically permutes αi , βi for i = 2, 3, 4. Thus, f cyclically
278
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permutes the curves C1,2 , C1,3 , C1,4 . This proves that the Jacobi relation yields the element [α1 ∧α2 ∧β2 −α1 ∧α2 ∧β1 , α1 ∧α3 ∧α4 ]+[α1 ∧α3 ∧β3 −α1 ∧α3 ∧β1 , α1 ∧α4 ∧α2 ] + [α1 ∧ α4 ∧ β4 − α1 ∧ α4 ∧ β1 , α1 ∧ α2 ∧ α3 ] in R2 (T ). Since terms like [α1 ∧ α2 ∧ β2 , α1 ∧ α3 ∧ α4 ] are in R2 (T ), this reduces to [α1 ∧α2 ∧β1 , α1 ∧α3 ∧α4 ]+[α1 ∧α3 ∧β1 , α1 ∧α4 ∧α2 ]+[α1 ∧α4 ∧β1 , α1 ∧α2 ∧α3 ]. In genus g ≥ 5, we may substitute α1 + α5 for α1 , and the difference yields [α5 ∧ α2 ∧ β1 , α1 ∧ α3 ∧ α4 ] + [α5 ∧ α3 ∧ β1 , α1 ∧ α4 ∧ α2 ] + [α5 ∧ α4 ∧ β1 , α1 ∧ α2 ∧ α3 ] plus terms like [α1 ∧ α2 ∧ β1 , α5 ∧ α3 ∧ α4 ] and [α5 ∧ α2 ∧ β1 , α5 ∧ α3 ∧ α4 ] which lie in R2 (T ). Permuting the indices, we show the following theorem. THEOREM 7.1 If g ≥ 5, the relation
[α1 ∧α4 ∧β5 , α2 ∧α3 ∧α5 ]+[α2 ∧α4 ∧β5 , α3 ∧α1 ∧α5 ]+[α3 ∧α4 ∧β5 , α1 ∧α2 ∧α5 ] = 0 holds in L2Q (Tg,1 ). Remark. This relation appears at least implicitly in work of Morita [Mo1], [Mo2], when he computed the image of the second Johnson homorphism. Remark. In [HS] (following Hain [Hn]) a presentation of the Malcev Lie algebra of Tg,1 is given. Theorem 7.1 is related to another quadratic relation R2 (see [HS, §3]) determined by the boundary of 6g,1 . Acknowledgment. The authors would like to thank Christoph Sorger for an introduction to the program Lie which served as a tool in understanding the IHX relation. References [GL]
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S. GAROUFALIDIS and J. LEVINE, Tree-level invariants of three-manifolds, Massey
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Gervais Unit´e Mixte de Recherche 6629 du Centre National de la Recherche Scientifique, Universit´e de Nantes, D´epartement de Math´ematiques, 2 rue de la Houssini`ere, 44072 Nantes CEDEX 03, France;
[email protected]; http://www.math.sciences.univ-nantes.fr/˜gervais Habegger Unit´e Mixte de Recherche 6629 du Centre National de la Recherche Scientifique, Universit´e de Nantes, D´epartement de Math´ematiques, 2 rue de la Houssini`ere, 44072 Nantes CEDEX 03, France;
[email protected]; http://www.math.sciences.univ-nantes.fr/˜habegger
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2,
HEATING OF THE AHLFORS-BEURLING OPERATOR: WEAKLY QUASIREGULAR MAPS ON THE PLANE ARE QUASIREGULAR STEFANIE PETERMICHL and ALEXANDER VOLBERG
Abstract We establish borderline regularity for solutions of the Beltrami equation f z −µf z¯ = 0 on the plane, where µ is a bounded measurable function, kµk∞ = k < 1. What is the 1,q minimal requirement of the type f ∈ Wloc which guarantees that any solution of the Beltrami equation with any kµk∞ = k < 1 is a continuous function? A deep result of 1,1+k+ε K. Astala says that f ∈ Wloc suffices if ε > 0. On the other hand, O. Lehto and T. Iwaniec showed that q < 1 + k is not sufficient. In [2], the following question was asked: What happens for the borderline case q = 1 + k? We show that the solution is still always continuous and thus is a quasiregular map. Our method of proof is based on a sharp weighted estimate of the Ahlfors-Beurling operator. This estimate is based on a sharp weighted estimate of a certain dyadic singular integral operator and on using the heat extension of the Bellman function for the problem. The sharp weighted estimate of the dyadic operator is obtained by combining J. Garcia-Cuerva and J. Rubio de Francia’s extrapolation technique and two-weight estimates for the martingale transform from [26]. Contents Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 0. Introduction: Main objects and results . . . . . . . . . . . . . . . . . . 282 1. The sharp weighted estimate for the Ahlfors-Beurling operator in L 2 (w d A) 287 2. The sharp weighted estimate for the Ahlfors-Beurling operator in L p (w d A) for p > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 3. The comparison of classical and heat A p -characteristics . . . . . . . . . 299 4. Injectivity at the critical exponent and regularity of solutions of the Beltrami equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2, Received 8 December 2000. Revision received 2 May 2001. 2000 Mathematics Subject Classification. Primary 42B20, 42C15, 42A50, 47B35, 47B38. Volberg’s work partially supported by National Science Foundation grant number DMS-9900375 and by United States–Israel Binational Science Foundation grant number 00030. 281
282
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
302
Notation We have the following notation: x = (x1 , x2 ), |x| = (x12 + x22 )1/2 ; B(x0 , r ) is the open ball, where B(x0 , r ) := {x ∈ R2 : |x − x0 | < r }; h f i B is the average of the function f over the ball B; x = (x1 , x2 ), k(x, t) := (1/πt) exp(−(|x|2 /t)) is the heat kernel on the plane; RR RR −1 Q heat w(x − y)k(y, t) dy1 y2 · w (x − y)k(y, t) dy1 y2 is the w,2 = supx∈R2 ,t>0 heat A2 characteristic of weight w (see Section 0); RR w(x − y)k(y, t) dy1 y2 Q heat = supx∈R2 ,t>0 w, p RR −(1/( p−1)) ·( w (x − y)k(y, t) dy1 y2 ) p−1 is the heat A p -characteristic of weight w (see Section 0); −1 Q class w,2 = sup B(x,r ) hwi B(x,r ) hw i B(x,r ) is the standard A2 -characteristic of weight w (see Section 0); −(1/( p−1)) i p−1 is the standard A -characteristic Q class p B(x,r ) ) w, p = sup B(x,r ) hwi B(x,r ) (hw of weight w (see Section 0); dyadic
Q w,2 = sup I,I ∈D hwi I hw−1 i I is the dyadic A2 -characteristic of weight w (see Section 1); dyadic
Q w, p = sup I,I ∈D hwi I (hw−(1/( p−1)) i I ) p−1 is the dyadic A p -characteristic of weight w (see Section 1); D is a collection of dyadic intervals (see Section 1).
0. Introduction: Main objects and results Our goal is to present a sharp estimate of the weighted Ahlfors-Beurling operator. This estimate is sufficient to prove that any weakly quasiregular map is quasiregular. Thus, we are interested in the following Ahlfors-Beurling operator (d A denotes area Lebesgue measure on C): Z Z 1 ϕ(ζ ) d A(ζ ) T ϕ(z) := π (ζ − z)2 understood as a Calder´on-Zygmund operator. Our starting point is the following theorem of Astala, Iwaniec, and E. Saksman [2]. THEOREM 0.1 Let µ ∈ L ∞ , kµk∞ = k < 1. Then the operators I − µT and I − T µ are invertible
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in L p (C, d A) for p ∈ (1 + k, 1 + (1/k)). Remark. In [2], the authors suggest the following problem: Prove that these operators have dense range for p = 1+(1/k) and are injective for p = 1+k. Actually, this problem has a long history and has reappeared in many papers on regularity of quasiconformal homeomorphisms and quasiregular maps. The L p -theory of quasiregular mappings was essentially formulated by B. Bojarski [6] and [7]. Later this subject came under intensive investigation. In particular, the best integrability of K -quasiconformal mappings and the (in a sense dual; see [25]) problem of minimal regularity of the quasiregular mappings are discussed in [16], [13] – [15], [17], [20], [21], [22], [24], and [25]. The best integrability result was finally established in [1]. Notice that the p − 1 problem for the operator T is still open. We recall that this problem consists of proving that kT k L p →L p = p − 1, p > 2. A discussion of the fantastically beautiful connections of this problem with the calculus of variations and C. Morrey’s problem can be found in [3]. Until recently the best result for all p appeared in [5], where probabilistic methods were used to obtain the estimate kT k L p →L p ≤ 4( p − 1), p > 2. Discussion of related results can be found in [4] and [30]. Recently, in [27] this estimate was slightly improved to kT k L p →L p ≤ 2( p − 1), p > 2. The main result of this paper may serve as yet another indication that this norm is p − 1. We establish the dense range/injectivity property in the present paper. Our principal tool is a sharp weighted estimate of the Ahlfors-Beurling operator. The main consequence (see [2]) is the above-mentioned geometric fact: Every weakly quasiregular map is quasiregular. We first quote several results and notions that we use. The symbol f = f µ represents the homeomorphic solution of the Beltrami equation f z¯ − µf z = 0,
f (z) = z + o(1),
z → ∞.
Here we consider compactly supported µ, µ ∈ L ∞ (C), kµk∞ = k < 1. The function µ is called the Beltrami coefficient of the Beltrami equation written above. The constant K = (1 + k)/(1 − k) has the geometric meaning of the bound on the distortion of infinitely small discs by the map f . (The resulting infinitely small ellipses have the ratio of their axes at most K .) The map f is called K -quasiconformal. The image of a disc by a K -quasiconformal homeomorphism is called a K -quasidisc. We need the following results from Astala, Iwaniec, and Saksman [2]. 0.2 Let f be a K -quasiconformal homeomorphism built by a certain µ as it is described above. Consider ω = | f z ◦ f −1 | p−2 for p ∈ (1 + k, 1 + (1/k)). THEOREM
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Then k(I − µT )−1 k L p →L p , k(I − T µ)−1 k L p →L p are bounded by C(k)λω, p , where λω, p := kT k L p (ω d A)→L p (ω d A) . THEOREM 0.3 Let f be a K -quasiconformal homeomorphism. Let k = (K + 1)/(K − 1), and let p ∈ (0, 1 + (1/k)). Then for any disc or K -quasidisc B, one has Z | f (B)| p/2 p 1 C(k) | f z | + | f z¯ | d A ≤ . |B| B 1 + (1/k) − p |B|
R Remark. Obviously, we have the same type of estimate for (1/|B|) B | f z | p d A and R p/2 for (1/|B|) B J f d A, where J f := | f z |2 − | f z¯ |2 is the Jacobian of the mapping f . Next, a very elegant result from [2] reduces the critical exponent cases p = 1 + (1/k), p = 1 + k to a weighted estimate of the Ahlfors-Beurling operator. This result is quoted for the convenience of the reader. THEOREM 0.4 Let k ∈ [0, 1), kµk∞ = k, f = f µ , ω = | f z ◦ f −1 | p−2 , p ∈ [2, 1 + (1/k)). If
kT k L p (ω d A)→L p (ω d A) ≤
C , 1 + a(1/k) − p
(0.1)
then I − µT, I − T µ have dense ranges in L 1+1/k (C) and are injective in L 1+k (C). Our main result implies the weighted estimate (0.1). Theorem 0.4 served as motivation for our main result. In its turn, injectivity at the critical exponent 1 + k proves that weakly quasiregular maps are quasiregular (see [2]). Proof The operator T is invertible in all L q , q ∈ (1, ∞). We notice that I − T µ = T (I − µT )T −1 . Thus, it is enough to prove that I − µT has closed range in L 1+(1/k) (C). Let p0 := 1 + (1/k). Take any ϕ ∈ C0∞ (C). By Theorem 0.1, ϕε := I − (1 − ε)µT
−1
ϕ ∈ L p0 (C).
By Theorem 0.2, kϕε k L p0 ≤ C(k)kT k L p0 (ωε d A)→L p0 (ωε d A) , where ωε = | f z ◦ f −1 | p−2 , and f denotes (as always) a homeomorphic solution of the Beltrami equation, but for Beltrami coefficient (1 − ε)µ instead of µ.
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Now assuming estimate (0.1), we conclude that kϕε k L p0 ≤
C C ≤ . 1 + (1/(1 − ε)k) − p0 ε
Thus, kεϕε k L p0 ≤ C1 < ∞,
ε → 0.
(0.2)
On the other hand, the definition of ϕε implies that (I − µT )ϕε = ϕ − εµϕε
(0.3)
and kϕε k L 2 ≤ C(k) < ∞,
∀ε ∈ [0, 1].
(0.4)
Now (0.2) and (0.4) imply that εϕε converge weakly to zero in L p0 . Keeping this in mind, we see that (0.3) implies that the range of I − µT is weakly dense in L p0 . So the range of I − µT is automatically dense in L p0 in the norm topology. We are now ready to formulate one of our main results. 0.5 Let k ∈ [0, 1), kµk∞ = k, f = f µ , ω = | f z ◦ f −1 | p−2 , p ∈ [2, 1 + (1/k)). Then THEOREM
kT k L p (ω d A)→L p (ω d A) ≤
C . 1 + (1/k) − p
This result is obtained as a corollary of the following theorems, which may be of interest in their own right. Let ω be any weight on R2 , and denote its heat extension into R3+ by ω(x, t) = ω(x1 , x2 , t): Z Z kx − yk2 1 ω(x, t) = ω(y) exp − dy1 dy2 . πt t R2 We define Q heat ω, p :=
sup
p−1 ω(x, t) ω−(1/( p−1)) (x, t) .
(x,t)∈R3+
The weights with finite Q heat ω, p are called A p -weights. There is an extensive theory of A p -weights (see, e.g., [29], [12]). The usual definition differs from the one above, but it describes the same class of weights. Actually, we say more about the relationship between the classical definition and ours. But first we state two more theorems, whose combined use gives Theorem 0.5.
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THEOREM 0.6 For any A p -weight w and any p ≥ 2, we have
kT k L p (w d A)→L p (w d A) ≤ C( p)Q heat ω, p . THEOREM 0.7 Let k ∈ [0, 1), kµk∞ = k, f = f µ , ω = | f z ◦ f −1 | p−2 , p ∈ [2, 1 + (1/k)). Then the weight ω belongs to A p and
Q heat ω, p ≤
C . 1 + (1/k) − p
Clearly, these last two results together imply Theorem 0.5. And Theorem 0.5 and Theorem 0.4 give positive answers to the Beltrami equation critical exponent questions discussed above and raised in [2]. At the end of our introduction, we discuss the connection between Q heat w, p and class Q w, p . Here Q class denotes the following supremum over all discs in the plane: w, p Z Z p−1 1 1 ω d A · ω−(1/( p−1)) d A . Q class := sup w, p |B(x, R)| B(x,R) B(x,R) |B(x, R)| B(x,R) Obviously, there exists a positive absolute constant a such that for any function w, heat a Q class w, p ≤ Q w, p .
Remark. The opposite inequality is easy to prove, too. We are grateful to F. Nazarov for indicating this to us. Originally, we did not have the following two theorems. This was not an obstacle to Theorem 0.5 because we still could estimate (see Theorem 0.7) Q heat ω, p for special weights ω. 0.8 There exists a finite absolute constant b such that THEOREM
class Q heat w, p ≤ bQ w, p .
In conjunction with our main Theorem 0.5, this gives us the following weighted estimate of the Ahlfors-Beurling operator, which answers another question of [2] positively. THEOREM 0.9 For any A p -weight w and any p ≥ 2, we have
kT k L p (w d A)→L p (w d A) ≤ C( p)Q class ω, p .
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In [2], it is proved that if k ∈ [0, 1), kµk∞ = k, f = f µ , ω = | f z ◦ f −1 | p−2 , and p ∈ [2, 1 + (1/k)), then the weight ω belongs to A p and Q class ω, p ≤
C . 1 + (1/k) − p
(0.5)
So instead of using Theorem 0.7, one can use Theorem 0.9 in conjunction with (0.5). As a result, for the special weight function ω = | f z ◦ f −1 | p−2 , where p ∈ [2, 1 + (1/k)), f := f µ , kµk∞ = k, and k ∈ [0, 1), we always get Q heat ω, p ≤
C . 1 + (1/k) − p
(0.6)
Theorem 0.6 then proves our main Theorem 0.5. Plan of the paper In Section 1, we prove Theorem 0.6 for p = 2. In Section 2, we prove Theorem 0.6 for p > 2. In Section 3, we give an easy proof of Theorem 0.8. In Section 4, we repeat (for the convenience of the reader) the reasoning of [2] that deduces the quasiregularity of weakly quasiregular maps from the injectivity of I −µT at the critical exponent. 1. The sharp weighted estimate for the Ahlfors-Beurling operator in L 2 (w d A) In this section, w is an arbitrary positive function on the plane. We want the estimate kT k L 2 (w d A)→L 2 (w d A) ≤ C Q heat w,2 .
(1.1)
As we see in Section 3, estimate (1.1) readily implies the estimate kT k L 2 (w d A)→L 2 (w d A) ≤ C Q class w,2 .
(1.2)
However, at this moment we do not know how to get the latter estimate without using the first. So now our immediate goal is to obtain (1.1). The operator T is given in the Fourier domain (ξ1 , ξ2 ) by the multiplier ξ12 ξ22 ζ¯ ξ1 ξ2 ζ¯ 2 (ξ1 − iξ2 )2 = − − 2i 2 . = = 2 2 2 2 2 2 2 ζ |ζ | ξ1 + ξ2 ξ1 + ξ2 ξ1 + ξ2 ξ1 + ξ22 Thus, T can be written as T = R12 − R22 −2i R1 R2 , where R1 , R2 are Riesz transforms on the plane (see [29] for their definition and properties). Another way of writing T is T = m 1 − im 2 , where m 1 , m 2 are Fourier multiplier operators. Notice that the multipliers themselves (as functions, not as multiplier operators) are connected by m 2 = m 1 ◦ ρ,
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where ρ is a π/4 rotation of the plane. So the multiplier operators are related by m 2 = Uρ m 1 Uρ−1 , where Uρ is an operator of ρ-rotation in the (x1 , x2 )-plane. But for any operator K , we have kUρ K Uρ−1 k L 2 (w d A)→L 2 (w d A) = kK k L 2 (w◦ρ −1 d A)→L 2 (w◦ρ −1 d A) . heat Combining this with the fact that Q heat w,2 = Q w◦ρ −1 ,2 for any rotation, we conclude
that we need the desired estimate (1.1) only for m 1 = R12 − R22 . Actually, we show that kRi2 k L 2 (w d A)→L 2 (w d A) ≤ C Q heat i = 1, 2. (1.3) w,2 , To prove (1.3) we fix, say, R12 and two test functions ϕ, ψ ∈ C0∞ . We use heat extensions. For f on the plane, its heat extension is given by the formula ZZ |x − y|2 1 f (y, t) := f (x) exp − d x1 d x2 , (y, t) ∈ R3+ . 2 πt t R We usually use the same letter to denote a function and its heat extension. LEMMA 1.1 RRR ∂ϕ ∂ψ Let ϕ, ψ ∈ C0∞ . Then the integral ∂ x1 · ∂ x1 d x 1 d x 2 dt converges absolutely and ZZ ZZZ ∂ϕ ∂ψ R12 ϕ · ψ d x1 d x2 = −2 · d x1 d x2 dt. (1.4) ∂ x1 ∂ x1
Proof The proof of this lemma is actually trivial. It is based on the well-known fact that a function is an integral of its derivative, and it also involves Parseval’s formula. Consider ϕ, ψ ∈ C0∞ and now ZZ ψ R12 ϕ d x1 d x2 ZZ
ξ12
ˆ ϕ(ξ ˆ 1 , ξ2 )ψ(−ξ 1 , −ξ2 ) dξ1 dξ2 ξ12 + ξ22 ZZ Z ∞ 2 2 ˆ 1 , ξ2 ) dξ1 dξ2 dt =2 e−2t (ξ1 +ξ2 ) ξ12 ϕ(ξ ˆ 1 , ξ2 )ψ(ξ 0 Z ∞ ZZ 2 2 −t (ξ12 +ξ22 ) ˆ = −2 iξ1 ϕ(ξ ˆ 1 , ξ2 )e−t (ξ1 +ξ2 ) iξ1 ψ(−ξ dξ1 dξ2 dt 1 , −ξ2 )e Z0 ∞ Z Z ∂ϕ ∂ψ = −2 (x1 , x2 , t) (x1 , x2 , t) d x1 d x2 dt ∂ x1 ∂ x1 Z0Z Z ∂ϕ ∂ψ = −2 (x1 , x2 , t) (x1 , x2 , t) d x1 d x2 dt. 3 ∂ x ∂ x1 1 R+ =
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We used Parseval’s formula twice above, and we also used the absolute convergence of the integrals ZZZ 2 2 ˆ 1 , ξ2 ) dξ1 dξ2 dt, e−2t (ξ1 +ξ2 ) ξ12 ϕ(ξ ˆ 1 , ξ2 )ψ(ξ R3+
ZZZ R3+
∂ψ ∂ϕ (x1 , x2 , t) (x1 , x2 , t) d x1 d x2 dt. ∂ x1 ∂ x1
For the first integral this is obvious. The absolute convergence of the second integral can be easily proved. We leave this as an exercise for the reader. Our next goal is to estimate the right-hand side of (1.4) from above. 1.2 For any ϕ, ψ ∈ C0∞ , and any positive function w on the plane, we have THEOREM
ZZZ
∂ϕ ∂ψ d x1 d x2 dt 3 ∂ x ∂ x 1 1 R+ ≤ AQ heat w,2
ZZ
|ϕ|2 w d x1 d x2 +
ZZ
|ψ|2
1 d x1 d x2 , w
where A is an absolute constant. In the proof we use the following key result. (In what follows, d 2 f denotes the Hessian form that is the second differential form of f .) THEOREM 1.3 For any Q > 1, define the domain D Q := {0 < (X, Y, x, y, r, s) : x 2 < X s, y 2 < Y r, 1 < r s < Q}. Let K be any compact subset of D Q . Then there exists a function B = B Q,K (X, Y, x, y, r, s) infinitely differentiable in a small neighborhood of K such that (1) 0 ≤ B ≤ 5Q(X + Y ), (2) −d 2 B ≥ |d x||dy|.
We prove Theorem 1.3 later. Now we use it to obtain the proof of Theorem 1.2. Proof of Theorem 1.2 Given a nonconstant smooth w that is constant outside some large ball, we consider Q = Q heat w,2 . We treat only the case w ∈ A2 , that is, Q < ∞, for otherwise there is nothing to prove. Consider two nonnegative functions ϕ, ψ ∈ C0∞ . Now take B = B Q,K , where a compact K remains to be chosen.
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We are interested in b(x, t) := B (ϕ 2 w)(x, t), (ψ 2 w−1 )(x, t), ϕ(x, t), ψ(x, t), w(x, t), w−1 (x, t) . This is a well-defined function because the choice of Q ensures that the 6-vector v, defined by v := (ϕ 2 w)(x, t), (ψ 2 w−1 )(x, t), ϕ(x, t), ψ(x, t), w(x, t), w−1 (x, t) , lies in D Q for any (x, t) ∈ R3+ . Also, we can fix any compact subset M of the open set R3+ and guarantee that for (x, t) ∈ M, the vector v lies in a compact K . In fact, notice that for our w and for compactly supported ϕ, ψ, the mapping (x, t) → v(x, t) maps compacts in R3+ to compacts in D Q . Now just take K large enough. We want to apply the Green formula to b(x, t). To do this, we introduce a Green function G(x, t), as in [11]: ∂ + 1 G = −δ0,1/2 in C(1, 1) = B(0, 1) × (0, 1), ∂t G=0 on ∂ 0 C(1, 1) = ∂ B(0, 1) × (0, 1), G=0 when t = 1. Here δ0,1/2 is a δ-function at the point (0, 1/2). It is important to keep in mind that G(x, 0) ≥ a 1 − kxk ,
(1.5)
where a is a positive absolute constant. We also need a Green function in the cylinder C(R, R 2 ) = B(0, R) × (0, R 2 ): ∂ + 1 G R = −δ0,R 2 /2 in C(R, R 2 ) = B(0, R) × (0, R 2 ), ∂t GR = 0 on ∂ 0 C(R, R 2 ) = ∂ B(0, R) × (0, R 2 ), R G =0 when t = R 2 . One can easily see that the following lemma holds. 1.4 Green functions in different-sized cylinders relate as follows: LEMMA
G R (x, t) =
1 x t G , . R R2 R2
We are ready to apply the Green formula to b(x, t).
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Let us first estimate b(0, R 2 /2) = B((ϕ 2 w)(0, R 2 /2), . . . , w −1 (0, R 2 /2)). Using the first property of Theorem 1.3, we get (x = (x1 , x2 ), as always) R2 R2 R 2 2 2 −1 b 0, ≤ 5Q heat (ϕ w) 0, + (ψ w ) 0, . w,2 2 2 2 Thus, ZZ 2|x|2 R2 5 2 2 (ϕ w)(x) exp − d x1 d x2 b 0, ≤ Q heat 2 π w,2 R2 R2 ZZ 2|x|2 5 heat 2 2 −1 + Q w,2 (ψ w )(x) exp − d x1 d x2 . π R2 R2 Now by the Green formula in C(R, R 2 ), ZZZ R2 ∂ b 0, =− + 1 G R (x, t)b(x, t) d x1 d x2 dt 2 C(R,R 2 )∩{t>ε} ∂t ZZZ ∂ = G R (x, t) − 1 b(x, t) d x1 d x2 dt ∂t C(R,R 2 )∩{t>ε} ZZ + b(x, ε)G R (x, ε) d x1 d x2 B(0,R)
∂b ∂G R GR − b ds dt + ∂n outer ∂ 0 C(R,R 2 )∩{t>ε} ∂n outer ZZZ ∂ = G R (x, t) − 1 b(x, t) d x1 d x2 dt ∂t C(R,R 2 )∩{t>ε} ZZ ZZ ∂G R R + b(x, ε)G (x, ε) d x1 d x2 + b ds dt B(0,R) ∂ 0 C(R,R 2 )∩{t>ε} ∂n inner ZZZ ∂ ≥ G R (x, t) − 1 b(x, t) d x1 d x2 dt. ∂t C(R,R 2 )∩{t>ε} ZZ
The last inequality is clear: the double integrals are both nonnegative because b is nonnegative and because G R is nonnegative and vanishes on the side boundary. Let us combine estimates of b(0, R 2 /2) into ZZ ZZ ZZZ ∂ 10 2 2 −1 R ϕ w + ψ w . G (x, t) − 1 b(x, t) ≤ ∂t π R2 C(R,R 2 )∩{t>ε} (1.6) Fix R and ε, and choose the compact set M = (x, t) : x ∈ clos(B(0, R)), ε ≤ t ≤ R 2 . The next calculation is simple, but it is key to the proof.
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LEMMA
1.5
Let v = (ϕ 2 w)(x, t), (ψ 2 w−1 )(x, t), ϕ(x, t), ψ(x, t), w(x, t), w−1 (x, t) , where all entries are heat extensions. If (x, t) ∈ M, then ∂ ∂v ∂v ∂v ∂v 2 − 1 b(x, t) = (−d 2 B) , + (−d B) , . ∂t ∂ x 1 ∂ x 1 R6 ∂ x 2 ∂ x 2 R6 Proof We compute the derivative with respect to time and the Laplacian of b using the chain rule and obtain the following: ∂v ∂ b = ∇ B, , ∂t ∂t R6 ∂v ∂v ∂v ∂v 2 1b = (d 2 B) , + (d B) , + (∇ B, 1v)R6 . ∂ x 1 ∂ x 1 R6 ∂ x 2 ∂ x 2 R6 Combining the two, we get ∂ ∂v ∂v ∂v ∂v ∂v 2 − 1 b = ∇ B, − 1v 6 − (d 2 B) , − (d B) , . R ∂t ∂t ∂ x 1 ∂ x 1 R6 ∂ x 2 ∂ x 2 R6 However, the first term is zero because all entries of the vector v are solutions of the heat equation. By Theorem 1.3 in M, −d 2 B ≥ |d x||dy|. For (x, t) ∈ M, Lemma 1.5 gives ∂ϕ ∂ψ ∂ϕ ∂ψ ∂ − 1 b(x, t) ≥ + . ∂t ∂ x1 ∂ x1 ∂ x2 ∂ x2
(1.7)
Combining (1.6) and (1.7), we get ZZ ∂ϕ ∂ψ ∂ϕ ∂ψ 10Q heat Z Z w,2 2 2 −1 G (x, t) ϕ w + ψ w . + ≤ ∂ x1 ∂ x1 ∂ x2 ∂ x2 π R2 M (1.8) Now it is time to use Lemma 1.4. So (1.8) implies ZZZ x t ∂ϕ ∂ψ ∂ϕ ∂ψ , 2 G + R ∂ x ∂ x ∂ x ∂ x R 1 1 2 2 M ZZ ZZ 10Q heat w,2 ≤ ϕ2w + ψ 2 w−1 , (1.9) π
ZZZ
R
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where M = {(x, t) : x ∈ clos(B(0, R)), ε ≤ t ≤ R 2 }. Let us fix any compact M0 in R3+ and choose R and ε in such a way that M0 ⊂ M. Restrict the integration in (1.9) to M0 , and let R → ∞. Taking into account that G(0, 0) = a > 0, we obtain Z Z Z ZZ 10Q heat Z Z ∂ϕ ∂ψ ∂ϕ ∂ψ w,2 ϕ2w + ψ 2 w−1 . + ≤ ∂ x1 ∂ x1 ∂ x2 ∂ x2 πa M0 But M0 is an arbitrary compact set in the upper half-space. Therefore, ZZZ
ZZ ∂ϕ ∂ψ ∂ϕ ∂ψ 10Q heat Z Z w,2 ϕ2w + ψ 2 w−1 . (1.10) + ≤ ∂ x1 ∂ x1 ∂ x2 ∂ x2 πa R3+
This immediately implies that ZZZ
∂ϕ ∂ψ ∂ϕ ∂ψ 20Q heat w,2 k f k L 2 (w) kψk L 2 (w−1 ) + ≤ 3 ∂ x ∂ x ∂ x ∂ x πa 1 1 2 2 R+
(1.11)
by substituting ϕ/t, tψ into (1.10) and minimizing over t. This proves Theorem 1.2.
Now we couple inequality (1.11) with Lemma 1.1, getting the right weighted estimates for R12 , R22 , R1 R2 , T . In particular, we have proved Theorem 0.6 for the case p = 2 up to the proof of Theorem 1.3. Proof of Theorem 1.3 We start with a much simpler “model” operator, Tσ . The logic is the following. We want to get a sharp weighted estimate of kTσ k L 2 (w)→L 2 (w) via the A2 -characteristic of w. Strangely enough, we first consider a two-weight estimate of kTσ k L 2 (u)→L 2 (v) with different weights u, v. In the paper of Nazarov, S. Treil, A. Volberg [26], one can find that this norm is attained on some “simple” test functions—and that this holds for every pair u, v. Thus, the same is true for u = v = w. However, on the family T of test functions, one can compute Nw,2 (Tσ ) := sup{kTσ tk L 2 (w) : t ∈ T , ktk L 2 (w) = 1}. It turns out that Nw,2 (Tσ ) ≈ Q class w,2 . J. Wittwer does this in [31] (see also [28]). Thus, we get kTσ k L 2 (w)→L 2 (w) = Nw,2 (Tσ ) ≈ Q class w,2 . This sharp one-weight estimate for the model operator (notice that its proof used a fact proved in two-weight theory) is used to construct the Bellman function for our operator. An application of the Green formula to this Bellman function is the content of the proof of Theorem 1.2.
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So let us show what the model operator is, what its sharp weighted estimate is, and how one obtains a special function (the Bellman function) from this estimate. Consider the family of dyadic singular operators Tσ : Tσ f = 6 I ∈D σ (I )( f, h I )h I . Here D is a dyadic lattice on R, h I is a Haar function associated with the dyadic interval I (h I is normalized in L 2 (R, d x)), and σ (I ) = ±1. We call the family Tσ the martingale transform. It is a dyadic analog of a Calder´on-Zygmund operator. The following are important questions about Tσ , the first one about two-weight estimates and the second one about one-weight estimates. (1) What are necessary and sufficient conditions for supσ kTσ k L 2 (u)→L 2 (v) < ∞? (2) What is the sharp bound on supσ kTσ k L 2 (w)→L 2 (w) in terms of w? How can one compute supσ kTσ k L 2 (w)→L 2 (w) ? These questions are dyadic analogs of notoriously difficult questions about classical Calder´on-Zygmund operators like the Hilbert transform, the Riesz transforms, and the Ahlfors-Beurling transform. The dyadic model is supposed to be easier than the continuous one. This turns out to be true. The answers to the questions above appeared in [26] and [31]. Moreover, these answers are key to answering questions about classical Calder´on-Zygmund operators. Strangely enough, the answer to the second question (which seems to be easier because it is about one-weight) seems to require the ideas from the two-weight case. Here is our explanation of this phenomenom. The necessary and sufficient conditions on (u, v) to answer the first question were given in [26]. They amount to the fact that supσ kTσ k L 2 (u)→L 2 (v) is almost attained on the family of simple test functions. This fact has beautiful consequences in the one-weight case. For then supσ kTσ k L 2 (w)→L 2 (w) is attainable (almost) on the family of simple test functions. One may try to compute supσ kTσ tk L 2 (w) for every element of this test family, thus getting a good estimate for the norm supσ kTσ k L 2 (w)→L 2 (w) . Test functions are rather simple, so this program can be carried out. This has been done in [31]. Here is the result. Recall that dyadic Q w,2 := sup hwi I hw−1 i I . I ∈D
THEOREM 1.6 For any A2 -weight w, we have the inequality dyadic
sup kTσ k L 2 (w)→L 2 (w) ≤ AQ w,2 . σ
Let us rewrite Theorem 1.6 as follows: dyadic sup 6 I ∈D σ (I )( f, h I )(g, h I ) ≤ AQ w,2 k f k L 2 (w) kgk L 2 (w−1 ) , σ (I )=±1
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or dyadic 6 I ∈D ( f, h I ) (g, h I ) ≤ AQ w,2 k f k L 2 (w) kgk L 2 (w−1 ) . This inequality is scaleless, so we write it as J ∈ D,
1 6 I ∈D ,I ⊂J h f i I− − h f i I+ hgi I− − hgi I+ |I | |J | dyadic
1/2
1/2
≤ AQ w,2 h f 2 wi J hg 2 w−1 i J . (1.12) Here I− , I+ are the left and the right halves of I , and h·il means averaging over l, as usual. Given a fixed J ∈ D and a number Q > 1, we wish to introduce the Bellman function of (1.12), B(X, Y, x, y, r, s) n 1 = sup 6 I ∈D ,I ⊂J h f i I− − h f i I+ hgi I− − hgi I+ |I | : |J | h f i J = x, hgi J = y, hwi J = r, hw−1 i J = s, dyadic
h f 2 wi J = X, hg 2 w−1 i J = Y, w ∈ A2
dyadic
, Q w,2
o ≤Q .
Obviously, the function B does not depend on J , but it does depend on Q. Its domain of definition is the following: R Q := 0 ≤ (X, Y, x, y, r, s), x 2 ≤ X s, y 2 ≤ Y r, 1 ≤ r s ≤ Q . By (1.12) it satisfies 0 ≤ B ≤ AQ X 1/2 Y 1/2 .
(1.13)
We prove that it also satisfies the following differential inequality. Denote v := (X, Y, x, y, r, s),
v− = (X − , Y− , x− , y− , r− , s− ),
v+ = (X + , Y+ , x+ , y+ , r+ , s+ ), and let v, v+ , v− lie in R Q such that v = 1/2(v− + v+ ). Then B(v) −
1 B(v+ ) + B(v− ) ≥ |x+ − x− ||y+ − y− |. 2
(1.14)
In fact, let f, g, w almost maximize B(v) (on the interval J ), let f + , g+ , w+ do this for B(v+ ), and let f − , g− , w− do this for B(v− ). The scalelessness of B allows us to put f + , g+ , w+ on J+ and f − , g− , w− on J− . Then we have “gargoyle” functions ( ( ( f + on J+ , g+ on J+ , w+ on J+ , F= G= W = f − on J− , g− on J− , w− on J− .
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Obviously, hFi J = 1/2(x+ + x− ) = x, hGi J = y, hW i J = r, hW −1 i J = s, hF 2 W i J = X, hG 2 W =1 i J = Y . These numbers together form the vector v. In other words, F, G, W compete with f, g, w in definition (1.12) of Bellman function B(v). By this definition, B(v) ≥
1 6 I ∈D ,I ⊂J hFi I− − hFi I+ hGi I− − hGi I+ |I |. |J |
But the almost optimality of f + , g+ , w+ on J+ and f − , g− , w− on J− gives us B(v+ ) ≥ −ε +
1 6 I ∈D ,I ⊂J+ hFi I− − hFi I+ hGi I− − hGi I+ |I |, |J+ |
B(v− ) ≥ −ε +
1 6 I ∈D ,I ⊂J− hFi I− − hFi I+ hGi I− − hGi I+ |I | |J− |
and
(recall that F = f ± on J± , G = g± on J± ). Combining these, we get 1 B(v+ ) + B(v− ) ≥ −ε + hFi J− − hFi J+ hGi J− − hGi J+ 2 = −ε + h f − i J− − h f + i J+ hg− i J− − hg+ i J+ = −ε + x− − x+ y− − y+ .
B(v) −
We are done with (1.14) because ε is an arbitrary positive number. Therefore, our B is a very concave function. We modify B to have its Hessian satisfy the conclusion of Theorem 1.3. To do that we fix a compact K in the interior of R Q , and we choose ε such that 100ε < dist(K , ∂ R Q ). Consider the convolution of B with (1/ε6 )ϕ(v/ε), v ∈ R6 , where ϕ is a bell-shaped infinitely differentiable function with support in the unit ball of R6 . It is now very easy to see that this convolution (we call it B K ,Q ) satisfies the following inequalities: 0 ≤ B K ,Q ≤ 6Q(X + Y ),
(1.15)
and for any vector ξ = (ξ1 , ξ2 , ξ3 , ξ4 , ξ5 , ξ6 ) ∈ R6 , −(d 2 B K ,Q ξ, ξ )R6 ≥ 2|ξ2 ||ξ3 |.
(1.16)
The factor 2 appears because B(v) − (1/2)(B(v+ ) + B(v− )) in (1.14) corresponds to −(1/2)d 2 B. Theorem 1.3 is completely proved.
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2. The sharp weighted estimate for the Ahlfors-Beurling operator in L p (w d A) for p > 2 Here again T is the Ahlfors-Beurling transform on the plane, and d A denotes Lebesgue measure on the plane. Now we prove that kT k L p (w d A)→L p (w d A) ≤ C( p)Q heat w, p .
(2.1)
In Section 3, it is shown how this implies kT k L p (w d A)→L p (w d A) ≤ C( p)Q class w, p .
(2.2)
Let us recall that for the weights on R, we can introduce dyadic
Q w, p
p−1
:= sup hwi I hw−(1/( p−1)) i I I ∈D
.
The weight ρ := w−(1/( p−1)) is called the dual weight, and dyadic
dyadic
0
(Q ρ, p0 ) p = (Q w, p ) p .
(2.3)
We need the following result of S. Buckley. (We formulate its dyadic version.) Let us denote M d f (x) = supx∈I,I ∈D h| f |i I . This is the dyadic maximal function. THEOREM 2.1 For any weight on the line R, dyadic
kM d k L p (w d x)→L p (w d x) ≤ C( p)Q ρ, p0 ,
1 < p < ∞.
We use it now to extend Theorem 1.6 to p > 2. THEOREM 2.2 For any weight on the line R, dyadic
sup kTσ k L p (w d x)→L p (w d x) ≤ C( p)Q w, p , σ
2 < p < ∞.
The proof that follows is word for word the extrapolation proof of Garcia-Cuerva and Rubio de Francia in [12]. We only need to be absolutely precise with exponents. Proof dyadic Let w ∈ A p . Let T stand for any of the Tσ ; let t = ( p − 2)/( p − 1). Then Z Z Z 2/ p p 2 |T f | w d x = sup |T f | uw d x ≤ sup |T f |2 vw d x. u≥0,kuk
≤1 0 L p /t (w d x)
v∈F
(2.4)
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Here F is a family of functions v of the following type: v = v(u) := u +
S2u S3u Su + + + ··· , 2kSk 4kSk2 8kSk3
where u runs over the unit ball of L p /t (w d x), the nonlinear operator S is given by the formula t S(u) := w−1 M d (u 1/t w) , 0
and kSk denotes kSk L p0 /t (w d x)→L p0 /t (w d x) . Now let us prove the following: (1) u ≤ v, (2) kvk L p0 /t (w d x) ≤ 2kuk L p0 /t (w d x) , dyadic
dyadic
dyadic
(3) vw ∈ A2 , Q vw,2 ≤ Q w, p . The first inequality is clear. To prove the second inequality, we notice that if kSk is finite, then the second inequality follows immediately. So let us prove this finiteness. Now dyadic kSk ≤ C( p)(Q w, p )t . (2.5)
Z
In fact, using Theorem 2.1 we get Z p0 0 0 (Su) p /t w d x = M d (u 1/t w) w1− p d x Z Z p0 −(1/( p−1)) d 1/t = M (u w) w dx = Z 0 0 0 dyadic 0 ≤ C( p)(Q w, p ) p u p /t w p w1− p d x Z 0 dyadic 0 = C( p)(Q w, p ) p u p /t w d x. Therefore, (2.5) is proved. Let us now check that
−1 dyadic huwi I S(u)w ≤ (Q w, p )1−t , I
p0 M d (u 1/t w) ρ d x
∀I ∈ D , u ≥ 0.
(2.6)
In fact, (1 − t)(1 − p) = 1. Using this and the pointwise estimate on (M d (u 1/t w))(x) ≥ hu 1/t wi I , ∀x ∈ I , we get
−t −(1/( p−1)) huwi I M d (u 1/t w) w−(1/( p−1)) I ≤ hu 1/t witI hwi1−t · hu 1/t wi−t iI . I I hw Therefore,
−t −(1/( p−1)) (1−t)(1− p) huwi I M d (u 1/t w) w−(1/( p−1)) I ≤ hwi1−t iI I hw dyadic
≤ (Q w, p )1−t .
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We are done with (2.6). Now at last we have our third estimate as a combination of (2.5), (2.6), and the fact that by its definition, S(v) ≤ 2kSkv: dyadic
∀v ∈ F ,
dyadic
Q vw,2 ≤ C( p)Q w, p .
(2.7)
Now using (2.7) and inequality kvk L p0 /t (w d x) ≤ 2kuk L p0 /t (w d x) ≤ 2, we get Z
|T f | p w d x
2/ p
Z
Z dyadic |T f |2 vw d x ≤ A(Q vw,2 )2 | f |2 vw d x v∈F Z 2/ p dyadic ≤ AC( p)(Q w, p )2 | f |pw d x kvk L p0 /t (w d x) ≤ sup
dyadic
≤ 2AC( p)(Q w, p )2 k f k2L p (w d x) . In the second inequality we used Theorem 1.6. Thus, dyadic kTσ k L p (w d x)→L p (w d x) ≤ 2AC( p)Q w, p , and Theorem 2.2 is completely proved. Theorem 1.3 has the following ( p > 2)-version. 2.3 p For any Q > 1, p > 2, define the domain D Q := {0 < (X, Y, x, y, r, s) : x p < THEOREM
0
p
0
X s p−1 , y p < Y r p −1 , 1 < r s p−1 < Q}. Let K be any compact subset of D Q . Then ( p)
there exists a function B = B Q,K (X, Y, x, y, r, s) infinitely differentiable in the small neighborhood of K such that 0 (1) 0 ≤ B ≤ C( p)Q X 1/ p Y 1/ p , (2) −d 2 B ≥ |d x||dy|. Proof The proof repeats verbatim the proof of Theorem 1.3. But now we base the considerations on Theorem 2.2 rather than Theorem 1.6. Now we are ready to prove our main estimate (2.1). The proof repeats the one given in the previous section for p = 2, but instead of b(x, t) we consider b( p) (x, t) := B ( p) v(x, t), where v(x, t) is the 6-vector 0 ( f p w)(x, t), (g p w−(1/( p−1)) )(x, t), f (x, t), g(x, t), w(x, t), w−(1/( p−1)) (x, t) .
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The rest is identical, but the use of b( p) explains why one needs to choose Q at −(1/( p−1)) (x, t)) p−1 . least as large as Q heat w, p = sup(x,t)∈R3+ w(x, t)(w Inequality (2.1) is completely proved. We have also proved our main Theorem 0.6. 3. The comparison of classical and heat A p -characteristics In this section we give an easy proof of the following theorem. 3.1 There exists a finite absolute constant b such that THEOREM
class Q heat w, p ≤ bQ w, p .
Proof Constants are denoted by the letters c, C; they may vary from line to line and even within the same line. We introduce the following notation. Bk denotes B(0, 2k ), k = R 0, 1, 2, . . . , h f i B stands for the average (1/|B|) B f d A, and f (B) stands for R RR h 2 B f d A. If B = B(0, r ), then h f i B stands for (1/(πr )) R2 f (x) exp − (kxk2 /(r 2 )) d x1 d x2 . LEMMA 3.2 Suppose that f and g, positive functions on the plane, are such that sup B h f i B hgi B = A. Then there exists a finite absolute constant c such that
h f i B hgihB ≤ c A for any disc B. Proof Scale invariance allows us to prove this for only one disc, B = B(0, 1). We start the estimate: A h f i B hgihB ≤ ch f i B 6k 22k exp(−22k−2 ) . h f i Bk On the other hand, h f i Bk > ch f i Bk−1 > · · · > ck h f i B . (Recall that B is the unit disc.) Plugging this into the inequality above, we get h f i B hgihB ≤ ch f i B 6k C k exp(−22k−2 )
A . h f iB
In other words, h f i B hgihB ≤ c A6k C k exp(−22k−2 ) = c A, and the lemma is proved.
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Now we want to prove Theorem 3.1. Fix B. Again, by scale invariance it is enough to consider B = B(0, 1). By the previous lemma, we know that h f i Bk hgihBk ≤ c A
(3.1)
for any k. Now h f ihB hgihB ≤ chgihB 622k exp(−22k−2 )h f i Bk ≤ chgihB 622k exp(−22k−2 )
cA hgihBk
.
The last inequality uses (3.1). On the other hand, hgihBk > chgihBk−1 > · · · > ck hgi B . (Recall that B is the unit disc.) Plugging this into the inequality above, we get h f ihB hgihB ≤ chgihB 6k C k exp(−22k−2 )
cA hgihB
.
In other words, h f ihB hgihB ≤ c2 A6k C k exp(−22k−2 ) = c2 A. Theorem 3.1 is completely proved. 4. Injectivity at the critical exponent and regularity of solutions of the Beltrami equation Theorem 0.4 was proved in [2]. It reduces the question of the injectivity of I − µT , I − T µ in L 1+k (C), k = kµk∞ < 1, to estimate (0.1): kT k L p (ω d A)→L p (ω d A) ≤
C , 1 + (1/k) − p
where ω = | f z ◦ f −1 | p−2 , f is a ((1 + k)/(1 − k))-quasiconformal homeomorphism, and p ∈ [2, 1 + (1/k)). This estimate has been proved twice in the present paper. We first proved the heat class estimate of the norm kT k via AQ heat w, p , then noticed that Q w, p ≈ Q w, p , and finally class noticed that (see [2]) Q w, p ≤ C/(1+(1/k)− p) for special ω’s that are certain powers of Jacobians of quasiconformal homeomorphisms. The second time, we started with heat the estimate of the norm kT k via AQ heat w, p . Then we directly computed Q w, p for ω’s that are certain powers of Jacobians of quasiconformal homeomorphism and again saw that Q heat w, p ≤ C/(1 + (1/k) − p) (see Theorem 0.7). Thus, we can be sure that injectivity holds. This fact is immediately applicable to so-called weakly quasiregular maps. Recall (see [2]) that the solution of Fz¯ − µFz = 0
in ⊂ C
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PETERMICHL and VOLBERG 1,q
can be called q-weakly quasiregular if it belongs to Wloc . Here µ ∈ L ∞ (), kµk∞ = k < 1. It is known (see [2]) that if q > 1 + k, then a q-weakly quasiregu1,2+ε lar map is actually in Wloc for a certain positive . In particular, it is quasiregular. Therefore, it is continuous, open, discrete, and so on. However, the proof of this “selfimprovement” fact is very subtle. It is based on Astala’s sharp estimates of smoothness of quasiconformal homeomorphisms, which were obtained as a solution of a famous problem of F. Gehring and E. Reich. In [2], one can find an easy example that shows this is no longer true for q < 1+k. Thus, the remaining question is about the critical exponent q = 1 + k. THEOREM
Let kµk∞
4.1 = k < 1. Then (1 + k)-weakly quasiregular maps are also quasiregular.
We repeat the argument from [2]. This is done for the sake of the convenience of the reader. Proof Choose ϕ ∈ C∞ 0 (). Set G = Fϕ. Then G z¯ − µG z = (ϕz¯ − µϕz )F.
(4.1)
Obviously, G is a Cauchy transform of the compactly supported function ψ := G z¯ . Then (4.1) can be rewritten as (F0 := (ϕz¯ − µϕz )F): (I − µT )ψ = F0 ∈ L 2 (C).
(4.2)
1,1+k In fact, F ∈ Wloc , so by Sobolev’s theorem (ϕz¯ −µϕz )F ∈ L ((2+2k)/(1−k)) (C), and (ϕz¯ − µϕz )F has compact support. So it is in L 2+ε (C). Looking at (4.2), we can see that µ = 0 outside of supp ϕ. Our equation (4.2) has an obvious solution as a Neumann series. It converges in L 2+ε (C) (we use the fact that kT k in L p is close to 1 when p is close to 2) and has its support in supp ϕ. So it is in L 1+k (C) as well. Let us call it ψ0 . Now we have two solutions of (4.2) in L 1+k (C) − ψ and ψ0 . However, we have already proved the injectivity of I − µT in L 1+k (C). Therefore, ψ = ψ0 . Thus, G z¯ = ψ ∈ L 2+ε (C). Therefore, the compactly supported function G is in W 1,2+ε (C). This means that G is quasiregular and, in particular, continuous, open, discrete, and so on. Theorem 4.1 is completely proved.
Acknowledgments. We are deeply grateful to Kari Astala, Eero Saksman, Michael Frazier, Tadeusz Iwaniec, Fedor Nazarov, and Jacob Plotkin for discussions and criticism.
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Petermichl Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540, USA;
[email protected]
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Volberg Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA;
[email protected]; current: Equipe d’Analyse, Universit´e Pierre et Marie Curie-Paris 6, 4 Place Jussieu, 75252 Paris CEDEX 05, France;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2,
IRREDUCIBILITY CRITERION FOR TENSOR PRODUCTS OF YANGIAN EVALUATION MODULES A. I. MOLEV
Abstract The evaluation homomorphisms from the Yangian Y(gln ) to the universal enveloping algebra U(gln ) allow one to regard the irreducible finite-dimensional representations of gln as Yangian modules. We give necessary and sufficient conditions for irreducibility of tensor products of such evaluation modules. 1. Introduction The Yangian Y(gln ) for the general linear Lie algebra gln is a deformation of the universal enveloping algebra U(gln [x]) in the class of Hopf algebras (see V. Drinfeld [6]). A theorem of Drinfeld (see [8]; see also V. Tarasov [24]) provides a complete description of finite-dimensional irreducible representations of Y(gln ) in terms of their highest weights. Recently, Arakawa [2] has found a character formula for each of these representations with the use of the Kazhdan-Lusztig polynomials (see also Vasserot [25] for the case of quantum affine algebras). M. Nazarov and Tarasov [22] (see also I. Cherednik [5]) have given an explicit construction for a class of so-called tame representations. However, the structure of the general finite-dimensional irreducible Y(gln )-module (with n ≥ 3) still remains unknown. In this paper we establish an irreducibility criterion for tensor products of the Yangian evaluation modules which thus solves this problem for a wide class of representations of Y(gln ). Let λ = (λ1 , . . . , λn ) be an n-tuple of complex numbers such that λi − λi+1 is a nonnegative integer for each i. Denote by L(λ) the irreducible finite-dimensional representation of the Lie algebra gln with the highest weight λ. For each a ∈ C, there is an evaluation homomorphism ϕa from the Yangian Y(gln ) to the universal enveloping algebra U(gln ) (see Section 2 for the definitions). Using ϕa , we make L(λ) into a Yangian module and denote it by L a (λ). We keep the notation L(λ) for the evaluation module L a (λ) with a = 0. The Hopf algebra structure on Y(gln ) allows DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2, Received 26 September 2000. Revision received 10 April 2001. 2000 Mathematics Subject Classification. Primary 17B37. Author’s work supported by the Australian Research Council.
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one to regard tensor products of the type L a1 (λ(1) ) ⊗ L a2 (λ(2) ) ⊗ · · · ⊗ L ak (λ(k) )
(1.1)
as Yangian modules. Our main result is a criterion of irreducibility of these modules. To formulate it we first note that the problem can be reduced to the particular case where all the parameters ai in (1.1) are equal to zero. This is done by using the composition of the module (1.1) with an appropriate automorphism of the Yangian (see Proposition 2.4). We give first an irreducibility criterion for the tensor product L(λ) ⊗ L(µ) of two evaluation modules. It is well known (see, e.g., [10] and Theorem 3.1) that this module is irreducible if the differences λi − µ j are not integers. Furthermore, for any c ∈ C, the simultaneous shifts λi 7 → λi + c and µ j 7→ µ j + c for all i and j do not affect the irreducibility of L(λ) ⊗ L(µ) (see Proposition 2.5). Thus, we may assume without loss of generality that all the entries of λ and µ are integers. We use the following definition. Two disjoint finite subsets A and B of Z are crossing if there exist elements a1 , a2 ∈ A and b1 , b2 ∈ B such that either a1 < b1 < a2 < b2 or b1 < a1 < b2 < a2 . Otherwise, A and B are called noncrossing. Given a highest weight λ with integer entries, introduce the following subset of Z: Aλ = {λ1 , λ2 − 1, . . . , λn − n + 1}. THEOREM 1.1 The module L(λ) ⊗ L(µ) is irreducible if and only if the sets Aλ \ Aµ and Aµ \ Aλ are noncrossing.
Using the argument of N. Kitanine, J. Maillet, and V. Terras [11], [15], Nazarov and Tarasov [23, Theorem 4.9] demonstrated that the irreducibility criterion for the multiple tensor product (1.1) can be obtained from the particular case of k = 2 tensor factors. Namely, the following binary property holds. Here λ(1) , . . . , λ(k) are n-tuples ( p) ( p) of complex numbers such that λi − λi+1 is a nonnegative integer for each i and p. BINARY PROPERTY
The module
L(λ(1) ) ⊗ L(λ(2) ) ⊗ · · · ⊗ L(λ(k) )
(1.2)
is irreducible if and only if all the modules L(λ( p) ) ⊗ L(λ(q) ) with p < q are irreducible. Note that the “only if” part of this assertion is implied by the well-known Proposition 2.6 (see Section 2). If the module (1.2) is irreducible, its highest weight is easy to
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find. Therefore, together with Theorem 1.1, the binary property allows one to determine whether a given irreducible Y(gln )-module can be realized in a tensor product (1.2). For the proof of Theorem 1.1, we use the Gelfand-Tsetlin bases of the gln modules L(λ) and L(µ). The key role is played by the formulas for the action of the Drinfeld generators of Y(gln ) in these bases, as well as by the quantum minor formulas for the Yangian lowering operators [16] (cf. Nazarov and Tarasov [20], [22]). In the case of the Yangian Y(gl2 ), our criterion coincides with the one obtained by V. Chari and A. Pressley [3], and it is also implicitly contained in Tarasov’s paper [24] (see also [17]). Nazarov and Tarasov [21] found a criterion of irreducibility of (1.1) in the case where each highest weight λ( p) has the form (α, . . . , α, β, . . . , β) with α − β ∈ Z+ . This generalized earlier results by T. Akasaka and M. Kashiwara [1], and A. Zelevinsky [26]. B. Leclerc, Nazarov, and J.-Y. Thibon [12] have found an irreducibility criterion for the induction products of evaluation modules over the affine Hecke algebras of type A with the use of the canonical bases (see also Leclerc and Thibon [13], Leclerc and Zelevinsky [14]). The application of the Drinfeld functor (see [7]; see also [2]) leads to an irreducibility criterion (equivalent to ours) for the Yangian modules (1.1), when the highest weights λ( p) satisfy some extra conditions. Namely, assuming that the λ( p) are partitions (we may do this without loss of generality), one should require that for all p < q the sum of the lengths of λ( p) and λ(q) does not exceed n. 2. Preliminaries We refer the reader to the expository papers [17] and [19], where the results on the structure theory and representations of the Yangians are collected. The Yangian Y(n) = Y(gln ) (see [6], [8]) is the complex associative algebra with (1) (2) the generators ti j , ti j , . . . , where 1 ≤ i, j ≤ n, and the defining relations ti j (u), tkl (v) = where
(1)
1 tk j (u)til (v) − tk j (v)til (u) , u−v
(2.1)
(2)
ti j (u) = δi j + ti j u −1 + ti j u −2 + · · · ∈ Y(n)[[u −1 ]] and u is a formal (commutative) variable. The Yangian Y(n) is a Hopf algebra with the coproduct 1 : Y(n) → Y(n) ⊗ Y(n) defined by n X 1 ti j (u) = tia (u) ⊗ ta j (u).
(2.2)
a=1
Given sequences a1 , . . . , ar and b1 , . . . , br of elements of {1, . . . , n}, the corresponding quantum minor of the matrix [ti j (u)] is defined by the following equivalent
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formulas: ar t ab11 ··· ··· br (u) =
=
X
sgn σ · taσ (1) b1 (u) · · · taσ (r ) br (u − r + 1)
(2.3)
sgn σ · ta1 bσ (1) (u − r + 1) · · · tar bσ (r ) (u).
(2.4)
σ ∈Sr
X σ ∈Sr
ar The series t ab11 ··· ··· br (u) is skew-symmetric under permutations of the indices ai or bi . The Poincar´e-Birkhoff-Witt theorem for the Yangian Y(n) (see, e.g., [19, Corollary 1.23]) implies that, given a subset of indices {a1 , . . . , ar } ⊆ {1, . . . , n}, the coefficients of the series tai a j (u) with i, j = 1, . . . , r generate a subalgebra of Y(n) isomorphic to Y(r ). The mapping 1··· b j ···n ti j (u) 7 → (−1)i+ j t 1···bi ··· n (−u) (2.5)
defines an algebra automorphism of Y(n); the hats indicate the indices to be omitted. The following proposition is proved in [18, Proposition 1.1] by using the Rmatrix form of the defining relations (2.1). PROPOSITION 2.1 We have the relations min{k,l} X a1 ··· ak c1 ···cl t b1 ···bk (u), t d1 ···dl (v) =
(−1) p−1 p! (u − v − k + 1) · · · (u − v − k + p) p=1 X a1 ··· c j ··· c j p ··· ak c1 ··· ai ··· ai ··· cl × t b1 ··· bk1 (u)t d1 ··· dl1 p (v) i 1 < ···
1
a1 ··· ak ··· bi p ··· dl (v)t b1 ··· d j1 ··· d j p ··· bk (u) .
Here the p-tuples of upper indices (ai1 , . . . , ai p ) and (c j1 , . . . , c j p ), respectively, are interchanged in the first summand on the right-hand side, while the p-tuples of lower indices (bi1 , . . . , bi p ) and (d j1 , . . . , d j p ) are interchanged in the second summand. We note the following particular case of these relations: l l X 1 X c1 ··· cl l tci b (u)t cd11···a···c (v) − t (v)t (u) . adi ···dl d1 ···b···dl u−v i=1 i=1 (2.6) This implies the well-known property of quantum minors: for any indices i, j, we have cl tci d j (u), t dc11··· (2.7) ··· dl (v) = 0.
l tab (u), t dc11···c ···dl (v) =
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We frequently use the following result proved in [22]. 2.2 The images of the quantum minors under the coproduct are given by X ar c1 ··· cr ··· ar 1 t ab11 ··· t ac11··· cr (u) ⊗ t b1 ··· br (u), ··· br (u) = PROPOSITION
c1 <···
summed over all subsets of indices {c1 , . . . , cr } from {1, . . . , n}. For m ≥ 1, introduce the series am (u), bm (u), and cm (u) by 1··· m−1,m+1 cm (u) = t 1··· (u). m (2.8) The coefficients of these series generate the algebra Y(n) (see [8]); they are called the Drinfeld generators. By a theorem of Drinfeld [8], every finite-dimensional irreducible representation of the Yangian Y(n) is a highest weight representation. That is, it contains a unique, up to a scalar factor, nonzero vector ζ (the highest vector) which is annihilated by all upper triangular elements ti j (u), i < j, and ζ is an eigenvector for the diagonal generators tii (u), tii (u) ζ = λi (u) ζ, i = 1, . . . , n. m am (u) = t 1··· 1··· m (u),
m bm (u) = t 1··· 1··· m−1,m+1 (u),
Here the λi (u) are formal series in u −1 with complex coefficients. We call the collection (λ1 (u), . . . , λn (u)) the highest weight of the representation. Equivalently, ζ is annihilated by b1 (u), . . . , bn−1 (u), and it is an eigenvector for each of the operators a1 (u), . . . , an (u) (see [8]), so that am (u) ζ = λ1 (u)λ2 (u − 1) · · · λm (u − m + 1) ζ,
m = 1, . . . , n.
If L is any Y(n)-module, then a nonzero element ζ ∈ L is called a singular vector if ζ is annihilated by all upper triangular generators ti j (u), i < j, and ζ is an eigenvector for the diagonal elements tii (u). Such a vector ζ generates a highest weight submodule in L. The following proposition is well known (see, e.g., [3], [4], [17]). 2.3 If L is an irreducible highest weight Y(n)-module and ζ ∈ L is annihilated by all operators ti j (u) with i < j, then ζ is proportional to the highest vector of L. PROPOSITION
Let the E i j , i, j = 1, . . . , n, denote the standard basis elements of the Lie algebra gln . For any a ∈ C, the mapping ϕa : ti j (u) 7 → δi j +
Ei j u−a
(2.9)
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defines an algebra epimorphism from Y(n) to the universal enveloping algebra U(gln ) so that any gln -module can be extended to a Y(n)-module via (2.9). In particular, let λ be an n-tuple of complex numbers λ = (λ1 , . . . , λn ) such that λi − λi+1 ∈ Z+ for all i (we call such n-tuples gln –highest weights). Consider the irreducible finitedimensional gln -module L(λ) with the highest weight λ with respect to the upper triangular Borel subalgebra. The corresponding Y(n)-module is denoted by L a (λ), and we call it the evaluation module. We keep the notation L(λ) for the module L a (λ) with a = 0. The coproduct 1 defined by (2.2) allows one to consider the tensor products (1.1) as Y(n)-modules. Let us denote by I the n-tuple (1, 1, . . . , 1). PROPOSITION 2.4 The Y(n)-module (1.1) is irreducible if and only if the module
L(λ(1) − a1 I ) ⊗ L(λ(2) − a2 I ) ⊗ · · · ⊗ L(λ(k) − ak I )
(2.10)
is irreducible. Proof Suppose (1.1) is irreducible. Let ξ ( p) denote the highest vector of the gln -module L(λ( p) ). We derive from (2.2) and (2.9) that ζ = ξ (1) ⊗ · · · ⊗ ξ (k) is the highest vector of the Y(n)-module (1.1) with the highest weight (λ1 (u), . . . , λn (u)), where (1) (k) λ λ λi (u) = 1 + i ··· 1 + i , u − a1 u − ak
i = 1, . . . , n.
(2.11)
Consider the automorphism of Y(n), ti j (u) 7→ f (u) ti j (u),
(2.12)
where f (u) is the formal series in u −1 given by f (u) = (1 − a1 u −1 ) · · · (1 − ak u −1 ). The composition of the module (1.1) with this automorphism is an irreducible Y(n)module e L with the highest weight (e λ1 (u), . . . , e λn (u)), where (k) (1) λ − a1 λ − ak e λi (u) = 1 + i ··· 1 + i , u u
i = 1, . . . , n.
(2.13)
On the other hand, the tensor product of the highest vectors of the gln -modules L(λ( p) − a p I ) is a singular vector of the Y(n)-module (2.10) with the weight given by (2.13). Therefore, e L is isomorphic to a subquotient of (2.10). However, these two modules have the same dimension, and hence, they are isomorphic. In particular, the module (2.10) is irreducible. The proof is completed by reversing the argument.
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PROPOSITION 2.5 Given c ∈ C, the simultaneous shifts ( p)
( p)
λi
7→ λi
+ c,
p = 1, . . . , k, i = 1, . . . n,
of the parameters of the module (1.2) do not affect its irreducibility. Proof If the module (1.2) is irreducible, then so is the module L c that is the composition of (1.2) with the automorphism of Y(n) given by ti j (u) 7→ ti j (u + c),
i, j = 1, . . . , n.
The highest weight of L c is (λc1 (u), . . . , λcn (u)) with (1) (k) λ λ λic (u) = 1 + i ··· 1 + i , u+c u+c
i = 1, . . . , n.
The proof is completed by repeating the argument of the proof of Proposition 2.4 with the use of the automorphism (2.12) of Y(n), where f (u) = (1 + cu −1 )k . The following proposition is well known (see, e.g., [3], [4], [10], [17]). 2.6 If the Y(n)-module (1.2) is irreducible, then any permutation of the tensor factors in (1.2) gives an isomorphic representation of Y(n). PROPOSITION
We use a version given in [16] (cf. [20]) of the construction of a basis of the gln module L(λ) which is originally due to I. Gelfand and M. Tsetlin [9]. We equip L(λ) with a Y(n)-module structure by using the epimorphism Y(n) → U(gln ),
ti j (u) 7 → δi j + E i j u −1
(2.14)
(see (2.9)). A pattern 3 (associated with λ) is a sequence of rows 3n , 3n−1 , . . . , 31 , where 3r = (λr 1 , . . . , λrr ) is the r th row from the bottom, the top row 3n coincides with λ, and the following betweenness conditions are satisfied: for r = 2, . . . , n, λri − λr −1,i ∈ Z+ ,
λr −1,i − λr,i+1 ∈ Z+ ,
for i = 1, . . . , r − 1.
(2.15)
For any pattern 3, introduce the vector ξ3 ∈ L(λ) by ξ3 =
→ rY −1 Y r =2,...,n i=1
τri (−λr −1,i − 1) · · · τri (−λri + 1)τri (−λri ) ξ,
(2.16)
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where ξ is the highest vector of L(λ) and ··· r τri (u) = u(u − 1) · · · (u − r + i + 1) t i+1 i ··· r −1 (u)
is the lowering operator (see also Section 4). The vectors ξ3 , where 3 runs over all patterns associated with λ, form a basis of L(λ). The τri (u) essentially coincide with the standard lowering operators arising from the transvector algebras (cf. [18]). We find from (2.14) that the operators Bm (u) = u(u − 1) · · · (u − m + 1) bm (u), Am (u) = u(u − 1) · · · (u − m + 1) am (u) in L(λ) are polynomials in u (see (2.8)). Their action in the basis {ξ3 } of L(λ) is given by the following formulas (see [16]). They can also be deduced from Lemmas 4.3 – 4.5 (see Section 4). We use the notation lri = λri − i + 1. PROPOSITION
2.7
We have Am (u) ξ3 = (u + lm1 ) · · · (u + lmm ) ξ3 , Bm (−lm j ) ξ3 = −
m+1 Y
(lm+1,i − lm j ) ξ3+δm j
(2.17) for j = 1, . . . , m,
i=1
where 3+δm j is obtained from 3 by replacing the entry λm j with λm j +1, and ξ3+δm j is supposed to be equal to zero if 3 + δm j is not a pattern. Applying the Lagrange interpolation formula, we can find the action of Bm (u) for any u. Note that the polynomial Bm (u) has degree m − 1 with the leading coefficient E m,m+1 . This therefore implies the Gelfand-Tsetlin formulas [9] for the action of the elements E m,m+1 : E m,m+1 ξ3 = −
m X (lm+1,1 − lm j ) · · · (lm+1,m+1 − lm j ) j=1
(lm1 − lm j ) · · · ∧ j · · · (lmm − lm j )
ξ3+δm j ,
(2.18)
where ∧ j indicates that the jth factor is skipped. We conclude this section with an equivalent form of the conditions of Theorem 1.1. Given complex gln –highest weights λ = (λ1 , . . . , λn ) and µ = (µ1 , . . . , µn ), we use the notation li = λi − i + 1,
m i = µi − i + 1,
i = 1, . . . , n.
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For a pair of indices i < j, we denote hl j , li i = {l j , l j + 1, . . . , li } \ {l j , l j−1 , . . . , li }, hm j , m i i = {m j , m j + 1, . . . , m i } \ {m j , m j−1 , . . . , m i }.
(2.19)
In particular, if λi = λi+1 = · · · = λ j , then hl j , li i = ∅. We assume now that λ and µ are gln –highest weights with integer entries. PROPOSITION 2.8 The sets Aλ \ Aµ and Aµ \ Aλ are noncrossing if and only if, for all pairs of indices 1 ≤ i < j ≤ n, we have
m j , m i 6∈ hl j , li i
l j , li 6 ∈ hm j , m i i.
or
(2.20)
Proof Let us write Cond(Aλ , Aµ ) for the condition that Aλ \ Aµ and Aµ \ Aλ are noncrossing. We use induction on n. In the case n = 2, the statement is obviously true. Let n ≥ 3. Suppose first that (2.20) holds. Set Aλ− = {l1 , . . . , ln−1 }
and
Aλ+ = {l2 , . . . , ln },
and define Aµ− and Aµ+ similarly. By the induction hypothesis, conditions Cond(Aλ− , Aµ− ) and Cond(Aλ+ , Aµ+ ) are both satisfied. If m 1 = l1 , then Cond(Aλ , Aµ ) obviously holds. We may assume without loss of generality that m 1 > l1 . Let Aλ− \ Aµ− = {li1 , . . . , lik },
Aµ− \ Aλ− = {m j1 , . . . , m jk },
where 1 ≤ i 1 < · · · < i k ≤ n and 1 = j1 < · · · < jk ≤ n. We must have for some a ∈ {1, . . . , k}, m ja+1 < lik < · · · < li1 < m ja , where the leftmost inequality is ignored when a = k. If 2 ≤ a ≤ k − 1, then together with Cond(Aλ+ , Aµ+ ) this clearly ensures Cond(Aλ , Aµ ). Similarly, this is also true when a = 1 and i 1 ≥ 2. So if a = 1, then the only case where both Cond(Aλ− , Aµ− ) and Cond(Aλ+ , Aµ+ ) hold but Cond(Aλ , Aµ ) does not is the one with the following inequalities between the elements of Aλ \ Aµ and Aµ \ Aλ : ln < m n < m jk < · · · < m j2 < lik < · · · < li2 < l1 < m 1 . However, in this case, m n ∈ hln , l1 i and l1 ∈ hm n , m 1 i, so that (2.20) is violated for i = 1 and j = n. An analogous argument shows that if a = k, then the only case where both Cond(Aλ− , Aµ− ) and Cond(Aλ+ , Aµ+ ) hold but Cond(Aλ , Aµ ) does not is ln < lik < · · · < li2 < m n < l1 < m jk < · · · < m j2 < m 1 .
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But then m n ∈ hln , l1 i and l1 ∈ hm n , m 1 i, which contradicts (2.20) again. Conversely, suppose Cond(Aλ , Aµ ) holds. This condition clearly implies both Cond(Aλ− , Aµ− ) and Cond(Aλ+ , Aµ+ ), and so, by the induction hypothesis, (2.20) holds for all pairs i < j with the possible exception of (i, j) = (1, n). If the latter condition fails, then we have ( ( m n ∈ hln , l1 i, ln ∈ hm n , m 1 i, or (2.21) l1 ∈ hm n , m 1 i m 1 ∈ hln , l1 i. However, in each of the two cases this contradicts Cond(Aλ , Aµ ). 3. Sufficient conditions Our aim in this section is to prove the following theorem. THEOREM 3.1 Let λ and µ be complex gln –highest weights. Suppose that for each pair of indices 1 ≤ i < j ≤ n, we have
m j , m i 6∈ hl j , li i
or
l j , li 6 ∈ hm j , m i i.
(3.1)
Then the Y(n)-module L(λ) ⊗ L(µ) is irreducible. Proof We give the proof as a sequence of lemmas. Let ξ and ξ 0 denote, respectively, the highest vectors of the gln -modules L(λ) and L(µ). Let N be a nonzero Y(n)-submodule of L(λ) ⊗ L(µ). A standard argument (see, e.g., [17]) shows that N must contain a singular vector ζ . The key part of the proof of the theorem is to show by induction on n that ζ = const · ξ ⊗ ξ 0 . (3.2) Then considering dual modules, we also show that the vector ξ ⊗ ξ 0 is cyclic. By Proposition 2.6, exchanging λ and µ if necessary, we may assume that for the pair (i, j) with i = 1 and j = n, the condition m 1 , m n 6∈ hln , l1 i
(3.3)
is satisfied. Consider the Gelfand-Tsetlin basis {ξ3 } of the gln -module L(λ) (see Section 2). The singular vector ζ is uniquely written in the form X ζ = ξ3 ⊗ η3 , (3.4) 3
summed over all patterns 3 associated with λ, and η3 ∈ L(µ).
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For the diagonal Cartan subalgebra h of gln , we denote by εi the basis vector of h∗ dual to the element E ii so that the n-tuple λ can be identified with the element λ1 ε1 + · · · + λn εn ∈ h∗ . We use a standard partial ordering on the weights of L(λ). Given two weights v, w ∈ h∗ , we write v w if w − v is a Z+ -linear combination of the simple roots εa − εa+1 . Equivalently, v w if and only if w−v =
n X
pa εa
(3.5)
a=1
with the conditions p1 , p1 + p2 , . . . , p1 + · · · + pn−1 ∈ Z+ ,
p1 + · · · + pn = 0.
The embedding U(gln ) ,→ Y(n),
(1)
E i j 7 → ti j ,
(3.6)
defines the natural U(gln )-module structure on L(λ) ⊗ L(µ). We usually identify the (1) operators E i j and ti j . The vector ζ is clearly a gln –singular vector. In particular, it is a weight vector. Since the basis {ξ3 } consists of weight vectors, each element η3 ∈ L(µ) in (3.4) is also a gln –weight vector. Moreover, all elements ξ3 ⊗ η3 in (3.4) have the same gln -weight. We denote the weight of the vector ξ3 , or the weight of the pattern 3, by w(3). It is well known (see [9]), and can be deduced, for example, from (2.17), that w(3) = w1 ε1 + · · · + wn εn ,
wk =
k X
λki −
i=1
k−1 X
λk−1,i .
(3.7)
i=1
We say that a pattern 3 occurs in the expansion (3.4) if η3 6= 0. Consider the set of patterns occurring in (3.4), and suppose that 30 is a minimal element of this set with respect to the partial ordering on the weights w(3). In other words, if 3 occurs in (3.4) and w(3) w(30 ), then w(3) = w(30 ). 3.2 The vector η30 coincides with ξ 0 , up to a constant factor. LEMMA
Proof We have bm (u) ζ = 0 for m = 1, . . . , n − 1. Therefore, by Proposition 2.2, X X c1 ··· cm m t 1··· c1 ··· cm (u) ξ3 ⊗ t 1··· m−1,m+1 (u) η3 = 0. c1 <···
(3.8)
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m Write the elements t 1··· c1 ··· cm (u) ξ3 as linear combinations of the basis vectors ξ3 , and m take the coefficient at ξ30 in the relation (3.8). The weight of the vector t 1··· c1 ··· cm (u) ξ3 (or, to be more precise, the weight of each of the coefficients of these series) equals
w0 = w(3) + ε1 + · · · + εm − εc1 − · · · − εcm . Therefore, w0 w(3), and w0 = w(3) if and only if ci = i for each i. Since 30 is a pattern of a minimal weight, the vector ξ30 can occur only in the expansion of m t 1··· 1··· m (u) ξ30 = am (u) ξ30 .
(3.9)
By (2.17), this implies m t 1··· 1··· m−1,m+1 (u) η30 = bm (u) η30 = 0
for each m. Thus, η30 is a singular vector of L(µ), and so η30 = const · ξ 0 . LEMMA 3.3 The pattern 30 is determined uniquely. Moreover, if a pattern 3 occurs in (3.4), then w(3) w(30 ).
Proof We use the fact that ζ is an eigenvector for the generators a1 (u), . . . , an (u). By Proposition 2.2, X X c1 ··· cm m am (u) ζ = t 1··· (3.10) c1 ··· cm (u) ξ3 ⊗ t 1··· m (u) η3 . c1 <···
This vector equals αm (u) ζ for a formal series αm (u). As we have seen in the proof of Lemma 3.2, the vector ξ30 can only occur in (3.10) in the expansion of (3.9). By (2.17) and Lemma 3.2, comparing the coefficients at ξ30 , we find that αm (u) =
0 ) · · · (u + l 0 ) (u + lm1 mm βm (u), u(u − 1) · · · (u − m + 1)
(3.11)
0 = λ 0 − i + 1 and the series β (u) is defined by a (u) ξ 0 = β (u) ξ 0 . where lmi m m m mi 0 , 1 ≤ i ≤ m < n, since Equation (3.11) uniquely determines the parameters lmi 0 −l0 0 lmi m,i+1 is a positive integer for each i. Thus, the pattern 3 is also determined uniquely. The second claim is now obvious.
LEMMA 3.4 For each entry λka of a pattern 3 occurring in (3.4), we have
λka − λ0ka ∈ Z+
for 1 ≤ a ≤ k ≤ n − 1.
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Proof If 3 occurs in (3.4), then w(3) w(30 ) by Lemma 3.3. We use induction on w(3). Fix 3 6 = 30 . Then there exists another pattern 30 occurring in (3.4) such that w(30 ) w(30 ) ≺ w(3), and for some m and some indices c1 < · · · < cm , m the expansion of t 1··· c1 ··· cm (u) ξ30 contains ξ3 with a nonzero coefficient. Indeed, if this is not the case, then considering the coefficient at ξ3 in (3.8), we come to the conclusion that bm (u) η3 = 0 for all m = 1, . . . , n − 1, and so η3 is, up to a constant, the highest vector of L(µ) (see the proof of Lemma 3.2). This implies that 3 and 30 must have the same weight. Due to Lemma 3.3, we have to conclude that 3 = 30 , a contradiction. m By (2.6), the operator t 1··· c1 ··· cm (u) can be represented as the commutator h i 1··· m m t 1··· c1 ··· cm (u) = · · · [t 1··· m (u), E mcm ], E m−1,cm−1 , . . . , E pc p , where p is the minimum of the indices i such that ci 6= i. Here, as before, we identify (1) m the elements E i j and ti j using the embedding (3.6). The operator t 1··· 1··· m (u) acts on the basis vectors ξ3 by scalar multiplication (see (2.17)). Furthermore, E i j with i < j is a commutator in the generators E k,k+1 with k = 1, . . . , n − 1. By the Gelfand-Tsetlin formulas (2.18), E k,k+1 ξ30 is a linear combination of the basis vectors ξ30 +δka , where a = 1, . . . , k. The proof is completed by the application of the induction hypothesis to the pattern 30 . LEMMA 3.5 The (n − 1)th row of the pattern 30 is (λ1 , . . . , λn−1 ).
Proof We prove the following property of 30 , which clearly implies the statement. For every r = 1, . . . , n − 1, we have the following: if i ≥ r and λ0n−1,i = λ0n−2,i−1 = · · · = λ0n−r,i−r +1 , then either λ0n−1,i = λi , or i ≥ r + 1 and λ0n−1,i = λ0n−2,i−1 = · · · = λ0n−r,i−r +1 = λ0n−r −1,i−r . Suppose the contrary, and let r take the minimum value for which the property fails. That is, there exists i ≥ r such that 30 := 30 + δn−1,i + · · · + δn−r,i−r +1 is a pattern. Since ζ is a singular vector, we have 1··· n−r t 1··· n−r −1,n (u) ζ = 0.
By Proposition 2.2, X
X
c1 <···
c ··· c
n−r n−r 1 t 1··· c1 ··· cn−r (u) ξ3 ⊗ t 1··· n−r −1,n (u) η3 = 0.
(3.12)
(3.13)
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The coefficient of the vector ξ30 ⊗ η30 in the expansion of the left-hand side of (3.13) must be zero. Let us determine which patterns 3 yield a nontrivial contribution to this n−r coefficient. Considering the weight of t 1··· c1 ··· cn−r (u) ξ3 , we come to the relation w(3) + ε1 + · · · + εn−r − εc1 − · · · − εcn−r = w(30 ) + εn−r − εn , and hence, w(3) − w(30 ) = εc1 + · · · + εcn−r − ε1 − · · · − εn−r −1 − εn . Since w(3) w(30 ) by Lemma 3.3, we obtain from (3.5) that ca = a for a = 1, . . . , n − r − 1, and cn−r ∈ {n − r, . . . , n}. Then w(3) = w(30 ) + εcn−r − εn . By Lemma 3.4, 3 should be obtained from 30 by increasing exactly one entry by 1 in each of the rows cn−r , cn−r + 1, . . . , n − 1. On the other hand, by the minimality of r and the betweenness conditions (2.15), the array 3 would not be a pattern unless cn−r = n − r or cn−r = n. Thus, the coefficient in question can have a contribution from only two summands in (3.13), namely, 1··· n−r −1,n n−r t 1··· 1··· n−r −1,n (u) ξ30 ⊗ t 1··· n−r −1,n (u) η30
(3.14)
n−r 1··· n−r t 1··· 1··· n−r (u) ξ30 ⊗ t 1··· n−r −1,n (u) η30 .
(3.15)
and We consider (3.14) first. By (2.6), 1··· n−r n−r t 1··· 1··· n−r −1,n (u) = t 1··· n−r (u), E n−r,n .
(3.16)
It has been observed above that the minimality of r and the betweenness conditions (2.15) imply that E cn ξ30 = 0 for n − r < c ≤ n − 1. Hence, using the relation h i E n−r,n = E n−r,n−r +1 , · · · E n−3,n−2 , [E n−2,n−1 , E n−1,n ] · · · , we get E n−r,n ξ30 = (−1)r −1 E n−1,n E n−2,n−1 · · · E n−r,n−r +1 ξ30 . Therefore, by (2.18), the expansion of E n−r,n ξ30 in terms of the basis vectors ξ3 contains ξ30 with a nonzero coefficient C. It is now convenient to use polynomial quantum minor operators defined by T
a1 ··· am b1 ··· bm (u)
am = u(u − 1) · · · (u − m + 1) t ab11 ··· ··· bm (u)
(see (2.3)). Using (2.17), we find from (3.16) that the coefficient of ξ30 in the expann−r sion of T 1··· 1··· n−r −1,n (u) ξ30 equals 0 0 C (u + ln−r,1 ) · · · ∧ j · · · (u + ln−r,n−r ),
j := i − r + 1,
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where C is a nonzero constant. For the second factor in (3.14), we find from (2.3) and Lemma 3.2 that T
1··· n−r −1,n 1··· n−r −1,n (u) η30
= (u + m 1 ) · · · (u + m n−r −1 )(u + m n + r ) η30 .
Consider now expression (3.15). By (2.17), we have T
1··· n−r 1··· n−r (u) ξ30
0 0 0 = (u + ln−r,1 ) · · · (u + ln−r, j + 1) · · · (u + ln−r,n−r ) ξ30 .
Since ζ is a gln –weight vector and w(30 ) = w(30 ) + εn−r − εn , the vector η30 is a 0 with w(M) = µ−ε 0 linear combination of the ξ M n−r +εn , where {ξ M } is the GelfandTsetlin basis of L(µ). This implies that the (n − r )th row of each of the patterns M is (µ1 , . . . , µn−r −1 , µn−r − 1). We therefore have E n−r,n η30 = const · ξ 0 , and so by (2.17) and (3.16), T
1··· n−r 1··· n−r −1,n (u) η30
= const · (u + m 1 ) · · · (u + m n−r −1 ) ξ 0 .
Combining the results of the above calculations and taking the coefficient of the vector ξ30 ⊗ η30 in (3.12), we obtain 0 0 C(u + ln−r,1 ) · · · ∧ j · · · (u + ln−r,n−r )(u + m 1 ) · · · (u + m n−r −1 )(u + m n + r ) 0 0 0 + const · (u + ln−r,1 ) · · · (u + ln−r, j + 1) · · · (u + ln−r,n−r )
× (u + m 1 ) · · · (u + m n−r −1 ) = 0. Deleting the common factors gives 0 C (u + m n + r ) + const · (u + ln−r, j + 1) = 0. 0 0 Put u = −ln−r, j − 1 in this relation. Since C is nonzero, we get m n = ln−r, j − r + 1 0 and thus m n = ln−1,i . By the betweenness conditions for 30 and 30 ,
λi − λ0n−1,i > 0
and
λ0n−1,i − λi+1 ≥ 0,
which implies that both differences li − m n and m n − li+1 are positive integers. Thus, m n ∈ hli+1 , li i ⊆ hln , l1 i, which contradicts (3.3). Therefore, our assumption that 30 is a pattern must be wrong. By Lemma 3.4, we can now conclude that all vectors ξ3 which occur in (3.4) belong to the U(gln−1 )-span of the highest vector ξ of L(λ). This span is isomorphic to the irreducible representation L(λ− ) of gln−1 with the highest weight λ− = (λ1 , . . . , λn−1 ). In particular, E nn ξ3 = λn ξ3 for each 3. Furthermore, if ζ is a linear combination 0 , then by Lemma 3.2, for the corresponding patterns we have of vectors ξ3 ⊗ ξ M w(3) + w(M) = w(30 ) + µ.
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0 = µ ξ 0 for all M, which implies that the (n − 1)th row of each Therefore, E nn ξ M n M 0 belongs to pattern M coincides with (µ1 , . . . , µn−1 ). In other words, each vector ξ M 0 the U(gln−1 )-span of ξ which is isomorphic to L(µ− ), where µ− = (µ1 , . . . , µn−1 ). Thus, ζ belongs to L(λ− ) ⊗ L(µ− ). (3.17)
By (2.2) and the defining relations (2.1), the Y(n − 1)-module structure on (3.17) coincides with the one obtained by restriction from Y(n) to the subalgebra generated by the ti j (u) with 1 ≤ i, j ≤ n − 1. The vector ζ is annihilated by the operators b1 (u), . . . , bn−2 (u). By the assumption of the theorem, for each pair (i, j) such that 1 ≤ i < j ≤ n − 1, condition (3.1) is satisfied. Therefore, the Y(n − 1)-module (3.17) is irreducible by the induction hypothesis, and we may finally conclude from Proposition 2.3 that (3.2) holds. Next, we derive a similar result for the singular lowest vectors under the assumptions of Theorem 3.1. As we pointed out, condition (3.3) can also be assumed due to Proposition 2.6. LEMMA 3.6 If ζ 0 ∈ L(λ) ⊗ L(µ) and ti j (u) ζ 0 = 0 for all 1 ≤ j < i ≤ n, then
ζ 0 = const · η ⊗ η0 ,
(3.18)
where η and η0 are the lowest vectors of L(λ) and L(µ), respectively. Proof Let ω be the permutation of the indices 1, . . . , n such that ω(i) = n − i + 1. The mapping Y(n) → Y(n), ti j (u) 7 → tω(i)ω( j) (u) (3.19) defines an automorphism of the Yangian Y(n). This follows easily from the defining relations (2.1). We equip the space L = L(λ) ⊗ L(µ) with another structure of the Y(n)-module which is obtained by pulling back through the automorphism (3.19). Denote this new representation by L ω . Similarly, the mapping U(gln ) → U(gln ),
E i j 7 → E ω(i)ω( j)
defines an automorphism of U(gln ). Denote by L(λ)ω the representation of U(gln ) obtained from L(λ) by pulling back through this automorphism, and extend it to Y(n) using (2.14). It follows from (2.2) that the Y(n)-module L ω is isomorphic to the tensor product L(λ)ω ⊗ L(µ)ω . The weight of the lowest vector η of L(λ) is (λn , . . . , λ1 ). Therefore η, when regarded as an element of L(λ)ω , is the highest vector of the weight λ. In particular, the U(gln )-module L(λ)ω is isomorphic to L(λ). Now, ζ 0 is a singular
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vector of the Y(n)-module L ω . By the claim proved above for the singular vectors, ζ 0 is, up to a constant factor, the tensor product of the highest vectors of L(λ)ω and L(µ)ω ; that is, (3.18) holds. To complete the proof of the theorem, we need to show that the submodule of L = L(λ) ⊗ L(µ) generated by the tensor product of the highest vectors ζ = ξ ⊗ ξ 0 coincides with L. For this we introduce a Y(n)-module structure on the space L ∗ dual to L. It follows immediately from the defining relations (2.1) that the mapping σ : Y(n) → Y(n),
ti j (u) 7 → ti j (−u)
defines an antiautomorphism of Y(n). Now L ∗ becomes a Y(n)-module if we set (y f )(v) = f σ (y)v , y ∈ Y(n), f ∈ L ∗ , v ∈ L . (3.20) Similarly, the dual space L(λ)∗ of the gln -module L(λ) can be regarded as a gln module with the action defined by (E i j f )(v) = f (−E i j v),
f ∈ L(λ)∗ , v ∈ L(λ).
We obtain easily from (2.2) that the Y(n)-module L ∗ is isomorphic to the tensor product L(λ)∗ ⊗ L(µ)∗ , where L(λ)∗ and L(µ)∗ are extended to Y(n) by (2.14). The vector ξ ∗ ∈ L(λ)∗ , dual to the highest vector ξ , is the lowest vector with the weight −λ. The highest weight of L(λ)∗ is therefore −λω = (−λn , . . . , −λ1 ). Thus, we have L ∗ ' L(−λω ) ⊗ L(−µω ).
(3.21)
If we assume that the vector ζ generates a proper submodule N in L, then its annihilator Ann N = f ∈ L ∗ f (v) = 0 for all v ∈ N is a nonzero submodule in L ∗ . Hence, Ann N must contain a vector ζ 0 that is annihilated by the generators ti j (u) with i > j. However, condition (3.3) remains satisfied when λ and µ are replaced, respectively, with −λω and −µω . So by Lemma 3.6, the vector ζ 0 must be, up to a constant factor, the tensor product of the lowest vectors of the representations L(−λω ) and L(−µω ). But the vector ξ ∗ ⊗ ξ 0 ∗ does not belong to Ann N . This results in a contradiction, and so the submodule generated by ζ must coincide with L. This completes the proof of Theorem 3.1. 4. Necessary conditions We keep using the notation (2.19). As in the previous section, we assume that λ and µ are complex gln –highest weights.
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THEOREM 4.1 Suppose that the Y(n)-module L(λ) ⊗ L(µ) is irreducible. Then for each pair of indices 1 ≤ i < j ≤ n, we have
m j , m i 6∈ hl j , li i
or
l j , li 6 ∈ hm j , m i i.
(4.1)
Proof The proof follows from a sequence of lemmas. We use induction on n. Given a gln – highest weight λ = (λ1 , . . . , λn ), we set λ− = (λ1 , . . . , λn−1 )
and
λ+ = (λ2 , . . . , λn ).
4.2 If the Y(n)-module L(λ)⊗L(µ) is irreducible, then Y(n−1)-modules L(λ− )⊗L(µ− ) and L(λ+ ) ⊗ L(µ+ ) are both irreducible. LEMMA
Proof We identify L(λ− ) and L(µ− ) with the U(gln−1 )-spans of the highest vectors ξ in L(λ) and ξ 0 in L(µ), respectively. Any generator E in of gln with i < n annihilates L(λ− ) and L(µ− ). Hence, by (2.2) and (2.14), the subspace L(λ− )⊗L(µ− ) of L(λ)⊗ L(µ) is invariant with respect to the action of the subalgebra Y(n − 1) of Y(n), and this action coincides with the one defined in Section 2. Suppose there is a nonzero submodule in L(λ− ) ⊗ L(µ− ) which does not contain the vector ξ ⊗ ξ 0 . Then this submodule contains a Y(n − 1)–singular vector ζ . However, ζ must also be a Y(n)–singular vector. Indeed, tin (u) ζ = 0 for any i < n, which easily follows from (2.1) and (2.2). This implies that L(λ) ⊗ L(µ) is not irreducible, a contradiction. Suppose now that the Y(n − 1)-submodule of L − = L(λ− ) ⊗ L(µ− ) generated by ξ ⊗ ξ 0 is proper. It follows from the defining relations (2.1) that the mapping τ : Y(n − 1) → Y(n − 1),
ti j (u) 7 → t ji (u)
defines an antiautomorphism of Y(n − 1). The dual space L −∗ becomes a Y(n − 1)module if we set (y f )(v) = f τ (y)v , y ∈ Y(n − 1), f ∈ L −∗ , v ∈ L − . We easily derive from (2.2) that the module L −∗ is isomorphic to L(µ− ) ⊗ L(λ− ). Since ξ ⊗ ξ 0 generates a proper submodule in L − , its annihilator in L −∗ is a nonzero submodule that does not contain the vector ξ 0 ⊗ ξ ∈ L(µ− ) ⊗ L(λ− ). However, the Y(n)-module L(µ) ⊗ L(λ) is irreducible by Proposition 2.6. Thus, our assumption
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leads again to a contradiction due to the previous argument. This proves that the Y(n− 1)-module L(λ− ) ⊗ L(µ− ) is irreducible. Now consider the Y(n − 1)-module L(λ+ ) ⊗ L(µ+ ). If the Y(n)-module L = L(λ) ⊗ L(µ) is irreducible, then so is the module L ∗ defined in (3.20). To complete the proof, we apply the isomorphism (3.21) and the above argument. Due to Lemma 4.2, if the Y(n)-module L(λ) ⊗ L(µ) is irreducible, then by the induction hypothesis, conditions Cond(Aλ− , Aµ− ) and Cond(Aλ+ , Aµ+ ) both hold. Therefore, by Proposition 2.8, conditions (4.1) are satisfied for all pairs (i, j) 6= (1, n). Suppose they are violated for the pair (1, n). Then, as was shown in the proof of Proposition 2.8, condition (2.21) should hold. Using Proposition 2.6 if necessary, we may assume that m n ∈ hln , l1 i and l1 ∈ hm n , m 1 i. Therefore, there exist indices p, q ∈ {1, . . . , n − 1} such that m n ∈ hl p+1 , l p i
l1 ∈ hm q+1 , m q i.
and
(4.2)
If p = n − 1, then by Cond(Aλ+ , Aµ+ ) we must have q = 1 (cf. the proof of Proposition 2.8). Thus, l1 ∈ hm 2 , m 1 i. Moreover, using also the condition Cond(Aλ− , Aµ− ), we conclude that there should exist indices r and s such that m 2 , . . . , m r ∈ {l2 , . . . , ls },
ls+1 , . . . , ln−1 ∈ {m r +1 , . . . , m n−1 },
as shown in the following picture:
s
ln
mn c
c s
c s
c
ln−1
m r +1 c s
mr c s s
ls+1
ls
c s
m2 c s s
m1 c
l1
In particular, this implies that li − m i ∈ Z+
for all i = 2, . . . , n − 1.
(4.3)
Now let p ≤ n − 2 in (4.2). Conditions Cond(Aλ− , Aµ− ) and Cond(Aλ+ , Aµ+ ) imply that l p−i+1 = m n−i for i = 1, . . . , p − 1 (4.4) while l1 ∈ hm n− p+1 , m n− p i,
(4.5)
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A. I. MOLEV
as illustrated:
s
ln
s
s
s
mn c
l p+1
m n−1 c s
c s
c s
m n− p+1 m n− p c c s s
lp
l2
c
c
m1 c
l1
Let L be a highest weight module over Y(n) generated by a vector ζ such that Tii (u) ζ = (u + λi )(u + µi ) ζ,
i = 1, . . . , n,
(4.6)
(r )
where Ti j (u) = u 2 ti j (u). We also suppose that the elements ti j with r ≥ 3 act trivially on L so that the operators Ti j (u) are polynomials in u. In other words, L is a module over the quotient algebra Y(n)/I , where I is the ideal generated by the (r ) elements ti j with r ≥ 3. Given sequences a1 , . . . , am and b1 , . . . , bm of elements of {1, . . . , n}, we denote am by T ab11 ··· ··· bm (u) the corresponding quantum minor defined, respectively, by (2.3) or (2.4), with ti j (u) replaced by Ti j (u). Similarly, we define the operators Bm (u) and Am (u) by (2.8) with the same replacement. For 1 ≤ a < r ≤ n, introduce the raising operators τar (v) and lowering operators τra (v) (see [16]) by τar (v) = T
1 ··· a 1 ··· a−1,r (v),
τra (v) = T
a+1 ··· r a ··· r −1 (v),
where v is a variable. We also set τra (v) ≡ 1 for r ≤ a. For any nonnegative integer k, introduce the product of the lowering operators Tra (v, k) = τra (v + k − 1) · · · τra (v + 1) τra (v)
if k ≥ 1,
(4.7)
while Tra (v, 0) ≡ 1. Suppose now that η ∈ L is a Y(n − 1)–singular vector. That is, η is annihilated by Ti j (u) for 1 ≤ i < j ≤ n − 1 and Tii (u) η = νi (u) η,
i = 1, . . . , n − 1,
(4.8)
for some polynomials νi (u) of degree two. The next three lemmas provide a basis for our calculations. LEMMA 4.3 We have the following relations in L:
Tii (u) Tna (v, k) η = νi (u) Tna (v, k) η
(4.9)
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if 1 ≤ i ≤ n − 1 and i 6= a, while Taa (u) Tna (v, k) η =
(u − v − k) νa (u) Tna (v, k) η u−v n X k νa (v) νa+1 (v − 1) · · · νc−1 (v − c + a + 1) + u−v c=a+1
× Tca (u) Tna (v + 1, k − 1) τnc (v − c + a) η.
(4.10)
Moreover, Ti,i+1 (u) Tna (v, k) η = 0
(4.11)
if 1 ≤ i < n − 1 and i 6= a, while for a < n − 1, n X k νa (v) νa+1 (v − 1) · · · νc−1 (v − c + a + 1) u−v
Ta,a+1 (u) Tna (v, k) η =
c=a+1
× Tc,a+1 (u) Tna (v + 1, k − 1) τnc (v − c + a) η.
(4.12)
In particular, if νa (−ρ) = 0 for some ρ, then Taa (u) Tna (−ρ, k) η =
(u + ρ − k) νa (u) Tna (−ρ, k) η, u+ρ
(4.13)
and for a < n − 1, Ta,a+1 (u) Tna (−ρ, k) η = 0.
(4.14)
Proof Note that the coefficients of Tna (v, k) are linear combinations of monomials in the ( j) generators tr s with a ≤ s ≤ n − 1. Suppose that i < a. We have Til (u) η = 0 for a ≤ l ≤ n − 1. Therefore, applying (2.1) to the commutators [Tii (u), Tr s (v)] and [Til (u), Tr s (v)], we conclude by an easy induction that Til (u) Tna (v, k) η = 0. This proves (4.9) and (4.11) for i < a. For i > a, both these relations are immediate from (2.7). Further, by (2.6), Taa (u) Tna (v, k) = +
1 u−v−k+1
u−v−k τ (v + k − 1) Taa (u) Tna (v, k − 1) u − v − k + 1 na n X c ··· n c−a−1 Tca (u) T aa ···b . ··· n−1 (v + k − 1) Tna (v, k − 1) (−1)
c=a+1
The subalgebra Ya of Y(n) generated by tr s (u) with a ≤ r, s ≤ n is naturally isomorphic to the Yangian Y(n −a +1). Applying the automorphism (2.5) to this subalgebra, we derive from the defining relations (2.1) T
a ···b c ··· n a ··· n−1 (v
+ 1) T
a+1 ··· n a ··· n−1 (v)
=T
a+1 ··· n a ··· n−1 (v
+ 1) T
a ···b c ··· n a ··· n−1 (v)
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for every c = a + 1, . . . , n. Hence, T
a ···b c ··· n a ··· n−1 (v
+ k − 1) Tna (v, k − 1) = Tna (v + 1, k − 1) T
a ···b c ··· n a ··· n−1 (v).
Note that Tca (u) commutes with τna (v + k − 1) by (2.7). Therefore, an easy induction on k gives u−v−k Tna (v, k) Taa (u) u−v n X Tca (u) Tna (v + 1, k − 1) T
Taa (u) Tna (v, k) = +
k u−v
a ···b c ··· n c−a−1 . a ··· n−1 (v) (−1)
(4.15)
c=a+1
The same argument proves the following counterpart of (4.15): for a < n − 1, Ta,a+1 (u) Tna (v, k) =
u−v−k Tna (v, k) Ta,a+1 (u) u−v n X k + Tc,a+1 (u) Tna (v + 1, k − 1) u−v ×T
c=a+1 a ···b c ··· n c−a−1 . a ··· n−1 (v) (−1)
(4.16)
By (2.4), T
a ···b c ··· n a ··· n−1 (v)
X
=
sgn σ · Tn,σ (n−1) (v − n + a + 1) · · · Tc+1,σ (c) (v − c + a)
σ ∈Sn−a
× Tc−1,σ (c−1) (v − c + a + 1) · · · Ta,σ (a) (v). Since Ti j (u) η = 0 for 1 ≤ i < j ≤ n − 1, we conclude from (4.8) that T
a ···b c ··· n a ··· n−1 (v) η
= τnc (v − c + a) νa (v) νa+1 (v − 1) · · · νc−1 (v − c + a + 1) η.
The proof is completed by using (4.15) and (4.16). 4.4 Let 1 ≤ a < n − 1. Then we have the relations E n−1,n Tna (v, k) = −k Tna (v, k − 1) T LEMMA
a+1 ··· n a ··· n−2,n (v
+ k − 1).
(4.17)
Moreover, T
a+1 ··· n a+1 ··· n−1 a ··· n−2,n (u) T a+1 ··· n−1 (u)
= τn−1,a (u) T
a+1 ··· n a+1 ··· n−1 a+1 ··· n (u) + τna (u) T a+1 ··· n−2,n (u).
(4.18)
IRREDUCIBILITY CRITERION FOR YANGIAN MODULES
Proof By (2.6), [E n−1,n , τna (v)] = −T mute, we can write
329
a+1 ··· n a ··· n−2,n (v). Since elements τna (u) and τna (v) com-
k X τna (v) · · · T E n−1,n Tna (v, k) = −
a+1 ··· n a ··· n−2,n (v+i−1) · · · τna (v+k−1).
(4.19)
i=1
Applying automorphism (2.5) to the subalgebra Ya introduced in the proof of Lemma 4.3, we derive from (2.1) that T
a+1 ··· n a ··· n−2,n (v) τna (v
+ 1) = τna (v) T
a+1 ··· n a ··· n−2,n (v
+ 1).
(4.20)
Together with (4.19), this proves (4.17) by an easy induction. To prove (4.18), consider the expression provided by Proposition 2.1 for the com··· n mutator [τn−1,a (u), T a+1 a+1 ··· n (v)]. Multiply both sides of the relation by u −v, and put u = v. The terms that do not vanish after this operation correspond to the maximum value of the summation parameter in the formula, which implies (4.18). We keep using the notation of Lemma 4.3. As before, ζ is the highest vector of L satisfying (4.6). We also regard L as a gln -module using (3.6). 4.5 We have the following relations in L: LEMMA
E in Tna (v, k) ζ = 0
if i < a.
(4.21)
If a < i ≤ n − 1, then E in Tna (v, k) ζ = (−1)n−i k
n Y
(v 0 + l j )(v 0 + m j ) τia (v + k − 1) Tna (v, k − 1) ζ,
j=i+1
(4.22) where
v0
= v + a + k − 1. Moreover, if 1 ≤ a ≤ n − 1, then
E an Tna (v, k) ζ = (−1)n−a−1 k × Tna (v + 1, k − 1) T
a ··· n−1 a ··· n−1 (v) − Tna (v, k
− 1) T
a+1 ··· n a+1 ··· n (v
+ k − 1) ζ. (4.23)
Proof The relation (4.21) follows from the fact that the coefficients of Tna (v, k) belong to the subalgebra Ya (see the proof of Lemma 4.3). We easily deduce from (2.6) that ··· n−1 n−a−1 a+1 ··· n E an , τna (v) = (−1)n−a−1 T aa ··· T a+1 ··· n (v). n−1 (v) − (−1)
330
A. I. MOLEV
Therefore, E an Tna (v, k) ζ = (−1)n−a−1 ×
k X
τna (v) · · · T
a ··· n−1 a+1 ··· n a ··· n−1 (v+i −1)−T a+1 ··· n
(v+i −1) · · · τna (v+k−1) ζ.
i=1
(4.24) Furthermore, by analogy with (4.20) we get T
a+1 ··· n a+1 ··· n
(v) τna (v + 1) = τna (v) T
a+1 ··· n a+1 ··· n
(v + 1).
Expression (4.24) now takes the form (−1)n−a−1
k X
τna (v) · · · τna (v + i − 2) T
a ··· n−1 a ··· n−1 (v
+ i − 1) Tna (v + i, k − i) ζ
i=1
− (−1)n−a−1 k Tna (v, k − 1) T
a+1 ··· n a+1 ··· n
(v + k − 1) ζ.
Similarly, again applying (2.5) to Ya , we bring the sum here by an easy induction to the form k X
+ i − 1) · · · τna (v) T (k − i + 1) τna (v + k − 1) · · · τna (v\
a ··· n−1 a ··· n−1 (v
+ i − 1)
i=1
\ − (k − i) τna (v + k − 1) · · · τna (v + i) · · · τna (v) T
a ··· n−1 a ··· n−1 (v
+ i) ζ,
n−1 which simplifies to k Tna (v + 1, k − 1) T aa ··· ··· n−1 (v) ζ , thus proving (4.23). ··· n By (2.6), if i > a, then [E in , τna (v)] = (−1)n−i T a+1 (v). As in the proof of a ···b i ··· n (4.17), this brings the left-hand side of (4.22) to the form
(−1)n−i k Tna (v, k − 1) T
a+1 ··· n (v a ···b i ··· n
+ k − 1) ζ.
Using (2.3), we get T
a+1 ··· n (v a ···b i ··· n
+ k − 1) ζ =
n Y
(v 0 + l j )(v 0 + m j ) τia (v + k − 1) ζ,
j=i+1
which completes the proof. We now consider the irreducible highest weight module V (λ, µ) over Y(n) generated by the highest vector ζ satisfying (4.6). It follows easily from (2.1) and the irre(r ) ducibility of V (λ, µ) that all elements ti j with r ≥ 3 act trivially in this module.
IRREDUCIBILITY CRITERION FOR YANGIAN MODULES
331
Furthermore, V (λ, µ) is isomorphic to the irreducible quotient of the submodule of L(λ) ⊗ L(µ) generated by the vector ξ ⊗ ξ 0 (see (2.2) and (2.14)). With the parameter p defined in (4.2), the numbers ki = li − m n− p+i ,
i = 1, . . . , p,
are positive integers by (4.3) and (4.4). Introduce the vector θ ∈ V (λ, µ) by 0 0 θ = Tn− p+1,1 (−λ1 , k1 ) Tn− p+2,2 (−λ2 , k2 ) · · · Tnp (−λ p , k p ) ζ,
(4.25)
where ζ is the highest vector of V (λ, µ), and Tra0 (v, ka ) denotes the derivative of the polynomial Tra (v, ka ) (see (4.7)). Our aim is to prove that the vector θ is zero. We do this in Lemma 4.7. The idea of the proof is to show that θ is annihilated by the operators Bi (u) for all i = 1, . . . , n − 1 and then apply Proposition 2.3 noting that θ is obviously not proportional to ζ . This works directly in the case p = 1. However, if p ≥ 2, then applying the operators Bi (u) to θ , we come to a more general problem— to prove that all vectors parametrized by a certain finite family of patternlike arrays 3 associated with λ are zero. We first prove a preliminary lemma that describes the properties of these vectors. The arrays 3 that arise in this way are called admissible. They are defined as follows. Each 3 is a sequence of rows 3r = (λr 1 , . . . , λrr ) with r = 1, . . . , n of the form described in Section 2. The top row 3n coincides with λ, and for all r the following conditions hold: λri − λr −1,i ∈ Z+
for i = 1, . . . , r − 1.
Each entry λri of 3 is equal to λi unless i = 2, . . . , p
and
r < n − p + i.
(4.26)
Moreover, we also require that if i ∈ {2, . . . , p}, then lii − m n− p+i ∈ Z+ ,
(4.27)
where we denote lri = λri − i + 1. This condition implies that 0 ≤ λri − λr −1,i ≤ ki for all i. By definition, only a part of an admissible array can vary with the remaining entries fixed, as illustrated:
λ1
λ p+1 ··· %%e e · · · λ p−1 %λn−1, p λ p+1 · · · e % ··· ··· e ··· % ··· e λ p+1 % λ1 %λn− p+1,2 · · · λee pp e %% λ1e · · · ··· % e % e % e λ1 λ22 e % λ e%
··· λ1
λp
λn λn−1
332
A. I. MOLEV
Given such an array, we set θ3 =
→ Y r =3,...,n
p Y
Teri (−λri , λri − λr −1,i ) ζ,
i=2
where the polynomials Teri (v, k) are defined by ( Tri0 (v, k) if r = n − p + i and k = ki , Teri (v, k) = Tri (v, k) otherwise.
(4.28)
Recall also that Tri (v, k) ≡ 1 if r ≤ i. Note that the factors in the brackets commute 0 = λ for all for any index r due to (2.7). We have ζ = θ30 for the array 30 with λri i r and i. Furthermore, θ = Tn− p+1,1 (−λ1 , k1 ) θ3 for the array 3 with lri = m n− p+i for all indices r and i satisfying (4.26). We define the weight w(3) of an admissible array 3 by (3.7). We use the ordering on the weights described in Section 3. Given 3, take the minimum index r = r (3) such that for some 2 ≤ a ≤ p the difference λra − λr −1,a is a positive integer. The following relations for the admissible arrays 3 with r (3) ≥ n − p + 2 are used in Lemma 4.7. 4.6 We have the following relations in V (λ, µ): LEMMA
Tii (u) θ3 = (u + λr −1,i )(u + µi ) θ3
for 1 ≤ i ≤ r − 1,
(4.29)
Ti j (u) θ3 = 0
for 1 ≤ i < j ≤ r − 1,
(4.30)
for r − 1 ≤ i < n,
(4.31)
Bi (u) θ3 =
i X
βi j (u, 3) θ3+δi j
j=1
where βi j (u, 3) are some polynomials in u, and we suppose that θ3 = 0 if 3 is not an admissible array. Proof Let 2 ≤ a ≤ p be the least index such that k = λra − λr −1,a > 0. Then θ3 = Tera (−λra , k) η,
η := θ30 ,
(4.32)
where 30 is the array obtained from 3 by increasing each entry λaa , . . . , λr −1,a by k. We use a (reverse) induction on the pairs (r, a) ordered lexicographically, with the base θ3 = ζ . Note that by the definition of admissible arrays we must have r ≤ n − p + a. Identify the subalgebra of Y(n) generated by ti j (u) with 1 ≤ i, j ≤ r with the Yangian Y(r ). By the induction hypothesis, η is a Y(r − 1)–singular vector such that
IRREDUCIBILITY CRITERION FOR YANGIAN MODULES
333
(4.8) holds with n replaced by r , where ( (u + λri )(u + µi ) if i ≤ a, νi (u) = (u + λr −1,i )(u + µi ) if i > a. Therefore, if Tera (−λra , k) = Tra (−λra , k), then (4.29) and (4.30) are immediate from Lemma 4.3. Suppose now that Tera (−λra , k) = Tra0 (−λra , k). Then r = n− p+a and k = ka . Therefore, λra = λa by (4.27). It is clear from Lemma 4.3 that (4.29) and (4.30) hold for i 6= a, so we may assume that i = a. In this case, due to (4.10) and (4.12), it suffices to show that Tra (−λa , ka ) η = 0 (4.33) and that for every c = a + 1, . . . , r , the polynomial νa (v) νa+1 (v − 1) · · · νc−1 (v − c + a + 1) Tra (v + 1, ka − 1) τr c (v − c + a) η (4.34) has zero of multiplicity at least two at v = −λa . Suppose first that p < n − 1, and consider Tra (−λa , ka ) η. By (2.3), we have X τra (v) = sgn σ · Tσ (a+1),a (v) · · · Tσ (r ),r −1 (v − r + a + 1). (4.35) σ ∈Sr −a
By the induction hypothesis, we have Tσ (r ),r −1 (u) η = 0 if σ (r ) < r − 1, while Tr −1,r −1 (u) η = (u + λr −1,r −1 )(u + µr −1 ) η. The factor u + µr −1 is zero if u = −λa − r + a + 1 by (4.4). Therefore, if v = −λa , then we may assume that σ (r ) = r in (4.35), which gives τra (−λa ) η = τr −1,a (−λa ) Tr,r −1 (−µr −1 ) η. Since Tr,r −1 (v) = τr,r −1 (v) is a lowering operator, we verify by an easy induction with the use of (2.7) and (4.13) that Tra (−λa , ka ) η = Tr −1,a (−λa , ka ) Tr,r −1 (−µr −1 , ka ) η.
We have ka = m r −1 − m r by (4.4), and so the equality Tra (−λa , ka ) η = 0 in the case p < n − 1 is implied by the fact that the vector e η = Tr,r −1 (−µr ) · · · Tr,r −1 (−µr −1 ) η
(4.36)
is zero in V (λ, µ). Since the Y(n)-module V (λ, µ) is irreducible, it is sufficient to show, due to Proposition 2.3, that the vector e η is annihilated by all operators Bi (u)
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A. I. MOLEV
with i = 1, . . . , n − 1. Since Tr,r −1 (u) commutes with the lowering operators τri (v), we may assume that the array 3 satisfies λri = λr −1,i for all i 6= a. In other words, η = θ30 is a Y(r )–singular vector such that Tii (u) η = (u + λri )(u + µi ) η
for i = 1, . . . , r.
By (4.11), e η is annihilated by the operators Ti,i+1 (u) for i = 1, . . . , r − 2, and by (4.9) and (4.13), e η is an eigenvector for the operators Tii (u) with i = 1, . . . , r − 1. We have Tr −1,r (u) = [Tr −1,r −1 (u), Er −1,r ]. By (4.23), Er −1,r e η = 0, and therefore Tr −1,r (u)e η = 0. On the other hand, if i ≥ r , then Bi (u) commutes with the elements Tr,r −1 (v) by (2.7). Hence, by the induction hypothesis, Bi (u) e η is a linear combination of the vectors Tr,r −1 (−µr ) · · · Tr,r −1 (−µr −1 ) θ30 +δi j with 2 ≤ j ≤ p. We conclude by induction on the weight of 30 that Bi (u) e η = 0, thus proving (4.33). Consider now the polynomial (4.34). Suppose first that c = r . Note that r −1 > a since p < n − 1. We have νa (v) = (v + λa )(v + µa ),
νr −1 (v) = (v + λr −1,r −1 )(v + µr −1 ).
(4.37)
However, µr −1 − r + a + 1 = λa by (4.4). This shows that the coefficient of η in (4.34) is divisible by (v + λa )2 . Now let a + 1 ≤ c < r . By (2.3), X τr c (−λa −c+a) = sgn σ ·Tσ (c+1),c (−λa −c+a) · · · Tσ (r ),r −1 (−λa −r +a+1). σ ∈Sr −c
We can repeat the argument that we have applied to expression (4.35) to show that the polynomial Tra (v + 1, ka − 1) τr c (v − c + a) η has zero at v = −λa . Together with (4.37), this completes the proof of (4.29) and (4.30) in the case p < n − 1. In the case p = n − 1, we have a = r − 1 and Tr,r −1 (−λr −1 , kr −1 ) η = Tr,r −1 (−µr ) · · · Tr,r −1 (−λr −1 ) η.
Note that operators Tr,r −1 (u) and Tr,r −1 (v) commute. Therefore, due to (4.3), it suffices to show that the vector (4.36) is zero. The argument used in the case p < n − 1 works here as well. For p = n − 1, the polynomial (4.34) equals νr −1 (v) Tr,r −1 (−λr −1 + 1, kr −1 + 1) η. If λr −1 = µr −1 , then νr −1 (v) = (v + λr −1 )2 . If λr −1 − µr −1 > 0, then Tr,r −1 (−λr −1 + 1, kr −1 + 1) η = Tr,r −1 (−µr ) · · · Tr,r −1 (−λr −1 + 1) η.
(4.38)
IRREDUCIBILITY CRITERION FOR YANGIAN MODULES
335
Using again the fact that the vector (4.36) is zero, we conclude that this vector is also zero. In both cases, (4.38) has zero of multiplicity at least two at v = −λr −1 , proving (4.29) and (4.30). To prove (4.31), we note that Bi (u) commutes with Tsa (v, k) for i ≥ s by (2.7). Therefore, it suffices to consider the case i = r − 1. We derive from (2.6) that Br −1 (u) = [Ar −1 (u), Er −1,r ]. Suppose first that p < n − 1. By (4.29) and (4.30), the operator Ar −1 (u) acts on θ3 as multiplication by a polynomial in u. So it suffices to prove that r −1 X Er −1,r θ3 = β j (3) θ3+δr −1, j , (4.39) j=1
where β j (3) are some constants. Write θ3 in the form (4.32), and assume that a < r − 1. We now use Lemma 4.4. By (4.17), we have ··· r 0 Er −1,r Tra (v, k), θ30 = Tra (v, k)Er −1,r θ30 −k Tra (v, k −1)Taa+1 ··· r −2,r (v +k −1)θ3 . (4.40) Further, (4.18) gives
T
a+1 ··· r a+1 ··· r −1 0 a ··· r −2,r (u) T a+1 ··· r −1 (u) θ3 a+1 ··· r = τr −1,a (u) T a+1 ··· r (u) θ30
··· r −1 By (2.6), T a+1 a+1 ··· r −2,r (u) = [T sis, we obtain
a+1 ··· r −1 a+1 ··· r −1 (u), r −1 X
Er −1,r θ30 =
+ τra (u) T
a+1 ··· r −1 0 a+1 ··· r −2,r (u) θ3 .
(4.41)
Er −1,r ]. Using the induction hypothe-
β j (30 ) θ30 +δr −1, j .
j=a+1
On the other hand, using (4.29) and (4.30), we derive from (2.3), T
a+1 ··· r −1 0 a+1 ··· r −1 (u) θ3
rY −1
=
(u + lr −1,i + a)(u + m i + a) θ30 ,
i=a+1
while T
a+1 ··· r 0 a+1 ··· r (u) θ3
=
r Y
(u + lri + a)(u + m i + a) θ30
i=a+1
since T T
a+1 ··· r a+1 ··· r (u)
a+1 ··· r 0 a ··· r −2,r (u) θ3
commutes with the lowering operators τr b (v). Thus, by (4.41), = (u + lrr + a)(u + m r + a)
rY −1 i=a+1
+
r −1 X j=a+1
(30 )
u + lri + a τ (u) θ30 u + lr −1,i + a r −1,a
βj τ (u) θ30 +δr −1, j . u + lr −1, j + a ra
(4.42)
336
A. I. MOLEV
Now put v = −λra and k = λra − λr −1,a into (4.40). The denominator u + lr −1,i + a in (4.42) becomes lr −1,i − lr −1,a at u = v + k − 1. Due to conditions (4.4) and (4.27), the difference lr −1,i − lr −1,a can only be zero if i = a + 1. Moreover, in this case, lr −1,a+1 = lr,a+1 = la+1 . Then 30 + δr −1,a+1 is not an admissible array, so that the summand with j = a + 1 does not occur in the sum in (4.42). The denominator u + lr −1,i + a with i = a + 1 does not occur in the product either since it cancels with u + lr,a+1 + a. Thus, the substitution u = v + k − 1 into (4.42) is well defined. Using the fact that τr −1,b (v) commutes with τra (u) if b ≥ a, we complete the proof of (4.39) for the case Tera (v, k) = Tra (v, k) in (4.32). Assume now that Tera (v, k) = Tra0 (v, k). Then, by (4.28), we must have r = n − p + a and k = ka = la − m r . Moreover, we also have λra = λa by (4.27). Take the derivative with respect to v in (4.40), and put v = −λa . Note that the factor u + m r + a in (4.42) vanishes at u = −λa + ka − 1. Furthermore, as has been shown above, Tra (−λa , ka ) θ30 +δr −1, j = 0 (see (4.33)). The application of the induction hypothesis finally proves (4.39) in the case a < r − 1. If a = r − 1 in (4.32), then η = θ30 is a Y(r )–singular vector, so that we may use the relation (4.23) to prove (4.39). The same relation applies in the case p = n − 1, where we also use the fact that the polynomial (4.38) has zero of multiplicity at least two at v = −λr −1 . Consider the vector θ ∈ V (λ, µ) defined in (4.25). LEMMA 4.7 The vector θ is zero.
Proof We prove by induction on the weight of 3 that Tn− p+1,1 (−λ1 , k1 ) θ3 = 0
(4.43)
for all admissible arrays 3 such that the parameter r = r (3) satisfies r ≥ n − p + 2. For the induction base we note that Tn− p+1,1 (−λ1 , k1 ) ζ = 0.
Indeed, using (2.7), we find that the vector on the left-hand side is annihilated by the operators Bi (u) with i = n − p + 1, . . . , n − 1. On the other hand, by (4.14), it is also annihilated by Ti,i+1 (u) with i = 1, . . . , n − p − 1. Further, Tn− p,n− p+1 (u) = [Tn− p,n− p (u), E n− p,n− p+1 ], and we find from (4.9) and (4.22) that it is also annihilated by E n− p,n− p+1 . By Proposition 2.3, the vector must be zero.
IRREDUCIBILITY CRITERION FOR YANGIAN MODULES
337
Suppose now that w(3) ≺ λ. Denote the left-hand side of (4.43) by e θ3 . We show that Bi (u) e θ3 = 0 for all i = 1, . . . , n − 1. By Lemma 4.6, the Y(n − p + 1)span of the vector θ3 is a highest weight module with the highest weight defined from (4.29) with r = n − p + 2. Exactly as above, we find that Tn− p,n− p+1 (u) θ3 is zero. Furthermore, the operators Bi (u) with i = n − p + 1, . . . , n − 1 commute with Tn− p+1,1 (−λ1 , k1 ). Therefore, by (4.31), Bi (u) e θ3 is a linear combination of the vectors Tn− p+1,1 (−λ1 , k1 ) θ3+δi j . If i ≥ n − p + 2, then the arrays 3 + δi j satisfy the condition r ≥ n − p + 2 on the parameter r = r (3) used in Lemma 4.6, and we complete the proof in this case by applying the induction hypothesis. If i = n − p + 1, then θ3+δn− p+1, j = τn− p+1, j (−λn− p+1, j − 1) θ30 for some array 30 for which the corresponding parameter r 0 = r (30 ) satisfies r 0 ≥ n − p + 2. However, τn− p+1, j (v) is permutable with Tn− p+1,1 (−λ1 , k1 ), which again ensures that Bi (u) e θ3 = 0 by the induction hypothesis. By (2.2) and (2.14), the operators Ti j (u) = u 2 ti j (u) in L(λ) ⊗ L(µ) are polynomials in u. Therefore, we can introduce the vector e θ ∈ L(λ) ⊗ L(µ) by 0 0 0 e θ = Tn− p+1,1 (−λ1 , k1 ) Tn− p+2,2 (−λ2 , k2 ) · · · Tnp (−λ p , k p ) (ξ ⊗ ξ )
(cf. (4.25)). Here ξ and ξ 0 are, respectively, the highest vectors of the gln -modules L(λ) and L(µ). 4.8 The vector e θ is nonzero. LEMMA
Proof Write the vector e θ in the form e θ=
X 3,M
0 c3,M ξ3 ⊗ ξ M ,
(4.44)
0 are the Gelfand-Tsetlin basis vectors in L(λ) and L(µ). It suffices where, ξ3 and ξ M to show that at least one coefficient c3,M is nonzero. We calculate these coefficients 0 = ξ0 for the case where ξ M is the highest vector ξ 0 of L(µ). That is, the (k, i)th M0 entry of the pattern M 0 coincides with µi for all k and i. We prove by a (reverse) induction on a that for any a = 2, . . . , p, we have X (a) 0 0 0 0 Tn− c3,M ξ3 ⊗ ξ M , (4.45) p+a,a (−λa , ka ) · · · Tnp (−λ p , k p ) (ξ ⊗ ξ ) = 3,M
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A. I. MOLEV
where, for each M occurring in this expansion, n−1 X
µ − w(M) =
qi (εi − εi+1 ),
qi ∈ Z+ ,
(4.46)
i=a (a)
and c3(a) ,M 0 6= 0 for the pattern 3(a) defined for each a = 1, . . . , p as follows. The (a)
entry λsi of 3(a) coincides with λi unless i = a, . . . , p and s < n − p + i. For (a) the entries with these s and i, we have λsi − i + 1 = m n− p+i . The betweenness conditions (2.15) for 3(a) are guaranteed by assumptions (4.3) and (4.4). Suppose that a ≤ p, and denote the left-hand side of (4.45) by θ (a) . By Proposition 2.2, we have X b1 ··· bn− p ··· r 1 τra (v) = T a+1 r := n − p + a, b1 ··· bn− p (v) ⊗ T a ··· r −1 (v), b1 <···
where the quantum minor operators in L(λ) and L(µ) are defined by formulas (2.3) and (2.4) and where the ti j (u) are replaced with the polynomial operators Ti j (u) = u ti j (u). If w is a weight of the gln -module L(µ), then w µ. Therefore, by the induction hypothesis, if b1 < a, then T
b1 ··· bn− p 0 a ··· r −1 (v) ξ M
=0
for each pattern M occurring in the expansion (4.45) for θ (a+1) . This proves (4.46). 0 which occur Furthermore, for any k ∈ Z+ , the tensor products of the form ξ3 ⊗ ξ M 0 in the expansion of Tra (v, k) θ (a+1) should have the form Tra (v, k) ξ3(a+1) ⊗ T
a ··· r −1 a ··· r −1 (v
+ k − 1) · · · T
a ··· r −1 0 a ··· r −1 (v) ξ .
(4.47)
The coefficient of ξ 0 equals k Y
(v + µa + j − 1)(v + µa+1 + j − 2) · · · (v + µr −1 + j − n + p).
j=1
Conditions (4.3) and (4.4) imply that for k = ka there is a unique factor in this product which vanishes at v = −λa . Therefore, the derivative of (4.47) with k = ka at v = −λa is, up to a nonzero constant factor, Tn− p+a,a (−λa , ka ) ξ3(a+1) ⊗ ξ 0 .
By definition (2.16), this coincides with ξ3(a) ⊗ ξ 0 , which proves (4.45). Similarly, the application of the operator Tn− p+1,1 (−λ1 , k1 ) to the vector θ (2) produces a linear combination (4.44). Here the coefficient c3(1) ,M 0 is the product of
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(2)
c3(2) ,M 0 and the factor k Y
(−λ1 + µ1 + j − 1)(−λ1 + µ2 + j − 2) · · · (−λ1 + µn− p + j − n + p)
j=1
with k = l1 − m n− p+1 which comes from the expansion of the coefficient of ξ 0 in (4.47) for a = 1. It remains to note that, by (4.5) (which holds for all 1 ≤ p ≤ n − 1), this factor is nonzero. If the Y(n)-module L(λ) ⊗ L(µ) is irreducible, then by (2.2) and (2.14) it is isomorphic to the highest weight module V (λ, µ). Lemmas 4.7 and 4.8 therefore imply that this contradicts assumption (4.2), thus proving Theorem 4.1. Theorem 1.1 is now implied by Theorems 3.1 and 4.1 due to Proposition 2.8. Acknowledgments. This work was completed during the author’s visit to the Erwin Schr¨odinger Institute, Vienna, for the Representation Theory Program, 2000. I would like to thank ESI and the organizers of the program, A. A. Kirillov and V. G. Kac, for the invitation. I am very much obliged to M. L. Nazarov for valuable discussions during the visit. He kindly informed me about the results of [12] and [23] prior to their publication, and the present form of Theorem 1.1 was inspired by the irreducibility criterion in [12]. Originally, the author obtained this result in the form of Theorems 3.1 and 4.1. References [1]
[2] [3] [4] [5] [6] [7]
T. AKASAKA and M. KASHIWARA, Finite-dimensional representations of quantum
affine algebras, Publ. Res. Inst. Math. Sci. 33 (1997), 839 – 867. MR 99d:17017 309 T. ARAKAWA, Drinfeld functor and finite-dimensional representations of Yangian, Comm. Math. Phys. 205 (1999), 1 – 18. MR 2001c:17011 307, 309 V. CHARI and A. PRESSLEY, Yangians and R-matrices, Enseign. Math. (2) 36 (1990), 267 – 302. MR 92h:17009 309, 311, 313 , Fundamental representations of Yangians and singularities of R-matrices, J. Reine Angew. Math. 417 (1991), 87 – 128. MR 92h:17010 311, 313 I. V. CHEREDNIK, A new interpretation of Gelfand-Tzetlin bases, Duke Math. J. 54 (1987), 563 – 577. MR 88k:17005 307 V. G. DRINFELD, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32, no. 1 (1985), 254 – 258. MR 87h:58080 307, 309 , Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl. 20 (1986), 62 – 64. MR 87m:22044 309
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, A new realization of Yangians and of quantum affine algebras, Soviet Math. Dokl. 36, no. 2 (1988), 212 – 216. MR 88j:17020 307, 309, 311 I. M. GELFAND and M. L. TSETLIN, Finite-dimensional representations of the group of unimodular matrices (in Russian), Dokl. Akad. Nauk SSSR 71 (1950), 825 – 828; English translation in Collected Papers, Vol. II, Springer, Berlin, 1988, 653 – 656. MR 89k:01042b 313, 314, 317 A. N. KIRILLOV and N. YU. RESHETIKHIN, The Yangians, Bethe ansatz and combinatorics, Lett. Math. Phys. 12, (1986) 199 – 208. MR 88a:81181 308, 313 N. KITANINE, J. M. MAILLET, and V. TERRAS, Form factors of the XXZ Heisenberg spin- 21 finite chain, Nuclear Phys. B 554 (1999), 647 – 678. MR 2001h:82016 308 B. LECLERC, M. NAZAROV, and J.-Y. THIBON, Induced representations of affine Hecke algebras and the canonical bases of quantum groups, preprint, arXiv:math.QA/0011074 309, 339 B. LECLERC and J.-Y. THIBON, Repr´esentations induites d’alg`ebres de Hecke affines et singularit´es de R-matrices, C. R. Acad. Sci. Paris Ser. I Math. 328 (1999), 1117 – 1122. MR 2000i:20007 309 B. LECLERC and A. ZELEVINSKY, “Quasicommuting families of quantum Pl¨ucker coordinates” in Kirillov’s Seminar on Representation Theory, ed. G. Olshanski˘ı, Amer. Math. Soc. Transl. Ser. 2 181, Amer. Math. Soc., Providence, 1998, 85 – 108. MR 99g:14066 309 J. M. MAILLET and V. TERRAS, On the quantum inverse scattering problem, Nuclear Phys. B 575 (2000), 627 – 644. MR 2001g:82045 308 A. I. MOLEV, Gelfand-Tsetlin basis for representations of Yangians, Lett. Math. Phys. 30 (1994), 53 – 60. MR 94m:17018 309, 313, 314, 326 , Finite-dimensional irreducible representations of twisted Yangians, J. Math. Phys. 39 (1998), 5559 – 5600. MR 99i:81106 309, 311, 313, 316 , Yangians and transvector algebras, preprint, arXiv:math.QA/9811115, to appear in Discrete Math. 310, 314 A. MOLEV, M. NAZAROV, and G. OLSHANSKI˘I, Yangians and classical Lie algebras, Russian Math. Surveys 51, no. 2 (1996), 205 – 282. MR 97f:17019 309, 310 M. NAZAROV AND V. TARASOV, Yangians and Gelfand-Zetlin bases, Publ. Res. Inst. Math. Sci. 30 (1994), 459 – 478. MR 96m:17033 309, 313 , On irreducibility of tensor products of Yangian modules, Internat. Math. Res. Notices 1998, 125 – 150. MR 99b:17014 309 , Representations of Yangians with Gelfand-Zetlin bases, J. Reine Angew. Math. 496 (1998), 181 – 212. MR 99c:17030 307, 309, 311 , On irreducibility of tensor products of Yangian modules associated with skew Young diagrams, Duke Math. J. 112 (2002), 343 – 378 308, 339 V. O. TARASOV, Irreducible monodromy matrices for the R-matrix of the X X Z -model and lattice local quantum Hamiltonians, Theoret. and Math. Phys. 63 (1985), 440 – 454. MR 87d:82022 307, 309 E. VASSEROT, Affine quantum groups and equivariant K -theory, Transform. Groups 3 (1998), 269 – 299. MR 99j:19007 307
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School of Mathematics and Statistics F07, University of Sydney, New South Wales 2006, Australia;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2,
ON IRREDUCIBILITY OF TENSOR PRODUCTS OF YANGIAN MODULES ASSOCIATED WITH SKEW YOUNG DIAGRAMS MAXIM NAZAROV and VITALY TARASOV
Abstract We study the tensor product W of any number of irreducible finite-dimensional modules V1 , . . . , Vk over the Yangian Y(gl N ) of the general linear Lie algebra gl N . For any indices i, j = 1, . . . , k, there is a canonical nonzero intertwining operator Ji j : Vi ⊗ V j → V j ⊗ Vi . It has been conjectured that the tensor product W is irreducible if and only if all operators Ji j with i < j are invertible. We prove this conjecture for a wide class of irreducible Y(gl N )-modules V1 , . . . , Vk . Each of these modules is determined by a skew Young diagram and a complex parameter. We also introduce the notion of a Durfee rank of a skew Young diagram. For an ordinary Young diagram, this is the length of its main diagonal. Introduction In this article we continue our study (see [NT1], [NT2]) of the finite-dimensional modules over the Yangian Y(gl N ) of the general linear Lie algebra gl N . The Yangian Y(gl N ) is a canonical deformation of the enveloping algebra U(gl N [u]) in the class of Hopf algebras (see [D1]). The unital associative algebra Y(gl N ) has an infinite family (s) of generators Ti j , where s = 1, 2, . . . and i, j = 1, . . . , N . In Section 4 we recall the definition of the Hopf algebra Y(gl N ) in terms of the generating series (1)
(2)
Ti j (u) = δi j + Ti j u −1 + Ti j u −2 + · · · .
(0.1)
The classification of the irreducible finite-dimensional Y(gl N )-modules has been given by V. Drinfeld [D2] by generalizing the results of V. Tarasov [T1], [T2]. But the structure of these modules still needs better understanding. For instance, the dimensions of these modules are not explicitly known in general. DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2, Received 8 December 2000. Revision received 4 April 2001. 2000 Mathematics Subject Classification. Primary 17B37. Nazarov’s work supported by the Engineering and Physical Sciences Research Council (United Kingdom) and by the European Commission under Training and Mobility Researchers grant number FMRX-CT97-0100. Tarasov’s work supported by Russian Fund for Fundamental Investigations grant number 99-01-00101 and by International Association (INTAS) grant number 99-01705. 343
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There is a distinguished family of irreducible finite-dimensional Y(gl N )-modules (see [C2], [O]). We call these modules elementary. Every elementary Y(gl N )-module is determined by a complex number z and by two partitions λ = (λ1 , λ2 , . . .) and µ = (µ1 , µ2 , . . .), such that λi > µi for any i = 1, 2, . . . . This module depends on λ and µ through the skew Young diagram ω = λ/µ (see Section 1). We denote the corresponding Y(gl N )-module by Vω (z); its explicit construction is given in Section 4. The vector spaces of the modules Vω (z) are the same for all values of z, and we denote this vector space by Vω . Suppose that for some positive integer M, the numbers of nonzero parts of λ and µ do not exceed M + N and M, respectively. Then the dimension of Vω equals the multiplicity of the irreducible gl M -module of highest weight (µ1 , . . . , µ M ) in the restriction to gl M of the the irreducible gl M+N module of highest weight (λ1 , . . . , λ M+N ). In particular, the space Vω is nonzero if and only if λi∗ − µi∗ 6 N for any index i = 1, 2, . . . . Here λ∗ = (λ∗1 , λ∗2 , . . .) and µ∗ = (µ∗1 , µ∗2 , . . .) are the partitions conjugate, respectively, to λ and µ. For any formal series f (u) ∈ C[[u −1 ]] with the leading term 1, the assignment Ti j (u) 7 → f (u) · Ti j (u) determines an automorphism of the algebra Y(gl N ). Let us denote this automorphism by α f . Further, there is a canonical chain of subalgebras Y(gl1 ) ⊂ · · · ⊂ Y(gl N ). The elementary modules are distinguished among all irreducible finite-dimensional Y(gl N )-modules V by the following fact. Consider the commutative subalgebra in Y(gl N ) generated by the centres of all algebras in that chain. This subalgebra is maximal commutative (see [C2], [NO]). The action of this subalgebra in V is semisimple if and only if the module V is obtained by pulling back through some automorphism α f from the tensor product Vω1 (z 1 ) ⊗ · · · ⊗ Vωk (z k )
(0.2)
of the elementary Y(gl N )-modules, for some skew Young diagrams ω1 , . . . , ωk and some complex numbers z 1 , . . . , z k such that z i − z j ∈ / Z when i 6= j. This fact was conjectured by I. Cherednik and proved by him in [C2] under certain extra conditions on the module V . In full generality this fact has been proved in [NT2]. Furthermore, if z i − z j ∈ / Z for all i 6 = j, the tensor product of Y(gl N )-modules (0.2) is always irreducible. Given the skew Young diagrams ω1 , . . . , ωk , it is natural to ask for which values of z 1 , . . . , z k ∈ C the Y(gl N )-module (0.2) is reducible. In [NT1] we answered this question in the case when each of the the diagrams ω1 , . . . , ωk has a rectangular shape. In that particular case, our answer confirmed the following general conjecture. For any indices i and j, consider the tensor product e Vω j (z j ) obtained via Vωi (z i ) ⊗ Vω j (z j ). Consider also the tensor product Vωi (z i ) ⊗ the opposite comultiplication on Y(gl N ). If z i − z j ∈ / Z, these two tensor products are irreducible and equivalent. Hence there is an invertible intertwining operator e Vω j (z j ), Ri j (z i , z j ) : Vωi (z i ) ⊗ Vω j (z j ) −→ Vωi (z i ) ⊗
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unique up to a nonzero multiplier from C. These multipliers can be chosen so that Ri j (z i , z j ) depends on z i and z j as a rational function of z i − z j (see Section 4). Now for any z i − z j ∈ C, denote by Ii j the leading coefficient of the expansion of the rational function Ri j (z i + u, z j ) into Laurent series in u near the origin u = 0. This coefficient is always an intertwining operator between the two tensor products. If z i − z j ∈ / Z, we simply have Ii j = Ri j (z i , z j ). It has been conjectured that the Y(gl N )-module (0.2) is irreducible if and only if for all i < j the operators Ii j are invertible (see, for instance, [CP]). We prove this conjecture in the present article (see Theorem 4.8). This result implies, in particular, that the Y(gl N )-module (0.2) is irreducible if and only if for all i < j the pairwise tensor products Vωi (z i )⊗Vω j (z j ) are irreducible. Given the skew Young diagrams ωi and ω j , it would be interesting to describe explicitly the set of all differences z i − z j ∈ Z, such that the module Vωi (z i ) ⊗ Vω j (z j ) is reducible. In the special case when both ωi and ω j are usual Young diagrams, this description has been given in [LNT] for sufficiently large N and independently in [M] for arbitrary N . The only if part of Theorem 4.8 is an easy and well-known fact. Our proof of the if part is based on the following general observation. Let A be any unital associative algebra over C, and let W be any finite-dimensional A-module. Denote by ρ the corresponding homomorphism A → End(W ). Suppose that the subalgebra ρ(A) ⊗ End(W ) ⊂ End(W ) ⊗ End(W ) = End(W ⊗ W ) contains the flip map x ⊗ y 7 → y ⊗ x in W ⊗ W . Then ρ(A) = End(W ) and the A-module W is irreducible (see Lemma 4.10). We employ this observation when A = Y(gl N ), and W is the tensor product (0.2). Moreover, to prove the irreducibility of (0.2) under the condition that all the operators Ii j with i < j are invertible, it suffices to show that Iii is proportional to the flip map in Vωi ⊗Vωi for each index i. For the details of this argument, see Proposition 4.14. We take the idea of this argument from the recent physical literature (see [KMT], [MT]). This argument extends to the b N ) in a straightforward way; we work with the Yangian quantum affine algebra Uq (gl Y(gl N ) in this article only to simplify the exposition. Moreover, this argument extends to other quantum affine algebras due to the existence of the universal R-matrix (see [FM], [K] for the precise formulation of the problem and known results in the general case). In this article we use the explicit realization of the elementary Y(gl N )-module Vω (z), introduced in [C2]. Denote λ1 − µ1 + λ2 − µ2 + · · · = n. Let n > 0. Consider the action of the symmetric group Sn in (C N )⊗n by permutations of the n tensor factors. Denote by πn the corresponding homomorphism C · S n → End (C N )⊗n . Then Vω is the subspace in (C N )⊗n defined as the image of the operator πn (Fω ) for a certain element Fω of the group ring C · S n . This element is constructed by a limiting
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process called the fusion procedure (see [C1]). Note that in [C1] most of the results were given without proofs. In Section 2 we give proofs of all properties of the element Fω which we need in this article. These proofs are new (cf. [N]). This choice of the realization of the Y(gl N )-module Vω (z) allows us to write down explicitly, for any z 0 ∈ C such that z − z 0 ∈ / Z, an invertible intertwining operator e Vω (z 0 ). R(z, z 0 ) : Vω (z) ⊗ Vω (z 0 ) −→ Vω (z) ⊗ This operator comes with a natural normalization. In particular, it is a rational function in z − z 0 with values in End(Vω ⊗ Vω ). Denote by Pω the flip map in Vω ⊗ Vω . We prove that the leading coefficient of the Laurent expansion in u near the origin u = 0 of the function R(z + u, z) equals c(ω)Pω u −d(ω) for a certain nonzero rational number c(ω) and a positive integer d(ω) (see Proposition 4.7). In the special case when ω is a usual Young diagram, this equality was proved in [DJKM]. Our proof of Proposition 4.7 employs a different method and is based on the results of Section 3. We call the number d(ω) the Durfee rank of the skew Young diagram ω. When ω is a usual Young diagram, d(ω) is the number of boxes on its main diagonal. In Section 1 we study the remarkable combinatorial properties of the numbers d(ω) for all skew Young diagrams ω. 1. Durfee rank of a skew Young diagram Let λ = (λ1 , λ2 , . . .) and µ = (µ1 , µ2 , . . .) be any two partitions. As usual, the parts are arranged in nonincreasing order: we have λ1 > λ2 > · · · > 0 and µ1 > µ2 > · · · > 0. We always suppose that λi > µi for any i = 1, 2, . . . . Consider the skew Young diagram ω = λ/µ. It can be defined as the set of pairs (i, j) ∈ Z2 i > 1, λi > j > µi . When µ = (0, 0, . . .), this is the Young diagram of the partition λ. We employ the standard graphic representation of Young diagrams on the plane R2 with the matrix-style coordinates (x, y). Here the coordinate x increases from top to bottom, while the coordinate y increases from left to right. The element (i, j) ∈ ω is represented by the unit box with the bottom right corner at the point (i, j) ∈ R2 . For example, in Figure 1 we represent the skew Young diagram ω for the partitions λ = (9, 9, 9, 7, 7, 3, 3, 3, 3, 0, 0, . . .) and µ = (5, 5, 3, 3, 3, 3, 2, 0, 0, . . .). We identify the elements of the set ω with the corresponding unit boxes on R2 . Let us call an element (i, j) ∈ ω left-convex if (i − 1, j) ∈ / ω and (i, j − 1) ∈ / ω. An element (i, j) ∈ ω is called left-concave if both (i − 1, j) ∈ ω and (i, j − 1) ∈ ω, but (i − 1, j − 1) ∈ / ω. Each of these two definitions has a natural interpretation in terms of boxes of the Young diagram ω. If λ 6 = µ = (0, 0, . . .), then there is only one left-convex element (1, 1) ∈ ω and no left-concave elements. Now consider those
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−→
−→ y
x
Figure 1
+ +
+
+
+
− +
+ −
+
+ −
Figure 2
diagonals of ω which contain a left-convex box. Let d be the total number of boxes on those diagonals. Let d 0 be the total number of boxes on the diagonals of ω which contain a left-concave box. The difference d −d 0 is called the Durfee rank of the skew diagram ω and is denoted by d(ω). If µ = (0, 0, . . .), then d(ω) = d is the number of boxes on the main diagonal {(i, i) ∈ Z2 | λi > i > 1 } of the Young diagram of λ, and our definition coincides with the usual one (see [A, Section 2.3]). For our exemplary skew Young diagram ω we mark by the + or − signs the diagonals, which contain, respectively, a left-convex or a left-concave box (see Figure 2). In this example the Durfee rank d(ω) equals 9 − 3 = 6. Our first observation is the following proposition. PROPOSITION 1.1 If the diagram ω is not empty, then d(ω) > 0.
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Proof For each i = 1, 2, . . . , let di be the number of boxes on that diagonal of ω which contains the leftmost box of the ith row. If the ith row of ω is empty, we set di = 0. Let di0 be the number of boxes on the next diagonal down. Then d(ω) = d1 − d10 + d2 − d20 + · · · .
(1.1)
By the definition of a skew Young diagram, we have di − di0 > 0 for every index i = 1, 2, . . . . But if i is the index of the last nonempty row of ω, then di = 1 and di0 = 0. Denote by `(ω) the number of nonempty rows in ω. The following proposition is a formula for d(ω). PROPOSITION 1.2 We have d(ω) = `(ω) − # (i, j) λ j − j = µi − i, i < j .
Proof We keep to the notation from the proof of Proposition 1.1. Further, for each i = 1, 2, . . . , put ei = 0 or ei = 1, depending on whether the ith row of ω is empty or not. By the definition of a skew Young diagram, we have the equalities di = #{ j | λ j − j > µi − i + 1 > µ j − j + 1} = #{ j | λ j − j > µi − i + 1, i 6 j} = #{ j | λ j − j > µi − i + 1, i < j} + ei , di0
= #{ j | λ j − j > µi − i > µ j − j + 1} = #{ j | λ j − j > µi − i, i < j}.
Therefore, di − di0 = ei − #{ j | λ j − j = µi − i, i < j}. Hence we obtain from (1.1) that d(ω) = (e1 + e2 + · · · ) − # (i, j) λ j − j = µi − i, i < j . By observing that e1 + e2 + · · · = `(ω), we now complete the proof. Let λ∗ = (λ∗1 , λ∗2 , . . .) and µ∗ = (µ∗1 , µ∗2 , . . .) be the partitions conjugate to λ and µ, respectively. Consider the corresponding skew Young diagram ω∗ = λ∗/µ∗ . It is obtained from the diagram ω ⊂ Z2 by transposition (i, j) 7→ ( j, i). Evidently, d(ω∗ ) = d(ω). Therefore, Proposition 1.2 can be reformulated as the following proposition.
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PROPOSITION 1.3 We have d(ω) = `(ω∗ ) − # (i, j) λ∗j − j = µi∗ − i, i < j .
By our definition, the Durfee rank d(ω) depends only on the shape of the diagram ω. That is, if a skew Young diagram ω[ is obtained from ω ⊂ Z2 by translation (i, j) 7→ (i +k, j +l) for a certain (k, l) ∈ Z2 , then d(ω[ ) = d(ω). Furthermore, suppose that a skew Young diagram ω\ is obtained from ω via the map (i, j) 7→ (k −i +1, l − j +1). The boxes of ω\ on the plane R2 are obtained from those of ω via rotation by 180◦ around the point (k/2, l/2). The main result of this section is the following theorem. 1.4 We have the equality d(ω\ ) = d(ω). THEOREM
Proof We have the inequalities λ1 − 1 > λ2 − 2 > · · · and µ1 − 1 > µ2 − 2 > · · · . So under the condition λ j − j = µi − i in Proposition 1.2, we have the equivalencies i < j ⇔ λi − i > λ j − j ⇔ λi − i > µi − i
⇔ λi > µi ,
i < j ⇔ µi − i > µ j − j ⇔ λ j − j > µ j − j ⇔ λ j > µ j . Denote by L (ω) the set of differences µi − i such that the ith row of ω is not empty. The set of differences λ j − j such that the jth row of ω is not empty is denoted by R (ω). By the above equivalencies, Proposition 1.2 can be restated as d(ω) = `(ω) − # L (ω) ∩ R (ω) . (1.2) We have `(ω\ ) = `(ω). We also have #(L (ω\ ) ∩ R (ω\ )) = #(L (ω) ∩ R (ω)) since L (ω\ ) = l − k − R (ω) −1,
R (ω\ ) = l − k − L (ω) −1
by the definition of ω\ . Thus we obtain Theorem 1.4 from formula (1.2). Let us call an element (i, j) ∈ ω right-convex if (i +1, j) ∈ / ω and (i, j +1) ∈ / ω. An element (i, j) ∈ ω is called right-concave if both (i +1, j) ∈ ω and (i, j +1) ∈ ω, but (i + 1, j + 1) ∈ / ω. Each of these two definitions has a natural interpretation in terms of boxes of the Young diagram ω. We could define the Durfee rank of ω using, respectively, the right-convex and right-concave boxes instead of the left-convex and left-concave boxes. Theorem 1.4 ensures that we would then get the same number, d(ω). Note that although the Young diagram of a nonempty partition always has only one left-convex box and no left-concave boxes, it may have several right-convex and right-concave boxes. So our definition of the Durfee rank of ω is quite natural.
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−→
−→ j
8 9
3 4
i
5 1 3 6 2 4 7
0 -3 -2 -1 -4 -3 -2 Figure 3
2. Fusion procedure for a skew Young diagram Take a nonempty skew Young diagram ω. Let n be the total number of boxes in ω. In this section we introduce a certain element Fω of the symmetric group ring C · S n . Under some extra conditions on the diagram ω, this element is used in Section 4 to determine a family of irredicible modules Vω (z) over the Yangian Y(gl N ). Here the parameter z is ranging over the complex field C. For the skew Young diagram ω[ obtained from ω by translation (i, j) 7 → (i + k, j + l) for a certain (k, l) ∈ Z2 , we have Vω[ (z) = Vω (z − k + l). Consider the column tableau of shape ω. It is obtained by filling the boxes of the diagram ω with the numbers 1, . . . , n consecutively by columns from left to right, downwards in every column. We denote this tableau by . Further, for each p = 1, . . . , n, put c p = j − i if the box (i, j) ∈ ω is filled with the number p in the column tableau . The difference j − i is called the content of the box (i, j) of the diagram ω. Our choice of the tableau provides an ordering of the collection of all contents of ω. In Figure 3, we show on the left the column tableau for the partitions λ = (5, 3, 3, 3, 3, 0, 0, . . .) and µ = (3, 3, 2, 0, 0, . . .). On the right we indicate the contents of all boxes of the diagram ω. In this example, n = 9 and the sequence of contents (c1 , . . . , c9 ) is (−3, −4, −2, −3, 0, −1, −2, 3, 4). For any two distinct numbers p, q ∈ {1, . . . , n}, let ( pq) be the transposition in the symmetric group Sn . Consider the rational functions of two complex variables u, v, with values in the group ring C · S n : ( pq) . u−v As direct calculation shows, these rational functions satisfy the equation f pq (u, v) = 1 −
f pq (u, v) f pr (u, w) f qr (v, w) = f qr (v, w) f pr (u, w) f pq (u, v)
(2.1)
(2.2)
for all pairwise distinct indices p, q, r . Evidently, for all pairwise distinct p, q, s, t, f pq (u, v) f st (z, w) = f st (z, w) f pq (u, v).
(2.3)
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For all distinct p, q, we also have f pq (u, v) f q p (v, u) = 1 −
1 . (u − v)2
(2.4)
Our construction of the element Fω ∈ C · S n is based on the following simple observation. Consider the rational function of u, v, w defined by the product at either side of (2.2). The factor f pr (u, w) in (2.2) has a pole at u = w. However, we have the following lemma. LEMMA 2.1 The restriction of the rational function (2.2) to the set of (u, v, w), such that u = v±1, is regular at u = w.
Proof Under the condition u = v ± 1, the product on the left-hand side of (2.2) can be written as ( pr ) + (qr ) 1 ∓ ( pq) · 1 − . v−w The latter rational function of v, w is manifestly regular at w = v ± 1. Now introduce n complex variables z 1 , . . . , z n . Equip the set of all pairs ( p, q), where 1 6 p < q 6 n, with the lexicographical ordering. Take the ordered product −→ Y
f pq (c p + z p , cq + z q )
(2.5)
( p,q)
over this set. Consider (2.5) as a rational function of the variables z 1 , . . . , z n with values in C · S n . Denote this rational function by Fω (z 1 , . . . , z n ). Let Dω be the vector subspace in Cn consisting of all tuples (z 1 , . . . , z n ) such that z p = z q whenever the numbers p and q appear in the same column of the tableau . The origin (0, . . . , 0) ∈ Cn belongs to Dω . The following statement goes back to [C1]. PROPOSITION 2.2 The restriction of the rational function Fω (z 1 , . . . , z n ) to the subspace Dω ⊂ Cn is regular at the point (0, . . . , 0).
Proof We provide an expression for the restriction of Fω (z 1 , . . . , z n ) to Dω which is manifestly regular at (0, . . . , 0). The factor f pq (c p + z p , cq + z q ) has a pole at z p = z q if and only if the numbers p and q stand on the same diagonal of the tableau . We then call the pair ( p, q) singular.
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NAZAROV and TARASOV
Let any singular pair ( p, q) be fixed. We write ( p, q) ≺ (s, t) if the pair ( p, q) precedes (s, t) in the lexicographical ordering, that is, if p < s or if p = s, q < t. In this ordering, the pair immediately before ( p, q) is ( p, q − 1). Moreover, the number q − 1 stands just above q in the tableau . Therefore, z q−1 = z q on Dω . We also have ( p, q) ≺ (q − 1, q) since p < q. By (2.2) and (2.3), the product over the pairs (s, t), −→ Y
f st (cs + z s , ct + z t ),
( p,q)≺(s,t)
is divisible on the left by the factor f q−1,q (cq−1 + z q−1 , cq + z q ). Restriction of this factor to z q−1 = z q is just 1 − (q − 1, q) since cq−1 = cq + 1. The element (1−(q −1, q))/2 ∈ C · S n is an idempotent. Now for every singular pair ( p, q), let us replace the two adjacent factors in (2.5), f p,q−1 (c p + z p , cq−1 + z q−1 ) f pq (c p + z p , cq + z q ), by f p,q−1 (c p + z p , cq−1 + z q−1 ) f pq (c p + z p , cq + z q ) f q−1,q (cq−1 + z q−1 , cq + z q )/2. This does not affect the values of the restriction of (2.5) to Dω . But the restriction to z q−1 = z q of the replacement product above is regular at z p = z q by Lemma 2.1. Due to Proposition 2.2, an element Fω ∈ C · S n can now be defined as the value at the point (0, . . . , 0) of the restriction to Dω of the function Fω (z 1 , . . . , z n ). Note that for n = 1, we have Fω = 1. For any n > 1, we have the following fact. PROPOSITION 2.3 The coefficient of Fω ∈ C · S n at the unit element of Sn is 1.
Proof For each r = 1, . . . , n − 1, let gr ∈ Sn be the adjacent transposition (r, r + 1). Let g0 ∈ Sn be the element of maximal length. Let us multiply the ordered product (2.5) by the element g0 on the right. Using the reduced decomposition g0 =
−→ Y
gq− p ,
( p,q)
we get the product −→ Y ( p,q)
gq− p −
1 . c p + z p − cq − z q
(2.6)
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It expands as a sum of the elements g ∈ Sn with coefficients from the field of rational functions of z 1 , . . . , z n valued in C. Since the decomposition (2.6) is reduced, the coefficient at g0 is 1. By the definition of Fω , this implies that the coefficient of Fω g0 ∈ C · S n at g0 ∈ Sn is also 1. In particular, Proposition 2.3 shows that Fω 6 = 0 for any nonempty diagram ω. Let us now denote by α the involutive antiautomorphism of the group ring C · S n defined by α(g) = g −1 for every g ∈ Sn . PROPOSITION 2.4 The element Fω ∈ C · S n is α-invariant.
Proof Any element of the group ring C · S n of the form f pq (u, v) is α-invariant. Applying the antiautomorphism α to an element of the form (2.5) just reverses the ordering of the factors corresponding to the pairs ( p, q). However, the initial ordering can then be restored by using the relations (2.2) and (2.3). Therefore, any value of the function Fω (z 1 , . . . , z n ) is α-invariant. So is the element Fω . The next property of the element Fω ∈ C · S n also follows easily from its definition. 2.5 Suppose the numbers r and r + 1 stand in the same column of . Then the element Fω ∈ C · S n is divisible by fr,r +1 (cr , cr +1 ) = 1 − (r, r + 1) on the left and on the right. THEOREM
Proof By Proposition 2.4, the divisibility of Fω by the element 1 − (r, r + 1) on the left is equivalent to the divisibility by the same element on the right. Let us prove the divisibility on the left. Using definition (2.5) along with the relations (2.2) and (2.3), we can write Fω (z 1 , . . . , z n ) = fr,r +1 (cr + zr , cr +1 + zr +1 )F(z 1 , . . . , z n )
(2.7)
for some rational function F(z 1 , . . . , z n ) valued in C · S n . By Proposition 2.2, the restriction of the function Fω (z 1 , . . . , z n ) to Dω is regular at (0, . . . , 0). But the restriction to Dω of the factor fr,r +1 (cr + zr , cr +1 + zr +1 ) in (2.7) is fr,r +1 (cr , cr +1 ). In particular, the value at (0, . . . , 0) of the restriction of Fω (z 1 , . . . , z n ) to Dω is divisible on the left by fr,r +1 (cr , cr +1 ) = 1 − (r, r +1). Let l ∈ {1, . . . , n − 1}. Denote by σl the embedding C · S n−l → C · S n defined by
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NAZAROV and TARASOV
σl : ( pq) 7→ ( p + l, q + l) for all distinct p, q = 1, . . . , n − l. Define a skew Young diagram ϕ as the shape of the tableau obtained from by removing each of the numbers 1, . . . , l. Consider the corresponding element Fϕ ∈ C · S n−l . PROPOSITION 2.6 The element Fω is divisible by σl (Fϕ ) on the left and on the right.
Proof Due to Proposition 2.4, the divisibility of Fω by the element σl (Fϕ ) on the left is equivalent to the divisibility by the same element on the right. We actually prove the divisibility on the right. By definition (2.5), we have Fω (z 1 , . . . , z n ) =
−→ Y
f pq (c p + z p , cq + z q ) · σl Fϕ (zl+1 , . . . , z n ) ,
(2.8)
( p,q)
where the pairs ( p, q) are ordered lexicographically and precede the pair (l + 1, l + 2). According to our proof of Proposition 2.2, the product over the pairs ( p, q) in (2.8) may be replaced by a certain rational function of z 1 , . . . , z n with the restriction to Dω regular at (0, . . . , 0), without affecting the values of the restriction. Now the divisibility of Fω by σl (Fϕ ) on the right follows from the decomposition (2.8) and the definition of the element Fϕ ∈ C · S n−l . Let m ∈ {1, . . . , n − 1}. Regard the group ring C · S m as a subalgebra in C · S n using the standard embedding Sm → Sn . Define a skew Young diagram ψ as the shape of the tableau obtained from by removing each of the numbers m +1, . . . , n. Consider the corresponding element Fψ ∈ C · S m . The following property of the element Fω is obtained by combining the arguments of Propositions 2.2 and 2.6. PROPOSITION 2.7 The element Fω ∈ C · S n is divisible by Fψ on the left and on the right.
Proof Let us introduce another expression for the restriction of Fω (z 1 , . . . , z n ) to Dω which is again manifestly regular at (0, . . . , 0). Reorder the pairs ( p, q) in the product (2.5) as follows; this reordering does not affect the values of the product due to (2.2) and (2.3). The pair (s, t) precedes the pair ( p, q) if t < q or if t = q, s < p. Let us now write (s, t) ≺ ( p, q) referring to this new ordering. Let any singular pair ( p, q) be fixed. In our new ordering, the pair immediately after ( p, q) is ( p + 1, q). Moreover, the number p + 1 stands just below p in the tableau . Therefore, z p = z p+1 on Dω . We also have ( p, +1) ≺ ( p, q) since p < q.
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By (2.2) and (2.3), the product −→ Y
f st (cs + z s , ct + z t )
(s,t)≺( p,q)
over the pairs (s, t) is divisible on the right by the factor f p, p+1 (c p +z p , c p+1 +z p+1 ). Restriction of this factor to z p = z p+1 is just 1 − ( p, p + 1) since c p+1 = c p − 1. The element (1−( p, p+1))/2 ∈ C · S n is an idempotent. Now for every singular pair ( p, q), let us replace the two adjacent factors in the reordered product (2.5), f pq (c p + z p , cq + z q ) f p+1,q (c p+1 + z p+1 , cq + z q ), by f p, p+1 (c p + z p , c p+1 + z p+1 ) f pq (c p + z p , cq + z q ) f p+1,q (c p+1 + z p+1 , cq + z q )/2. This does not affect the values of the restriction of (2.5) to Dω . But the restriction to z p = z p+1 of the replacement product is regular at z p = z q by Lemma 2.1. Due to Proposition 2.4, the divisibility of Fω by the element Fψ on the left is equivalent to the divisibility by the same element on the right. Let us now prove the divisibility on the left. In our new ordering of the factors in (2.5), we have Fω (z 1 , . . . , z n ) = Fψ (z 1 , . . . , z m ) ·
−→ Y
f pq (c p + z p , cq + z q ),
(2.9)
( p,q)
where ( p, q) (m −1, m). By the above argument, in (2.9) the product over the pairs ( p, q) may be replaced by a certain rational function of z 1 , . . . , z n with the restriction to Dω regular at (0, . . . , 0), without affecting the values of the restriction. Now the divisibility of Fω by Fψ on the left follows from the decomposition (2.9) and the definition of the element Fψ ∈ C · S m . We can now give a relatively short proof of the main result of this section (cf. [N]). 2.8 Suppose the numbers p < q stand next to each other in the same row of the tableau . Let r be the number at the bottom of the column of containing p. Then the element Fω is divisible on the left by the product THEOREM
←− Y s= p,...,r
−→ Y t=r +1,...,q
f st (cs , ct ) .
(2.10)
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NAZAROV and TARASOV
Proof It suffices to prove Theorem 2.8 only in the particular case q = n. For q < n, the required statement then follows by Proposition 2.7 applied to m = q. Furthermore, it suffices to consider only the case p = 1. For p > 1, the required statement then follows by Proposition 2.6 applied to l = p − 1. Assume that p = 1 and q = n. Then the skew Young diagram ω consists of two columns only. Further, every row of ω but one consists of a single box. There is also a row of two boxes, which in the tableau are filled with numbers 1 and n. So there is only one box on every diagonal of ω. The function Fω (z 1 , . . . , z n ) is then regular at (0, . . . , 0) and takes the value −→ Y
−→ Y
←− Y
f st (cs , ct ) =
s=1,...,n−1 t=s+1,...,n −→ Y
s=1,...,r
×
s=1,...,r −1
−→ Y t=s+1,...,r
−→ Y
f st (cs , ct )
t=r +1,...,n −→ Y
f st (cs , ct ) ·
s=r +1,...,n
−→ Y
f st (cs , ct ) ;
t=s+1,...,n
the last equality is obtained by using the relations (2.2) and (2.3). So the element Fω is indeed divisible on the left by the product (2.10) with p = 1 and q = n. 3. Leading term near the origin Take any two nonempty skew Young diagrams ω and ω0 . Let n and n 0 be the numbers of boxes in ω and ω0 , respectively. In this section we introduce a certain rational function Fωω0 (u) of one complex variable u with the values in the symmetric group ring C · S n+n 0 . Let z and z 0 be any two complex numbers. In Section 4 the leading term of the Laurent expansion of the function Fωω0 (u) at the point u = z − z 0 determines an intertwining operator of Y(gl N )-modules e Vω0 (z 0 ). Vω (z) ⊗ Vω0 (z 0 ) −→ Vω (z) ⊗ Here the tilde refers to the opposite comultiplication on the Hopf algebra Y(gl N ). In the present section we compute the leading term of the Laurent expansion of Fωω0 (u) with ω = ω0 near the origin u = 0. This result is used in Section 4. Let us regard C · S n as a subalgebra in C · S n+n 0 via the standard embedding Sn → Sn+n 0 . In particular, Fω ∈ C · S n may be regarded as an element of C · S n+n 0 . Consider also the embedding C · S n 0 → C · S n+n 0 defined by ( pq) 7→ ( p + n, q + n) for distinct p, q = 1, . . . , n 0 . In this section the image in C · S n+n 0 of any element F ∈ C · S n 0 under the latter embedding is denoted by F ∨ . Let c1 , . . . , cn and c10 , . . . , cn0 0 be the sequences of contents corresponding to the skew diagrams ω and ω0 . Here we order the contents using the column tableaux.
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357
Define the function Fωω0 (u) as the ordered product −→ Y
Fω ·
−→ Y
f p,q+n (c p + u, cq0 ) · Fω∨0 .
(3.1)
q=1,...,n 0
p=1,...,n
Note that the so-defined rational function Fωω0 (u) may have poles only at u ∈ Z. 3.1 The function Fωω0 (u) can be written as either of the following products: PROPOSITION
←− Y
−→ Y
p=1,...,n
Fω Fω∨0 ·
f p,q+n (c p + u, cq0 ) · Fω Fω∨0 ,
q=1,...,n 0 −→ Y ←− Y p=1,...,n
f p,q+n (c p + u, cq0 ) .
(3.2)
(3.3)
q=1,...,n 0
Proof Using definition (2.5) along with the relations (2.2) and (2.3), we obtain the equality of the rational functions in z 1 , . . . , z n and u, Fω (z 1 , . . . , z n ) ·
=
←− Y p=1,...,n
−→ Y p=1,...,n −→ Y
−→ Y
f p,q+n (c p + z p + u, cq0 )
q=1,...,n 0
f p,q+n (c p + z p + u, cq0 ) · Fω (z 1 , . . . , z n ).
q=1,...,n 0
By the definition of the element Fω ∈ C · S n , this equality implies that products (3.1) and (3.2) are also equal to each other. Similarly, we can prove the equality of products (3.1) and (3.3). Now consider the function Fωω (u). For any diagram ω[ obtained from ω ⊂ Z2 by a translation, we have Fω[ ω[ (u) = Fωω (u). Our preliminary result on Fωω (u) is the following proposition. 3.2 The order of the pole of Fωω (u) at u = 0 does not exceed d(ω). PROPOSITION
Proof We use the expression for d(ω) given by Proposition 1.3. Our argument is similar to that of Proposition 2.2 but more elaborate (cf. [N, Theorem 4.1]).
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NAZAROV and TARASOV
The function Fωω (u) is defined as the ordered product Fω ·
−→ Y
p=1,...,n
−→ Y
f p,q+n (c p + u, cq ) · Fω∨ .
(3.4)
q=1,...,n
The factor f p,q+n (c p + u, cq ) in (3.4) has a pole at u = 0 if and only if the numbers p and q stand on the same diagonal of the tableau . This pole is simple. Again, we then call the pair ( p, q) singular. Any singular pair ( p, q) belongs to one of the following three types: (i) the number p is not at the bottom of its column in ; (ii) p is at the bottom of its column in , but q is not leftmost in its row; (iii) p is at the bottom of its column in , and q is leftmost in its row. Observe that the total number of singular pairs of type (iii) is exactly d(ω). Indeed, let p be at the bottom of the jth column. Then c p = j − λ∗j . Consider the diagonal of the tableau containing p. Any number q on this diagonal, except maybe the first number, has in a neighbour on the left. As usual, here we are reading the diagonal from top left to bottom right. Suppose q is the first on the diagonal. The total number of such pairs ( p, q) equals the number of the columns, which is `(ω∗ ). But if q has a neighbour on the left, say, in the ith row, then i − µi∗ > cq . If we had i − µi∗ > cq , the number q could not be the first on its diagonal. So we must have i − µi∗ = cq = c p = j − λ∗j . Here we also have i < j. Thus the total number of singular pairs (iii) equals `(ω∗ ) − # (i, j) λ∗j − j = µi∗ − i, i < j = d(ω). We prove that when we estimate the order of the pole of (3.4) at u = 0 from above, the singular pairs (i) and (ii) do not count. Proposition 3.2 then follows. Step I. Using only the relations (2.3), we can rewrite the ordered product (3.4) as Fω ·
−→ Y q=1,...,n
−→ Y
f p,q+n (c p + u, cq ) · Fω∨ .
(3.5)
p=1,...,n
Consider any singular pair ( p, q) of type (i). In the ordered product (3.5), the factor f p,q+n (c p + u, cq ) is immediately followed by factor f p+1,q+n (c p+1 + u, cq ). Note that here c p = cq . The numbers p and p + 1 stand in the same column of the tableau . By Theorem 2.5, the element Fω is divisible on the right by f p, p+1 (c p , c p+1 ) = 1− ( p, p+1) = f p, p+1 (c p + u, c p+1 + u). Again using the relations (2.2) and (2.3), we obtain that the product of all factors in (3.5) preceding f p,q+n (c p + u, cq ) is also divisible by f p, p+1 (c p + u, c p+1 + u) on
TENSOR PRODUCTS OF YANGIAN MODULES
359
the right. But the product f p, p+1 (c p + u, c p+1 + u) f p,q+n (c p + u, cq ) f p+1,q+n (c p+1 + u, cq ) has no pole at u = 0 by Lemma 2.1. Thus when we estimate the order of the pole of (3.5) at u = 0 from above, the singular pairs (i) do not count. Step II. Now consider the diagonal of containing the number n. Let q1 , . . . , qd be all the entries of on that diagonal. We assume that q1 < · · · < qd so that qd = n. The singular pairs (n, q2 ), . . . , (n, qd ) are all of type (ii). The singular pair (n, q1 ) is of type (ii) or (iii). In case (ii), let the index k range over 1, . . . , d. In case (iii), the index k ranges over 2, . . . , d. For every such k, let pk be the number standing in next to the left of qk . Of course, for k > 1 we have pk = qk−1 + 1. However, to include the possible case k = 1 in our argument, we still use the notation pk . Let rk be the number standing in the tableau at the bottom of the column containing pk . The number at the top of the column containing qk is then rk + 1. By Theorem 2.8, the element Fω ∈ C · S n is divisible on the left by the product ←− Y
−→ Y
s= pk ,...,rk
f st (cs , ct ) =
t=rk +1,...,qk
−→ Y
t=rk +1,...,qk
←− Y
f st (cs , ct ) .
(3.6)
s= pk ,...,rk
For distinct indices k, the elements of C · S n defined by (3.6) pairwise commute. Let F be the element of C · S n obtained by multiplying the elements (3.6) over all k. We can write Fω = F G for some G ∈ C · S n . We can also write Fω∨ = F ∨ G ∨ in C · S 2n . Put r = rd for short. Using relations (2.3), we can rewrite the product (3.5) as Fω ·
−→ Y
q=1,...,n
−→ Y
f p,q+n (c p + u, cq )
p=1,...,r −→ Y
×
q=1,...,n
−→ Y
f p,q+n (c p + u, cq ) · Fω∨ . (3.7)
p=r +1,...,n
Denote by Q the sequence obtained from the sequence 1, . . . , n by transposing its segments pk , . . . , rk and rk +1, . . . , qk for every possible index k. Using the relations (2.2) and (2.3) and the decomposition Fω∨ = F ∨ G ∨ , we can rewrite (3.7) as Fω ·
−→ Y q=1,...,n
−→ Y
f p,q+n (c p + u, cq )
p=1,...,r ∨
×F ·
−→ Y q∈Q
−→ Y p=r +1,...,n
f p,q+n (c p + u, cq ) · G ∨ . (3.8)
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NAZAROV and TARASOV
As in the product (3.5), here for every singular pair ( p, q) of type (i), the factor f p,q+n (c p + u, cq ) is still followed by the factor f p+1,q+n (c p+1 + u, cq ). Let us now take any singular pair (n, qk ) of type (ii). In the second line of the ordered product (3.8), consider the sequence of factors −→ Y
−→ Y
q=qk , pk
f p,q+n (c p + u, cq ) .
(3.9)
p=r +1,...,n
Note that c p = cq in (3.9) only if ( p, q) = (n, qk ). The rightmost factor in the product (3.6) is f pk qk (c pk , cqk ). Due to (2.2) and (2.3), the product of all factors in (3.8) preceding the sequence (3.9) is divisible by f pk +n,qk +n (c pk , cqk ) on the right. But the entire product −→ Y
f pk +n,qk +n (c pk , cqk ) ·
q=qk , pk
=
−→ Y
f p,q+n (c p + u, cq )
p=r +1,...,n
−→ Y
q= pk ,qk
−→ Y
f p,q+n (c p + u, cq )
p=r +1,...,n−1
× f n, pk +n (cn + u, c pk ) f n,qk +n (cn + u, cqk ) f pk +n,qk +n (c pk , cqk ) has no pole at u = 0 due to Lemma 2.1. Thus when we estimate the order of the pole of (3.8) at u = 0 from above, any singular pair (n, qk ) of type (ii) does not count. Step III. By rewriting the second line of the product (3.7) to obtain (3.8), we exclude from our count all singular pairs ( p, q) of type (ii) with p = n. If the skew Young diagram ω consists of only one column, there are no singular pairs of type (ii) and the proof finishes on Step I. Suppose that the diagram ω has more than one column. Now let ψ be the skew Young diagram obtained from ω by removing the last nonempty column. Let m be the total number of boxes in ψ. Due to Proposition 2.7, the element Fω∨ is divisible by Fψ∨ on the left. But we also have −→ Y q=1,...,n
−→ Y
f p,q+n (c p + u, cq ) · Fψ∨
p=r +1,...,n
=
Fψ∨
·
−→ Y q=m,...,1,m+1,...,n
−→ Y
f p,q+n (c p + u, cq )
p=r +1,...,n
(see Proposition 3.1). Hence the entire product in the second line of (3.7) is divisible on the left by Fψ∨ . Now in an argument similar to that of Step II, we exclude from our count all singular pairs ( p, q) of type (ii) with p = m. Continuing this argument, we
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361
exclude from our count all singular pairs of type (ii). All singular pairs of type (i) can still be excluded from our count, as was done in Step I. For the skew Young diagram ω, let us introduce the rational function of u, Y −1 n 1 h ω (u) = . · 1− u (u − c p + cq )2
(3.10)
16 p
Recall once again that here the contents c1 , . . . , cn of ω are ordered by using the column tableau . Now consider the skew Young diagram ω∗ conjugate to ω, and the corresponding rational function h ω∗ (u). In general, we do not have the equality h ω∗ (u) = (−1)n · h ω (−u). But the following proposition is always true. PROPOSITION 3.3 The leading term of the Laurent expansion at u = 0 of the function h ω∗ (u) coincides with that of the function (−1)n · h ω (−u).
Proof Consider the sequence of numbers obtained by reading the entries of the column tableau in the natural way, that is, by rows downwards, from left to right in every row. This sequence can be also obtained from the sequence 1, . . . , n by a certain permutation g ∈ Sn . In general, the permutation g is nontrivial. The sequence of contents corresponding to the diagram ω∗ is −cg(1) , . . . ,−cg(n) . So (−1)n · h ω∗ (−u) =
−1 n u
·
Y 16 p
1−
1 . (u − cg( p) + cg(q) )2
Dividing the right-hand side of this equality by that of (3.10), we get the product of
1−
1
−1 1 · 1 − (u + c p − cq )2 (u − c p + cq )2
(3.11)
over all pairs ( p, q) such that p < q but g −1 ( p) > g −1 (q). For any such pair, the number q stands above and to the right of p in the tableau . In particular, then c p − cq 6 = ±1. But then the factor (3.11) is regular at u = 0 and takes the value 1 at that point. The next proposition provides another expression for the function h ω (u) with the arbitrary skew Young diagram ω = λ/µ. Consider the infinite product Y u − λi + µi Y (u + λi − µ j − i + j)(u + µi − λ j − i + j) · , u (u + λi − λ j − i + j)(u + µi − µ j − i + j) i
i< j
(3.12)
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NAZAROV and TARASOV
where i, j = 1, 2, . . . . Here the first fraction equals 1 for all but a finite number of indices i. The second fraction equals 1 for all but a finite number of indices i and j. Hence the product (3.12) determines a rational function of the variable u. PROPOSITION 3.4 The rational function (3.12) is equal to h ω∗ (u) · (1 − u)n .
Proof For each i = 1, 2, . . . denote by h i (u) the product of the factors 1−
u − c p + cq − 1 u − c p + cq +1 1 = · u − c p + cq u − c p + cq (u − c p + cq )2
(3.13)
over p < q such that the numbers p and q stand in the ith column of the tableau . If this column is empty, we put h i (u) = 1. For the numbers p and q, we have c p − cq = q − p. The total number of boxes in this column is λi∗ − µi∗ . So by direct calculation, ∗ ∗ (u − λi∗ + µi∗ ) u λi −µi −1 . h i (u) = ∗ ∗ (u − 1)λi −µi Further, for each i, put ai = i − µi∗ − 1 and bi = i − λi∗ . The numbers ai and bi are the contents, respectively, of the top and the bottom boxes of the ith column of the diagram ω if this column is not empty. If it is empty, λi∗ = µi∗ and bi = ai + 1. Now for i < j, denote by h i j (u) the product of all those factors (3.13), where the numbers p and q stand, respectively, in the ith and the jth columns of . If at least one of these two columns is empty, we put h i j (u) = 1. By direct calculation, h i j (u) =
u − ai + b j − 1 u − bi + a j +1 · . u − ai + a j u − bi + b j
Note that we also have λ∗1 − µ∗1 + λ∗2 − µ∗2 + · · · = n. So by definition (3.10), h ω (u) · (1 − u)n u − 1 n Y Y = · h i (u) · h i j (u) u i
=
Y u
− λi∗
+ µi∗
i< j
u·
i
Y (u + µi∗ − λ∗j − i + j)(u + λi∗ − µ∗j − i + j) i< j
(u + µi∗ − µ∗j − i + j)(u + λi∗ − λ∗j − i + j)
.
Replacing here the diagram ω by its conjugate ω∗ , we get Proposition 3.4. The total number of indices i, such that λi 6 = µi , is the number `(ω) of nonempty rows in the skew Young diagram ω. For any i < j, we also have the inequalities λi − i > µ j − j,
λi − i > λ j − j,
µi − i > µ j − j.
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363
Therefore, the order of the pole of the rational function (3.12) at u = 0 is `(ω) − # (i, j) λ j − j = µi − i, i < j = d(ω) = d(ω∗ ) (see Proposition 1.2). Now Proposition 3.4 implies that the order of the pole of the function h ω (u) at u = 0 is also d(ω). Denote by c(ω) the coefficient at u −d(ω) in the Laurent expansion of h ω (u) at u = 0. By Proposition 3.3, we have c(ω∗ ) = c(ω) · (−1)n+d(ω) .
(3.14)
The following proposition is the main result of this section. We employ this result in Section 4. THEOREM 3.5 The leading term of the Laurent expansion at u = 0 of Fωω (u) is
c(ω) · (1, n + 1)(2, n + 2) · · · (n, 2n) · Fω Fω∨ u −d(ω) .
(3.15)
Proof We use induction on n, the total number of boxes in ω. If n = 1, then Fω = 1,
Fω∨ = 1,
and
Fωω (u) = 1 − (1, 2)/u.
But for n = 1 we have h ω (u) = −1/u by definition (3.10). So the statement of Theorem 3.5 is obvious in this case. From now on we assume that n > 1. Suppose that the diagram ω consists of one column only. Then the diagram ω∗ consists of only one row. It follows from Proposition 2.3 and Theorems 2.5 and 2.7 that X X Fω = sgn(g) · g and Fω∗ = g, g∈Sn
g∈Sn
where sgn(g) = ±1 depending on whether the permutation g is even or odd. Now denote by β the involutive automorphism of the group ring C · S 2n defined by β(g) = sgn(g) · g for every g ∈ S2n . Then we have β Fωω (u) = Fω∗ ω∗ (−u) by definition (3.4). On the other hand, the image of (3.15) under β is (−1)n c(ω) · (1, n + 1)(2, n + 2) · · · (n, 2n) · Fω∗ Fω∨∗ u −d(ω) . Using equality (3.14), we see that the statements of Theorem 3.5 for ω and ω∗ are equivalent in the case when ω consists of one column only. We note that then d(ω) = 1, but we did not use the last equality here. We only used Fω∗ = β(Fω ). From now on we assume that there is more than one column in the skew Young diagram ω. Let δ be the skew Young diagram consisting of the first nonempty column
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NAZAROV and TARASOV
of ω. Let ϕ be the diagram obtained from ω by removing this column; the diagram ϕ is also nonempty. Each of the diagrams δ and ϕ has fewer than n boxes. We assume that the statement of Theorem 3.5 is valid for each of the diagrams δ and ϕ instead of for ω. Step I. Let l be the number of boxes in the column δ. As in Section 2, denote by σl the embedding C · S n−l → C · S n defined by σl : ( pq) 7 → ( p + l, q + l) for all distinct p, q = 1, . . . , n − l. Using (3.4) with (2.3), we can write Fωω (u) as Fω · A(u)B(u)C(u)D(u) · Fω∨ ,
(3.16)
where A(u), B(u), C(u), D(u) are, respectively, ordered products of the factors f p,q+n (c p + u, cq ) over p, q = 1, . . . , l; p = 1, . . . , l
q = l + 1, . . . , n;
and
p = l + 1, . . . , n
and
q = 1, . . . , l;
p, q = l + 1, . . . , n. The pairs ( p, q) in each of these four products are ordered lexicographically. By Propositions 2.6 and 2.7, the element Fω ∈ C · S n is divisible on the left and on the right by σl (Fϕ ) and Fδ . The element Fδ /l! ∈ C · Sl is an idempotent. In general, there may be no nonzero idempotents in the subset C · Fϕ ⊂ C · S n−l . But there is always an element Iϕ ∈ C · S n−l divisible on the left by Fϕ , such that Iϕ Fϕ = Fϕ . By Proposition 2.4 applied to Fϕ , this follows from the semisimplicity of C · S n−l . Due to Proposition 2.3, the element Jϕ = α(Iϕ ) is divisible by Fϕ on the right, and we have Fϕ Jϕ = Fϕ . Using Proposition 3.1, we can now rewrite (3.16) as Fω · Fδ A(u)Fδ∨ · Fδ B(u)σl (Iϕ )∨ · σl (Jϕ )C(u)D(u)Fω∨ · (l!)−3 .
(3.17)
Step II. By the inductive assumption for δ, the factor Fδ A(u)Fδ∨ in (3.17) has a simple pole at u = 0, and the corresponding residue coincides with that of h δ (u) · (1, n + 1) · · · (l, n + l) · Fδ Fδ∨ .
(3.18)
Further, Fδ B(u)σl (Fϕ )∨ is conjugate by the involution (l + 1, n + l + 1) · · · (n, 2n) to F(u) = Fδ ·
−→ Y p=1,...,l
−→ Y
f pq (c p + u, cq ) · σl (Fϕ ).
(3.19)
q=l+1,...,n
The function F(u) is regular at u = 0, and F(0) = Fω . This follows by setting z 1 = · · · = zl = u in Proposition 2.2. In particular, the factor Fδ B(u)σl (Iϕ )∨ in
TENSOR PRODUCTS OF YANGIAN MODULES
365
(3.17) is regular at u = 0. We show that the order of the pole at u = 0 of the factor σl (Jϕ )C(u)D(u)Fω∨ in (3.17) does not exceed d(ω) − 1. This allows us to replace the factor Fδ A(u)Fδ∨ in (3.17) by the product (3.18), without affecting the coefficient at u −d(ω) in the Laurent expansion at u = 0. Independently of Proposition 3.2, this also implies that the order of the pole at u = 0 of Fωω (u) does not exceed d(ω). It is the argument of Proposition 3.2 that we use here. In particular, for any numbers p and q standing on the same diagonal of , let us keep calling the pair ( p, q) singular. Any singular pair belongs to one of the types (i), (ii), (iii), as described in the beginning of the proof of Proposition 3.2. Note that the number l stands at the bottom of the first column in the tableau . Consider the product −→ Y
σl (Jϕ )C(u)D(u)Fω∨ = σl (Jϕ ) ·
p=l+1,...,n
−→ Y
f p,q+n (c p + u, cq ) · Fω∨ .
q=1,...,n
Here the factor f p,q+n (c p + u, cq ) has a pole at u = 0 if and only if the pair ( p, q) is singular. The factor σl (Jϕ ) is divisible on the right by σl (Fϕ ). By the argument of Proposition 3.2, when we estimate the order of pole of the above product at u = 0 from above, the singular pairs of types (i) and (ii) do not count. The total number of the singular pairs ( p, q) of type (iii) is d(ω). For any singular pair ( p, q) of type (iii), we have p > l, except for the pair ( p, q) = (l, l). Hence in the above product, the number of singular pairs ( p, q) of type (iii) is exactly d(ω) − 1. Step III. Thus the coefficient at u −d(ω) in the Laurent expansion at u = 0 of the function Fωω (u) coincides with that of the function h δ (u)Fω · (1, n + 1) · · · (l, n + l) · Fδ Fδ∨ · Fδ B(u)σl (Iϕ )∨ × σl (Jϕ )C(u)D(u)Fω∨ · (l!)−3 = h δ (u)Fω · (1, n + 1) · · · (l, n + l) × Fδ B(u)σl (Iϕ )∨ · C(u)D(u)Fω∨ · (l!)−1 .
(3.20)
Without affecting that coefficient, we can also replace in (3.20) the constant factor Fω by the function F(−u) (see (3.19)). The function F(−u) can be also written as σl (Fϕ ) ·
←− Y p=1,...,l
←− Y
f pq (c p , cq + u) · Fδ
q=l+1,...,n
(see Proposition 3.1). By exchanging the indices p, q and using the commutation relations (2.3), the last product can be rewritten as σl (Fϕ ) ·
←− Y p=l+1,...,n
←− Y q=1,...,l
f q p (cq , c p + u) · Fδ .
(3.21)
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NAZAROV and TARASOV
Let us now denote by E(u) the ordered product of the factors f q+n, p (cq , c p + u) over p = l +1, . . . , n
q = 1, . . . , l,
and
the pairs ( p, q) being taken in the reversed lexicographical order. Using the relations (2.4) repeatedly, we obtain the equality n l Y Y E(u)C(u) = 1− p=l+1 q=1
1 . (u + c p − cq )2
(3.22)
Denote by h(u) the rational function of u at the right-hand side of this equality. The product (3.21) is conjugate by the involution (1, n + 1) · · · (l, n + l) ∈ S2n to σl (Fϕ )E(u)Fδ∨ . By the above argument, the coefficient at u −d(ω) in the Laurent expansion at u = 0 of the function (3.20) coincides with that of the function h δ (u)F(−u) · (1, n + 1) · · · (l, n + l) · Fδ B(u)σl (Iϕ )∨ · C(u)D(u)Fω∨ · (l!)−1 = h δ (u) · (1, n + 1) · · · (l, n + l) · Fδ B(u)σl (Iϕ )∨ × σl (Fϕ )E(u)Fδ∨ C(u)D(u)Fω∨ · (l!)−1 = h δ (u)h(u) · (1, n + 1) · · · (l, n + l) · Fδ B(u)σl (Iϕ )∨ · σl (Fϕ )D(u)Fω∨ ; here we also use divisibility of the product of Fω∨ and of the product C(u)D(u)Fω∨ by Fδ∨ on the left, along with equality (3.22). Step IV. The element Fω∨ is also divisible on the left by σl (Fϕ )∨ . By the inductive assumption for the diagram ϕ, the leading term in the Laurent expansion at u = 0 of σl (Fϕ )D(u)σl (Fϕ )∨ coincides with that of the function h ϕ (u) · (l + 1, n + l + 1) · · · (n, 2n) · σl (Fϕ )σl (Fϕ )∨ . Therefore, by the result of Step III, the coefficient at u −d(ω) in the Laurent expansion at u = 0 of the function Fωω (u) coincides with that of the function h δ (u)h(u)h ϕ (u) · (1, n + 1) · · · (l, n + l) × Fδ B(u)σl (Iϕ )∨ · (l + 1, n + l + 1) · · · (n, 2n) · σl (Fϕ )Fω∨ = h δ (u)h(u)h ϕ (u) · (1, n + 1) · · · (l, n + l) × (l + 1, n + l + 1) · · · (n, 2n) · F(u)Fω∨ (see definition (3.19)). But here we have h δ (u)h(u)h ϕ (u) = h ω (u) by (3.10). The statement of Theorem 3.5 for the skew Young diagram ω now follows from the equality F(0) = Fω . This equality has already been obtained in Step II.
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367
We complete this section with one remark on the ordered product ←− Y
−→ Y
p=1,...,n
f p,q+n (c p + u, cq0 )
(3.23)
q=1,...,n 0
for two nonempty skew Young diagrams ω and ω0 . This product appears in (3.2). Denote the product (3.23) by G ωω0 (u); this is a rational function of u which may have poles only at u ∈ Z. By Proposition 3.1, we have Fωω0 (u) = G ωω0 (u)Fω Fω∨0 . Let us now extend the notation F ∨ used earlier in this section for the elements G ∈ C · S n 0 as follows. For any G ∈ C · S n+n 0 , denote by G ∨ the element gGg −1 conjugate to G by the permutation from Sn+n 0 , g : (1, . . . , n 0 , n 0 +1, . . . , n + n 0 ) 7 → (n + 1, . . . , n + n 0 , 1, . . . , n). 3.6 We have the equality PROPOSITION
0
G ωω0 (u)G ∨ ω0 ω (−u)
=
n Y n Y
1−
p=1 q=1
1 . (u + c p − cq0 )2
Proof By the definition of the function G ω0 ω (u), we have G∨ ω0 ω (−u) = =
←− Y
−→ Y
p=1,...,n 0 q=1,...,n −→ Y ←− Y p=1,...,n
f p+n,q (c0p − u, cq ) f q+n, p (cq0 , c p + u) ;
q=1,...,n 0
here the second equality has been obtained by exchanging the indices p, q and using the relations (2.3). Now the required statement can be derived from the definition (3.23) of the function G ωω0 (u) by using the relations (2.4). 4. Irreducibility of Yangian modules We begin this section by recalling the definition of the Yangian of the general linear Lie algebra gl N . This is the unital associative algebra Y(gl N ) over C, with the infinite (s) family of generators Ti j , where s = 1, 2, . . . and i, j = 1, . . . , N . The defining relations can be written in terms of the generating series (0.1) as (u − v) · Ti j (u), Tkl (v) = Tk j (u)Til (v) − Tk j (v)Til (u).
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NAZAROV and TARASOV
The Yangian Y(gl N ) is a Hopf algebra. Coproduct 1 : Y(gl N ) → Y(gl N ) ⊗ Y(gl N ) is given by N X 1 Ti j (u) = Tik (u) ⊗ Tk j (u). k=1
e : Y(gl N ) → Y(gl N ) ⊗ Y(gl N ) is the composition of the The opposite coproduct 1 coproduct 1 and subsequent transposition of tensor factors in Y(gl N ) ⊗ Y(gl N ). For e V the Yangian modany two Y(gl N )-modules U and V , we denote by U ⊗ V and U ⊗ ules, where the action of Y(gl N ) on the tensor product of vector spaces is determined e , respectively. via the coproduct 1 and the opposite coproduct 1 Our definition implies that for any z ∈ C, the assignment Ti j (u) 7→ Ti j (u − z) defines an automorphism of the Hopf algebra Y(gl N ). The formal series in (u − z)−1 should be re-expanded in u −1 . Denote this automorphism by τz . We also need a matrix form of the definition of the Hopf algebra Y(gl N ). Let us introduce the following notation. Consider the tensor product of any unital associative algebras A1 ⊗ · · · ⊗ An . For p = 1, . . . , n, let ι p : A p → A1 ⊗ · · · ⊗ An be the embedding as the pth tensor factor: ι p (X ) = 1⊗( p−1) ⊗ X ⊗ 1⊗(n− p) ,
X ∈ Ap .
For X ∈ A p and Y ∈ Aq , where p 6 = q, let us write (X ⊗ Y )( pq) = ι p (X )ιq (Y ). For any Z ∈ A p ⊗ Aq , determine the element Z ( pq) ∈ A1 ⊗ · · · ⊗ An by linearity. For any vector spaces U1 , . . . , Un , we identify the algebras End(U1 ⊗ · · · ⊗ Un ) and End(U1 ) ⊗ · · · ⊗ End(Un ). Now take the vector space C N = U . For i, j = 1, . . . , N , let E i j ∈ End(U ) be the matrix units. Let P : x ⊗ y 7→ y ⊗ x be the flip map in U ⊗ U . Put R(u) = u − P. Identifying End(U ⊗ U ) = End(U ) ⊗ End(U ), we can write N X
R(u) = u −
E i j ⊗ E ji .
(4.1)
i, j=1
Combine all series (0.1) into a series T (u) with coefficients in Y(gl N ) ⊗ End(U ): T (u) =
N X
Ti j (u) ⊗ E ji .
(4.2)
i, j=1
Then the defining relations of the algebra Y(gl N ) are equivalent to the relation T (01) (u)T (02) (v)R (12) (u − v) = R (12) (u − v)T (02) (v)T (01) (u)
(4.3)
for the series with coefficients in Y(gl N ) ⊗ End(U ) ⊗ End(U ), where we label the tensor factors by the indices 0, 1, 2.
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369
Now label the tensor factors in Y(gl N ) ⊗ Y(gl N ) ⊗ End(U ) by 1, 2, 3 as usual. Put
T [k] (u) = T (k3) (u),
k = 1, 2.
(4.4)
Then we have the equality (1 ⊗ id) T (u) = T [2] (u)T [1] (u).
(4.5)
We refer the reader to the survey [MNO] for more details on the definition of the Yangian Y(gl N ) and for a description of its basic properties. Note that the series denoted by T (u) in [MNO] differs from our series (4.2) by transposition E i j 7→ E ji in the second tensor factor. Let ω = λ/µ be a nonempty skew Young diagram. Let n be the number of boxes in ω. Take the vector space U ⊗n . The symmetric group Sn acts naturally in this vector space, permuting the tensor factors. Denote by πn the corresponding homomorphism C · S n → End(U ⊗n ). In Section 2 we define a certain element Fω ∈ C · S n . Let the subspace Vω ⊂ U ⊗n be the image of the operator πn (Fω ). PROPOSITION 4.1 We have Vω 6= {0} if and only if the length of every column of ω does not exceed N .
The results of the present article do not depend on this proposition. Hence we omit the proof and refer to [C1] and [NT1] instead. Now take two nonempty skew Young diagrams ω and ω0 with n and n 0 boxes, respectively. Let c1 , . . . , cn and c10 , . . . , cn0 0 be the corresponding sequences of contents, ordered according to the column tableaux. Consider the rational function G ωω0 (u) with the values in C · S n+n 0 , defined by (3.23). For any value of u ∈ / Z, the operator πn+n 0 (G ωω0 (u)) is well defined, and by Proposition 3.1, it preserves the subspace Vω ⊗ Vω0 ⊂ U ⊗n ⊗ U ⊗n = U ⊗(n+n ) . 0
0
Let Rωω0 (u) be the value of the restriction of πn+n 0 (G ωω0 (u)) to Vω ⊗ Vω0 ; Rωω0 (u) is a rational function of u valued in End(Vω ) ⊗ End(Vω0 ). Suppose that each of the vector spaces Vω and Vω0 is nonzero (see Proposition 4.1). Then by Proposition 3.6, 0
(21) Rωω0 (u)Rω0 ω (−u)
n Y n Y = 1− p=1 q=1
1
. (u + c p − cq0 )2
(4.6)
Take one more nonempty skew Young diagram ω00 . Let n 00 be its number of boxes. 4.2 We have the equality of rational functions of u, v with values in the tensor product PROPOSITION
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NAZAROV and TARASOV
End(Vω ) ⊗ End(Vω0 ) ⊗ End(Vω00 ), (12)
(13)
(23)
(23)
(13)
(12)
Rωω0 (u − v)Rωω00 (u)Rω0 ω00 (v) = Rω0 ω00 (v)Rωω00 (u)Rωω0 (u − v).
(4.7)
Proof Relation (4.7) for the functions Rωω0 (u), Rωω00 (u), Rω0 ω00 (u) follows from the respective relation for the functions G ωω0 (u), G ωω00 (u), G ω0 ω00 (u). The latter relation is an equality of rational functions of u, v with the values in the algebra C · S n+n 0 +n 00 . That equality follows from (2.2) and (2.3); we do not write it here. By applying the homomorphism πn+n 0 +n 00 to both sides of that equality and taking the restriction to 0 00 Vω ⊗ Vω0 ⊗ Vω00 ⊂ U ⊗(n+n +n ) , we obtain (4.7). Denote by ε the Young diagram with one box (1, 1). Then Vε = U . Note that Rεε (u) = R(u)/u. When ω = ω0 = ω00 = ε, equality (4.7) is the Yang-Baxter equation for the Yangian Y(gl N ). We now define a family of modules over the algebra Y(gl N ); we call these modules elementary. Take any z ∈ C. Consider the rational function Rωε (z − u) of the variable u. This function takes values in the algebra End(Vω ) ⊗ End(U ) and is regular at u = ∞. Moreover, at u = ∞ it takes the value 1. Here we keep to the assumption Vω 6= {0}. Consider the Laurent expansion at u = ∞, Rωε (z − u) = 1 +
N X
(s)
u −s E i j (ω, z) ⊗ E ji .
i, j=1 (s)
Here each tensor factor E i j (ω, z) is a certain element of the algebra End(Vω ). 4.3 The assignment for all s = 1, 2, . . . and i, j = 1, . . . , N , PROPOSITION
(s)
(s)
Ti j 7 → E i j (ω, z),
(4.8)
defines a Y(gl N )-module structure on the vector space Vω . Proof (s) We have to verify that the operators E i j (ω, z) satisfy the defining relations of Y(gl N ). Let us rewrite definition (4.8) in a matrix form. It is T (u) 7→ Rωε (z − u).
(4.9)
In view of the defining relation (4.3), we have to verify the equality (01) (02) (02) (01) Rωε (z − u)Rωε (z − v)R (12) (u − v) = R (12) (u − v)Rωε (z − v)Rωε (z − u)
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of rational functions in u, v with the values in End(Vω ) ⊗ End(U ) ⊗ End(U ), where the tensor factors are labelled by the indices 0, 1, 2. Changing these labels to 1, 2, 3, respectively, and taking into account that R(u) = u Rεε (u), we verify that (12) (13) (23) (23) (13) (12) Rωε (z − u)Rωε (z − v)Rεε (u − v) = Rεε (u − v)Rωε (z − v)Rωε (z − u).
But the latter equality follows from Proposition 4.2 when ω0 = ω00 = ε. The Y(gl N )-module defined by Proposition 4.3 is denoted by Vω (z) and called an elementary module. Note that for any z ∈ C, the Y(gl N )-module Vω (z) is obtained from the module Vω (0) by pulling back through the automorphism τz of the algebra Y(gl N ). Let us now give another description of the elementary Y(gl N )-module Vω (z). First consider the module Vε (z). The vector space of this module is U = C N , and the (s) action of the algebra Y(gl N ) in this space is determined by the assignment Ti j 7→ z s E i j (see (4.1)). 4.4 The action of algebra Y(gl N ) in the tensor product Vε (c1 + z) ⊗ · · · ⊗ Vε (cn + z) preserves the subspace Vω ⊂ U ⊗n . The restriction of this action to Vω gives exactly the Y(gl N )-module Vω (z). PROPOSITION
Proof Let us write the definition of the action of the algebra Y(gl N ) in Vε (z) in matrix form. It is T (u) 7→ Rεε (z − u). Then due to (4.5), the action of Y(gl N ) in the tensor product Vε (c1 + z) ⊗ · · · ⊗ Vε (cn + z) is determined by the assignment T (u) 7→
←− Y
( p,n+1) Rεε (c p + z − u).
p=1,...,n
The right-hand side of this assignment can be rewritten as ←− Y
πn+1 f p,n+1 (c p + z − u, 0) = πn+1 G ωε (z − u) .
p=1,...,n
We have already shown that πn+1 G ωε (z − u) preserves the subspace Vω ⊗ Vε in U ⊗n ⊗ U . This means that the action of Y(gl N ) in Vε (c1 + z) ⊗ · · · ⊗ Vε (cn + z) preserves the subspace Vω ⊂ U ⊗n . Restriction of this action to Vω is determined exactly by the assignment T (u) 7 → Rωε (z − u) because of the definition of Rωε (z − u). But the last assignment coincides with the matrix definition of the Y(gl N )-module Vω (z).
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For any z ∈ C, let us expand the rational function Rωω0 (u) into the Laurent series at u = z. Let (u − z)−aωω0 (z) Iωω0 (z) be the leading term of this expansion. This defines an integer aωω0 (z) and a nonzero linear operator Iωω0 (z) in Vω ⊗ Vω0 . Again, here we keep to the assumption that Vω , Vω0 6 = {0}. The rational function Rωω0 (u) is regular at u = z for any z ∈ / Z. Then by (4.6) this function also does not vanish at u = z for any z ∈ / Z. Thus, for any z ∈ / Z, we have aωω0 (z) = 0 and Iωω0 (z) = Rωω0 (z). Furthermore, equality (4.6) implies the following proposition. 4.5 (21) For any z ∈ C, the product Iωω0 (z)Iω0 ω (−z) ∈ End(Vω ⊗ Vω0 ) is proportional to the identity map. In particular, the maps Iωω0 (z) and Iω0 ω (−z) are invertible simultaneously. PROPOSITION
The basic property of the linear operator Iωω0 (z) is given by the next proposition. PROPOSITION 4.6 For any z, z 0 ∈ C, the element Iωω0 (z − z 0 ) ∈ End(Vω ⊗ Vω0 ) is an intertwiner of the e Vω0 (z 0 ). Yangian modules Vω (z) ⊗ Vω0 (z 0 ) −→ Vω (z) ⊗
Proof Formula (4.7) implies the following equality of rational functions in u, v: (12)
(23)
(23)
(12)
(13) (13) Rωω0 (z − v)Rωε (z − u)Rω0 ε (v − u) = Rω0 ε (v − u)Rωε (z − u)Rωω0 (z − v).
Taking the leading terms of the Laurent expansions of both sides at v = z 0 , we get (12)
(23)
(23)
(12)
(13) (13) Iωω0 (z − z 0 )Rωε (z − u)Rω0 ε (z 0 − u) = Rω0 ε (z 0 − u)Rωε (z − u)Iωω0 (z − z 0 ).
The latter equality implies Proposition 4.6 (see (4.5) and (4.9)). Recall that the positive integer d(ω) is the Durfee rank of the nonempty skew Young diagram ω (see Section 1). The nonzero rational number c(ω) is the value of the function u d(ω) h ω (u) at u = 0 (see (3.10)). Denote by Pω the flip map in Vω ⊗ Vω , so that Pω (x ⊗ y) = y ⊗ x for any x, y ∈ Vω . The main result of the previous section, Theorem 3.5, can be now restated as follows. PROPOSITION 4.7 We have aωω (0) = d(ω) and Iωω (0) = c(ω)Pω .
Proof The function Fωω (u) with the values in C · S 2n can be written as G ωω (u)Fω Fω∨ .
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Hence the required statement follows from Theorem 3.5 by the definition of the operator Iωω (0) in Vω ⊗ Vω . Our Theorem 1.4 also has an interpretation in terms of the Y(gl N )-module Vω (z). We omit the details and refer to [C2, Proposition 2.5] instead. Now take k > 1 nonempty skew Young diagrams ω1 , . . . , ωk . We assume that each of the vector spaces Vω1 , . . . , Vωk is not zero (see Proposition 4.1). Take also k arbitrary complex numbers z 1 , . . . , z k . Consider the elementary Y(gl N )-modules Vω1 (z 1 ), . . . , Vωk (z k ). The next theorem is the main result of the present paper. 4.8 The Y(gl N )-module Vω1 (z 1 ) ⊗ · · · ⊗ Vωk (z k ) is irreducible if and only if all the intertwiners Iωi ω j (z i −z j ) with 1 6 i < j 6 k are invertible. THEOREM
There is an immediate corollary; we state it as a theorem because of its importance. 4.9 The Y(gl N )-module Vω1 (z 1 )⊗· · ·⊗Vωk (z k ) is irreducible if and only if all the modules Vωi (z i ) ⊗ Vω j (z j ) with 1 6 i < j 6 k are irreducible. THEOREM
We prove Theorem 4.8 in the remainder of this section. Before giving the proof, let us make a few remarks. Applying Theorem 4.8 when k = 1, 2, we obtain, in particular, the following properties of the elementary Y(gl N )-modules: (a) the Y(gl N )-module Vω (z) is irreducible; (b) the Y(gl N )-module Vω (z) ⊗ Vω0 (z 0 ) is irreducible, provided z − z 0 ∈ / Z; 0 e 0 (c) if Vω (z) ⊗ Vω (z ) is irreducible, then it is equivalent to Vω (z) ⊗ Vω0 (z 0 ). Properties (a), (b), and (c) are well known (see, for instance, [C2], [CP], [NT2]). But we would like to emphasize here that Theorem 4.8 gives each of them a new independent proof. Using Theorem 4.8 together with Proposition 4.7, we obtain for any k = 2, 3, . . . a remarkable new property of the elementary Y(gl N )-modules: (d) the kth tensor power of the Y(gl N )-module Vω (z) is irreducible. Conversely, property (d) for k = 2 implies that the operator Iωω (0) in the vector space Vω ⊗ Vω is proportional to the flip map Pω (see Proposition 4.6). The only if part of Theorem 4.8 is an easy and well-known statement. For the sake of completeness, we give the proof of this statement at the end of this section. Our proof of the if part is based on the following general observation. Let A be any unital associative algebra over C, and let W be any finite-dimensional A-module. Denote by ρ the corresponding homomorphism A → End(W ). Let PW ∈ End(W ⊗ W ) be the flip map.
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LEMMA 4.10 If PW ∈ ρ(A) ⊗ End(W ), then the A-module W is irreducible.
Proof Let 1 ∈ End(W ) be the unit element, by our assumption 1 ∈ ρ(A). For every element X ∈ End(W ), we have 1 ⊗ X ∈ ρ(A) ⊗ End(W ). Then also X ⊗ 1 = PW · (1 ⊗ X ) · PW ∈ ρ(A) ⊗ End(W ). Hence X ∈ ρ(A) for every X ∈ End(W ). So we actually have ρ(A) = End(W ). Now take the algebra A = Y(gl N ) and its module W = Vω1 (z 1 ) ⊗ · · · ⊗ Vωk (z k ). Under the assumption that all the intertwiners Iωi ω j (z i −z j ) with 1 6 i < j 6 k are invertible, we show that in the above notation, PW ∈ ρ Y(gl N ) ⊗ End(W ). (4.10) The if part of Theorem 4.8 then follows by Lemma 4.10. We split the proof of (4.10) into four propositions below. Let ω be a nonempty skew Young diagram with n boxes. We continue to assume that the subspace Vω ⊂ U ⊗n is nonzero. Let Endω (U ⊗n ) ⊂ End(U ⊗n ) be the stabilizer of Vω . Let χ : Endω (U ⊗n ) → End(Vω ) bω (u) with cobe the canonical homomorphism χ(X ) = X |Vω . Introduce the series T ⊗n efficients in the tensor product Y(gl N ) ⊗ End(U ) = Y(gl N ) ⊗ (End(U ))⊗n , bω (u) = T (01) (u + c1 ) · · · T (0n) (u + cn ). T Here we labelled the factors in Y(gl N ) ⊗ (End(U ))⊗n by the indices 0, 1, . . . , n. PROPOSITION 4.11 bω (u) are in Y(gl N ) ⊗ Endω (U ⊗n ). Coefficients of the series T
Proof Consider the rational function Fω (z 1 , . . . , z n ), defined in Section 2 as the product (2.5). Relation (4.3) implies that T (01) (u + c1 + z 1 ) · · · T (0n) (u + cn + z n ) · πn Fω (z 1 , . . . , z n ) = πn Fω (z 1 , . . . ,z n ) · T (0n) (u + cn + z n ) · · · T (01) (u + c1 + z 1 ). Hence, by Proposition 2.2, bω (u)πn (Fω ) = πn (Fω )T (0n) (u + cn ) · · · T (01) (u + c1 ). T
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bω (u) Due to Proposition 4.11, we can apply the homomorphism id ⊗χ to the series T and obtain in this way a series with coefficients in Y(gl N ) ⊗ End(Vω ). We denote the latter series by Tω (u). Henceforth, we use the following generalization of notation (4.4). Given a vector space V and an element X ∈ Y(gl N ) ⊗ End(V ), we write X [k] = X (k3) ∈ Y(gl N ) ⊗ Y(gl N ) ⊗ End(V ),
k = 1, 2.
PROPOSITION 4.12 We have 1 Tω (u) = Tω[2] (u)Tω[1] (u).
Proof bω (u)) = T bω[2] (u)T bω[1] (u). By applying the homomorFormula (4.5) implies that 1(T phism id ⊗ id ⊗χ to this equality, the proposition follows. Let γ : Y(gl N ) → End(Vω ) be the defining homomorphism of the module Vω (z). Take one more skew Young diagram ω0 ; we again assume that Vω0 6= {0}. 4.13 We have (γ ⊗ id) Tω0 (u) = Rωω0 (z − u). PROPOSITION
Proof Let n 0 be the number of boxes in the diagram ω0 . Consider the canonical homomorphism 0 χ 0 : Endω0 (U ⊗n ) → End(Vω0 ). It is straightforward to verify that each (γ ⊗ id)(Tω0 (u)) and Rωω0 (z − u) equals (χ ⊗ χ 0 ) ◦ πn+n 0 G ωω0 (z − u) . Now consider the formal power series in u −1 , TW (u) = Tω(01) (u + z 1 ) · · · Tω(0k) (u + z k ) ∈ Y(gl N ) ⊗ End(W ). k 1 Here we identify Y(gl N ) ⊗ End(W ) with Y(gl N ) ⊗ End(Vω1 ) ⊗ · · · ⊗ End(Vωk ), and we label the tensor factors by 0, 1, . . . , k. The next proposition completes the proof of the if part of Theorem 4.8. 4.14 The image (ρ ⊗ id)(TW (u)) is a rational function in u with values in ρ Y(gl N ) ⊗ End(W ) ⊂ End(W ) ⊗ End(W ). PROPOSITION
The leading coefficient in its Laurent expansion at u = 0 is proportional to PW .
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Proof It follows from Proposition 4.12 that 1 TW (u) = TW[2] (u)TW[1] (u). Using this relation and Proposition 4.13, we get −→ Y
(ρ ⊗ id) TW (u) =
←− Y
j=1,...,k
Rω(i,i ωj+k) (z − z − u) . i j j
(4.11)
i=1,...,k
Here we regard End(W ) ⊗ End(W ) as the 2k-fold tensor product End(Vω1 ) ⊗ · · · ⊗ End(Vωk ) ⊗ End(Vω1 ) ⊗ · · · ⊗ End(Vωk ).
(4.12)
The right-hand side of equality (4.11) is a rational function in u by definition. Denote a=
k X
aωi ω j (z i − z j ).
i, j=1
Then the product of the leading terms of the Laurent expansions at u = 0 of all factors in the product (4.11) equals (−u)−a multiplied by −→ Y
←− Y
j=1,...,k
Iω(i,i ωj+k) (z − z ) i j . j
(4.13)
i=1,...,k (i,i+k)
(i,i+k)
By Proposition 4.7, any factor Iωi ω j (0), corresponding to i = j, equals Pωi to a nonzero multiplier. Using this result, we can rewrite (4.13) as k Y i=1
Pω(i,i+k) · i
Iω( j+k,i) (z j − z i )Iω(i,i ωj+k) (z i − z j ) j ωi j
Y
up
(4.14)
16i< j6k (i,i+k)
up to a nonzero multiplier. Here all factors Pωi pairwise commute. By Proposition 4.5, each of the factors in (4.14) corresponding to 1 6 i < j 6 k is proportional to the identity map in (4.12). In particular, these factors also pairwise commute. Moreover, each of these factors is nonzero because the operators Iωi ω j (z i − z j ) with 1 6 i < j 6 k are invertible by our assumption. So the entire product (4.13) is nonzero and proportional to k Y Pω(i,i+k) = PW . i i=1
Thus the leading term in the Laurent expansion of the function (ρ ⊗ id)(TW (u)) near the origin u = 0 equals (−u)−a PW up to a nonzero factor from C.
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We complete this section with the proof of the only if part of Theorem 4.8. Consider the Y(gl N )-module W = Vω1 (z 1 ) ⊗ · · · ⊗ Vωk (z k ). Suppose that there is a noninvertible intertwiner Iωi ω j (z i − z j ) with 1 6 i < j 6 k. Assume that the pair (i, j) is lexicographically minimal with this noninvertibility property. The operator (i,i+1) Iω(ii j) ω j (z i − z j ) · · · Iωi ωi+1 (z i − z i+1 )
is nonzero and noninvertible. By Proposition 4.6, this is an intertwiner of Yangian modules W −→ Vω1 (z 1 ) ⊗ · · · ⊗ Vωi−1 (z i−1 ) e Vωi+1 (z i+1 ) ⊗ · · · ⊗ Vω j (z j ) ⊗ Vω j+1 (z j+1 ) ⊗ · · · ⊗ Vωk (z k ). ⊗ Vωi (z i ) ⊗ So the Y(gl N )-module W is reducible. This proves the only if part of Theorem 4.8. Acknowledgments. It is our pleasure to thank N. Kitanine for a stimulating discussion and for bringing the results of [KMT] and [MT] to our attention. This work was done when the authors stayed at the Mathematisches Forschungsinstitut Oberwolfach, with support from the Volkswagen-Stiftung. While finishing this article, the second author stayed at the Max-Planck-Institut f¨ur Mathematik in Bonn, Germany. References [A]
G. ANDREWS, The Theory of Partitions, Encyclopedia Math. Appl. 2,
[CP]
V. CHARI and A. PRESSLEY, Fundamental representations of Yangians and
Addison-Wesley, Reading, Mass., 1976. MR 58:27738 347 singularities of R-matrices, J. Reine Angew. Math. 417 (1991), 87 – 128. MR 92h:17010 345, 373 [C1] I. CHEREDNIK, Special bases of irreducible representations of a degenerate affine Hecke algebra, Functional Anal. Appl. 20 (1986), 76 – 78. MR 87m:22031 346, 351, 369 [C2] I. CHEREDNIK, A new interpretation of Gelfand-Tzetlin bases, Duke Math. J. 54 (1987), 563 – 577. MR 88k:17005 344, 345, 373 [DJKM] E. DATE, M. JIMBO, A. KUNIBA, and T. MIWA, “A quantum trace formula for R-matrices” in Selected Topics in Conformal Field Theory and Statistical Mechanics, ed. H. S. Song, Kyohak Yunkusa, Korea, 1990, 87 – 111. 346 [D1] V. DRINFELD, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32 (1985), 254 – 258. MR 87h:58080 343 [D2] V. DRINFELD, A new realization of Yangians and of quantum affine algebras, Soviet Math. Dokl. 36 (1988), 212 – 216. MR 88j:17020 343 [FM] E. FRENKEL and E. MUKHIN, Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras, Comm. Math. Phys. 216 (2001), 23 – 57. MR 2002c:17023 345
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[K]
M. KASHIWARA, On level zero representations of quantized affine algebras, preprint,
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N. KITANINE, J.-M. MAILLET, and V. TERRAS, Form factors of the XXZ Heisenberg spin- 21 finite chain, Nuclear Phys. B 554 (1999), 647 – 678. MR 2001h:82016
[LNT]
B. LECLERC, M. NAZAROV, and J.-Y. THIBON, “Induced representations of affine
arXiv:math.QA/0010293 345
345, 377
[MT] [M] [MNO] [N]
[NO] [NT1]
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[O]
[T1] [T2]
Hecke algebras and canonical bases of quantum groups” to appear in Studies in Memory of Issai Schur, ed. A. Joseph et al., Birkh¨auser, Boston, 2002. 345 J.-M. MAILLET and V. TERRAS, On the quantum inverse scattering problem, Nuclear Phys. B 575 (2000), 627 – 644. MR 2001g:82045 345, 377 A. I. MOLEV, Irreducibility criterion for tensor products of Yangian evaluation modules, Duke Math. J. 112 (2002), 307 – 341. 345 A. MOLEV, M. NAZAROV, and G. OLSHANSKI˘I, Yangians and classical Lie algebras, Russian Math. Surveys 51 (1996), 205 – 282. MR 97f:17019 369 M. NAZAROV, “Yangians and Capelli identities” in Kirillov’s Seminar on Representation Theory, Amer. Math. Soc. Transl. Ser. 2 181, Amer. Math. Soc., Providence, 1997, 139 – 164. MR 99g:17033 346, 355, 357 M. NAZAROV and G. OLSHANSKI, Bethe subalgebras in twisted Yangians, Comm. Math. Phys. 178 (1996), 483 – 506. MR 98d:17018 344 M. NAZAROV and V. TARASOV, On irreducibility of tensor products of Yangian modules, Internat. Math. Res. Notices 1998, 125 – 150. MR 99b:17014 343, 344, 369 M. NAZAROV and V. TARASOV, Representations of Yangians with Gelfand-Zetlin bases, J. Reine Angew. Math. 496 (1998), 181 – 212. MR 99c:17030 343, 344, 373 G. I. OLSHANSKI˘I, “Representations of infinite-dimensional classical groups, limits of enveloping algebras, and Yangians” in Topics in Representation Theory, Adv. Soviet Math. 2, Amer. Math. Soc., Providence, 1991, 1 – 66. MR 92g:22039 344 V. TARASOV, Structure of quantum L-operators for the R-matrix of the X X Z -model, Theoret. and Math. Phys. 61 (1984), 1065 – 1072. MR 87c:82023 343 V. TARASOV, Irreducible monodromy matrices for an R-matrix of the X X Z -model, and lattice local quantum Hamiltonians, Theoret. Math. Phys. 63 (1985), 440 – 454. MR 87d:82022 343
Nazarov Department of Mathematics, University of York, Heslington, York YO10 5DD, England;
[email protected] Tarasov Steklov Institute of Mathematics, 27 Fontanka, Saint Petersburg 191011, Russia;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2,
EXTERIOR ALGEBRA METHODS FOR THE MINIMAL RESOLUTION CONJECTURE DAVID EISENBUD, SORIN POPESCU, FRANK-OLAF SCHREYER, and CHARLES WALTER
Abstract If r ≥ 6, r 6= 9, we show that the minimal resolution conjecture (MRC) fails for a √ general set of γ points in Pr for almost (1/2) r values of γ . This strengthens the result of D. Eisenbud and S. Popescu [EP1], who found a unique such γ for each r in the given range. Our proof begins like a variation of that of Eisenbud and Popescu, but uses exterior algebra methods as explained by Eisenbud, G. Fløystad, and F.O. Schreyer [EFS] to avoid the degeneration arguments that were the most difficult part of the Eisenbud-Popescu proof. Analogous techniques show that the MRC fails for linearly normal curves of degree d and genus g when d ≥ 3g − 2, g ≥ 4, re-proving results of Schreyer, M. Green, and R. Lazarsfeld. 1. Introduction From the Hilbert function of a homogeneous ideal I in a polynomial ring S over a field k, one can compute a lower bound for the graded Betti numbers βi, j = dimk ToriS (S/I, k) j because these numbers are the graded ranks of the free modules in the minimal free resolution of I . The graded Betti numbers are upper semicontinuous in families of ideals with constant Hilbert function, and in many cases this lower bound is achieved by the general member of such a family. The statement that it is achieved for a particular family T is called the minimal resolution conjecture (MRC) for T , following A. Lorenzini, who made the conjecture for the family of ideals of sufficiently general sets of γ points in Pr (all r, γ ). The MRC for a general set of γ points in Pr has received considerable attention (see Eisenbud and Popescu [EP1] for full references and discussion). In particular, it is known that the MRC is satisfied if r ≤ 4 (see F. Gaeta [G1], [G2], A. Geramita DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 2, Received 26 December 2000. 2000 Mathematics Subject Classification. Primary 13D02, 16E05, 14F05; Secondary 13C10, 14N05, 14M05. Eisenbud’s work supported by National Science Foundation grant number DMS-9810361. Popescu’s work supported by National Science Foundation grant number DMS-0083361. 379
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and Lorenzini [GL], E. Ballico and Geramita [BG], C. Walter [W], F. Lauze [L]) or γ r (see A. Hirschowitz and C. Simpson [HS]), but computer evidence produced by Schreyer, extended also by M. Boij in his 1994 thesis and by S. Beck and M. Kreuzer in [BK], suggested that it might fail for certain examples starting with 11 points in P6 . Indeed, Eisenbud and Popescu [EP1] proved that if r ≥ 6, r 6= 9, then the MRC fails for a general set of √ γ = r + (3 + 8r + 1)/2 points in Pr . This family of counterexamples to the minimal resolution conjecture begins with Schreyer’s suggested 11 points in P6 , and it seems to have contained all the examples in print in 1999. However, the thesis of Boij, completed in 1994 but published only in [B], contains computations suggesting one further counterexample: 21 points in P15 . In this paper we simplify and extend the idea of Eisenbud and Popescu to prove the existence of infinitely many new counterexamples, including the one suggested by Boij. We prove the following. 1.1 The MRC fails for the general set of γ points in Pr whenever r ≥ 6, r 6= 9, and √ √ r + 2 + r + 2 ≤ γ ≤ r + (3 + 8r + 1)/2, THEOREM
and also when (r, γ ) = (8, 13) or (15, 21). √ √ Thus we get about ( 2 − 1) r counterexamples in Pr for each r ≥ 6, r 6 = 9. Our proof not only gives more, but it also avoids the subtle degeneration argument used by Eisenbud and Popescu. In the presentation below we skip some of the steps presented in Eisenbud and Popescu [EP1], and we fully treat only the new ideas, so it may be helpful to the reader to explain their strategy and the point at which it differs from ours. Their proof can be divided into three steps, starting from a set 0 of γ general points in Pr . Step I. Their crucial first step is to consider the Gale transform of 0 ⊂ Pr , which is a set 0 0 of γ points in Ps with γ = r + s + 2 determined “naively” as follows. If the columns of the ((r + 1) × γ )-matrix M represent the points of 0, then a ((s + 1) × γ )matrix N representing 0 0 is given by the transpose of the kernel of M (see [EP1] for a precise definition). The free resolution of the canonical modules of the homogeneous coordinate rings of 0 and 0 0 are related, and using this relation and the fact that (under our numerical hypotheses) the ideal of 0 0 is not 2-regular, they construct a map φ0 from a certain linear free complex F• (µ) to the dual of the resolution of the ideal I0
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381
of 0. For the r and γ considered in Theorem 1.1, straightforward arithmetic shows that the injectivity of the top degree component of φ0 would force the graded Betti numbers of I0 to be too large for the MRC to hold. (We recall the definition of F• (µ) in Sec. 3.) Step II. They show that φ0 is an injection of complexes if F• (µ) has a property somewhat weaker than exactness called linear exactness or irredundancy (see Def. 2.1). Step III. For a generic set of points 0, they show that F• (µ) is irredundant by a subtle degeneration argument: they degenerate the general set 0 to lie on a special curve C, and the argument finishes with a study of a refined stability property of the tangent bundle of the projective space restricted to C in a certain embedding of C connected with the Gale transform of the points. In our new proof, Step I remains unchanged, but Step II is replaced by a more precise statement, which requires us to verify a weaker condition in Step III; this weak condition can be verified without any degeneration argument, making the replacement for Step III much simpler. More precisely, the injectivity of φ0 is needed only for a sufficiently large irredundant quotient of the complex F• (µ), and we give a criterion for this weaker condition using exterior algebra methods. The new criterion is checked in Step III with an argument inspired by Green’s proof in [Gr2] of the linear syzygy conjecture. The generality of 0 enters via work of Kreuzer [K] showing that certain multiplication maps of the canonical module of the cone over 0 are 1-generic in the sense of Eisenbud [E]. In Section 2 we introduce and study the irredundancy of linear complexes using the Bernˇste˘ın-Gel’fand-Gel’fand (BGG) correspondence. In Section 3 we construct the complex whose irredundancy is the key to the failure of the MRC. Finally, in Section 4 we briefly explain how Gale duality allows this theory to be applied to sets of points, and we give the arithmetic part of the proof. As another application of our techniques, we recover the result on curves of Schreyer (unpublished) and of Green and Lazarsfeld [GrL] (see Th. 4.1). Notation. Throughout this paper k denotes an arbitrary field. Let V be a finitedimensional vector space over k. We work with graded modules over S := Sym(V ) and E := ∧V ∗ . We think of elements of V as having degree 1 and elements of V ∗ as having degree −1. We write m for the maximal ideal generated by V in S.
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2. Linear complexes and exterior modules In this section we illustrate the exterior algebra approach to linear free resolutions with some basic ideas used in the rest of the paper, and with a new proof of the linear rigidity theorem of Eisenbud and Popescu [EP1]. Definition 2.1 A complex of graded free S-modules F• :
···
φ3
F2
φ2
F1
φ1
F0
is a called a standard linear free complex if Fi is generated in degree i for all i. (Note that all the differentials φi are then represented by matrices of linear forms.) A linear free complex is any twisted (or shifted) standard linear free complex (i.e., a right-bounded complex of free modules Fi , so that for some integer s the generators of Fi are all in degree s + i for all i). We say that a linear free complex F• is irredundant (see [EFS]) or linearly exact (see [EP1]) if the induced maps Fi+1 /mFi+1 → mFi /m2 Fi are monomorphisms for i > 0. One sees immediately that a standard linear free complex F• as above is irredundant if and only if Hi (F• )i = 0 for all i > 0. As explained in [EFS], irredundancy can be conveniently rephrased via the BGG correspondence as follows. If P is a graded E-module, we define a linear free Scomplex F• = L(P) with free modules Fi = S ⊗k Pi generated in degree i and differentials X φi : Fi → Fi−1 , 1 ⊗ p 7→ xi ⊗ ei p ∈ S ⊗ Pi−1 , where {xi } and {ei } are (fixed) dual bases of V and V ∗ . Up to twisting and shifting, every linear free complex of S-modules arises from a unique E-module in this way. The linear complex L(P) is standard if Pi = 0 for all i < 0. Given a graded E-module P, we write P ∗ for the dual graded E-module P ∗ = Homk (P, k). We take k in degree zero, so that (P ∗ )i is the dual vector space to P−i . PROPOSITION
2.2
If F• = L(P) :
···
- F2
- F1
- F0
is a standard linear free complex of S-modules, then F• is irredundant if and only if P ∗ is generated as an E-module in degree zero.
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383
Thus any linear free complex F• = L(P) as in Proposition 2.2 has a unique maximal irredundant quotient F•0 , which is functorial in F• , constructed as follows. If Q is the E-submodule of P ∗ generated by P0∗ , then F•0 := L(Q ∗ ). Alternatively, F•0 = L(P/N ), where N is the submodule with graded pieces Ni = {n ∈ Pi | ∧i V ∗ · n = 0}. Observe that we have F00 = F0 . The next lemma is the basic tool that allows us to deduce that a minimal free resolution is large. A weaker version of the result was implicit in [EP1]. 2.3 Suppose that α• : F• → G • is a map from an irredundant standard linear free complex to a minimal free complex. If α0 is a split monomorphism, then αi is a split monomorphism for all i ≥ 0. LEMMA
Proof By induction, it suffices to prove that α1 is a split monomorphism. Since both F• and G • are minimal (i.e., the differentials are represented by matrices of elements of m), there is a commutative diagram F1 /mF1
- mF0 /m2 F0
α1 ? ? G 1 /mG 1 - mG 0 /m2 G 0
whose vertical maps are induced by α• and whose horizontal maps are induced by the differentials of F• and G • . Since α0 makes F0 a summand of G 0 , the right-hand vertical map is a monomorphism. Because F• is irredundant, the top map is a monomorphism. It follows that the map α 1 : F1 /mF1 → G 1 /mG 1 induced by α1 is a monomorphism, and since G 1 is free, this implies that α1 is split. Lemma 2.3 may be applied to give a lower bound on Betti numbers. PROPOSITION 2.4 Suppose that α• : F• → G • is a map from a standard linear complex to a minimal free resolution. There exists a map β• : F• → G • which is homotopic to α• and such that β• factors through the maximal irredundant quotient F•0 of F• . Further, if α0 : F0 → G 0 is a split monomorphism, then the rank of G i is at least as big as the rank of Fi0 for all i.
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Proof Let γ• : F• → F•0 be the natural map to the maximal irredundant quotient. The image of F10 in F00 = F0 is contained in that of F1 , so there is a map β1 : F10 → G 1 lifting α0 : F00 → G 0 . Since G • is acyclic, we may continue to lift, and so inductively we get a map of complexes β•0 : F•0 → G • . We take β• = β•0 γ• , and thus we have β0 = α0 . Since F• is a free complex and G • is acyclic, this implies that α• is homotopic to β• . The second statement follows by applying Lemma 2.3 to the map β•0 . In [EP1] a rigidity result for irredundancy is required. The proof given there is just a reference to the result on the rigidity of Tor proved by M. Auslander and D. Buchsbaum in [AB]. The exterior method yields a novel proof of this result (which we do not need in the sequel). 2.5 Let R be a graded commutative or anticommutative ring, and let M be a graded Rmodule that is generated by M0 . If R 0 ⊂ R is a graded subring such that R10 · M0 = M1 , then M is generated by M0 as an R 0 -module also. PROPOSITION
Proof We show by induction on n that R10 · Mn = Mn+1 , the initial case n = 0 being the hypothesis. If R10 · Mn−1 = Mn , then R10 · Mn = R10 · (R1 Mn−1 ) = R1 · (R10 Mn−1 ) = R1 · Mn = Mn+1 , where the second equality holds by the (anti)commutativity assumption and the third by the induction hypothesis. 2.6 (Linear rigidity) Let S = k[x0 , . . . , xr ] be a polynomial ring, and let COROLLARY
F• :
···
- F2
- F1
- F0
be an irredundant standard linear free complex. Let S 0 = S/I be a graded quotient of S. The complex G • := S 0 ⊗ S F• is irredundant if and only if H1 (G • ) is zero in degree 1. Proof The irredundancy of G • involves information about only S00 and S10 , so we may assume that S 0 is a polynomial ring S 0 = Sym(V 0 ) with V 0 = V /I1 . Write E 0 = ∧(V 0∗ ) ⊂ E = ∧(V ∗ ) for the corresponding exterior algebras. If F• = L(P), then the complex G • over S 0 corresponds to the same graded vector space P, regarded as an E 0 -module by restriction of scalars. Consider now the truncated complex K• :
···
- 0
- 0
- G1
- G 0.
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The hypothesis H1 (G • )1 = 0 implies that H1 (K • )1 = 0 and hence, by Proposition 2.2, that P1∗ = V 0∗ · P0∗ . On the other hand, P ∗ is generated as an E-module by P0∗ since F• is an irredundant standard linear complex. So, by Proposition 2.5, we also have P ∗ = E 0 · P0∗ . By Proposition 2.2, this suffices. Remark 2.7 A similar “linear rigidity” result for complexes over the exterior algebra can also be deduced from Proposition 2.5 applied this time for R a polynomial ring. 3. The complexes F• (µ) We apply Proposition 2.2 and Lemma 2.3 to a family of linear complexes F• (µ) (which were called E −1 (µ) in [EP1]). It is convenient here to define them in terms of the functor L. Let U , V , and W be finite-dimensional vector spaces of dimensions u, v, and w, P P respectively. Let A := l ∧l W ∗ ⊗Syml (U ∗ ), and let Q := l ∧l+1 W ∗ ⊗Syml (U ∗ ). Then A is an anticommutative graded algebra, and Q is a graded A-module. If µ : W ⊗ U → V is a pairing, then the dual µ∗ : V ∗ → W ∗ ⊗ U ∗ extends to a map of algebras µ˜ : E = ∧(V ∗ ) → A. We regard Q as an E-module via µ, ˜ and we consider the corresponding linear free complex F• (µ) := L(Q ∗ ) over S = Sym(V ): F• (µ) :
0 → ∧w W ⊗ Dw−1 (U ) ⊗ S(−w + 1) → · · · → ∧2 W ⊗ U ⊗ S(−1) → W ⊗ S,
where Dm (U ) denotes the m-graded piece of the divided power algebra. Though we do not need the formula, it is easy to give the differentials explicitly: δl (µ) : Fl (µ) → Fl−1 (µ) is the composite of the tensor product of the diagonal maps of the exterior and divided powers algebras, ∧l+1 W ⊗ Dl U ⊗ S(−l)
- ∧l W ⊗ W ⊗ Dl−1 U ⊗ U ⊗ S(−l),
and the map induced by the pairing µ, ∧l W ⊗ W ⊗ Dl−1 U ⊗ U ⊗ S(−l)
- ∧l W ⊗ Dl−1 U ⊗ S(−l + 1).
Note that if V = W ⊗ U and µ is the identity map, then Q is generated as an E-module by Q 0 , so the complex F• (µ) is irredundant by Proposition 2.2. The proof in [EP1] shows that F• (µ) remains exact when µ is specialized to a certain pairing coming from the canonical module of a generic set of γ points in Pr for suitable γ , r . Here we are instead interested in knowing whether F• (µ) has an irredundant quotient complex with the same first and last terms as F• (µ). This condition turns out to be a lot easier to analyze!
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Following Eisenbud [E], we say that µ : W ⊗U → V is 1-generic if µ(a ⊗ b) 6 = 0 for all nonzero vectors a ∈ W and b ∈ U . We say that µ is geometrically 1-generic if the induced pairing µ ⊗ k is 1-generic, where k is the algebraic closure of k. If µ is geometrically 1-generic, then by a classic and elementary result of Hopf the dimensions u, v, w of the three vector spaces satisfy the inequality v ≥ u + w − 1. The main result of this section is the following. THEOREM 3.1 If the pairing µ : W ⊗ U → V is geometrically 1-generic and if W 6= 0 and U 6= 0, then F• (µ) and its maximal irredundant quotient F•0 (µ) have the same last 0 term Fw−1 = Fw−1 ; that is,
F•0 (µ) :
∧w W ⊗ Dw−1 (U ) ⊗ S(−w + 1) - F0 - ··· - F0 w−2 1
- W ⊗ S.
Proof We use notation as above. Let Q 0 be the submodule of Q generated by Q 0 , and set P = Q ∗ , P 0 = Q 0∗ , so that F• = L(P) and F•0 = L(P 0 ). By Proposition 2.2, the conclusion of the theorem is equivalent to the statement that the E-multiplication map - P−w+1 m : E w−1 ⊗k P0 w w w w w w w w w w−1 ∗ ∗ - w ∗ ∧ V ⊗W ∧ W ⊗ Sym (U ∗ ) w−1
is surjective. Via the natural identification W ∗ ⊗ ∧w W ∼ = ∧w−1 W , this is equivalent to the map ∗ - Sym m 0 : ∧w−1 V ∗ ⊗k ∧w−1 W w−1 (U ) induced by µ being surjective. A straightforward computation shows that the image of m 0 is the space generated by the ((w − 1) × (w − 1))-minors of the linear map of free Sym(U ∗ )-modules µ¯ :
W ⊗ Sym(U ∗ )
- V ⊗ Sym(U ∗ )(1)
associated to µ, and the next lemma shows that under the 1-genericity assumption these minors span Symw−1 (U ∗ ). This proves the theorem. 3.2 If µ : W ⊗ U → V is a geometrically 1-generic pairing of finite-dimensional vector spaces and if d is an integer such that d ≤ dim W , then the (d × d)-minors of the associated linear map µ¯ : W ⊗ Sym(U ∗ ) → V ⊗ Sym(U ∗ )(1) of free Sym(U ∗ )modules span Symd (U ∗ ). LEMMA
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Proof It is enough to prove the lemma for an algebraically closed base field k, so that the Nullstellensatz holds. Also, u, v, w continue to be the dimensions of the three vector spaces. We first show that, for an arbitrary subspace W 0 ⊂ W of dimension d, the (d ×d )minors of the restricted map µ¯ 0 :
W 0 ⊗ Sym(U ∗ )
- V ⊗ Sym(U ∗ )(1),
regarded as polynomial functions on U , have a common zero only at the origin 0 ∈ U . For µ¯ 0 is essentially a (v × d )-matrix whose entries are linear functions on U , so its (d × d )-minors have a common zero at a point b ∈ U if and only if the matrix’s columns, when evaluated at b, are not linearly independent. But this means that there is a 0 6 = a ∈ W 0 such that µ(a ⊗ b) = 0, which gives b = 0 by 1-genericity. - V 0 with A general position argument shows that, for a general projection V 0 dim V = d + u − 1, the only common zero of the (d × d )-minors of the composite map µ¯ 00 : W 0 ⊗ Sym(U ∗ ) - V 0 ⊗ Sym(U ∗ )(1) is still 0 ∈ U . (One may argue that the dual of µ¯ 0 may be regarded as a surjective map - O d (1) of vector bundles on Pu−1 , and a rank d vector bundle generated Ov by global sections on a (u − 1)-dimensional space can always be generated by just d + u − 1 of them (see, e.g., Eisenbud and E. Evans [EE] or J.-P. Serre [S]).) The expected codimension of the locus defined by the (d ×d )-minors of the ((d + u−1)×d )-matrix representing µ¯ 00 is u, which is equal to the actual codimension. Thus the Eagon-Northcott complex resolving the minors of µ¯ 00 is exact, and it follows that there are d+u−1 linearly independent such minors. Since this is also the dimension d of Symd (U ∗ ), we see that the minors span Symd (U ∗ ), as required. We say that a map of complexes α• : F• → G • is a degreewise split injection if each αi : Fi → G i is a split injection (this is weaker than being a split injection of complexes); in this case, we say that F• is a degreewise direct summand of G • . The irredundant complexes F•0 (µ) arise in geometric situations as follows. THEOREM 3.3 Let L , L 0 be two line bundles on a scheme X over a field k, and let L 00 := L ⊗ L 0 . Let W ⊂ H0 (L), U ⊂ H0 (L 0 ) be nonzero finite-dimensional linear series such that the multiplication 0µ : W ⊗U (L 00 )
is geometrically 1-generic. Let V ⊂ H0 (L 00 ) be a finite-dimensional linear series L containing W · U . Let S := Sym(V ), and let M ⊂ n≥0 H0 (L ⊗ L 00 ⊗n ) be a finitely
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generated graded S-submodule such that W ⊂ M0 . Then the maximal irredundant quotient F•0 (µ) :
∧w W ⊗ Dw−1 (U ) ⊗ S(−w + 1) - F0 - ··· - F0 w−2 1
- W⊗S
of the linear free complex F• (µ) injects as a degreewise direct summand of the minimal free resolution of M. (Here, as above, w = dim W .) Proof Let G • be the minimal free resolution of M over S. The hypotheses on M imply that G 0 = (W ⊗ S) ⊕ L with L a free S-module. We can construct inductively a morphism α• : F• (µ) → G • such that α0 : F0 (µ)/mF0 (µ) → G 0 /mG 0 is a monomorphism, as a lifting ···
···
- ∧3 W ⊗ D2 U ⊗ S(−2) - ∧2 W ⊗ U ⊗ S(−1) - W⊗S .. .. ∩ .. . .. α1 ... ..? ..? ? - G2 - G1 - (W ⊗ S) ⊕ L -M
For instance, α1 exists since the composition ∧2 W ⊗U ⊗ S(−1) → W ⊗ S → M vanishes. So by Lemma 2.3, α• induces a degreewise split inclusion F•0 (µ) ⊂ - G • with F•0 (µ) having the form given in Theorem 3.1 (since µ is geometrically 1-generic). It is easy to see that if X is a geometrically reduced and irreducible scheme, then any pairing µ : W ⊗U - V induced by multiplication of sections as in Theorem 3.3 is geometrically 1-generic. The hypothesis sometimes holds for reducible or nonreduced schemes too, as in the case we use for the proof of Theorem 1.1. The following example shows that the complexes F• (µ) and F•0 (µ) are not always the same. Example 3.4 If we take U = W = H0 (OP1 (2)) with µ the multiplication map to V = H0 (OP1 (4)), that is, - V = k[s, t]4 , µ : k[s, t]2 ⊗ k[s, t]2 = W ⊗ U then µ is geometrically 1-generic because k[s, t] is a domain and remains so over L the algebraic closure of k. The module M = n≥0 H0 (OP1 (2) ⊗ OP1 (4n)) of Theorem 3.3 is the graded module associated to the line bundle OP1 (2 points) on P1 , regarded as a module over the homogeneous coordinate ring S of P4 via the embedding of P1 as the rational normal quartic C. This line bundle is ωC (1), the twist of the canonical line bundle by the hyperplane line bundle, so the minimal resolution of
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M is the dual of the minimal resolution of the homogeneous coordinate ring of C, suitably shifted. It has the form G• :
0
- S 6 (−2)
- S(−4)
- S 8 (−1)
- S3
- M
- 0.
On the other hand, F• has the form F• (µ) :
- S 6 (−2)
0
- (S 3 ⊗ S 3 )(−1)
δ1 (µ) -
S3,
and F1 = S 9 (−1) cannot inject into G 1 = S 8 (−1). The columns of the matrix of δ1 (µ) have exactly one k-linear dependence relation, so the maximal irredundant quotient of F• has the form F•0 :
0
- S 6 (−2)
- S 8 (−1)
- S3.
So, in particular, F20 = F2 , in accordance with Theorem 3.1. By Lemma 2.3, any map from F• to G • lifting the identity on F0 is a degreewise split monomorphism on F•0 ; in fact, F•0 is isomorphic to the linear strand of G • in this case. 4. Failure of the MRC Before proving Theorem 1.1, we present the bare bones of Gale duality. The reader who wishes to go further into this rich and beautiful subject can consult Eisenbud and Popescu [EP1], [EP2] or I. Dolgachev and D. Ortland [DO]. Let 0 ⊂ Pr be a set of γ = r +s +2 points. The linear forms on Pr define a linear series V ⊂ H0 (O0 (1)). The orthogonal complement V ⊥ ⊂ H0 (O0 (1))∗ is identified by Serre duality with a linear series (ω0 )−1 ⊂ H0 (K 0 (−1)) (where K 0 denotes the canonical sheaf of 0). Under mild hypotheses on 0, the linear series (ω0 )−1 is very ample and gives an embedding whose image 0 0 ⊂ Ps is the Gale transform of 0. As a set it is well defined only modulo linear changes of coordinates on Ps and Pr . Concretely, if the columns of the ((r + 1) × γ )-matrix M represent the points of 0 ⊂ Pr and if N is a (γ × (s + 1))-matrix whose columns are a basis of ker(M), then the rows of N represent the points of 0 0 ⊂ Ps . In the proof of Theorem 1.1 we need four facts about the Gale transform, which the reader may easily verify (or find in [EP1], [EP2]). • The subset 0 0 ⊂ Ps is defined for a general 0. In fact, it is defined as long as no subset containing all but two of the points of 0 lies in a hyperplane of Pr . • The Gale transform of 0 0 is 0. • There is a natural identification of V = H0 (OPr (1)) with (ω0 0 )−1 . • 0 is general if and only if 0 0 is general. More formally, a GL(s + 1)-invariant Zariski open condition on 0 0 induces a GL(r + 1)-invariant Zariski open condition on 0.
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Proof of Theorem 1.1 Let s ≥ 3 be an integer, and set r =
s+1 2 + δ.
( 1 s 0≤δ≤ − 2 2
Suppose that if s ≤ 4, if s ≥ 5,
and let 0 ⊂ Pr be a set of γ (r, s) = r + s + 2 points, in linearly general position. We show that the MRC fails for 0. From this statement it is easy to solve for γ in terms of r , obtaining the range given in Theorem 1.1. Let I0 be the homogeneous ideal of 0 ⊂ Pr , and let ω0 = ExtrS−1 I0 , S(−r − 1) be the canonical module. The free resolution of ω0 is, up to a shift in degree, the dual of the resolution of S/I0 . We assume that 0 is a general set of points such that the following hold: its Gale transform 0 0 ⊂ Ps is defined and is in linearly general position, 0 imposes independent conditions on quadrics, and 0 0 is not contained in any quadric (note that s+2 < γ (r, s) ≤ r +2 2 2 ). 0 Since γ (r, s) > s+2 2 , it follows that 0 does not impose independent conditions on quadrics of Ps , and thus H1 (I0 0 (2)) 6 = 0. We set U := (ω0 0 )−2 = Homk H1 (I0 0 (2)), k 6 = 0. (In this setting we have dim U = δ + 1.) Write W = H0 (OPs (1)), and V = H0 (OPr (1)). Multiplication induces a natural pairing µ:
W ⊗U
- V = (ω0 0 )−1 .
We now carry out the checks needed to apply Theorem 3.3. First, 0 0 ⊂ Ps is in linearly general position, so by Kreuzer [K], the pairing µ is 1-generic over any field and thus geometrically 1-generic. Second, 0 ⊂ Pr imposes independent conditions on quadrics, so (ω0 )−d = 0 for d ≥ 2. Hence ω0 (−1) is a finitely generated graded L S-submodule of n≥0 H0 (K 0 (n − 1)) containing W = (ω0 )−1 in degree zero. Therefore the irredundant linear free quotient F•0 (µ) of the complex F• (µ) injects onto a degreewise split direct summand of the minimal free resolution of ω0 (−1), which is the dual of the minimal free resolution of (S/I0 )(r + 2). Hence the true graded Betti number β(r −s),(r −s+2) for S/I0 satisfies s+δ 0 β(r −s),(r −s+2) ≥ rank Fs (µ) = > 0. δ
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e(r −s),(r −s+2) is found in [EP1], or The corresponding expected graded Betti number β it can be computed by taking the Hilbert series for 0, P j X t2 j bjt i = dim(S/I0 )i · t = 1 + (r + 1)t + γ (r, s) · , 1−t (1 − t)r +1 i
and calculating that (−1)
r −s
br −s+2
(2δ + 4 − s 2 + s) · = 2 (s − s + 2δ + 4)
s+1 2 +δ . s
This number is the alternating sum of the graded Betti numbers of degree r − s + 2, in this case β(r −s),(r −s+2) − β(r −s+1),(r −s+2) . Thus the expected graded Betti number is e(r −s),(r −s+2) = max (−1)r −s br −s+2 , 0 . β e(r −s),(r −s+2) = 0, which is equivalent to 0 ≤ δ ≤ The MRC certainly fails when β s s 2 − 2. The MRC also fails for s = 3, 4 and δ = 2 − 1 because s+δ e(r −s),(r −s+2) >β δ in those cases. (For s = 4 this is Boij’s example.) Straightforward computation shows e(r −s),(r −s+2) . that there are no other cases where s+δ >β δ Schreyer observed around 1983 (unpublished) that the MRC fails for linearly normal curves of degree d > g 2 − g, g ≥ 4. A result of Green and Lazarsfeld [GrL] implies that this bound can be improved to d > 3g − 3. We can use the technique developed above to give a new proof of the relevant part of the Green-Lazarsfeld result, recovering the failure of the MRC in a new way. THEOREM 4.1 Let C be a smooth curve of genus g ≥ 2, let L ∈ Picd (C) be a line bundle on C of degree d ≥ 2g + 2, and denote by ϕ|L| : C → Pd−g the embedding defined by the complete linear system |L|. (a) If H0 (L ⊗ ωC−1 ) 6 = 0 (i.e., equivalently, if the curve ϕ|L| (C) ⊂ Pd−g has a (d − 2g + 1)-secant Pd−2g−1 ), then the maximal irredundant quotient of the free linear complex M F• := L ∧l+1 H0 (ωC ) ⊗ Dl H0 (L ⊗ ωC−1 ) l≥0
injects as a degreewise direct summand of the minimal free resolution of the L Sym(H0 (L))-module n≥0 H0 (ωC ⊗ L n ). In particular, ϕ|L| (C) ⊂ Pd−g does not satisfy the property Nd−2g .
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(b)
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If g ≥ 4 and d > 3g − 3, then the MRC fails for ϕ|L| (C) ⊂ Pd−g .
Remark 4.2 By Green [Gr1], ϕ|L| (C) ⊂ Pd−g satisfies the property Nd−2g−1 (i.e., the embedding is projectively normal, the homogeneous ideal Iϕ|L| (C) is generated by quadrics, and all syzygies in the first (d − 2g − 2) steps in its minimal free resolution are linear). On the other hand, if C is nonhyperelliptic, then property Nd−2g fails for ϕ|L| (C) ⊂ Pd−g if and only if the curve has a (d − 2g + 1)-secant (d − 2g − 1)-plane, by Green and Lazarsfeld [GrL, Th. 2]. Theorem 4.1(a) spells out explicitly this failure. Proof of Theorem 4.1 By hypothesis, d ≥ 2g + 2, so ϕ|L| (C) ⊂ Pd−g is arithmetically Cohen-Macaulay; its homogeneous ideal Iϕ|L| (C) is generated by quadrics and is 2-regular. Now Theorem 3.3, applied to the geometrically 1-generic pairing µ:
H0 (ωC ) ⊗ H0 (L ⊗ ωC−1 ) → H0 (L),
yields the complex F• = F• (µ) whose maximal irredundant quotient injects as a L degreewise direct summand of the minimal free resolution of n≥0 H0 (ωC ⊗ L n ), which is the dual of the minimal free resolution of S/Iϕ|L| (C) (d − g + 1). Here and in the sequel, S = Sym(H0 (L)). It follows that the graded Betti number βd−2g,d−2g+2 of S/Iϕ|L| (C) satisfies the inequality βd−2g,d−2g+2 ≥ dim Symg−1 H (L 0
⊗ ωC−1 )
d − 2g + 1 = . g−1
In particular, βd−2g,d−2g+2 ≥ 1, and so ϕ|L| (C) ⊂ Pd−g does not satisfy property Nd−2g , which finishes the proof of part (a). Since the embedding ϕ|L| (C) ⊂ Pd−g is arithmetically Cohen-Macaulay, its hyperplane section 0 ⊂ Pd−g−1 has the same graded Betti numbers as ϕ|L| (C). Thus, as in [EP1] or the proof of Theorem 1.1, we obtain the following for the alternated sum of graded Betti numbers of degree d − 2g + 2 for S/Iϕ|L| (C) : βd−2g,d−2g+2 − βd−2g+1,d−2g+2
(d − g 2 + g) d − g − 1 = . (d − 2g + 2) g−1
If 3g − 2 ≤ d ≤ g 2 − g, the MRC predicts that βd−2g,d−2g+2 = 0, while from (a) we get βd−2g,d−2g+2 ≥ 1; hence the MRC fails for d in the above range. For d ≥ g 2 − g + 1, the expected βd−2g+1,d−2g+2 is zero while the expected βd−2g,d−2g+2 is always larger than d−2g+1 g−1 , and so the linear complex in (a) does not account for the failure of the MRC.
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1 For g ≥ 4, there exist a gg−1 = |OC (D)| on C. The pairing η : H0 OC (D) ⊗ H0 L ⊗ OC (−D) → H0 (L)
is 1-generic and thus η defines a (2 × (d − 2g + 2))-matrix with linear entries in Pd−g whose (2 × 2)-minors vanish in the expected codimension (see [E]). Therefore the Eagon-Northcott complex resolving the (2 × 2)-minors of this matrix is a linear exact complex of length d − 2g + 1 which injects as a degreewise direct summand in the top strand of the resolution of Iϕ|L| (C) . In particular, βd−2g+1,d−2g+2 ≥ (d − 2g + 1) > 1, so the MRC also fails for ϕ|L| (C) when d ≥ g 2 − g + 1. This concludes the proof of part (b). Acknowledgments. It is a pleasure to thank Tony Iarrobino and the Mathematisches Forschungsinstitut Oberwolfach, Germany. Iarrobino, while reviewing Boij [B] for the American Mathematical Society MathSciNet, noticed that Boij’s example of 21 points in P15 was not in the Eisenbud-Popescu list. The authors of the present paper were at that moment attending a conference at Oberwolfach. Iarrobino triggered our collaboration by asking Eisenbud and Popescu whether their methods could encompass the new example. The joy and success we had in answering his question owe much to the marvelous facilities and atmosphere in the Black Forest. References [AB]
M. AUSLANDER and D. A. BUCHSBAUM, Codimension and multiplicity, Ann. of
[BG]
E. BALLICO and A. V. GERAMITA, “The minimal free resolution of the ideal of s general points in P3 ” in Proceedings of the 1984 Vancouver Conference in
Math. (2) 68 (1958), 625 – 657. MR 20:6414 384
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Algebraic Geometry, CMS Conf. Proc. 6, Amer. Math. Soc., Providence, 1986, 1 – 10. MR 87j:14079 380 S. BECK and M. KREUZER, “How to compute the canonical module of a set of points” in Algorithms in Algebraic Geometry and Applications (Santander, Spain, 1994), Progr. Math. 143, Birkh¨auser, Basel, 1996, 51 – 78. MR 97g:13044 380 ˇ ˘IN, I. M. GEL’FAND, and S. I. GEL’FAND, Algebraic vector bundles on I. N. BERNSTE n P and problems of linear algebra (in Russian), Funktsional. Anal. i Prilozhen. 12, no. 3 (1978), 66 – 67; English translation in Funct. Anal. Appl. 12, no. 3 (1978), 212 – 214. MR 80c:14010a M. BOIJ, Artin level modules, J. Algebra 226 (2000), 361 – 374. MR 2001e:13024 380, 393 I. DOLGACHEV and D. ORTLAND, Point Sets in Projective Spaces and Theta Functions, Ast´erisque 165, Soc. Math. France, Montrouge, 1988. MR 90i:14009 389 D. EISENBUD, Linear sections of determinantal varieties, Amer. J. Math. 110 (1988), 541 – 575. MR 89h:14041 381, 385, 393
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D. EISENBUD and E. G. EVANS JR., Generating modules efficiently: Theorems from
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D. EISENBUD, G. FLØYSTAD, and F.-O. SCHREYER, Sheaf cohomology and free
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D. EISENBUD and S. POPESCU, Gale duality and free resolutions of ideals of points,
algebraic K -theory, J. Algebra 27 (1973), 278 – 305. MR 48:6084 387 resolutions over exterior algebras, preprint, arXiv:math.AG/0104203 379, 382
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[K] [L] [Lo1] [Lo2] [S]
[W]
Invent. Math. 136 (1999), 419 – 449. MR 2000i:13014 379, 380, 382, 383, 384, 385, 389, 391, 392 , The projective geometry of the Gale transform, J. Algebra 230 (2000), 127 – 173. MR 2001g:14083 389 F. GAETA, Sur la distribution des degr´es des formes appartenant a` la matrice de l’id´eal homog`ene attach´e a` un groupe de N points g´en´eriques du plan, C. R. Acad. Sci. Paris 233 (1951), 912 – 913. MR 13:524d 379 , A fully explicit resolution of the ideal defining N generic points in the plane, preprint, 1995. 379 A. V. GERAMITA and A. LORENZINI, “The Cohen-Macaulay type of n + 3 points in Pn ” in The Curves Seminar at Queen’s (Kingston, Ontario, 1989), Vol. VI, Queen’s Papers in Pure and Appl. Math. 83, Queen’s Univ., Kingston, Ontario, 1989, exp. no. F. MR 91c:14058 380 M. GREEN, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), 125 – 171. MR 85e:14022 392 , The Eisenbud-Koh-Stillman conjecture on linear syzygies, Invent. Math. 136 (1999), 411 – 418. MR 2000j:13024 381 M. GREEN and R. LAZARSFELD, Some results on the syzygies of finite sets and algebraic curves, Compositio Math. 67 (1988), 301 – 314. MR 90d:14034 381, 391, 392 A. HIRSCHOWITZ and C. SIMPSON, La r´esolution minimale de l’id´eal d’un arrangement g´en´eral d’un grand nombre de points dans Pn , Invent. Math. 126 (1996), 467 – 503. MR 97i:13015 380 M. KREUZER, On the canonical module of a 0-dimensional scheme, Canad. J. Math. 46 (1994), 357 – 379. MR 95d:13021 381, 390 F. LAUZE, Rang maximal pour TPn , Manuscripta Math. 92 (1997), 525 – 543. MR 98f:14016 380 A. LORENZINI, Betti numbers of points in projective space, J. Pure Appl. Algebra 63 (1990), 181 – 193. MR 91e:14045 , The minimal resolution conjecture, J. Algebra 156 (1993), 5 – 35. MR 94g:13005 J.-P. SERRE, Modules projectifs et espaces fibr´es a` fibre vectorielle, S´eminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot 1957/58, fasc. 2, Secr´etariat Math., Paris, exp. 23. MR 31:1277 387 C. H. WALTER, The minimal free resolution of the homogeneous ideal of s general points in P4 , Math. Z. 219 (1995), 231 – 234. MR 96f:13024 380
EXTERIOR ALGEBRA METHODS FOR THE MRC
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Eisenbud Department of Mathematics, University of California, Berkeley, Berkeley, California 94720, USA;
[email protected] Popescu Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794, USA;
[email protected]; and Department of Mathematics, Columbia University, New York, New York 10027, USA Schreyer Fachbereich Mathematik, Universit¨at Bayreuth, D-95440 Bayreuth, Germany;
[email protected] Walter Laboratoire J.-A. Dieudonn´e, Unit´e Mixte de Recherche 6621 du Centre National de la Recherche Scientifique, Universit´e de Nice-Sophia Antipolis, 06108 Nice CEDEX 02, France;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 3,
OSCILLATORY INTEGRAL OPERATORS WITH LOW-ORDER DEGENERACIES ALLAN GREENLEAF and ANDREAS SEEGER
Abstract We prove sharp L 2 -estimates for oscillatory integral and Fourier integral operators for which the associated canonical relation C ⊂ T ∗ L × T ∗ R projects to T ∗ L and to T ∗ R with corank 1 singularities of type ≤ 2. This includes two-sided cusp singularities. Applications are given to operators with one-sided swallowtail singularities such as restricted X-ray transforms for well-curved line complexes in five dimensions. 1. Introduction Let L , R be open sets in Rd . This paper is concerned with L 2 -bounds for oscillatory integral operators Tλ of the form Z Tλ f (x) = eıλ8(x,z) σ (x, z) f (z) dz, (1.1) where 8 ∈ C ∞ ( L × R ) is real-valued, σ ∈ C0∞ ( L × R ), and λ is large. We also write Tλ ≡ Tλ [σ ] to indicate the dependence on the symbol σ . The decay in λ of the L 2 -operator norm of Tλ is determined by the geometry of the canonical relation (1.2) C = (x, 8x , z, −8z ) : (x, z) ∈ L × R ⊂ T ∗ L × T ∗ R , specifically, by the behavior of the projections π L : C → T ∗ L and π R : C → T ∗R : π L : (x, z) 7 → x, 8x (x, z) , π R : (x, z) 7 → z, −8z (x, z) ; (1.3) DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 3, Received 26 July 2000. Revision received 3 May 2001. 2000 Mathematics Subject Classification. Primary 35S30; Secondary 42B99, 47G10. Greenleaf’s research supported in part by National Science Foundation grant number DMS 9877101. Seeger’s research supported in part by National Science Foundation grant number DMS 9970042. 397
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here 8x and 8z denote the partial gradients with respect to x and z. Note that rank Dπ L = rank Dπ R is equal to d + rank 8x z and that the determinants of Dπ L and Dπ R are equal to h(x, z) := det 8x z (x, z). (1.4) If C is locally the graph of a canonical transformation, that is, if h(x, z) 6= 0, then kTλ k = O(λ−d/2 ) (see L. H¨ormander [15], [16]). If the projections have singularities, then there is less decay in λ, and in various specific cases the decay has been determined. In dimension d = 1, D. Phong and E. Stein [21] obtained a complete description of the L 2 -mapping properties for the case of real-analytic phase functions. Similar results for C ∞ -phases (which, however, missed the endpoints) and related L p -estimates for averaging operators in the plane are in [24]. The bounds for oscillatory integral operators in one dimension, with C ∞ -phases, have recently been substantially improved by V. Rychkov [22], so that many endpoint estimates are now available in the C ∞ -category. Such general results are not known in higher dimensions even under the assumption of rank 8x z ≥ d − 1. We list some known cases. If both projections π L and π R have fold (S1,0 ) singularities, then kTλ k = O(λ−(d−1)/2−1/3 ) (see [17], [19], [5]). If only one of the projections has fold singularities, then by [8] we have kTλ k = O(λ−(d−1)/2−1/4 ); this is sharp if the other projection is maximally degenerate (see [12]) but can be improved when that projection satisfies some finitetype condition (for sharp results of this sort, see A. Comech [3]). This one-sided behavior comes up naturally when studying restricted X-ray transforms (see [6], [11], [14]). In [9] the authors began a study of the case of higher one-sided Morin (S1r ,0 ) singularities, which are the stable singularities of corank 1, and it was shown under suitable additional (“strongness”) assumptions that such estimates can be deduced from sharp estimates for two-sided S1r −1 ,0 -singularities. Thus the authors were able to prove that if one projection is a Whitney cusp, that is, of type S1,1,0 , then kTλ k = O(λ−(d−1)/2−1/6 ); again, this is sharp if the other projection is maximally degenerate. It is conjectured that if one of π L or π R has S1r ,0 -singularities, then kTλ k = O(λ−(d−1)/2−1/(2r +2) ) (for the discussion of some model cases where this is satisfied and sharp, see [9]). Here we take up the case r = 3; such mappings are commonly referred to as swallowtail singularities. In order to prove this result, it is crucial to get a sharp result for operators with two-sided cusp singularities. THEOREM
We have the following. (i) Suppose that the only singularities of one of the projections (π L or π R ) are Whitney folds, Whitney cusps, or swallowtails. Then kTλ k L 2 →L 2 =
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O(λ−(d−1)/2−1/8 ) for λ ≥ 1. Suppose that the only singularities of both projections π L and π R are Whitney folds or Whitney cusps. Then kTλ k L 2 →L 2 = O(λ−(d−1)/2−1/4 ) for λ ≥ 1.
A slightly weaker result than (ii) was recently obtained by Comech and S. Cuccagna [4], who proved for two-sided cusp singularities the bound kTλ k L 2 →L 2 ≤ Cε λ−(d−1)/2−1/4+ε with Cε → ∞ as ε → 0. We prove somewhat more general results about operators of the same “type” but with the stability assumptions weakened. To formulate the hypotheses, we review the definition of kernel vector fields for a map. Fix n-dimensional manifolds M, N and points P0 ∈ M and Q 0 ∈ N . Let f : M → N be a C ∞ -map with f (P0 ) = Q 0 . Let U be a neighborhood of P. A vector field V is a kernel field for the map f on U if V is smooth on U and if D f P V = det(D f P )W f (P) for P ∈ U ; here W is a smooth vector field on N defined near Q 0 = f (P0 ) and det(D f P ) is calculated with respect to any local systems of coordinates. Suppose now that rank D f (P0 ) ≥ n − 1. Then there are a neighborhood of P and e is another kernel field on U , a nonvanishing kernel vector field V for f on U . If V e then V = αV − det(D f )W in some neighborhood of P0 for some vector field W and smooth function α. This is easy to see by an elementary calculation. Indeed, we may choose coordinates x = (x 0 , xn ) on M and y = (y 0 , yn ) on N , vanishing at P0 and Q 0 , respectively, so that Dx0 f = (A, b) and Dxn f = (ct , d), where A is an invertible ((n − 1) × (n − 1))-matrix, b and c are vectors in d ∈ R, and A, b, c, d depend smoothly on x. Define the vector field V by V = ∂xn − hA−1 b∂x 0 i. Then clearly D f (V ) = (d − ct A−1 b)∂ yn and det D f = (d − ct A−1 b) det A; thus V is a kere) = det(D f )Z with e = hβ 0 ∂x 0 i + βn ∂xn so that D f (V nel field. Now assume that V 0 0 0 0 Z = Z + γn ∂ yn , Z = hγ ∂ y 0 i; here β = (β , βn ) are smooth functions of x, and γ = (γ 0 , γn ) are smooth functions of y. Then, at any x, Aβ 0 + bβn = det(D f )β 0 ; e = βn V + (det D f )Z 0 , as therefore β 0 = −A−1 bβn + det D f A−1 γ 0 , and thus V claimed. Definition Suppose that M and N are smooth n-dimensional manifolds and that f : M → N is a smooth map with dim ker(D f ) ≤ 1 on M. We say that f is of type k at P if there is a nonvanishing kernel field V near P so that V j (det D f ) P = 0 for j < k but V k (det D f ) P 6= 0. From the previous discussion, it is clear that this definition does not depend on the choice of the nonvanishing kernel field. If one assumes that D f drops rank simply on
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the singular variety {det D f = 0} (i.e., if ∇ det D f 6 = 0), then the definition agrees with the one proposed by Comech [3]. THEOREM 1.1 Suppose that both π L and π R are of type ≤ 2 on C . Then for λ ≥ 1,
Tλ 2 2 = O(λ−(d−1)/2−1/4 ). L →L
1.2 Suppose that Dπ L drops rank simply on the singular variety {det Dπ L = 0}, and suppose that π L is of type ≤ 3 on C . Then for λ ≥ 1,
Tλ 2 2 = O(λ−(d−1)/2−1/8 ). L →L THEOREM
Of course, the analogous statement holds with π L replaced by π R in Theorem 1.2. As a corollary of both theorems, we obtain the sharp endpoint estimate for two-sided cusp and one-sided swallowtail singularities stated above. Remark. The estimates in Theorem 1.1 and Theorem 1.2 are stable under small perturbations of 8 and σ in the C ∞ -topology. The above theorems imply sharp L 2 -Sobolev estimates for Fourier integral operators (see [8]). Let C ⊂ T ∗ L \ {0} × T ∗ R \ {0}, and let F ∈ I µ ( L , R ; C) (see [15] for the definition and [8] for the reduction of smoothing estimates for Fourier integral operators to decay estimates for oscillatory integral operators). As a corollary of Theorems 1.1 and 1.2, one obtains the following theorem. 1.3 We have the following. (i) If both π L and π R are of type ≤ 2, then F maps L 2α,comp to L 2α−µ−1/4,loc . (ii) If one projection (π L or π R ) is of type ≤ 3 and the rank of its differential drops only simply, then F maps L 2α,comp to L 2α−µ−3/8,loc . THEOREM
Remarks. (1) As an example of Theorem 1.3(i), consider, as in [9, §6], a family of curves in R4 of the form γx (t) = exp(t X + t 2 Y + t 3 Z + t 4 W )(x) for smooth vector fields X, Y, Z , W on R4 such that both of the sets of vectors n o 1 1 1 X, Y, Z ∓ [X, Y ], W ∓ [X, Z ] + [X, [X, Y ]] 6 4 24
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are linearly independent at each point x. Then the generalized Radon transform Z R f (x) = f γx (t) χ(t) dt, χ ∈ C0∞ (R), R
belongs to I −1/2 (R4 , R4 ; C) with the canonical relation C a two-sided cusp; that is, both π L and π R are Whitney cusps, and hence it follows from Theorem 1.3(i) that R : L 2α,comp → L 2α+1/4,loc for all α ∈ R, generalizing the well-known fact for the translation-invariant family γx (t) = x + (t, t 2 , t 3 , t 4 ). (2) Consider the translation-invariant families of curves in R3 , γx1 (t) = x + 2 (t, t , t 4 ) and γx2 (t) = x + (t, t 3 , t 4 ). Then {γx1 } is associated with a canonical relation, C 1 , which is a two-sided cusp, while {γx2 } is associated with a canonical relation, C 2 , for which both projections are type 2, but not Whitney cusps. In fact, the singular variety of C 2 is not smooth; it is a union of two intersecting hypersurfaces, and det(Dπ L ) and det(Dπ R ) vanish of order 2 at the intersection and simply away from that intersection. Averaging operators associated with any (not necessarily translation-invariant) sufficiently small C ∞ -perturbation of either {γx1 } or {γx2 } still has both projections of type 2, and hence map L 2α,comp → L 2α+(1/4),loc . (3) As an instance of Theorem 1.3(ii), let R be the restricted X-ray transform associated to a well-curved line complex C in R5 (see [9, §5] for the definition). Then π R has (at most) swallowtail singularities, and R maps L 2comp (R5 ) into L 21/8,loc (C). As an example, consider a curve α → γ (α) in R4 with γ 0 , γ 00 , γ 000 , and γ (4) being linearly independent at each α, and consider the X-ray transform associated to the rigid 5-dimensional line complex consisting of lines {`x 0 ,α : x 0 ∈ R4 , α ∈ R} in R5 , where `x 0 ,α = {(x 0 + tγ (α), t), t ∈ R}, and perturbations of this example. For the rigid case, the projection π L is a blowdown in the sense of [12] or [14]; that is, it exhibits a maximal degeneracy; this behavior, however, is not invariant under small perturbations and is not required for Theorem 1.3 to apply. (4) As an example of a restricted X -ray transform in R4 which is not well-curved in the sense of [9], consider the situation in the previous example, but with γ replaced by one of the curves γ (1) (α) = (α, α 2 , α 4 ) or γ (2) (α) = (α, α 3 , α 4 ) in R3 . For both examples, π R satisfies a type 3 condition with det dπ R vanishing simply; however, the singularity of dπ R for the canonical relation associated to the second line complex (defined by γ (2) ) is not of swallowtail type. Again R and perturbations thereof map L 2comp (R4 ) into L 21/8,loc (C). (5) For conormal operators in two dimensions, the condition of type ≤ k for π L corresponds to a left finite-type condition of order k + 2 in the terminology of [23], and the condition of (exact) type k corresponds to the type (1, k + 1) condition in the terminology of [24].
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2. Bounds for operators with two-sided type 2 conditions We decompose the operator according to the size of det 8x z following Phong and Stein [20], who used this decomposition to estimate operators with fold singularities. Various extensions and refinements are in [23], [21], [5], [10], [3], and [4]; in fact, we use the key estimate in [4] as the first step in our proof of Theorem 1.1. As in that work (see also [23], [3]), we need to localize VL h and V R h, where VL and V R are nonvanishing kernel vector fields for π L and π R , respectively. We may suppose that the support of σ is small and choose coordinates x = (x 0 , xd ), z = (z 0 , z d ) in Rd−1 × R vanishing at a reference point P 0 = (x 0 , z 0 ) so that 8x 0 z 0 (P 0 ) = Id−1 , 0 0
8xd z 0 (P 0 ) = 0,
8x 0 z d (P 0 ) = 0.
0 0
x z := 8−1 = (8t )−1 . Representatives for the kernel Write 8z x := 8−1 x 0 z 0 and 8 z0 x 0 x 0 z0 vector fields are then given by 0 0
V R = ∂ x d − 8 x d z 0 8z x ∂ x 0 , 0 0
VL = ∂z d − 8z d x 0 8 x z ∂z 0
(2.1)
(see [2] and the discussion in the introduction). Let K be a fixed compact set in L × R which contains the support of σ in its interior. Let A0 ≥ 102d so that k8kC 5 (K ) ≤ 10−2d A0 .
(2.2)
We also assume that |VL2 h| ≥ A−1 1 ,
|V R2 h| ≥ A−1 1 ,
(2.3)
for some A1 ≥ 1. After additional localization, we may assume that σ is supported on a set of small diameter ε; for later use we choose −2 ε = 10−1 min{A−2 0 , A1 }.
(2.4)
Let β0 ∈ C ∞ (R) be an even function supported in (−1, 1) and equal to one in (−1/2, 1/2). Let β(s) ≡ β1 (s) = β0 (s/2) − β0 (s), and for j ≥ 1, let β j (s) = β1 (21− j s) = β0 (2− j s) − β0 (2− j+1 s). √ We may assume that λ is large. Let `0 = [log2 ( λ)], that is, the largest integer ` such that 2` ≤ λ1/2 . Then let σ j,k,l (x, z) = σ (x, z)β 2l h(x, z) β j 2l/2 V R h(x, z) βk 2l/2 VL h(x, z) , 0 σ j,k,` (x, z) = σ (x, z)β0 2`0 h(x, z) β j 2`0 /2 V R h(x, z) βk 2`0 /2 VL h(x, z) ; (2.5) 0 thus if j, k > 0, then on the support of σ j,k,l we have |h| ≈ 2−l , |VL h| ≈ 2k−l/2 , |V R h| ≈ 2 j−l/2 .
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Our main technical result sharpens estimates given in [4]; we use here, as throughout, the notation A . B to denote inequalities A ≤ C B with constants C independent of λ, j, k, l. THEOREM 2.1 We have the following estimates: √ (i) for 0 < l < `0 = [log2 ( λ)],
kTλ [σ j,k,l ]k L 2 →L 2 . λ−(d−1)/2 min{2l/2 λ−1/2 ; 2−(l+ j+k)/2 };
(2.6)
(ii) 0 kTλ [σ j,k,` ]k L 2 →L 2 . λ−(d−1)/2−1/4 2−( j+k)/2 . 0
(2.7)
Given Theorem 2.1, we can deduce Theorem 1.2 by simply summing estimates (2.6) P 0 and (2.7). The bound j,k kTλ [σ j,k,` ]k ≤ λ−(d−1)/2−1/4 is immediate. Moreover, 0 X X kTλ [σ j,k,l ]k2 ≤ I + I I, √ 0≤l≤log2 ( λ) 0≤ j,k≤l/2
where I ≤
X
X
√ 0≤l≤log2 ( λ)
2l/2 λ−d/2
j,k j+k≤log2 (λ2−2l )
. λ−(d−1)/2−1/4
X
(λ2−2l )−1/4 [1 + log(λ2−2l )]2 . λ−(d−1)/2−1/4
√ 0≤l≤log2 ( λ)
and II ≤
X √
2−l/2 λ−(d−1)/2
0≤l≤log2 ( λ)
. λ−(d−1)/2
X
2−( j+k)/2
j,k j+k≥log2 (λ2−2l )
X √ 0≤l≤log2 ( λ)
2−l/2 (2l λ−1/2 ) 1 + log2 (λ2−2l ) . λ−(d−1)/2−1/4 .
We make some preliminary observations needed in the proof of Theorem 2.1. In what follows we always make the following assumption. Assumption: k ≤ j. For k ≥ j, apply the corresponding estimates for the adjoint of Tλ [σ j,k,l ]. For the proof of Theorem 2.1, we may assume, by the known result for one-sided folds (see [8]), that 2k−l/2 ≤ 2 j−l/2 ≤ ε, where ε is as in (2.4).
(2.8)
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Affine changes of variables Before starting with estimates, we wish to mention the effect of changes of variables on (2.5). Set x = x(u) and z = z(v), and let 9(u, v) = 8(x(u), z(v)). Let h(x, z) = det 8x z and e h(u, v) = det 9uv (u, v); then Dx Dz e h(u, v) = h x(u), z(v) det det . Du Dv P P P eR = eL = If V R = si (x, z)∂xi , VL = ti (x, z)∂zi and V σi (u, v)∂u i , V P eR (g(x(u), z(v)) and VL g(x(u), z(v)) = τi (u, v)∂vi , then V R g(x(u), z(v)) = V eL (g(x(u), z(v)) if and only if σE = (Dx/Du)−1 sE and τE = (Dz/Dv)−1 tE. V In particular, if our changes of variables are affine and of the form x(u) = x 0 + (u 0 + a 0 u d , u d ),
z(v) = z 0 + (v 0 + b0 vd , vd )
(2.9)
with constant vectors a 0 , b0 ∈ Rd−1 and P = (x 0 , z 0 ) and if VL and V R are eR = ∂u d − (a t + 8x z 0 8z 0 x 0 )∂u 0 and V eL = of the form (2.1), then we have V d 0 0 ∂vd − (bt + 8z d x 0 8x z )∂v 0 , where all coefficient functions are evaluated at (x 0 , z 0 ) + 0 0 (u 0 + a 0 u d , u d , v 0 + b0 vd , vd ). Thus, by choosing a 0 = −8x z (P)8z 0 xd (P), b0 = 0 0 eR )0,0 = ∂u d , (V eL )0,0 = ∂vd . −8z x (P)8x 0 z d (P), we achieve that (V Localization We perform various localizations to small boxes in (x, z)-space. Let P = (x 0 , z 0 ) ∈ L × R ⊂ RdL ×RdR , let a ∈ RdL and b ∈ RdR be vectors with 1 ≤ kak∞ , kbk∞ ≤ 2, and let πa⊥ , πb⊥ be the orthogonal projections to the orthogonal complement of Ra in RdL and Rb in RdR , respectively. Suppose that 0 < γ1 ≤ γ2 < 1 and 0 < δ1 ≤ δ2 ≤ ε, and let B Pa,b (γ1 , γ2 , δ1 , δ2 ) = (x, z) : |πa⊥ (x − x 0 )| ≤ γ1 , |hx − x 0 ai| ≤ γ2 , |πb⊥ (z − z 0 )| ≤ δ1 , |hz − z 0 bi| ≤ δ2 . (2.10) Definition We say that χ ∈ C0∞ is a normalized cutoff function associated to B Pa,b (γ1 , γ2 , δ1 , δ2 ) if it is supported in B Pa,b (γ1 , γ2 , δ1 , δ2 ) and satisfies the (natural) estimates ⊥ (π ∇x )m L ha∇x in L (π ⊥ ∇z )m R hb∇z in R χ(x, z) ≤ γ −m L γ −n L δ −m R δ −n R a b 1 2 1 2 whenever m L + n L ≤ 10d, m R + n L ≤ 10d. Here (πa⊥ ∇x )n L stands for any differential operator hE u 1 ∂x i · · · hE u n L ∂x i, where the vectors u 1 , . . . , u n L are unit vectors perpendicular to a. We denote by Z Pa,b (γ1 , γ2 , δ1 , δ2 ) the class of all normalized cutoff functions associated to B Pa,b (γ1 , γ2 , δ1 , δ2 ).
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We often localize to boxes of the form (2.10) and consider Tλ [ζ σ j,k,l ], where ζ is a cutoff function that is controlled by an absolute constant times a normalized cutoff function in the above sense. Suppose now that P = (x 0 , z 0 ), that our change of variable is as in (2.9), and that a = (a 0 , 1), b = (b0 , 1). Suppose that ζ is a normalized cutoff function associated to B Pa,b (γ1 , γ2 , δ1 , δ2 ). Let e ζ (u, v) = ζ (x(u), z(v)). Then e ζ is supported in ed ,ed 0 0 e ˜ ˜ ˜ B(0,0) (γ˜1 , γ˜2 , δ1 , δ2 ) with γ˜ = (1 + |a |)γ , δ = (1 + |b |)δ, and there is a positive constant C (independent of γ , δ) such that C −1e ζ is a normalized cutoff function ed ,ed e ˜ ˜ associated to B(0,0) (γ˜1 , γ˜2 , δ1 , δ2 ). Changing variables as in (2.9) in the expression for the operator Tλ [ζ σ j,k,l ] yields Z Tλ [ζ σ j,k,l ] f z(v) = e ζ (u, v)e σ j,k,l eıλ9(u,v) dv eRe eL e with e σ j,k,l (u, v) = σ (x(u), z(v))β(2l e h(u, v))β j (2l/2 V h(u, v))βk (2l/2 V h(u, v)) e and h(u, v) = det 9uv (u, v). Basic estimates We now give estimates for various pieces localized to (thin) boxes which are usually longer in the directions of the kernel fields VL and V R . In order to formulate our results, we start with a definition. Definition Let P = (x 0 , z 0 ) ∈ L × R , and let 0 0 a P = − 8x z (P)8z 0 xd (P), 1 ,
0 0 b P = − 8z x (P)8x 0 z d (P), 1 .
(2.11)
Define, for fixed j, k, l,
a ,b A P (γ1 , γ2 , δ1 , δ2 ) := sup Tλ [ζ σ j,k,l ] : ζ ∈ Z P P P (γ1 , γ2 , δ1 , δ2 ) ,
0 A P0 (γ1 , γ2 , δ1 , δ2 ) := sup Tλ [ζ σ j,k,` ] : ζ ∈ Z Pa P ,b P (γ1 , γ2 , δ1 , δ2 ) . 0
(2.12) (2.13)
√ Here `0 = [log2 ( λ)]. The main estimate in Comech and Cuccagna [4] applies to operators whose kernels are localized to boxes B Pa P ,b P (2−l , 2− j−l/2 , 2−l , 2−k−l/2 ). This result is formulated in (2.14) of the following proposition. The constants implicit in the inequalities below do not depend on j, k, l. 2.2 We have the following. PROPOSITION
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For 2l ≤ λ1/2 , sup A P (2−l , 2− j−l/2 , 2−l , 2−k−l/2 ) . 2l/2 λ−d/2
(2.14)
P
and sup A P (2−l , 2− j−l/2 , 2−l , 2−k−l/2 ) . 2−(l+ j+k)/2 λ−(d−1)/2 .
(2.15)
P
(ii)
√ Let l = [log2 ( λ)] = `0 . Then sup A P0 (2−`0 , 2− j−`0 /2 , 2−`0 , 2−k−`0 /2 ) . λ−(d−1)/2−1/4 2−( j+k)/2 . (2.16) P
Proposition 2.2 is the starting point in our proof and is extended via orthogonality arguments. The basic steps are contained in Propositions 2.3 – 2.5. In√what follows, N denotes an integer less than or equal to 10d − 1 and l = [log2 ( λ)] = `0 . Then the following estimates hold uniformly for j, k, l. PROPOSITION 2.3 We have the following. (i) For 2l ≤ λ1/2 , k ≤ j ≤ l/2,
sup A P (2 j+k−l , 2k−l/2 , 2 j+k−l , 2k−l/2 ) P
. sup A P (2−l , 2− j−l/2 , 2−l , 2−k−l/2 ) P
+ 2−l(2d−1)/2 2−( j+k)/2 (2−2l λ)−N /2 . (ii)
(2.17)
For k ≤ j ≤ `0 /2, sup A P0 (2 j+k−`0 , 2k−`0 /2 , 2 j+k−`0 , 2k−`0 /2 ) P
. sup A P0 (2−`0 , 2− j−`0 /2 , 2−`0 , 2−k−`0 /2 ) P
+ λ−d/2+1/4 2−( j+k)/2 .
(2.18)
PROPOSITION 2.4 We have the following. (i) For 2l ≤ λ1/2 , k ≤ j ≤ l/2,
sup A P (2 j−l/2 , 2 j−l/2 , 2k−l/2 , 2k−l/2 ) P
. sup A P (2 j+k−l , 2k−l/2 , 2 j+k−l , 2k−l/2 ) P
+ 2( j+k)(d−1) 2k 2−l(2d−1)/2 (2 j+k−2l λ)−N /2 .
(2.19)
INTEGRAL OPERATORS WITH LOW-ORDER DEGENERACIES
(ii)
407
For k ≤ j ≤ `0 /2, sup A P0 (2 j−`0 /2 , 2 j−`0 /2 , 2k−`0 /2 , 2k−`0 /2 ) P
. sup A P0 (2 j+k−`0 , 2k−`0 /2 , 2 j+k−`0 , 2k−`0 /2 ) P
+ 2( j+k)(2(d−1)−N )/2+k λ−(2d−1)/4−N /2 .
(2.20)
2.5 We have the following. (i) For 2l ≤ λ1/2 , k ≤ j ≤ l/2, PROPOSITION
kTλ [σ j,k,l ]k . sup A P (2 j−l/2 , 2 j−l/2 , 2k−l/2 , 2k−l/2 ) P
+ 2( j+k−l)d/2 (λ2k−3l/2 )−N /2 . (ii)
(2.21)
For k ≤ j ≤ `0 /2, 0 kTλ [σ j,k,` ]k . sup A P0 (2 j−`0 /2 , 2 j−`0 /2 , 2k−`0 /2 , 2k−`0 /2 ) 0 P
+ 2( j+k)d/2−k N /2 λ−d/2−N /8 .
(2.22)
Taking these estimates for granted, we can give the following proof. Proof of Theorem 2.1 Observe that since k ≤ j ≤ l/2 and 2l ≤ λ1/2 , the quantities 2−l(2d−1)/2 2−( j+k)/2 (2−2l λ)−N /2 , 2( j+k)(d−1)/2+k−l(2d−1)/2 (2 j+k−2l λ)−N /2 , and 2( j+k−l)d/2 (λ2k−3l/2 )−N /2 are all dominated by a constant times λ−(d−1)/2 min{2−(l+ j+k)/2 , 2l/2 λ−1/2 }, and a combination of the first parts of Propositions 2.3 – 2.5 gives
Tλ [σ j,k,l ] . sup A P (2−l , 2− j−l/2 , 2−l , 2−k−l/2 ) P
+ λ−(d−1)/2 min{2−(l+ j+k)/2 , 2l/2 λ−1/2 }. We estimate the quantities A P (2−l , 2− j−l/2 , 2−l , 2−k−l/2 ) by Proposition 2.2, and (2.5) follows; (2.6) is proved in the same way, using instead (2.18), (2.20), and (2.22).
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GREENLEAF and SEEGER
3. Proofs of the Propositions Preliminaries We begin by stating two elementary lemmas that are used several times in the proof of Propositions 2.3 – 2.5. LEMMA 3.1 PM Suppose that ζ ∈ Z Pa P ,b P (γ1 , γ2 , δ1 , δ2 ). Then ζ = i=1 ci ζi , where ζi ∈ Z Q i (εγ1 , a P ,b P εγ2 , εδ1 , εδ2 ) with Q i ∈ B P (γ1 , γ2 , δ1 , δ2 ) and such that |ci |, M ≤ Cε (independent of the specific choice of ζ and γ , δ, P).
Proof The proof is immediate. LEMMA 3.2 Let Q = (x Q , z Q ) ∈ B Pa P ,b P (γ1 , γ2 , δ1 , δ2 ), where γ1 ≤ γ2 ≤ ε, δ1 ≤ δ2 ≤ ε. Suppose that 0 < e γ1 ≤ e γ2 ≤ ε−1 γ2 , 0 < e δ1 ≤ e δ2 ≤ ε−1 δ2 , and assume that
min
ne γ1 e δ1 o ≥ max{γ2 , δ2 }. , γ2 e e δ2
(3.1)
Then there are positive constants C, C1 (independent of γ , e γ , δ, e δ , P, Q) such that a Q ,b Q a P ,b P −1 e e for ζ ∈ Z Q (e γ1 , e γ2 , δ1 , δ2 ), we have C1 ζ belonging to Z Q (Ce γ1 , Ce γ2 , Ce δ1 , e C δ2 ). Proof Observe that |a P − a Q | + |b P − b Q | . max{γ2 , δ2 }. a P ,b P Then the relevant geometry is that, by assumption (3.1), the boxes B Q (e γ1 , e γ2 , a Q ,b Q e δ1 , e δ2 ) and B (e γ1 , e γ2 , e δ1 , e δ2 ) are contained in fixed dilates of each other. The Q
asserted estimates are easy to check. We denote by η a (C0∞ (R))-function that is supported in (−1, 1) and satisP ∞ d−1 ))-function χ is defined by fies n∈Z η(· − n) Q≡ 1. Moreover, the (C 0 (R d−1 χ (x1 , . . . , xd−1 ) = i=1 η(xi ). In the proofs of Propositions 3.3 – 3.5, we use dilates and translates of η and χ to decompose a suitable cutoff function ζ as X ζ = ζX Z . (3.2) X,Z ∈Zd
INTEGRAL OPERATORS WITH LOW-ORDER DEGENERACIES
409
The definition of ζ X Z depends on the particular geometry and is given by (3.12), (3.25), and (3.30) in the three respective cases. We then employ orthogonality arguments to estimate the operator norm of Tλ [ζ σ j,k,l ] in terms of the operator norms of TX Z := Tλ [ζ X Z σ j,k,l ]. This is done by using the Cotlar-Stein lemma (see [25, Chap. VII, §2]). We then have ∗ . to estimate the kernels of TX∗ W Te X Z and TX Z TY e Z The kernel of TX∗ W Te X Z is given by Z H (w, z) ≡ H X W e (w, z) = e−ıλ(8(x,w)−8(x,z)) κ X W e (3.3) XZ X Z (x, w, z) d x, where κX W e X Z (x, w, z) = ζ e X Z (x, z)ζ X W (x, w)σ j,k,l (x, z)σ j,k,l (x, w).
(3.4)
The kernel of TX Z TY∗e is given by Z Z K (x, y) ≡ K X Z Y e eıλ(8(x,z)−8(y,z)) ω X Z Y e Z (x, y) = Z (z, x, y) dz,
(3.5)
with ωX Z Y e Z (z, x, y) = ζ X Z (x, z)ζY e Z (y, z)σ j,k,l (x, z)σ j,k,l (y, z).
(3.6)
Our localizations ζ X Z always have the property that the supports of ζ X W and ζ e XZ are disjoint whenever |X i − e X i | ≥ 3 for some i ∈ {1, . . . , d}. Moreover, the supports e of ζ X Z and ζY e Z are disjoint whenever |Z i − Z i | ≥ 3 for some i ∈ {1, . . . , d}. This implies that TX∗ W Te XZ = 0
if |X − e X |∞ ≥ 3,
TX Z TY∗e =0 Z
if |Z − e Z |∞ ≥ 3.
(3.7)
In what follows, we split variables X and Z in Zd as X = (X 0 , X d ) and Z = Z d ). The geometric meaning of this splitting depends on the particular situation in Propositions 3.3 – 3.5. The main orthogonality properties always follow from either the localization properties of the operator in terms of h, VL h, or V R h, or by an integration by parts with respect to the directions orthogonal to a P or b P . To describe this, we assume that a P = ed , b P = ed at a suitable reference point, a situation that we are always able to achieve by an affine change of variables as described in §2. If 8x 0 (x, w) 6= 8x 0 (x, z) for all x with (x, w, z) ∈ supp κ X W e X Z , then we may integrate by parts with respect to the x 0 variables; specifically, we have Z N H (w, z) = (i/λ) e−ıλ(8(x,w)−8(x,z) )L N[κ X W e (3.8) X Z ](x, w, z) d x, (Z 0 ,
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GREENLEAF and SEEGER
where the differential operator L is defined by 8 0 (x, z) − 8 0 (x, w) x x L g = divx 0 g . |8x 0 (x, z) − 8x 0 (x, w)|2
(3.9)
Similar formulas hold for the z 0 integration by parts for the integral defining K (x, y). We give a proof of estimates (2.17), (2.19), and (2.21), and the proof of (2.18), (2.20), and (2.22) is similar. Here we note that the lower bound on |h| in the localization (2.5) is used in the proof of estimate (2.14); however, it is not needed for the proof of Propositions 2.3 – 2.5. Remarks on the proof of Proposition 2.2 In order to prove (2.14), it suffices, by Lemma 3.1, to estimate A P (ε2−l , ε2− j−l/2 , ε2−l , ε2−k−l/2 ) for small ε. By an affine change of variable, as discussed in (2.9), we may assume that P = (0, 0) and that 8z 0 xd (0, 0) = 0, 8x 0 z d (0, 0) = 0; thus 8x 0 z 0 is close to the identity Id−1 on the support of ζ , and the quantities |8z 0 xd (x, z)| and |8x 0 z d (x, z)| are bounded by A0 ε2−k−l/2 for (x, z) ∈ supp ζ (recall that k ≤ j). Moreover, a P = ed , b P = ed ; thus ζ is, up to a constant, a normalized cutoff function associated to a box where |x 0 |, |z 0 | ≤ ε2−l , |xd | ≤ 2− j−l/2 , |z d | ≤ 2−k−l/2 . This puts us in the same situation as in the proof of [4, (3.6)]. If A P (ε2−l , ε2− j−l/2 , ε2−l , ε2−k−l/2 ) does not vanish identically, then the function |h(x, z)| is comparable to 2−l on the box B Pa P ,b P (ε2−l , ε2− j−l/2 , ε2−l , ε2−k−l/2 ). Set S = Tλ [ζ σ j,k,l ]. For j ≥ k (assumed here), the kernel of SS ∗ can be estimated using integration by parts, and all the details of this argument are provided in [4]. Estimate (2.15) is more standard, but we sketch the argument for completeness. We may assume that (VL ) P = ∂z d and (V R ) P = ∂xd , and then “freezing” xd , z d , we may write Z 0 S f (x , xd ) = S xd ,z d f (·, z d ) (x 0 ) dz d . R
S xd ,z d
Each is an oscillatory integral operator of the form (1.1) in Rd−1 , and the mixed Hessian of the phase function has maximal rank d − 1; however, the amplitudes have less favorable differentiability properties. Note that each (x 0 , z 0 ) differentiation causes a blowup of O(2l ) = O(λ1/2 ). These estimates for the amplitudes are analogous to the differentiability properties of symbols of type (1/2, 1/2), and in this situation the classical bound remains true; one can combine H¨ormander’s argument in [16] with almost-orthogonality arguments in the proof of the Calder´on-Vaillancourt theorem for pseudo-differential operators in [2]. (See also [11] for related but somewhat different arguments for Fourier integral operators associated to canonical graphs.) Here it follows that the L 2 -operator norm of S xd ,z d is O(λ−(d−1)/2 ) uniformly in xd , z d . From the definition of σ j,k,l , we see that there are intervals I and J of length
INTEGRAL OPERATORS WITH LOW-ORDER DEGENERACIES
411
O(2− j−l/2 ) and O(2−k−l/2 ), respectively, so that S xd ,z d = 0 unless xd ∈ I and z d ∈ J . Thus from applications of Minkowski’s and Cauchy-Schwarz’s inequalities it follows that kSk . 2−( j+l/2)/2 2−(k+l/2)/2 λ−(d−1)/2 . Equation (2.16) is proved in the same way. Proof of Proposition 2.3 Fix P. By Lemma 3.1, it suffices to estimate kTλ [ζ σ j,k,l ]k, where ζ belongs to a ,b Z P P P (ε2 j+k−l , ε2k−l/2 , ε2 j+k−l , ε2k−l/2 ), with norm independent of P. By an affine change of variable, as discussed in (2.9), we may assume that P = (0, 0) and that 8z 0 xd (0, 0) = 0, 8x 0 z d (0, 0) = 0; hence k8x 0 z 0 − I k ≤ 2−d ,
(3.10)
|8z 0 xd (x, z)| + |8x 0 z d (x, z)| ≤ A0 ε2
k−l/2
,
(3.11)
for (x, z) ∈ supp ζ . Moreover, a P = ed , b P = ed ; thus ζ is, up to a constant, a normalized cutoff function associated to a box where |x 0 |, |z 0 | ≤ ε2 j+k−l , |xd |, |z d | ≤ 2k−l/2 . For X, Z ∈ Zd , let ζ X Z (x, z) = ζ (x, z)χ(2l ε−1 x 0 − X 0 )η(2 j+l/2 ε−1 xd − X d ) × χ(2l ε−1 z 0 − Z 0 )η(2k+l/2 ε−1 z d − Z d ),
(3.12)
and let TX Z = Tλ [ζ σ j,k,l ]. By (3.10) and (3.11) and Lemma 3.2, there are posia ,b Q
tive constants C, C1 such that C1−1 ζ X Z belongs to Z Q Q C2−k−l/2 ). Thus
(C2−l , C2− j−l/2 , C2−l ,
kTX Z k . sup A P (2−l , 2− j−l/2 , 2−l , 2−k−l/2 ),
(3.13)
P
and it remains to show almost orthogonality of the pieces TX Z . By our localization, the orthogonality properties (3.7) are satisfied. Therefore assertion (2.17) follows from
∗
T Te . 2−l(2d−1)− j−k λ2−2l |W 0 − Z 0 | −N XW X Z if 2A0 |W 0 − Z 0 | ≥ |Wd − Z d | and |W 0 − Z 0 | ≥ C1 ,
TX∗ W Te XZ = 0
if 2A0 |W 0 − Z 0 | < |Wd − Z d | and |Wd − Z d | ≥ C1
(3.14)
(3.15)
(for suitable C1 1) and
TX Z T ∗ . 2−l(2d−1)− j−k λ2−2l |X 0 − Y 0 | −N Ye Z if 2A0 |X 0 − Y 0 | ≥ |X d − Yd | and |X 0 − Y 0 | ≥ C1 ,
(3.16)
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GREENLEAF and SEEGER
TX Z TY∗e =0 Z
if 2A0 |X 0 − Y 0 | < |X d − Yd | and |X d − Yd | ≥ C1 .
(3.17)
We now show (3.15) and (3.14). The kernel H of TX∗ W Te X Z is given by (3.3) and (3.4). In order to see (3.15), pick points (x, w) ∈ supp ζ X W and (x, z) ∈ supp ζ e XZ, and also assume that (x, w) and (x, z) belong to supp σ j,k,l (if there are no two such points, then TX∗ W Te X Z = 0). By definition of σ j,k,l , we have |h(x, z) − h(x, w)| ≤ 2−l+2 .
(3.18)
Also, for all (x, z˜ ) ∈ supp ζ , we have |h z d (x, z˜ ) − VL h(x, z˜ )| ≤ A0 ε2k−l/2 ,
(3.19)
so that |h z d (x, z˜ )| ≥ 2k−l/2−2 . Note that ε2−k−l/2 (|Wd − Z d | − 2) ≤ |wd − z d | ≤ 2−k−l/2 (|Wd − Z d | + 2)ε and that |w0 − z 0 | ≤ C0 (|W 0 − Z 0 | + 2)2−l ε. Therefore |h(x, w) − h(x, z)| ≥ |h z d (x, z)||wd − z d | − A0 |w0 − z 0 | ≥ 2k−l/2−4 ε2−k−l/2 |Wd − Z d | = ε2−l−2 |Wd − Z d |
(3.20)
if |Wd − Z d | ≥ max{2A0 |W 0 − Z 0 |∞ , 2A0 } and |Wd − Z d | ≥ 2A0 . Observe that (3.18) and (3.20) can hold simultaneously only when |Wd − Z d | stays bounded; this implies (3.15). Now we assume that |Wd − Z d | ≤ 2A0 |W 0 − Z 0 |∞ , and we show (3.14) if |W 0 − Z 0 | ≥ C1 for sufficiently large C1 . We perform integration by parts with respect to the x 0 variables in (3.3), using (3.8) and (3.9). Now in view of (3.10) and (3.11), we have |8x 0 (x, w) − 8x 0 (x, z)| ≥ |8x 0 z 0 (x, z)(w0 − z 0 )| − A0 ε2k−l/2 |wd − z d | − A0 |w − z|2 ≥ ε2−l |W 0 − Z 0 |
(3.21)
if |W 0 − Z 0 | ≥ C1 for suitable C1 . Moreover, the x-derivatives of 8(x, w) − 8(x, z) are O(2−l |W 0 − Z 0 |), and differentiating the symbol causes a blowup of O(2l ) for each differentiation. Thus for |W 0 − Z 0 | ≥ C1 , −N −2l |L N (κ X W e |W 0 − Z 0 | . X Z )| . λ2 Taking into account the x support thus yields the estimate |H (w, z)| . 2−l(d−1) 2−l/2− j λ2−2l |W 0 − Z 0 |
−N
.
INTEGRAL OPERATORS WITH LOW-ORDER DEGENERACIES
413
R R By Schur’s test, we have to bound supw |H (w, z)| dz and supz |H (w, z)| dw. Since the integrals are extended over sets of measure O(2−l(d−1) 2−l/2−k ), we obtain bound (3.14). We still have to estimate the kernel K given by (3.5) and (3.6). Note that |h xd (x, z˜ ) − V R h(x, z˜ )| ≤ A0 ε2k−l/2 , so that |h xd (x, z)| ≈ 2 j−l/2 (recall that j ≥ k). Thus in place of (3.20) we have |h(x, z) − h(y, z)| ≥ 2 j−l/2 |xd − yd | − A0 |x 0 − y 0 |,
(3.22)
and in place of (3.21) we have |8z 0 (x, z) − 8z 0 (y, z)| ≥ |8z 0 x 0 (y, z)(x 0 − y 0 )| − A0 ε2k−l/2 |xd − yd | − A0 |x − y|2 . (3.23) Since |xd − yd | ≈ |X d − Yd |, we proceed as before to obtain (3.16) and (3.17). Proof of Proposition 2.4 We continue to use the same notation as in the previous proof, although our localizations are with respect to different (larger) boxes. By Lemma 3.1, it suffices to estimate the operator norm of Tλ [ζ σ j,k,l ], where now ζ ∈ Z P (ε2 j−l/2 , ε2 j−l/2 , ε2k−l/2 , ε2k−l/2 ). Again we may assume that by an affine change of variable P = (0, 0) and that 8z 0 xd , 8x 0 z d vanish at (0, 0). It follows that |8z 0 xd (x, z)| + |8x 0 z d (x, z)| ≤ A0 ε2 j−l/2 ,
(x, z) ∈ supp ζ,
(3.24)
and again a P = ed , b P = ed . For X, Z ∈ Zd , we now define ζ X Z (x, z) = ζ (x, z)χ(2− j−k+l ε−1 x 0 − X 0 )η(2−k+l/2 ε−1 xd − X d ) × χ(2− j−k+l ε−1 z 0 − Z 0 )η(2−k+l/2 ε−1 z d − Z d ),
(3.25)
and we set TX Z = Tλ [ζ X Z σ j,k,l ]. In view of (3.24), Lemma 3.1, and Lemma 3.2, kTX Z k ≤ sup A P (2 j+k−l , 2k−l/2 , 2 j+k−l , 2k−l/2 ). P
To show the orthogonality, observe that (3.7) remains valid. Moreover, the width of the smaller boxes in the z d direction is comparable to the z d -width of the original boxes, namely, ≈ 2k−l/2 . This shows that TX∗ W Te XZ = 0 for sufficiently large C1 .
if |Wd − Z d | ≥ C1
(3.26)
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GREENLEAF and SEEGER
This estimate is complemented by
∗
T Te . 22( j+k)(d−1) 22k 2−l(2d−1) (λ2 j+k−2l |W 0 − Z 0 |)−N XW X Z if |W 0 − Z 0 | ≥ C1
(3.27)
for large C1 . To see (3.27), we integrate by parts with respect to x 0 . Our kernel is still given by (3.3) and (3.4). To perform the integration by parts, we may assume that |Wd − Z d | ≤ C1 by (3.26). We now see from (3.24) that |8x 0 (x, w) − 8x 0 (x, z)| ≥ |8x 0 z 0 (x, z)(w0 − z 0 )| − A0 ε2 j−l/2 |wd − z d | − A0 |w − z|2 , but |w0 − z 0 | ≈ |W 0 − Z 0 |ε2 j+k−l , and |wd − z d | ≤ 2k−l/2 ε|Wd − Z d | ≤ Cε2k−l/2 . Thus, if |W 0 − Z 0 | is sufficiently large, we have the lower bound |8x 0 (x, w) − 8x 0 (x, z)| & 2 j+k−l |W 0 − Z 0 | N for (x, w, z) ∈ supp κ X W e X Z . Therefore, analyzing L (κ X W e X Z ) as in the proof of Proposition 3.3, we see that −N j+k−2l |L N (κ X W e |W 0 − Z 0 | . X Z )| . 2
From this we get the pointwise bound −N |H (w, z)| . 2( j+k−2l)(d−1) 2k−l/2 λ2 j+k−2l |W 0 − Z 0 | . For Schur’s test we have to integrate this in x or y over a set of measure 2( j+k−l)(d−1) 2k−l/2 , and we obtain in fact a slightly better estimate than (3.27). Next, it remains to show that
TX Z T ∗ . 22( j+k)(d−1) 22k 2−l(2d−1) λ2 j+k−2l |X 0 − Y 0 | −N e YZ if 2A0 |X 0 − Y 0 | ≥ |X d − Yd | and |X 0 − Y 0 | ≥ C1
(3.28)
and TX Z TY∗e =0 Z
if 2A0 |X 0 − Y 0 | < |X d − Yd | and |X d − Yd | ≥ C1 .
(3.29)
The proof of these estimates is similar to the proof of the corresponding estimates in Proposition 2.3. Estimate (3.22) continues to hold, and estimate (3.23) is replaced by the weaker estimate |8z 0 (x, z) − 8z 0 (y, z)| ≥ |8z 0 x 0 (y, z)(x 0 − y 0 )| − A0 ε2 j−l/2 |xd − yd | − A0 |x − y|2 , which, however, still gives the asserted bound since |xd − yd | ≈ 2k−l/2 ε|X d − Yd | and |x 0 − y 0 | ≈ 2−l |X 0 − Y 0 |.
INTEGRAL OPERATORS WITH LOW-ORDER DEGENERACIES
415
Proof of Proposition 2.5 We may assume that the support of ζ is small (i.e., contained in a ball of radius ε). By Lemma 3.1, it suffices to estimate the operator norm of Tλ [ζ σ j,k,l ], where now ζ ∈ Z P (ε, ε, ε, ε). By affine changes of variables, we may assume that P = (0, 0) and that 8z 0 xd , 8x 0 z d vanish at (0, 0). Thus |8z 0 xd (x, z)| + |8x 0 z d (x, z)| ≤ A0 ε,
(x, z) ∈ supp ζ.
For X, Z ∈ Zd , we now consider TX Z = Tλ [ζ σ j,k,l ] with ζ X Z (x, z) = ζ (x, z)χ(2− j+l/2 ε−1 x 0 − X 0 )η(2− j+l/2 ε−1 xd − X d ) × χ(2−k+l/2 ε−1 z 0 − Z 0 )η(2−k+l/2 ε−1 z d − Z d ),
(3.30)
and again kTX Z k . sup P A P (2 j+k−l , 2k−l/2 , 2 j+k−l , 2k−l/2 ). For the orthogonality of the pieces we now use, besides (3.7), assumptions (2.3). By our choice of ε, we have |VL2 h − ∂z d VL h| ≤ A0 ε ≤
A1 10
and similarly |V R2 h − ∂xd V R h| ≤
A1 . 10
Thus 4 · 2k−l/2 ≥ |VL h(x, w) − VL h(x, z)| ≥ (2A1 )−1 |wd − z d | − A0 |w0 − z 0 |, 4 · 2 j−l/2 ≥ |V R h(x, z) − V R h(y, z)| ≥ (2A1 )−1 |xd − yd | − A0 |x 0 − y 0 |. This shows that TX∗ W TX˜ Z = 0
if |Wd − Z d | ≥ C,
(3.31)
TX Z TY∗Z˜
if |X d − Yd | ≥ C,
(3.32)
=0
for C = 10A0 A1 . Now assume that |Wd − Z d | ≤ C1 . Then if (x, w, z) ∈ supp ζ X W ζ X Z , we have |8x 0 (x, w) − 8x 0 (x, z)| ≥ |8x 0 z 0 (x, z)(w0 − z 0 )| − A0 ε|wd − z d | − A0 |w − z|2 , but now |w0 − z 0 | ≈ |W 0 − Z 0 |2k−l/2 and |wd − z d | ≤ 2C2k−l/2 . Thus for large |W 0 − Z 0 | we have the lower bound |8x 0 (x, w) − 8x 0 (x, z)| ≥ 2k−l/2 |W 0 − Z 0 |, and it follows that k−3l/2 |L N [κ X W e |W 0 − Z 0 |)−N . X Z ](x, w, z)| . (λ2
416
GREENLEAF and SEEGER
Consequently, |H (w, z)| . 2( j−l/2)d (λ2k−3l/2 |W 0 − Z 0 |)−N . To apply Schur’s test, we observe that for fixed z the w integral is extended over a set of measure O(2(k−l/2)d ) (likewise for fixed w the z integral). We obtain the bound −N kTX∗ W TX˜ Z k . 2( j+k−l)d λ2k−3l/2 |W 0 − Z 0 | (3.33) if |W 0 − Z 0 | ≥ C 0 . By a similar argument, −N kTX Z TZ∗Z˜ k . 2( j+k−l)d λ2 j−3l/2 |X 0 − Y 0 | .
(3.34)
The asserted estimate (2.21) now follows from combining (3.31) – (3.34) and the estimate for the individual pieces. 4. One-sided type three singularities In this section we discuss the proof of Theorem 1.2. The reasoning is very close to that given by the authors in [9], but the assumptions there are somewhat different. We thus only sketch the proof and refer the reader to [8] and [9] for details of some of the arguments. First we need an extension of Theorem 1.1 to oscillatory integral operators of the form Z Z Tµ f (x) = f (y) eıµψ(x,y,ϑ) a(x, y, ϑ) dϑ dy, where the frequency variable ϑ lives in an open set 2 ⊂ R N and we assume that a ∈ C0∞ ( L × R ×2). It is assumed that ψ is a nondegenerate phase function in the sense of H¨ormander [15] (but not necessarily homogeneous); that is, ∇ϑi (∇x,y,ϑ 9), i = 1, . . . , N , are linearly independent. The canonical relation Cψ ⊂ T ∗ L × T ∗ R is given by Cψ = (x, ψx , y, −ψ y ) : ψϑ = 0 . 4.1 Suppose that the projections π L : Cψ → T ∗ L , π R : Cψ → T ∗ R are of type ≤ 2. Then kTµ k L 2 →L 2 = O(µ−(d+N −1)/2−1/4 ), µ → ∞. This estimate is stable under small perturbations of ψ and a in the C ∞ -topology. LEMMA
The reduction to the situation in Theorem 1.2 involves canonical transformations on T ∗ L and T ∗ R and then, as in [15], an application of the method of stationary phase to reduce the number of frequency variables (see [8] for details). The following lemma deals with phase functions 8(x, z) without frequency variables. 4.2 Let 8 be a real-valued phase function defined near (x 0 , z 0 ), and assume that LEMMA
INTEGRAL OPERATORS WITH LOW-ORDER DEGENERACIES
417
∇x (det 8x z (x 0 , z 0 )) 6= 0 and that | det 8x z (x 0 , z 0 )| + |V R det 8x z (x 0 , z 0 )| ≤ c|∇x (det 8x z (x 0 , z 0 ))|. Let M > 0. Then if c is sufficiently small, there are neighborhoods 0L of x0 , 0R of z 0 , neighborhoods U and V of (x0 , ∇x 8(x 0 , z 0 )) in T ∗ L , a canonical transformation χ : U → V , and a unitary operator Uλ , such that the following statements hold if σ is supported in 0L × 0R . (i) If Tλ is the integral operator with kernel σ (x, z)eıλ8(x,z) , then Uλ Tλ = Sλ + Rλ ,
(ii)
where Sλ is an integral operator with kernel τ (x, z)eıλ9(x,z) and kRλ k L 2 →L 2 = O(λ−M ). If C8 = {(x, 8x , z, −8z ), (x, z) ∈ supp σ }, then for C9 = (x, 9x , z, −9z ), (x, z) ∈ supp τ we have C9 ⊂ (χ(x, ξ ), z, ζ ) : (x, ξ, z, ζ ) ∈ C8 .
(iii)
We have ∇z (det 9x z ) 6= 0 for (x, z) ∈ supp τ .
Proof This proof can be extracted from the arguments in [9, §4]. Proof of Theorem 1.2 We work with Tλ as in (1.1) where (x, z) is close to the origin, and the origin lies on the singular surface {(x, z) : det 8x z = 0}. We may assume, after a change of variable in z, that 8(0, z) = 0, 8x 0 z 0 (0, z) = I, 8x 0 z d (0, z) = 0 (4.1) (cf. the proof of Lemma 2.7 in [9]), and by a change of variable in x we may also assume that 8z 0 xd (0, 0) = 0. We assume that π L is of type ≤ 2 and that ∇x,z (det 8x z ) 6 = 0 where det 8x z vanishes. If 8xd xd z d 6= 0 or 8xd z d z d 6 = 0 or 8xd z d z d z d 6= 0, then we have a fold or cusp singularity and better results than the one claimed in Theorem 1.2 were proved in [8] and [9]. Therefore, assume that 8xd xd z d , 8xd z d z d , and 8xd z d z d z d are small. By (4.1) and Lemma 4.2, we may assume the more restrictive assumption that ∇z (det 8x z ) 6= 0, which near the origin is equivalent to ∇z (8xd z d ) 6 = 0, again by (4.1). After a rotation, we may assume that 8xd z d z 1 (0, 0) 6 = 0.
(4.2)
418
GREENLEAF and SEEGER
We now consider the operator Tλ Tλ∗ and estimate it by the slicing technique in [9] (also familiar from the proof of Strichartz estimates). Now Tλ Tλ∗ f (x) = R K xd ,yd [ f (·, yd )](x 0 ) dyd , where the kernel of K xd ,yd as an integral operator acting on functions in Rd−1 is given by Z 0 0 K xd ,yd (x 0 , y 0 ) = eıλ[8(x ,xd ,z)−8(y ,yd ,z)] σ (x, z)σ (y, z) dz. The computation in [9] shows that after rescaling the estimation is reduced to showing ± that two integral operators Hµ± ≡ Hµ,γ ,c with kernel Z ± Hµ± (u, v) = eı9 (u,v,z;γ ,c) bγ ,c (u, v, z) dz are bounded on L 2 (Rd−1 ) with norm O(µ−(d−1)−1/4 ), µ ≥ 1. Here bγ ,c is C0∞ and 9 ± (u, v, z; γ , c) = hu − v8x 0 (0, c, z)i ± 80xd (0, c, z) + r± (u, v, z, γ , c), where γ and c are small parameters and r± (u, v, z, 0, c) ≡ 0. The dependence of 9 ± , r± , and b on γ and c is smooth, and the bounds have to be uniform for small γ , c. For this it remains to show that the operators Hµ± are oscillatory integral operators with two-sided type 2 singularities to which we can apply Lemma 4.1 in d − 1 dimensions (with d frequency variables). It suffices to check the type 2 condition at γ = 0, c = 0. Condition (4.2) guarantees that 9 ± is indeed a nondegenerate phase function with critical set Crit9 ± = (u, v, z) : ∇z 9 ± = 0 0 0 0 0 = (u, v, z) : v = u ∓ 8x z 8z 0 xd , 8xd z d − 8xd z 0 8z x 8x 0 z d = 0 where the 8 derivatives are evaluated at (0, z). At x = 0, the second equation becomes 8xd z d (0, z) = 0, and by (4.2) we may solve this equation, expressing z 1 as a function z 1± of z˜ = (z 00 , z d ) with z 00 = (z 2 , . . . , z d−1 ). Set G ± (˜z ) = 8x 0 (0, z 1± (˜z ), z˜ ) 0 0 and B ± (˜z ) = 8x z (0, z 1± (˜z ), z˜ )8z 0 xd (0, z 1± (˜z ), z˜ ); then the canonical relation for vanishing γ , c is given by C9 ± = u, G ± (˜z ), u ∓ B ± (˜z ), G ± (˜z ) , which is parametrized by the coordinates (u, z˜ ). The derivative of the projection to T ∗ L in these coordinates is given by ∂z 1 DG ± = 8x 0 z 1 + 8x 0 z˜ , ∂ z˜ e = ∂/∂z d and by (4.1) we see that its determinant equals (−1)d−1 ∂z 1± /∂z d and that V Pd−1 L e e is a kernel vector field for the left projection. Moreover, V R = VL + i=1 ci (z)∂/∂u i ,
INTEGRAL OPERATORS WITH LOW-ORDER DEGENERACIES
419
eL and V eR coincide when acting on the determinant. By implicit differentiso that V ation, we see that ∂zkd z 1± − 8−1 z d xd z 1 8xd z dk+1 belongs to the ideal generated by 8xd z j , d eL and V eR are of type ≤ k − 1 if one of the j ≤ k. From this one deduces that V
derivatives 8x z j , j ≤ k + 1, does not vanish. We apply this for k = 3 to conclude d d the proof. References [1]
´ and R. VAILLANCOURT, A class of bounded pseudo-differential A.-P. CALDERON
[2]
A. COMECH, “Integral operators with singular canonical relations” in Spectral Theory,
operators, Proc. Nat. Acad. Sci. USA 69 (1972), 1185 – 1187. MR 45:7532
[3]
[4]
[5] [6] [7] [8]
[9] [10] [11] [12]
[13] [14]
[15]
Microlocal Analysis, Singular Manifolds, Math. Top. 14, Akademie, Berlin, 1997, 200 – 248. MR 99f:58198 402, 410 , Optimal regularity of Fourier integral operators with one-sided folds, Comm. Partial Differential Equations 24 (1999), 1263 – 1281. MR 2000m:35190 398, 400, 402 A. COMECH and S. CUCCAGNA, Integral operators with two-sided cusp singularities, Internat. Math. Res. Notices 2000, 1225 – 1242. MR 2002c:35284 399, 402, 403, 405, 410 S. CUCCAGNA, L 2 estimates for averaging operators along curves with two-sided k-fold singularities, Duke Math. J. 89 (1997), 203 – 216. MR 99f:58199 398, 402 I. M. GELFAND and M. I. GRAEV, Line complexes in the space Cn , Funct. Anal. Appl. 2 (1968), 219 – 229. MR 38:6522 398 M. GOLUBITSKY and V. GUILLEMIN, Stable Mappings and Their Singularities, Grad. Texts in Math. 14, Springer, New York, 1973. MR 49:6269 A. GREENLEAF and A. SEEGER, Fourier integral operators with fold singularities, J. Reine Angew. Math. 455 (1994), 35 – 56. MR 95h:58130 398, 400, 403, 416, 417 , Fourier integral operators with cusp singularities, Amer. J. Math. 120 (1998), 1077 – 1119. MR 99g:58120 398, 400, 401, 416, 417, 418 , On oscillatory integral operators with folding canonical relations, Studia Math. 132 (1999), 125 – 139. MR 2000g:58040 402 A. GREENLEAF and G. UHLMANN, Nonlocal inversion formulas for the X-ray transform, Duke Math. J. 58 (1989), 205 – 240. MR 91b:58251 398, 410 , Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms, I, Ann. Inst. Fourier (Grenoble) 40 (1990), 443 – 466 MR 91k:58126; II, Duke Math. J. 64 (1991), 415 – 444. MR 93b:58146 398, 401 , Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, J. Funct. Anal. 89 (1990), 202 – 232. MR 91i:58146 V. GUILLEMIN, Cosmology in (2 + 1)-dimensions, Cyclic Models, and Deformations of M2,1 , Ann. of Math. Stud. 121, Princeton Univ. Press, Princeton, 1989. MR 91k:58140 398, 401 ¨ L. HORMANDER , Fourier integral operators, I, Acta Math. 127 (1971), 79 – 183.
420
[16] [17]
[18] [19] [20] [21] [22] [23] [24] [25]
GREENLEAF and SEEGER
MR 52:9299 398, 400, 416 , Oscillatory integrals and multipliers on FL p , Ark. Mat. 11 (1973), 1 – 11. MR 49:5674 398, 410 R. MELROSE and M. TAYLOR, Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle, Adv. Math. 55 (1985), 242 – 315. MR 86m:35095 398 B. MORIN, Formes canoniques des singularit´es d’une application diff´erentiable, C. R. Acad. Sci. Paris 260 (1965), 5662 – 5665. MR 31:5212 Y. PAN and C. D. SOGGE, Oscillatory integrals associated to folding canonical relations, Colloq. Math. 60/61 (1990), 413 – 419. MR 92e:58210 398 D. H. PHONG and E. M. STEIN, Radon transforms and torsion, Internat. Math. Res. Notices 1991, 49 – 60. MR 93g:58144 402 , The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997), 105 – 152. MR 98j:42009 398, 402 V. RYCHKOV, Sharp L 2 bounds for oscillatory integral operators with C ∞ phases, Math. Z. 236 (2001), 461 – 489. CMP 1 821 301 398 A. SEEGER, Degenerate Fourier integral operators in the plane, Duke Math. J. 71 (1993), 685 – 745. MR 94h:35292 401, 402 , Radon transforms and finite type conditions, J. Amer. Math. Soc. 11 (1998), 869 – 897. MR 99f:58202 398, 401 E. M. STEIN, Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton Univ. Press, Princeton, 1993. MR 95c:42002 409
Greenleaf Department of Mathematics, University of Rochester, Rochester, New York 14627, USA;
[email protected] Seeger Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 3,
A KOHNO-DRINFELD THEOREM FOR QUANTUM WEYL GROUPS VALERIO TOLEDANO LAREDO
Abstract Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W . In [MTL], we introduced a new, W -equivariant flat connection on h with simple poles along the root hyperplanes and values in any finite-dimensional g-module V . It was conjectured in [TL] that its monodromy is equivalent to the quantum Weyl group action of the generalised braid group of type g on V obtained by regarding the latter as a module over the quantum group U~ g. In this paper, we prove this conjecture for g = sln . 1. Introduction One of the many virtues of quantum groups is their ability to describe the monodromy of certain first-order systems of Fuchsian partial differential equations (PDEs). If U~ g is the Drinfeld-Jimbo quantum group of the complex, simple Lie algebra g, the universal R-matrix of U~ g yields a representation of Artin’s braid group on n strings Bn on the n-fold tensor product V ⊗n of any finite-dimensional U~ g-module V . A fundamental and paradigmatic result of T. Kohno and V. Drinfeld establishes the equivalence of this representation with the monodromy of the Knizhnik-Zamolodchikov equations for g with values in V ⊗n (see [Dr3], [Dr4], [Dr5], [Ko2]). G. Lusztig (and, independently, A. Kirillov and N. Reshetikhin, and Y. So˘ıbelman) realised that U~ g also yields representations of another braid group, namely, the generalised braid group Bg of Lie-type g (see [Lu2], [KR], [So]). Whereas the R-matrix representation is a deformation of the natural action of the symmetric group Sn on n-fold tensor products, these representations of Bg deform the action of (a finite extension of) the Weyl group W of g on any finite-dimensional g-module V. The aim of this paper is to show that these quantum Weyl group representations describe the monodromy of the flat connection introduced in [MTL] and, independently, in [FMTV]. More precisely, we realise Bg as the fundamental group of the orbit space hreg /W of the set of regular elements of a Cartan subalgebra h of g under DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 3, Received 23 November 2000. Revision received 8 May 2001. 2000 Mathematics Subject Classification. Primary 17B37; Secondary 16W30, 20F36, 32S40, 81R50. Author’s work partially supported by the Research Institute for Mathematical Sciences of Kyoto University. 421
422
VALERIO TOLEDANO LAREDO
the action of W (see [Br]). Then one can define a flat vector bundle (V , ∇κ ) with fibre V over hreg /W (see [MTL]). The connection ∇κ depends upon a parameter ~ ∈ C, and it was conjectured in [TL] that, when ~ is regarded as a formal variable, its monodromy is equivalent to the quantum Weyl group action of Bg on V . This conjecture was checked in [TL] for a number of pairs (g, V ) including vector representations of classical Lie algebras and adjoint representations of simple Lie algebras. In the present paper, we prove this conjecture for g = sln , so that Bg = Bn . The proof relies on the Kohno-Drinfeld theorem for U~ slk via the use of the dual pair (glk , gln ). Our main observation is that the duality between glk and gln derived from their joint action on the space Mk,n of (k × n)-matrices exchanges ∇κ for sln and the Knizhnik-Zamolodchikov connection for slk , thus acting as a simple-minded integral transform. This shows the equivalence of the monodromy representation of ∇κ for sln with a suitable R-matrix representation for U~ slk . The proof is completed by noting that the duality between U~ glk and U~ gln exchanges the R-matrix representation of U~ slk with the quantum Weyl group representation of U~ sln . This may be schematically summarised by the following diagram: ∇KZ , slk 6
Mk,n
- ∇κ , sln 6
KD ? R ∨ , U~ slk
~ Mk,n
? - q W, U~ sln
The structure of the paper is as follows. In Section 2 we give the construction of the connection ∇κ following [MTL]. We show in Section 3 that the duality between glk and gln identifies the Knizhnik-Zamolodchikov connection for n-fold tensor products of symmetric powers of the vector representation of slk and the connection ∇κ for sln . In Section 4 we recall the definition of the Drinfeld-Jimbo quantum groups U~ glk and U~ gln , and in Section 5 we show how they jointly act on the quantum (k × n)-matrix space. The corresponding R-matrix and quantum Weyl group representations of Bn are shown to coincide in Section 6. Finally, Section 7 contains our main result. 2. Flat connections on hreg The results in this section are due to J. Millson and V. Toledano Laredo [MTL]. They were obtained independently by De Concini around 1995 (unpublished). Let g be a complex, simple Lie algebra with Cartan subalgebra h and root system R ⊂ h∗ . Let
A KOHNO-DRINFELD THEOREM FOR QUANTUM WEYL GROUPS
423
S hreg = h \ α∈R Ker(α) be the set of regular elements in h, and let V be a finitedimensional g-module. We shall presently define a flat connection on the topologically trivial vector bundle hreg × V over hreg . For this purpose we need the following simple flatness criterion due to Kohno [Ko1]. Let B be a complex, finite-dimensional vector space, and let A = {Hi }i∈I be a finite collection of hyperplanes in B determined by the linear forms φi ∈ B ∗ , i ∈ I . LEMMA 2.1 Let F be a finite-dimensional vector space, and let {ri } ⊂ End(F) be a family indexed by I . Then X dφi ∇=d− ri (2.1) φi i∈I
defines a flat connection on (B \ A ) × F if and only if, for any subset J ⊆ I maximal T for the property that j∈J H j is of codimension 2, the following relations hold for any j ∈ J : h X i rj, r j 0 = 0. (2.2) j 0 ∈J
For any α ∈ R, choose root vectors eα ∈ gα , f α ∈ g−α such that [eα , f α ] = h α = α ∨ , and let hα, αi κα = (eα f α + f α eα ) ∈ U g (2.3) 2 be the truncated Casimir operator of the sl2 (C)-subalgebra of g spanned by eα , h α , f α . Note that κα does not depend upon the particular choice of eα and f α and that κ−α = κα . THEOREM 2.2 The one-form
∇κh = d − h
X dα α 0
α
κα = d −
h X dα κα 2 α
(2.4)
α∈R
defines, for any h ∈ C, a flat connection on hreg × V . Proof By Lemma 2.1, we must prove that for any rank 2 subsystem R0 ⊆ R determined by the intersection of R with a 2-dimensional subspace in h∗ , the following holds for any α ∈ R0+ = R0 ∩ R + : h i X κα , κβ = 0. (2.5) β∈R0+
424
VALERIO TOLEDANO LAREDO
This may be proved by an explicit computation by considering in turn the cases where R0 is of type A1 × A1 , A2 , B2 , or G 2 , but it is more easily settled by the following elegant observation of A. Knutson [Kn]. Let g0 ⊆ g be the semisimple Lie algebra with root system R0 , let h0 ⊂ h be its Cartan subalgebra, and let C0 ∈ Z (U g0 ) be P its Casimir operator. Then β∈R + κβ − C0 lies in U h0 , so that (2.5) holds since κα 0 commutes with h0 . Let G be the complex, connected, and simply connected Lie group with Lie algebra g, let T be its torus with Lie algebra h, let N (T ) ⊂ G be the normaliser of T , and let W = N (T )/T be the Weyl group of G. Let Bg = π1 (hreg /W ) be the generalised braid group of type g, and let σ : Bg → N (T ) be a homomorphism compatible with Bg
σ-
N (T ) Z Z Z Z ~ ? Z W
(2.6)
p
(see [Ti]). We regard Bg as acting on V via σ . Let hg → hreg be the universal cover reg − of hreg and hreg /W . 2.3 ∗ The one-form p ∗ ∇κh defines a Bg -equivariant flat connection on hg reg × V = p (hreg × V ). It therefore descends to a flat connection on the vector bundle PROPOSITION
V
- hg reg × Bg V
(2.7) ? hreg /W
which is reducible with respect to the weight space decomposition of V . Proof • g ) ⊗ V is given by γ → (γ −1 )∗ ⊗ σ (γ ). The action of Bg on • (hg reg , V ) = (h reg Thus, if γ ∈ Bg projects onto w ∈ W , we get, using p · γ −1 = w−1 · p, γ p ∗ ∇κh γ −1 = d −
hX ∗ dp wα/ p ∗ wα ⊗ σ (γ )κα σ (γ )−1 . 2
(2.8)
α∈R
Since κα = (hα, αi/2)(eα f α + f α eα ) is independent of the choice of the root vectors eα and f α , Ad(σ (γ ))κα = κwα and (2.8) is equal to p ∗ ∇κh , as claimed. The connection p ∗ ∇κh is flat by Theorem 2.2 and commutes with the fibrewise action of h because each κα is of weight zero.
A KOHNO-DRINFELD THEOREM FOR QUANTUM WEYL GROUPS
425
Thus, for any homomorphism σ : Bg → N (T ) compatible with (2.6), Proposition 2.3 yields a monodromy representation ρhσ : Bg → GL(V ) which permutes the weight spaces compatibly with W . By standard ordinary differential equation (ODE) theory, ρhσ depends analytically on the complex parameter h and, when h = 0, is equal to the action of Bg on V given by σ . We record for later use the following elementary proposition. PROPOSITION 2.4 Let γ ∈ Bg = π1 (hreg /W ), and let e γ : [0, 1] → hreg be a lift of γ . Then
ρhσ (γ ) = σ (γ )P (e γ ),
(2.9)
where P (e γ ) ∈ GL(V ) is the parallel transport along e γ for the connection ∇κh on hreg × V . Proof Let e γ : [0, 1] → hg γ so that e γ (1) = γ −1e γ (0). Then, since the reg be a lift of γ and e ∗ h connection on p (hreg × V ) is the pullback of ∇κ , and the connection on ( p ∗ (hreg × V ))/Bg is the quotient of p ∗ ∇κh , we find that ρhσ (γ ) = P (γ ) = σ (γ )P (e γ ) = σ (γ )P (e γ ).
(2.10)
By [Br], Bg is presented on generators Ti , i = 1, . . . , n, labelled by a choice of simple roots αi of R with relations Ti T j Ti · · · = T j Ti T j · · ·
(2.11)
for any 1 ≤ i < j ≤ n, where each side of (2.11) has a number of factors equal to the order of si s j in W and sk ∈ W is the orthogonal reflection across the hyperplane Ker(αk ). Ti projects onto si ∈ W . An explicit choice of representatives of T1 , . . . , Tn in π1 (hreg /W ) may be given as follows. Let t ∈ hreg lie in the fundamental Weyl chamber so that ht, αi > 0 for any α ∈ R+ . Note that for any simple root αi , the intersection tαi = t − (1/2)ht, αi iαi∨ of the affine line t + C · αi∨ with Ker(αi ) does not lie in any other root hyperplane Ker(β), β ∈ R \ {αi }. Indeed, if htαi , βi = 0, then 2ht, βi = ht, αi ihαi∨ , βi = ht, β − si βi,
(2.12)
whence ht, βi = −ht, si βi, a contradiction since si permutes positive roots different from αi . Now let D be an open disc in t + C · αi∨ of center tαi such that its closure D does not intersect any root hyperplane other than Ker(αi ). Consider the path γi :
426
VALERIO TOLEDANO LAREDO
[0, 1] → t + C · αi∨ from t to si t determined by the requirement that γi |[0,1/3]∪[2/3,1] be affine and lie in t + R · α ∨ \ D, γi (1/3), γi (2/3) ∈ ∂ D, and that γi |[1/3,2/3] be a semicircular arc in ∂ D, positively oriented with respect to the natural orientation of t +C·αi . Then, the image of γi in hreg /W is a representative of Ti in π1 (hreg /W, W t) (see [Br]). 3. Knizhnik-Zamolodchikov equations and dual pairs We show in this section that the joint action of glk and gln on the space Mk,n of (k×n)matrices identifies the connection ∇κ for g = sln and the Knizhnik-Zamolodchikov ∗ ) = C[x , . . . , x ] be the algebra of polynomial connection for slk . Let S(Mk,n kn 11 ∗ ) by functions on Mk,n . The group GLk (C) × GLn (C) acts on S(Mk,n (gk , gn ) p(x) = p(gkt xgn )
(3.1)
∗ ) of S(M ∗ ), d ∈ N, invariant. and leaves the homogeneous components S d (Mk,n k,n ∗ The decomposition of S(Mk,n ) under GLk (C) × GLn (C) is well known (see, e.g., [Zh, Sec. 132], which we follow closely, or [Mc, Sec. 1.4]∗ ). Let Nk , Nn be the groups of (k × k)– and (n × n)–upper triangular unipotent matrices, respectively.
LEMMA
3.1
We have ∗ Nk ×Nn S(Mk,n ) = C[11 , . . . , 1min(k,n) ],
(3.2)
where 1l (x) = det(xi j )1≤i, j≤l is the l th principal minor of the matrix x. Proof ∗ ) be the subset of matrices x Assume for simplicity that k ≤ n. Let D ⊂ S(Mk,n such that 1i (x) 6= 0 for i = 1, . . . , k. By the Gauss decomposition, any x ∈ D is conjugate under Nkt × Nn to a unique (k × n)-matrix d(x) with the same principal minors as x, diagonal principal (k × k)-block, and the remaining columns equal to zero. Consider now the (k × n)-matrix 11 (x) 12 (x) 13 (x) · · · 1k−1 (x) 1k (x) 0 · · · 0 −1 0 0 ··· 0 0 0 ··· 0 0 −1 0 · · · 0 0 0 · · · 0 . m(x) = .. .. .. .. .. .. .. . . . ··· . . . ··· . 0
0
0
···
−1
0
0 ···
0 (3.3)
∗I
am grateful to M. Vergne for pointing out that decomposition (3.4) was known long before [Zh] and the work of R. Howe [Ho] and to M. Brion for providing the reference (see [Mc]).
A KOHNO-DRINFELD THEOREM FOR QUANTUM WEYL GROUPS
427
Since 1i (m(x)) = 1i (x), 1 ≤ i ≤ k, m(x) is also conjugate to d(x), and therefore ∗ ) is invariant to x, under Nkt × Nn . Thus, by density of D , a polyonomial p ∈ S(Mk,n under Nk × Nn if and only if it is a function of m(x) and therefore if and only if it is a polynomial in 11 , . . . , 1k . Let Y p ⊂ N p be the set of Young diagrams with at most p rows. For λ ∈ Y p , set Pp ( p) |λ| = i=1 λi , and let Vλ be the irreducible representation of GL p (C) of highest weight λ. 3.2 As (GLk × GLn )-modules, THEOREM
∗ S d (Mk,n )∼ =
M λ∈Ymin(k,n) |λ|=d
(k)
(n)
Vλ ⊗ Vλ .
(3.4)
Proof Assume again that k ≤ n for simplicity. By Lemma 3.1, the highest weight vectors for ∗ ) are the polynomials in 1 , . . . , 1 which the action of GLk (C)×GLn (C) on S(Mk,n k 1 (k)
(n)
are eigenvectors for the torus of GLk (C)×GLn (C). Since 1l is of weight $l ⊕$l , ( p) where $l is the lth fundamental weight of GL p (C), the highest weight vectors are mk 1 the monomials 1m 1 · · · 1k with corresponding pair of Young diagrams (λ, λ) where λ = (m 1 + · · · + m k , m 2 + · · · + m k , . . . , m k ).
(3.5)
Thus, (3.4) holds since 1l is a homogeneous function of degree l. As a glk -module, ∗ S(Mk,n ) = C[x11 , . . . , xk1 ] ⊗ · · · ⊗ C[x1n , . . . , xkn ]
(3.6)
(k)
e , 1 ≤ i < j ≤ n, defined by and is therefore acted upon by the glk -intertwiners ij e(k) = ij
X
1⊗(i−1) ⊗ X a ⊗ 1⊗( j−i−1) ⊗ X a ⊗ 1⊗(n− j) ,
(3.7)
a
where {X a }, {X a } are dual bases of glk with respect to the pairing hX, Y i = tr(X Y ). ∗ ) is acted upon by the operators κ (n) , On the other hand, as a gln -module, S(Mk,n ij 1 ≤ i < j ≤ n, where (n) κi j = eα f α + f α eα (3.8) is the truncated Casimir operator of the sl2 (C)-subalgebra of gln corresponding to the ( p) root α = θi − θ j . Let e1 , . . . , e p be the canonical basis of C p , and let E ab ec = δbc ea ,
428
VALERIO TOLEDANO LAREDO ( p)
1 ≤ a, b ≤ p, be the corresponding basis of gl p with dual basis E ba . Let 1 ≤ i < j ≤ n; then we have the following proposition. PROPOSITION 3.3 ∗ ): The following holds on S(Mk,n (k)
(n)
(n)
(n)
e =κ −E −E . 2 ij ij ii jj
(3.9)
Proof e(k) acts on S(M ∗ ) as By (3.7), ij k,n e(k) = ij
X
xai ∂bi xbj ∂a j ,
(3.10)
1≤a,b≤k
where xr c and ∂r c are the operators of multiplication by and derivation with respect to xr c . On the other hand, given that the sl2 (C)-triple {eα , h α , f α } corresponding to the (n) (n) (n) (n) ∗ ): root α = θi − θ j of gln is {E i j , E ii − E j j , E ji }, the following holds on S(Mk,n X (n) κi j = xai ∂a j xbj ∂bi + xbj ∂bi xai ∂a j . (3.11) 1≤a,b≤k
Subtracting, we find that (k)
(n)
e −κ =− 2 ij ij
(n)
X
(n)
δab xai ∂bi + δab xbj ∂a j = −E ii − E j j ,
(3.12)
1≤a,b≤k
as claimed. (n)
Let λ ∈ Ymin(k,n) , and let Vλ be the corresponding simple GLn (C)-module. By (n) Theorem 3.2, Vλ may be identified with the subspace of vectors of highest weight λ ∗ ). Denote by ι : V (n) → S(M ∗ ) the corresponding for the action of glk on S(Mk,n λ k,n (n)
gln -equivariant embedding, and let µ = (µ1 , . . . , µn ) ∈ Nn be a weight of Vλ . LEMMA 3.4 (n) (n) The embedding ι maps the subspace Vλ [µ] ⊂ Vλ of weight µ onto the subspace µ Mλ of vectors of highest weight λ for the action of glk on
S µ Ck = S µ1 Ck ⊗ · · · ⊗ S µn Ck ⊂ C[x11 , . . . , xk1 ] ⊗ · · · ⊗ C[x1n , . . . , xkn ], (3.13) where S µ j Ck is the space of polynomials in x1 j , . . . , xk j which are homogeneous of degree µ j . The corresponding isomorphism M (n) M Vλ [ν] ∼ Mλν (3.14) = ν∈Sn µ
ν∈Sn µ
A KOHNO-DRINFELD THEOREM FOR QUANTUM WEYL GROUPS
429
L is equivariant with respect to Sn , which acts on ν∈Sn µ S ν Ck by permuting the tensor factors and on Vλ by regarding Sn as the subgroup of permutation matrices of GLn (C). Proof µ (n) ∗ ) of The equality ι(Vλ [µ]) = Mλ holds because S µ Ck is the subspace of S(Mk,n (n)
weight µ for the gln -action since E ii xrmj = δi j mxrmj . The Sn -equivariance stems from the fact that the permutation of the tensor factors in S • Ck ⊗ · · · ⊗ S • Ck ∼ = S • (Ck ⊗ Cn ) is given by the action of Sn ⊂ GLn (C) on Cn . Let Dn = (z 1 , . . . , z n ) ∈ Cn z i = z j for some 1 ≤ i < j ≤ n , and let X n = Cn \ Dn . Regard n X n o Cn0 = (z 1 , . . . , z n ) ∈ Cn zj = 0 j=1
as the Cartan subalgebra of diagonal matrices in sln , and X n0 = Cn0 \ Dn as the set of its regular elements. Since the inclusion X n0 ⊂ X n is a homotopy equivalence, π1 (X n ) ∼ = π1 (X n0 ) = Bn are generated by T1 , . . . , Tn−1 with Ti T j = T j Ti
if |i − j| ≥ 2,
Ti Ti+1 Ti = Ti+1 Ti Ti+1 m, i = 1, . . . , n − 1.
(3.15) (3.16)
(k)
Define i j ∈ Endglk (S µ Ck ) by (3.7) where {X a }, {X a } are now dual bases of slk and extend the connection (2.4) to X n in the obvious way. The following theorem is the main result of this section. 3.5 µ The function f : X n → Mλ ⊂ S µ1 Ck ⊗ · · · ⊗ S µn Ck is a horizontal section of the Knizhnik-Zamolodchikov connection X dz i − dz j (k) h0 ∇KZ = d − h0 i j (3.17) zi − z j THEOREM
1≤i< j≤n
Q (n) if and only if the Vλ [µ]-valued function g = f · 1≤i< j≤n (z i − z j )h(µi +µ j +2µi µ j /k) is a horizontal section of X dz i − dz j (n) ∇κh = d − h κ , (3.18) zi − z j i j 1≤i< j≤n
where h 0 = 2h.
430
VALERIO TOLEDANO LAREDO
Proof Pk (k) Let 1(k) = i=1 E ii be the generator of the centre of glk so that, in obvious notation, (k) (k) e = +(1/k)πi (1(k) )π j (1(k) ). The operators 2 e(k) and κ (n) both act on M µ ∼ λ = ij ij ij ij (n)
Vλ [µ], and, by Proposition 3.3, their restrictions differ by −µi − µ j . The claim follows since, for any 1 ≤ l ≤ n, πl (1(k) ) acts on S µ Ck as multiplication by µl . Remark. When k = 2 and λ is of the form (|µ|/2, |µ|/2, 0, . . . , 0), where |µ| = P i=1 µi , Theorem 3.5 is a representation-theoretic analogue of the coincidence between the Kapovich-Millson bending flows on the space of n-gons in R3 with side lengths µ1 , . . . , µn (see [KM]) and the Gel’fand-Cetlin flows on the Grassmannian Gr2 (Cn ) (see [GS]) observed by J.-C. Hausmann and Knutson in the context of Gel’fandMcPherson duality (see [HK]). I am grateful to J. Millson for a careful explanation of this coincidence. Remark. An interesting relation between ∇κ and the Knizhnik-Zamolodchikov connection was recently noted by G. Felder, Y. Markov, V. Tarasov, and A. Varchenko in [FMTV], where a variant of the connection (2.4) is independently introduced and studied. One of the main results of [FMTV] is that, for any simple Lie algebra g, the connection ∇κ with values in a tensor product V1 ⊗ · · · ⊗ Vn of n simple g-modules is, when supplemented by suitable dynamical parameters, bispectral to (i.e., commutes with) the Knizhnik-Zamolodchikov connection for g with values in the same n-fold tensor product. An analogous result is obtained in [TV] for a difference analogue of the connection ∇κ . By comparison, Theorem 3.5 can only hold for g = sln , since it relies on the “coincidence” of the regular Cartan of gln with the configuration space of n ordered points in C and asserts the equality of the two connections. 0
h and ∇ h , we To relate the monodromy representations of Bn corresponding to ∇KZ κ 0 need to specify how these induce flat connections on X n /Sn and X n /Sn , respectively. h 0 , we let S act on the fibre For ∇KZ n M M Mλν ⊂ S ν Ck (3.19) ν∈Sn µ
ν∈Sn µ
by permuting the tensor factors, and we take the quotient connection. For ∇κh , we use the construction of Proposition 2.3 and the homomorphism σ : Bn → SLn (C) given
A KOHNO-DRINFELD THEOREM FOR QUANTUM WEYL GROUPS
431
by (n)
(n)
(n)
T j −→ exp(E j, j+1 ) exp(−E j+1, j ) exp(E j, j+1 ) 1 .. . 1 0 1 = −1 0 1
..
, (3.20)
. 1
where the off-diagonal terms are the ( j, j + 1) and ( j + 1, j) entries. A direct computation, or [Ti, Th. 3.3], shows that assignment (3.20) does indeed extend to a homomorphism Bn → SLn (C). Choose the generators T1 , . . . , Tn−1 of Bn as at the end of Section 2. COROLLARY 3.6 Let µ be a weight of Vλ , and let M πκh : Bn → GL Vλ [ν] ,
0
h πKZ : Bn → GL
ν∈Sn µ
M ν∈Sn µ
Mλν
(3.21)
be the monodromy representations of the braid group Bn corresponding to the connections (3.18) and (3.17), respectively. Then, for any j = 1, . . . , n − 1, 2h πKZ (T j ) = πκh (T j ) · e
(n)
(n)
(n)
(n)
−πi h(E j j +E j+1 j+1 +2E j j E j+1 j+1 /k)
(n)
· eiπ E j j .
Proof
(3.22)
(n)
Let s j ∈ SLn (C) be the right-hand side of (3.20) so that s j = ( j j + 1) · eiπ E j j in h 0 , P h denote parallel transport for ∇ h 0 and ∇ h , respectively. Then, GLn (C). Let PKZ κ κ KZ by Theorem 3.5 and Proposition 2.4, the following holds on Mλν ∼ = Vλ [ν]: 2h 2h πKZ (T j ) = ( j j + 1)PKZ (T j )
= ( j j + 1)e−πi h(ν j +ν j+1 +2ν j ν j+1 /k) Pκh (T j ) (n)
= s j Pκh (T j )eiπ E j j e = πκh (T j )e as claimed.
(n)
(n)
(n)
(n)
(n)
−πi h(E j j +E j+1 j+1 +2E j j E j+1 j+1 /k) (n)
(n)
(n)
−πi h(E j j +E j+1 j+1 +2E j j E j+1 j+1 /k) iπ E (n) jj
e
,
(3.23)
432
VALERIO TOLEDANO LAREDO
4. The quantum group U~ gl p In this and the following sections, we work over the ring C[[~]] of formal power series in the variable ~. All tensor products of C[[~]]-modules are understood to be completed in the ~-adic topology. For p ∈ N, let ai j = 2δi j − δ|i− j|=1 , 1 ≤ i, j ≤ p, be the entries of the Cartan matrix of type A p−1 , and let U~ gl p be the corresponding Drinfeld-Jimbo quantum group (see [Dr1], [Ji1]), that is, the algebra over C[[~]] topologically generated by elements E i , Fi , i = 1, . . . , p − 1, and Di , i = 1, . . . , p, subject to the q-Serre relations [Di , D j ] = 0, [Di , E j ] = (δi j − δi j+1 )E j , [E i , F j ] = δi j 1−ai j
X
X k=0
[Di , F j ] = −(δi j − δi j+1 )F j , e~Hi
− e−~Hi
e~ − e−~
,
(4.2) (4.3)
1 − ai j 1−a −k E ik E j E i i j = 0, k
∀i 6 = j,
(4.4)
1 − ai j 1−a −k (−1) Fik F j Fi i j = 0, k
∀i 6= j,
(4.5)
(−1)k
k=0 1−ai j
(4.1)
k
where Hi = Di − Di+1 and, for any n ≥ k ∈ N, en~ − e−n~ , e~ − e~ [n]! = [n][n − 1] · · · [1], [n]! n = . k [k]![n − k]! [n] =
(4.6) (4.7) (4.8)
U~ gl p is a topological Hopf algebra with coproduct 1 and counit ε given by 1(Di ) = Di ⊗ 1 + 1 ⊗ Di , 1(E i ) = E i ⊗ e
~Hi
+ 1 ⊗ Ei ,
(4.9) (4.10)
1(Fi ) = Fi ⊗ 1 + e−~Hi ⊗ Fi ,
(4.11)
ε(E i ) = ε(Fi ) = ε(Di ) = 0.
(4.12)
and Note that I = D1 + · · · + D p is central, so that U~ gl p ∼ = U~ sl p ⊗ C[I ][[~]] as Hopf algebras, where the coproduct on C[I ][[~]] is given by 1(I ) = I ⊗ 1 + 1 ⊗ I and where U~ sl p ⊂ U~ gl p is the closed Hopf subalgebra generated by E i , Fi , and Hi , i = 1, . . . , p − 1.
A KOHNO-DRINFELD THEOREM FOR QUANTUM WEYL GROUPS
433
By a finite-dimensional representation of U~ gl p , we shall mean a U~ gl p -module that is topologically free and finitely generated over C[[~]] and on which I acts semisimply with eigenvalues in C. Choose an algebra isomorphism φ : U~ gl p → U gl p [[~]] mapping each Di onto E ii (see [Dr2, Prop. 4.3]), and let V be a finitePp dimensional gl p -module on which 1( p) = i=1 E ii acts semisimply. Then U gl p [[~]] acts on V [[~]] and the latter becomes, via φ, a finite-dimensional representation of U~ gl p . Conversely, we have the following proposition. 4.1 Let V be a finite-dimensional representation of U~ gl p , and let V = V /~V be the corresponding gl p -module. Then, as U~ gl p -modules, PROPOSITION
V ∼ = V [[~]].
(4.13)
Proof Since I is diagonalisable on V and commutes with U~ gl p , we may assume that it acts on V as multiplication by a scalar λ ∈ C. Since V is topologically free, V ∼ = V [[~]] as C[[~]]-modules so that V is a deformation of the finite-dimensional sl p -module V . Since sl p is simple, H 1 (sl p , End(V )) = 0 and V is isomorphic, as sl p and therefore as gl p -module, to the trivial deformation of V . Thus, V ∼ = V [[~]] as U gl p [[~]] and therefore as U~ gl p -modules. 4.2 Let U, V be finite-dimensional gl p -modules on which 1( p) acts semisimply. If U ⊗ V decomposes as M U ⊗V ∼ NW W (4.14) = COROLLARY
W
for some gl p -modules W and multiplicities N W ∈ N, then, as U~ gl p -modules, M U [[~]] ⊗ V [[~]] ∼ N W W [[~]]. (4.15) = W
Proof By (4.14), both sides of (4.15) have the same specialisation at ~ = 0 and are therefore isomorphic by Proposition 4.1. 5. The dual pair (U~ glk , U~ gln ) ∗ ) of funcWe need the analogue of Theorem 3.2 in the setting of the algebra S~ (Mk,n tions on quantum (k × n)-matrix space. With the exception of Theorems 5.4 and 5.5, ∗ ) is the althis section follows [Ba, Sec. 1.5] (see also [Ga]). By definition, S~ (Mk,n gebra over C[[~]] topologically generated by elements X i j , 1 ≤ i ≤ k, 1 ≤ j ≤ n,
434
VALERIO TOLEDANO LAREDO
with relations X kl X i j X i j X kl = e−~ X kl X i j X kl X i j − (e~ − e−~ )X k j X il
if k > i and l < j or k < i and l > j, if k > i and l = j or k = i and l > j, if k > i and l > j. (5.1) ∗ ) is N-graded by decreeing that each X is of degree 1, and we denote its S~ (Mk,n ij ∗ ), d ∈ N. For any (k ×n)-matrix m with entries homogeneous components by S~d (Mk,n m i j ∈ N, set m
m
m
m
m kn m 11 Xm = X 11 · · · X k1k1 · · · X 1n1n · · · X kn m kn m 11 = X 11 · · · X 1n1n · · · X k1k1 · · · X kn .
(5.2)
By the commutation relations (5.1), the Xm , with m ∈ Mk,n (N) such that |m| = P d ∗ i, j m i j = d, span S~ (Mk,n ). THEOREM 5.1 (Parshall-Wang) The monomials Xm , m ∈ Mk,n (N), are linearly independent over C[[~]]. In particular, ∗ ). the set {Xm }|m|=d is a C[[~]]-basis of S~d (Mk,n
Proof This is proved in [PW, Th. 3.5.1] for k = n and over the field C(q) of rational functions of q = e~ rather than over C[[~]]. However, the proof works equally well for k 6= n and, as remarked in [PW], over C[[~]]. ∗ ) is a module algebra over U gl ⊗ U gl . This may As in the classical case, S~ (Mk,n ~ k ~ n be seen in the following way. For any l ∈ N, one readily checks that the assignment
Xi j →
l X
X il 0 ⊗ X l 0 j
(5.3)
l 0 =1 ∗ ) → S (M ∗ ) ⊗ extends uniquely to an algebra homomorphism 1kln : S~ (Mk,n ~ k,l ∗ ) such that, for any l, m ∈ N, the following diagram commutes: S~ (Ml,n ∗ S~ (Mk,n )
1kln
1kmn ? ∗ ∗ S~ (Mk,m ) ⊗ S~ (Mm,n ) ∗ ) S~ (Mk,k
- S~ (M ∗ ) ⊗ S~ (M ∗ ) k,l l,n
1 ⊗ 1lmn 1klm
(5.4)
? - S~ (M ∗ ) ⊗ S~ (M ∗ ) ⊗ S~ (M ∗ ) m,n k,l l,m ⊗1
∗ ) are topological bialgebras with comultiplicaIn particular, and S~ (Mn,n tions 1kkk and 1nnn , respectively, and counit ε(X i j ) = δi j . Moreover, the maps 1kkn
A KOHNO-DRINFELD THEOREM FOR QUANTUM WEYL GROUPS
435
∗ ) the structure of an S (M ∗ )-S (M ∗ )-bicomodule algebra, and 1knn give S~ (Mk,n ~ ~ n,n k,k ∗ ) and S (M ∗ ) each homogeneous component of which is invariant under S~ (Mk,k ~ n,n d d d ∗ ∗ ∗ since 1kln (S~ (Mk,n )) ⊂ S~ (Mk,l ) ⊗ S~ (Ml,n ). ∗ ) We need a columnwise (resp., rowwise) description of the coaction of S~ (Mk,k ∗ ∗ (resp., S~ (Mn,n )) on S~ (Mk,n ). Consider the quantum k- and n-dimensional planes, ∗ ) and S (M ∗ ). By commutation relations (5.1) and that is, the algebras S~ (Mk,1 ~ 1,n ∗ ) via the maps Theorem 5.1, these may be embedded as subalgebras of S~ (Mk,n ∗ ∗ c j : S~ (Mk,1 ) → S~ (Mk,n ),
ri :
∗ S~ (M1,n )
→
∗ S~ (Mk,n ),
c j (X i1 ) = X i j ,
(5.5)
ri (X 1 j ) = X i j ,
(5.6)
∗ ) is a left algebra comodule over with 1 ≤ i ≤ k, 1 ≤ j ≤ n. By (5.4), S~ (Mk,1 ∗ ), and S (M ∗ ) is a right algebra comodule over S (M ∗ ). S~ (Mk,k ~ ~ n,n 1,n
5.2 ∗ )-comodules, As left, N-graded S~ (Mk,k LEMMA
∗ ∗ ⊗n S~ (Mk,n )∼ ) = S~ (Mk,1
(5.7)
via the map 8 : p1 ⊗ · · · ⊗ pn → c1 ( p1 ) · · · cn ( pn ). Similarly, as right, N-graded ∗ )-comodules, S~ (Mn,n ∗ ∗ ⊗k S~ (Mk,n )∼ ) (5.8) = S~ (M1,n via 9 : q1 ⊗ · · · ⊗ qk → r1 (q1 ) · · · rk (qk ). Proof The map 8 clearly preserves the grading and, by Theorem 5.1, restricts to a C[[~]]linear isomorphism of homogeneous components since it bijectively maps the mono∗ )⊗n onto the basis Xm of S (M ∗ ). The fact that 8 itself is an mial basis of S~ (Mk,1 ~ k,n ∗ )⊗k or S (M ∗ ) is the isomorphism follows easily because any element of S~ (M1,n ~ k,n convergent sum of its homogeneous components. As is readily checked, the diagram ∗ ∗ S~ (Mk,k ) ⊗ S~ (Mk,1 )
1 ⊗ cj - S~ (M ∗ ) ⊗ S~ (M ∗ ) k,k k,n 6 1kkn
6
1kk1 ∗ S~ (Mk,1 )
cj
- S~ (M ∗ ) k,n
(5.9)
436
VALERIO TOLEDANO LAREDO
is commutative for any 1 ≤ j ≤ n, and therefore so is ∗ ∗ (S~ (Mk,k ) ⊗ S~ (Mk,1 ))⊗n
µ(n) ⊗ 8
∗ ∗ S~ (Mk,k ) ⊗ S~ (Mk,n )
6 1kkn
6
1⊗n kk1 ∗ ⊗n S~ (Mk,1 )
8
(5.10)
- S~ (M ∗ ) k,n
∗ )⊗n → S (M ∗ ) is the n-fold multiplication. This proves where µ(n) : S~ (Mk,k ~ k,k (5.7). The proof of (5.8) is identical. ∗ ). For p = k, n, consider the We turn now to the action of U~ glk ⊗ U~ gln on S~ (Mk,n vector representation of U~ gl p , that is, the module V = C p [[~]] with basis e1 , . . . , e p and action given by
Di = E ii ,
E i = E ii+1 ,
Fi = E i+1i ,
(5.11)
where E ab ec = δbc ea . Let e1 , . . . , e p ∈ V ∗ be the dual basis of e1 , . . . , e p , let U~ gl◦p be the restricted dual of U~ gl p , and let ti j ∈ U~ gl◦p be the matrix coefficient defined by ti j (x) = hei , xe j i. (5.12) PROPOSITION 5.3 The assignment X i j → ti j extends uniquely to a bialgebra morphism κ p : ∗ ) → U gl◦ . S~ (M p, ~ p p
Proof We need to check that the ti j satisfy relations (5.1), that is, that when evaluated on 1(x), x ∈ U~ gl p , ti j ⊗ tkl tkl ⊗ ti j = e−~ tkl ⊗ ti j tkl ⊗ ti j − (e~ − e−~ )tk j ⊗ til
if k > i and l < j or k < i and l > j, if k > i and l = j or k = i and l > j, if k > i and l > j. (5.13)
Let R ∨ = σ · R ∈ End(V ⊗ V ), where σ ∈ GL(V ⊗ V ) is the flip and R is the universal R-matrix of U~ gl p acting on V ⊗ V . Then (see [Ji2], [CP, Sec. 8.3.G]) p X R = e~ E ii ⊗ E ii + i=1
X 1≤i6 = j≤ p
E ii ⊗ E j j +(e~ −e−~ )
X 1≤i< j≤ p
E i j ⊗ E ji , (5.14)
A KOHNO-DRINFELD THEOREM FOR QUANTUM WEYL GROUPS
so that the matrix entries of R ∨ are δ ~ e jl ∨ Rik, = e~ − e−~ jl 0
if i = l and k = j, if i = j, k = l, and j > l, otherwise.
437
(5.15)
From (5.15), one readily checks that both sides of (5.13) coincide when evaluated on any A ∈ End(V ⊗ V ) commuting with R ∨ . Since R ∨ is a U~ gl p -intertwiner, κ extends to an algebra morphism that respects the counit and coproduct since 1(ti j ) = Pp q=1 tiq ⊗ tq j . 5.4 ∗ ) the structure of an The maps κ p , p = k, n, of Proposition 5.3 give S~ (Mk,n algebra module over U~ glk ⊗ U~ gln with invariant homogeneous components ∗ ), d ∈ N. S~d (Mk,n The maps 8, 9 of Lemma 5.2 yield isomorphisms
THEOREM
(1)
(2)
∗ ∗ ⊗n S~ (Mk,n )∼ ) = S~ (Mk,1
(3)
and
∗ ∗ ⊗k S~ (Mk,n )∼ ) = S~ (M1,n
(5.16)
as N-graded U~ glk - and U~ gln -modules, respectively. ( p) ( p) ( p) The action of the generators E q , Fq , q = 1, . . . , p − 1, and Dq , q = m 1, . . . , p, of U~ gl p , p = k, n, in the monomial basis X , m ∈ Mk,n (N), is given by (k)
Di
(k)
Ei
Xm = Xm =
n X j=1 n X
m i j Xm ,
(k)
Xm =
n X
n Y
[m i+1 j ]
e~(m i j 0 −m i+1 j 0 ) Xm+εi j −εi+1 j ,
(5.18)
j 0 = j+1
j=1
Fi
(5.17)
[m i j ]
j−1 Y
e−~(m i j 0 −m i+1 j 0 ) Xm−εi j +εi+1 j ,
(5.19)
j 0 =1
j=1
where (εab )cd = δac δbd , and (n)
D j Xm = (n) Ej
m
X =
k X i=1 k X
m i j Xm ,
(n)
Xm =
k X i=1
k Y
[m i j+1 ]
e~(m i 0 j −m i 0 j+1 ) Xm+εi j −εi j+1 ,
(5.21)
i 0 =i+1
i=1
Fj
(5.20)
[m i j ]
i−1 Y i 0 =1
e−~(m i 0 j −m i 0 j+1 ) Xm−εi j +εi j+1 .
(5.22)
438
VALERIO TOLEDANO LAREDO
Proof ∗ ) given by τ (X ) = X , we Using the transposition anti-involution τ on S~ (Mk,k ij ji ∗ ∗ ) ⊗ S (M ∗ ) and may regard S~ (Mk,n ) as a right algebra comodule over S~ (Mk,k ~ n,n ∗ ) ⊗ U gl → C[[~]], m = k, n, given by therefore, via the pairings h·, ·i : S~ (Mm,m ~ m Proposition 5.3 as a left algebra module over U~ glk ⊗ U~ gln . This proves (1) and (2). ∗ ), Explicitly, for x (m) ∈ U~ glm , m = k, n, and p ∈ S~ (Mk,n
x (k) p = x (k) ⊗ 1, τ ⊗ 1 · 1kkn ( p) ,
x (n) p = 1 ⊗ x (n) , 1knn ( p) .
(5.23) (5.24)
Using (5.23) and (5.11), one gets (k)
Di
Xi0 j (k) Ei X i 0 j (k) Fi X i 0 j
= δii 0 X i j ,
(5.25)
= δi+1i 0 X i j ,
(5.26)
= δii 0 X i+1 j .
(5.27)
Using the algebra module property x( pq) = µ(1(x) p ⊗ q), where x ∈ U~ glk , ∗ ), and µ : S (M ∗ )⊗2 → S (M ∗ ) is multiplication, (4.9) – (4.11) p, q ∈ S~ (Mk,n ~ ~ k,n k,n and commutation relations (5.1) show by induction on m ∈ N that (k)
X im0 j = δii 0 m X imj ,
(5.28)
(k)
m−1 X im0 j = δi+1i 0 [m] X i j X i+1 j,
(5.29)
(k)
X im0 j = δii 0 [m] X im−1 X i+1 j . j
(5.30)
Di Ei Fi
∗ (1) = id, 1(a+1) = Let 1(a) : U~ glk → U~ gl⊗a k , a ∈ N , be recursively defined by 1 ⊗(a−1) (a) 1 ⊗ id ·1 . Then, by (4.9) – (4.11), (a)
1
(k) Di (k)
1(a) E i
(k)
1(a) Fi
=
=
=
a X b=1 a X b=1 a X
(k)
⊗ 1⊗(a−b) ,
(k)
⊗ (e~Hi )⊗(a−b) ,
1⊗(b−1) ⊗ Di 1⊗(b−1) ⊗ E i (k)
(k)
(k)
(e−~Hi )⊗(b−1) ⊗ Fi
⊗ 1⊗(a−b) .
(5.31)
(5.32)
(5.33)
b=1
Formulae (5.17) – (5.19) now follow from the algebra module property and (5.31) – (5.33). The proof of (5.20) – (5.22) is similar. The following result is proved in [Ba] and [Ga] for the quantum groups Uq glk , Uq gln by a different method.
A KOHNO-DRINFELD THEOREM FOR QUANTUM WEYL GROUPS
439
THEOREM 5.5 ∗ ) decomposes as For any d ∈ N, the U~ glk ⊗ U~ gln -module S~d (Mk,n ∗ S~d (Mk,n )∼ =
M λ∈Ymin(k,n) |λ|=d
(k)
(n)
Vλ [[~]]⊗Vλ [[~]].
(5.34)
Proof ∗ ) has no torsion, and it is therefore a topologically free By Theorem 5.1, S~d (Mk,n ∗ )/~S d (M ∗ ) is the (gl ⊕ gl )C[[~]]-module. Moreover, by (5.17) – (5.22), S~d (Mk,n k n k,n ~ ∗ d module S (Mk,n ). The conclusion follows from Theorem 3.2 and Proposition 4.1. 6. Braid group actions on quantum matrix space ∗ ) We compare in this section two actions of the braid group Bn on the algebra S~ (Mk,n of functions of quantum (k × n)-matrix space. The first is the R-matrix representation ∗ ) as the U gl -module S (M ∗ )⊗n . The second is obtained by regarding S~ (Mk,n ~ k ~ k,1 ∗ ) viewed as a U gl -module. We the quantum Weyl group action of Bn on S~ (Mk,n ~ n shall show that these representations essentially coincide, thus extending to the q∗ ) via the setting the fact that the symmetric group Sn acts on (S • Ck )⊗n ∼ = S(Mk,n permutation matrices in GLn (C). More precisely, for any 1 ≤ j ≤ n, let R ∨j be the universal R-matrix of U~ glk ∗ ) ∼ S (M ∗ )⊗n , and let S be acting on the j and j + 1 tensor copies of S~ (Mk,n = ~ k,1 j the quantum Weyl group element of U~ gln corresponding to the simple root α j = θ j − θ j+1 . We shall show that R ∨j = S j · e (n)
(n)
(n)
(n)
−~(D j +D j D j+1 /k)
(n)
· eiπ D j ,
(6.1)
(n)
where D1 , . . . , Dn are the generators of the Cartan subalgebra of U~ gln . The proof of (6.1) is based upon the following observation, which we owe to B. Feigin. Both ∗ )⊗n , so that sides of (6.1) only act upon the j and j + 1 tensor copies of S~ (Mk,1 ∗ )⊗2 ∼ S (M ∗ ). Since both sides its proof reduces to a computation in S~ (Mk,1 = ~ k,2 ∗ intertwine the action of U~ glk on S~ (Mk,2 ), it suffices to compare them on highest weight vectors. These and the action of R ∨j are computed in Section 6.1. The action of S j is computed in Section 6.2. Remark. It is easy to check that neither action of Bn is compatible with the algebra ∗ ), so that (6.1) cannot be proved by merely checking it on the structure of S~ (Mk,n generators X i j . This stems from the fact that quantum Weyl group operators are not grouplike.
440
VALERIO TOLEDANO LAREDO
6.1. R-matrix action on singular vectors ∗ ). For any d ∈ N, let S~d Ck be the homogeneous component of degree d of S~ (Mk,1 By (5.17) – (5.19), S~d Ck is a deformation of the dth symmetric power S d Ck of the vector representation of glk . Let µ1 , µ2 ∈ N; then we have the following lemma. LEMMA 6.1 As U~ glk -modules, µ µ S~ 1 Ck ⊗ S~ 2 Ck ∼ =
min(µ 1 ,µ2 ) M
(k)
Vµ1 +µ2 −i [[~]],
(6.2)
i
i=0 (k)
where V a is the irreducible representation of glk with highest weight (a, b, 0, b
µ1 ,µ2
. . . , 0). The corresponding highest weight vectors vi µ ,µ vi 1 2
are given by
i X µ −i+a i−a µ −a a a i = (−1) e~a(µ2 −a+1) X 111 X 21 ⊗ X 122 X 22 . a
(6.3)
a=0
Proof Decomposition (6.2) follows from the Pieri rules for glk and Corollary 4.2. Fix i ∈ µ µ {0, . . . , min(µ1 , µ2 )}. By (5.17), any v ∈ S~ 1 Ck ⊗ S~ 2 Ck of weight (µ1 + µ2 − i, i, 0, . . . , 0) is of the form v=
i X
µ −i+a
ca X 111
µ −a
i−a X 21 ⊗ X 122
a X 22
(6.4)
a=0
for some constants ca ∈ C. By (4.10) and (5.18), 1(E j )v = 0 for any j ≥ 2 so that v is a highest weight vector if and only if 1(E 1 )v =
i X
µ −i+a+1
ca [i − a]e~(µ2 −2a) X 111
µ −a
i−a−1 X 21 ⊗ X 122
a X 22
a=0
+
i X
µ −i+a
ca [a]X 111
µ −a+1
i−a X 21 ⊗ X 122
a−1 X 22
(6.5)
a=0
is equal to zero. This yields ca = −ca−1 e~(µ2 −2a+2) [i − a + 1]/[a] and therefore i ~a(µ2 −a+1) ca = (−1)a e c0 , (6.6) a whence (6.3).
A KOHNO-DRINFELD THEOREM FOR QUANTUM WEYL GROUPS
441 µ
µ
Let R be the universal R-matrix of U~ slk , and let R ∨ = σ · R : S~ 1 Ck ⊗ S~ 2 Ck → µ µ S~ 2 Ck ⊗ S~ 1 Ck be the corresponding U~ slk -intertwiner, where σ is the permutation of the tensor factors. PROPOSITION 6.2 L µ2 k µ µ µ The following holds on S~ 1 Ck ⊗ S~ 2 Ck S~ C ⊗ S~ 1 Ck : µ1 ,µ2
R ∨ vi
µ2 ,µ1
= (−1)i e~((µ1 −i)(µ2 −i)−i−µ1 µ2 /k) vi
.
(6.7)
Proof For any 1 ≤ i ≤ k − 1, let si = (i i + 1) ∈ Sk be the ith elementary transposition, and let j k k l k m w0 = (1 k)(2 k − 1) · · · (6.8) 2 2 be the longest element of Sk . Consider the following reduced expression for w0 : w0 = sk−1 · · · s1 sk−1 · · · s2 · · · · · · sk−1 sk−2 sk−1 = si1 · · · sik(k−1)/2 ,
(6.9)
and let β j = si1 · · · si j−1 (θi j − θi j +1 ) be the associated enumeration of the positive roots of slk so that β1 = θk−1 − θk ,
β2 = θk−2 − θk , βk = θk−2 − θk−1 ,
..., ..., .. .
βk−1 = θ1 − θk , β2k−3 = θ1 − θk−1 ,
βk(k−1)/2 = θ1 − θ2 . (6.10) Let E β j , Fβ j ∈ U~ slk , 1 ≤ j ≤ k(k − 1)/2, be the corresponding quantum root vectors so that E β j = E i and Fβ j = Fi whenever β j is the simple root αi (see [Lu1, Prop. 1.8]). Then (see [KR], [LS], [Ro], [CP, Th. 8.3.9]) k−1 k(k−1)/2 X Y H i ⊗ Hi expq (q − q −1 )E β j ⊗ Fβ j , R = exp ~ i=1
(6.11)
j=1
k−1 k−1 where {H i }i=1 ⊂ h is the basis of the Cartan subalgebra of slk dual to {Hi }i=1 with ~ respect to the pairing hX, Y i = tr(X Y ), q = e , X xn expq (x) = q n(n−1)/2 , (6.12) [n]! n≥0
and the product in (6.11) is taken so that the factor expq ((q −q −1 )E β j ⊗ Fβ j ) is placed µ ,µ to the left of expq ((q − q −1 )E β j 0 ⊗ Fβ j 0 ) whenever j > j 0 . To compute R vi 1 2 , note that for any positive root β 6= θ1 − θ2 and 0 ≤ a ≤ µ1 , µ −a
E β X 111
a X 21 =0
(6.13)
442
VALERIO TOLEDANO LAREDO µ
since, by (5.17), (µ1 − a, a, 0, . . . , 0) + β is not a weight of S~ 1 Ck . Thus, using (5.17) – (5.19), µ1 ,µ2
Rvi
k−1 X µ ,µ = exp ~ H i ⊗ Hi expq (q − q −1 )E 1 ⊗ F1 vi 1 2 i=1
X
=
(−1)a e~a(µ2 −a+1)
0≤a≤i 0≤n≤i−a e~n(n−1)/2 (e~
· ·
− e−~ )n
[n]! µ −i+a+n i−a−n X 111 X 21
i ~((µ1 −i+a+n)(µ2 −a−n)+(i−a−n)(a+n)−µ1 µ2 /k) e a
[i − a]![µ2 − a]! [i − a − n]![µ2 − a − n]! µ −a−n
⊗ X 122
a+n X 22 ,
(6.14)
which, upon setting α = a + n, yields e~((µ1 −i)(µ2 −i)−i−µ1 µ2 /k) i X i ~(i−α)(µ1 −i+α+1) µ2 µ1 −i+α i−α µ −α α · (−1)α e Sα X 11 X 21 ⊗ X 122 X 22 , (6.15) α α=0
where Sαµ = e~α(µ−α+1) α X α [µ − α + n]! ~(α−n)(µ−α+n+1)+~n(n−1)/2 ~ · (−1)n e (e − e−~ )n . n [µ − α]! n=0
(6.16) µ
We claim that Sα = 1 for any α ≤ µ ∈ N, so that (6.7) holds. Indeed, using α α − 1 −~a α − 1 ~(α−a) = δα>a e + δa>0 e , a a a−1
(6.17)
one readily finds that µ−1
µ
Sαµ = e2~(µ−α+1) Sα−1 − (e2~(µ−α+1) − 1)Sα−1 , µ
µ
whence Sα = 1 by induction on α since S0 = 1 for any µ ∈ N. 6.2. Quantum Weyl group action on singular vectors Let E, F, H be the standard generators of U~ sl2 .
(6.18)
A KOHNO-DRINFELD THEOREM FOR QUANTUM WEYL GROUPS
443
LEMMA 6.3 ∗ ): The following holds in S~ (Mk,2 µ1 +1,µ2 −1
,
(6.19)
µ1 −1,µ2 +1
,
(6.20)
µ1 ,µ2
= [µ2 − i] vi
F vi
µ1 ,µ2
= [µ1 − i] vi
H
=
E vi
µ ,µ vi 1 2
µ ,µ (µ1 − µ2 ) vi 1 2 .
(6.21)
Proof By (5.21), µ1 ,µ2
E vi
i X µ −i+a+1 i−a µ −a−1 a a i = (−1) e~a(µ2 −a+1) [µ2 − a]e~(i−2a) X 111 X 21 ⊗ X 122 X 22 a a=0
+
=
i X i ~a(µ2 −a+1) µ −i+a i−a+1 µ −a a−1 (−1)a e [a]X 111 X 21 ⊗ X 122 X 22 a
a=0 i X
(−1)a
a=0
i ~a(µ2 −a+1) µ −i+a+1 i−a µ −a−1 a e [µ2 − a]e~(i−2a) X 111 X 21 ⊗ X 122 X 22 a
i X µ −i+a+1 i−a µ −a−1 a a i (−1) e~(a+1)(µ2 −a) [i − a]X 111 X 21 ⊗ X 122 − X 22 a a=0
= [µ2 − i] =
i X
i ~a(µ2 −a) µ1 +1−i+a i−a µ −1−a a (−1) e X 11 X 21 ⊗ X 122 X 22 a a
a=0 µ +1,µ2 −1 [µ2 − i]vi 1 .
(6.22)
444
VALERIO TOLEDANO LAREDO
Similarly, by (5.22), µ1 ,µ2
F vi
=
i X
(−1)a
a=0
+
i X a=0
i ~a(µ2 −a+1) µ −i+a−1 i−a µ −a+1 a e [µ1 − i + a]X 111 X 21 ⊗ X 122 X 22 a
i ~a(µ2 −a+1) (−1) e [i − a]e−~(µ1 −µ2 −i+2a) a a
µ −i+a
· X 111 =
i X
(−1)a
a=0
−
i X a=0
µ −a
i−a−1 X 21 ⊗ X 122
a+1 X 22
i ~a(µ2 −a+1) µ −i+a−1 i−a µ −a+1 a e [µ1 − i + a]X 111 X 21 ⊗ X 122 X 22 a
i ~(a−1)(µ2 −a+2) e [a]e−~(µ1 −µ2 −i+2a−2) (−1) a a
µ −i+a−1
µ −a+1
i−a a X 21 ⊗ X 122 X 22 i X µ −1−i+a i−a µ +1−a a a i (−1) e~a(µ2 −a+2) X 111 X 21 ⊗ X 122 = [µ1 − i] X 22 a
· X 111
=
a=0 µ −1,µ2 +1 [µ1 − i]vi 1 .
(6.23)
Finally, (6.21) follows from (5.20). Now let S = expq −1 (q −1 Eq −H ) expq −1 (−F) expq −1 (q Eq H )q H (H +1)/2
(6.24)
be the generator of the quantum Weyl group of U~ sl2 (see [Lu3], [KR], [So]; we use the form given in [Ka], [Sa]), where q = e~ and the q-exponential is defined by (6.12). 6.4 ∗ ): The following holds in S~ (Mk,2 PROPOSITION
µ1 ,µ2
Svi
µ2 ,µ1
= (−1)µ1 −i q (µ1 −i)(µ2 −i+1) vi
.
(6.25)
Proof µ−i−k,i+k Fix µ, i ∈ N with 2i ≤ µ. By Lemma 6.3, the vectors u k = vi with 0 ≤ k ≤
A KOHNO-DRINFELD THEOREM FOR QUANTUM WEYL GROUPS
445
µ − 2i satisfy E u k = [k] u k−1 ,
(6.26)
F u k = [µ − 2i − k] u k+1 ,
(6.27)
H u k = (µ − 2i − 2k) u k
(6.28)
and therefore span the indecomposable U~ sl2 -module of dimension µ−2i +1. Moreover, [µ − 2i − k]! k [µ − 2i − k]! µ−2i−k uk = F u0 = E u µ−2i . (6.29) [µ − 2i]! [µ − 2i]! Since Ad(S)H = −H , Su 0 is proportional to u µ−2i and, by (6.24), is therefore equal to F µ−2i (−1)µ−2i q µ−2i u 0 = (−1)µ q µ−2i u µ−2i . (6.30) [µ − 2i]! Next, using Ad(S)F = −q −H E and (6.29), we find that S uk =
[µ − 2i − k]! Ad(S)F k Su 0 = (−1)µ−k q (k+1)(µ−2i−k) u µ−2i−k . [µ − 2i]! µ1 ,µ2
Thus, setting µ = µ1 + µ2 so that vi µ1 ,µ2
S vi
(6.31)
= u µ2 −i , we find that µ2 ,µ1
= (−1)µ1 −i q (µ1 −i)(µ2 −i+1) vi
,
(6.32)
as claimed. 6.3. Identification of R and quantum Weyl group actions Fix 1 ≤ j ≤ n, and let R ∨j be the universal R-matrix of U~ glk acting on the j and ∗ )⊗n . Let S be the element of the quantum Weyl group j + 1 tensor copies of S~ (Mk,1 j of U~ gln corresponding to the simple root θ j − θ j+1 . THEOREM 6.5 ∗ ): The following holds on S~ (Mk,n
R ∨j = S j · e
(n)
(n)
(n)
−~(D j +D j D j+1 /k)
(n)
· eiπ D j .
(6.33)
Proof j (n) (n) (n) (n) Let U~ gl2 ⊂ U~ gln be the Hopf subalgebra generated by E j , F j , D j , D j+1 . By j
(5.20) – (5.22), U~ gl2 only acts upon the variables X i j , X i j+1 , 1 ≤ i ≤ k. Thus, j ∗ ) , U~ gl2 , and therefore S j , only acts on the j and j + 1 tensor copies S~ (Mk,1 j ∗ ⊗n ∼ S (M ∗ ). Since this is also the case of R ∨ , the ∗ ) S~ (Mk,1 = ~ k,n j+1 of S~ (Mk,1 ) j j
∗ ) ⊗ proof of (6.33) reduces to a computation on the (U~ glk ⊗U~ gl2 )-module S~ (Mk,1 j
446
VALERIO TOLEDANO LAREDO
∗ ) ∗ ∼ S~ (Mk,1 j+1 = S~ (Mk,2 ). Both sides of (6.33) clearly commute with U~ glk , so it suffices to check their equality on the singular vectors of M µ µ ∗ ∗ S~ (Mk,1 ) j ⊗ S~ (Mk,1 ) j+1 = S~ 1 Ck ⊗ S~ 2 Ck , (6.34) µ1 ,µ2 ∈N
µ ,µ
µ
µ
that is, on the vectors vi 1 2 ∈ S~ 1 Ck ⊗ S~ 2 Ck of Lemma 6.1. By Propositions 6.2 and 6.4, µ ,µ µ ,µ R ∨j vi 1 2 = S j · e−~(µ1 +µ1 µ2 /k) (−1)µ1 vi 1 2 , (6.35) (n)
whence (6.33) since, for any 1 ≤ l ≤ n, Dl
∗ ). gives the N-grading on S~ (Mk,1 l
Remark. The coincidence of the two representations of Bn studied above was also noted by P. Baumann [Ba, Prop. 12] in the special case when both actions are reµ µ ∗ ) where µ = · · · = µ = 1. stricted to the subspace S~ 1 Ck ⊗· · ·⊗ S~ n Ck of S~ (Mk,n n 1 7. Monodromic realisation of the quantum Weyl group of U~ sln The following is the main result of this paper. It was conjectured for any simple Lie algebra g by De Concini around 1995 (unpublished) and, independently, in [TL]. We prove it here for g = sln (C). THEOREM 7.1 Let g = sln (C), let V be a finite-dimensional g-module, let µ be a weight of V , and let M Vµ = V [ν] (7.1) ν∈W µ
be the direct sum of the weight spaces of V corresponding to the Weyl group orbit of µ. Let πκh : Bg → GL(V µ [[h]]) be the monodromy representation of the connection X dα d −h πV (κα ) (7.2) α α>0
obtained by regarding h as a formal variable. Let πW : Bg → GL(V µ [[h]]) be the quantum Weyl group action obtained by regarding V [[h]] as a U~ g-module with ~ = 2πi h. Then πκh and πW are equivalent. Proof We may assume that V is irreducible with highest weight λ = (λ1 , . . . , λn ), where P (n) λi ∈ N. Regard V as a gln -module by letting 1(n) = nj=1 E j j act as multiplication Pn by |λ| = i=1 λi , and fix some k ≥ n. By Lemma 3.4, V [ν] is isomorphic to the space Mλν of singular vectors of weight λ for the diagonal action of glk on ∗ S ν Ck = S ν1 Ck ⊗ · · · ⊗ S νn Ck ⊂ S(Mk,n ),
(7.3)
A KOHNO-DRINFELD THEOREM FOR QUANTUM WEYL GROUPS
447
and, by Corollary 3.6, the monodromy representation of the Knizhnik-Zamolodchikov L connection (3.17) on ν∈Sk Mλν and πκh are related by 0
h πKZ (T j ) = πκh (T j )e
(n)
(n)
(n)
(n)
−πi h(E j j +E j+1 j+1 +2E j j E j+1 j+1 /k) iπ E (n) jj
e
,
(7.4)
where h 0 = 2h. h 0 to the R-matrix represenWe now use the Kohno-Drinfeld theorem to relate πKZ tation of Bn corresponding to the action of U~ glk on S ν Ck [[~]]. For this purpose, let φ : U~ glk → U glk [[~]] be an algebra isomorphism whose reduction mod ~ is the iden(k) (k) tity and which acts as the identity on the Cartan subalgebras, that is, φ(Di ) = E ii , 1 ≤ i ≤ k (see [Dr2, Prop. 4.3]). Then U~ glk act on each S ν j Ck [[~]] via φ and on S ν1 Ck [[~]] ⊗ · · · ⊗ S νn Ck [[~]] = S ν1 Ck ⊗ · · · ⊗ S νn Ck [[~]] = S ν Ck [[~]]
(7.5)
via the n-fold coproduct 1(n) : U~ glk → U~ gl⊗n k recursively defined by 1(1) = id, (a+1)
1
(7.6)
= 1 ⊗ id
⊗(a−1)
(a)
◦1
,
a ≥ 1,
(7.7)
where 1 = 1(2) is given by (4.9) – (4.11). Let 10 be the standard, cocommutative coproduct on U glk so that U glk acts (n) on S ν Ck via 10 : U glk → U gl⊗n k defined as in (7.6) – (7.7) with 1 replaced by 10 . Since 1 = 10 + o(~) and H 1 (slk , slk ⊕ slk ) = 0, there exists a twist F = ⊗(2) 1 ⊗ 1 + o(~) ∈ U glk [[~]] such that, for any x ∈ U~ glk , φ ⊗ φ ◦ 1(x) = F10 φ(x) F −1 . (7.8) It follows that the actions of U~ glk and U glk [[~]] on S ν Ck are related by (n)
φ ⊗(n) ◦ 1(n) (x) = F (n) 10
−1 φ(x) F (n) ,
x ∈ U~ glk ,
(7.9)
where F (n) ∈ U gl⊗n k [[~]] is recursively defined by F (1) = 1, F
(a+1)
(7.10)
= F ⊗1
⊗(a−1)
· 10 ⊗ id
⊗(a−1)
(F
(a)
),
a ≥ 1.
(7.11)
ν Now let Mλν ⊂ S ν Ck and M~,λ ⊂ S ν Ck [[~]] be the subspaces of vectors of highest weight λ for the actions of glk and U~ glk , respectively. We need the following lemma.
7.2 ν We have F (n) Mλν [[~]] = M~,λ . LEMMA
448
VALERIO TOLEDANO LAREDO
Proof Let S ν Ck (λ) ⊂ S ν Ck and S ν Ck [[~]](λ) ⊂ S ν Ck [[~]] be the isotypical components of types Vλ and Vλ [[~]] for glk and U~ glk , respectively. The subspace F (n) S ν Ck (λ)[[~]] is invariant under U~ glk by (7.9), and its reduction mod ~ is equal to S ν Ck (λ) since F (n) = 1⊗(n) + o(~). Thus, by Proposition 4.1, F (n) S ν Ck (λ)[[~]] is isomorphic to S ν Ck (λ)[[~]] and therefore contained in S ν Ck [[~]](λ). Since this holds for any λ, the inclusion is an equality. Now noting that F (n) is equivariant for the action of the Cartan subalgebras of glk and U~ glk since 10 ◦ φ(h) = φ ⊗ φ ◦ 1(h) for any h ∈ h, we get F (n) Mλν [[~]] = F (n) S ν Ck (λ)[λ][[~]] = F (n) S ν Ck (λ)[[~]] [λ] = S ν Ck [[~]](λ)[λ] ν = M~,λ ,
(7.12)
as claimed L Summarising, we have an action of Bn on ν∈Sn µ Mλν via the monodromy of the L ν via the Knizhnik-Zamolodchikov connection and an action of Bn on ν∈Sn µ M~,λ R-matrix representation of U~ slk . Drinfeld’s theorem (see [Dr3], [Dr4], [Dr5]) asserts that the twist F may be chosen so that M M ν F (n) : Mλν [[~]] −→ M~,λ (7.13) ν∈Sn µ
ν∈Sn µ
is Bn -equivariant, so that for any 1 ≤ j ≤ n − 1, h πKZ (T j ) = F (n) 0
−1
R ∨j F (n) ,
(7.14)
where R ∨j is the universal R-matrix for U~ slk acting on the j and j + 1 copies of L ν k 0 ν∈Sn µ S C [[~]] and ~ = πi h . ∗ ∗ ))⊗n be the algebra of functions on quantum Now let S~ (Mk,n ) ∼ = (S~ (Mk,1 (k × n)-matrix space defined in Section 5, and, for any ν ∈ Nn , let S~ν Ck = S~ν1 Ck ⊗ · · · ⊗ S~νn Ck
(7.15)
be the space of polynomials that are homogeneous of degree ν j in the variables X 1 j , . . . , X k j , for any 1 ≤ j ≤ n. By (5.17) – (5.19), S~ν Ck /~S~ν Ck is the glk -module S ν Ck , so that, by Proposition 4.1, we may identify S~ν Ck with S ν Ck [[~]] as U~ glk modules. By Theorem 5.5 and the fact that the N-grading on the jth tensor factor of ∗ ) ∼ (S (M ∗ ))⊗n is given by the action of D (n) , the space M ν is isoS~ (Mk,n = ~ 1,k j ~,λ (n)
(n)
morphic to the subspace Vλ [ν][[~]] ⊂ Vλ [[~]] of weight ν. Now using Theorem
A KOHNO-DRINFELD THEOREM FOR QUANTUM WEYL GROUPS
449
6.5, we find that πκh (T j ) = F (n) ·e
−1
· Sj · e (n)
(n)
(n)
(n)
−~(D j +D j D j+1 /k)
(n)
(n)
(n)
πi h(E j +E j+1 +2E j E j+1 /k)
(n)
· eiπ D j · F (n) (n)
· eiπ E j ,
(7.16)
where S j = πW (T j ) is the quantum Weyl group element of U~ gln corresponding to L (n) (n) the simple root α j = θ j − θ j+1 . Since for any l, Ell = Dl on V µ ∼ = ν∈Sk Mλν , we get
where ρ (n) = (1/2) of sln .
(n)
−1
= F (n)
−1
= F (n)
−1 ~ρ (n) /2
P
j=1 (n
Sje
(n)
S j e−~H j e
(n)
− 2 j + 1)D j
(n)
−~/2(D j −D j+1 )
πκh (T j ) = F (n)
/2
F (n)
F (n)
S j e−~ρ
(n) /2
F (n) ,
(7.17)
is the half-sum of the positive (co)roots
Remark. Theorem 7.1 is proved in [TL] for the following pairs (g, V ), where g is a simple Lie algebra and V is an irreducible, finite-dimensional representation: (1) g = sl2 and V is any irreducible representation; (2) g = sln and V is a fundamental representation; (3) g = son and V is the vector or a spin representation; (4) g = spn and V is the defining vector representation; (5) g = e6 , e7 and V is a minuscule representation; (6) g = g2 and V is the 7-dimensional representation; (7) g is any simple Lie algebra and V ∼ = g is its adjoint representation. Acknowledgments. This paper was begun at the Research Institute for Mathematical Sciences (RIMS) of Kyoto University. I am very grateful to M. Kashiwara for his invitation to spend the summer of 1999 at RIMS and to RIMS for its hospitality. During my stay, I greatly benefitted from very stimulating and informative discussions with M. Kashiwara and B. Feigin. I also wish to express my gratitude to A. D’Agnolo, P. Baumann, B. Enriquez, J. Millson, R. Rouquier, and P. Schapira for innumerable, useful, and friendly conversations. References [Ba] [Br]
P. BAUMANN, The q-Weyl group of a q-Schur algebra, preprint, 1999. 433, 438, 446 E. BRIESKORN, Die Fundamentalgruppe des Raumes der regul¨aren Orbits einer
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Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, California 94720-5070, USA;
[email protected]; permanent: Institut de Math´ematiques de Jussieu, Unit´e Mixte de Recherche 7586, Case 191, Universit´e Pierre et Marie Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 3,
SYMPLECTIC LEAVES OF COMPLEX REDUCTIVE POISSON-LIE GROUPS MILEN YAKIMOV
Abstract All factorizable Lie bialgebra structures on complex reductive Lie algebras were described by Belavin and Drinfeld. We classify the symplectic leaves of all related Poisson-Lie groups. A formula for their dimensions is also proved. Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Lie bialgebras and Poisson-Lie groups . . . . . . . . . . . . . . . 1.1. Lie bialgebras . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Symplectic leaves of Poisson-Lie groups . . . . . . . . . . . 1.3. The passage from double cosets to symplectic leaves . . . . . 2. Lie algebra structure of certain doubles and duals . . . . . . . . . . 2.1. Description of D(g) and of the embeddings g, g∗ ⊂ D(g) . . . 2.2. Description of D(g+ ) and of the embeddings g+ , g− → D(g+ ) 2.3. Another approach to the structure of D(g+ ) . . . . . . . . . 3. Factorizable reductive Lie bialgebras and Poisson-Lie groups . . . . 3.1. Belavin-Drinfeld classification . . . . . . . . . . . . . . . . 3.2. Related Poisson-Lie groups . . . . . . . . . . . . . . . . . 3.3. Induction on factorizable complex reductive Lie bialgebras and Poisson-Lie groups . . . . . . . . . . . . . . . . . . . . . 4. Symplectic leaves in G − . . . . . . . . . . . . . . . . . . . . . . 4.1. Double cosets of i + (G + ) in D(G + ) . . . . . . . . . . . . . 4.2. Structure of the group 6− = i + (G + ) ∩ i − (G − ) ⊂ D(G + ) . . 4.3. Summary of the results on the symplectic leaves of G − . . . . 5. Symplectic leaves of G . . . . . . . . . . . . . . . . . . . . . . . 5.1. G r double cosets of D(G) . . . . . . . . . . . . . . . . . . 5.2. The group 6 = d(G) ∩ G r . . . . . . . . . . . . . . . . . 5.3. Summary of the results on the symplectic leaves of G . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
453 455 455 458 460 461 461 463 464 465 465 468 472 474 475 489 492 493 493 502 503
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 3, Received 4 May 2000. Revision received 1 May 2001. 2000 Mathematics Subject Classification. Primary 22E10; Secondary 17B62, 53D17 Author’s work supported by National Science Foundation grant numbers DMS-9603239 and DMS-9400097. 453
454
6. Symplectic leaves of the dual Poisson-Lie group of G References . . . . . . . . . . . . . . . . . . . . . . .
MILEN YAKIMOV
. . . . . . . . . . . . . . . . . . . . .
507 507
0. Introduction The structure of symplectic leaves of a Poisson manifold M naturally carries information about the geometry of the Poisson bivector field. Its importance to dynamical systems is related to the fact that the phase space of any Hamiltonian system on M is a symplectic leaf of M. When the Poisson structure can be quantized algebraically, it also has applications to the representation theory of the corresponding quantized algebra of functions. Roughly speaking, the Poisson ideals of functions vanishing on closures of symplectic leaves are deformed to primitive ideals of the quantized algebra, but the actual relation is subtle. In the 1970s and 1980s, a detailed study of the primitive ideals of universal enveloping algebras of Lie algebras and their relation to coadjoint orbits was done. Nice accounts of it can be found in the review articles [17] and [25] and in the book [5]. For standard (simple) Poisson-Lie groups and their quantized algebras of functions, this relation was examined more recently by Y. Soibelman [31], T. Hodges and T. Levasseur [10], [11], [12], and A. Joseph [18], [19], [20] (see also [23], [13]). In the early 1980s, A. Belavin and V. Drinfeld classified all non-skew-symmetric r -matrices on complex simple Lie algebras. The same procedure describes all factorizable Lie bialgebra structures on complex reductive Lie algebras (see [9]). The set of symplectic leaves on the corresponding Poisson-Lie groups is well understood for the standard structure (see [4], [10], [14]), on the corresponding compact real forms (see [31], [24], [22]), and in the cases of the simplest twistings of the standard r -matrix by elements of the second tensor power of the (fixed) Cartan subalgebra (see [23], [13]). At the same time, very little is known for any other Poisson structure in this list. The goal of this article is to present a unified description of the sets of symplectic leaves of the complex reductive Poisson-Lie groups associated to all r -matrices from Belavin and Drinfeld’s list. We also prove a formula for the dimension of each leaf. In the rest of this introduction, we give a brief description of our results. Any Poisson-Lie group G of the discussed class has two canonical Poisson-Lie subgroups G ± . Their Lie algebras are the spans of the second and the first components of the associated r -matrix. The groups G ± lie inside two parabolic subgroups P± of G containing two opposite Borel subgroups B± of G (P± ⊃ B± ). Denote by H the maximal torus of G which is the intersection of B+ and B− . Then G ± differs from P± only on a part of the maximal torus H. Let W, W1 , and W2 denote the Weyl groups of G and of the Levi factors L 1 , L 2 of P± , containing H. Throughout this paper we call a subgroup L of G a standard Levi subgroup of G if it is the Levi factor of a parabolic subgroup P of G such that P ⊃ B+ and
COMPLEX REDUCTIVE POISSON-LIE GROUPS
455
L ⊃ H. The tangent Lie algebras of these subgroups of G are called standard Levi subalgebras of Lie G. The choice of maximal torus H of G is clear in each particular case. The main ingredient in the classification theorem for the leaves of G − is a choice of minimal length representative v in W for a coset from W/W1 . (By v˙ we denote a representative of it in the normalizer N (H ) of H in G.) For each such element v, the maximal subalgebra of Lie(L 1 ) which is stable under Adv˙ is a standard Levi subalgebra of g. The corresponding reductive subgroup of G is denoted by G v . The second main ingredient in the classification is a choice (for each fixed v) of an orbit of the twisted conjugation action of the derived subgroup (G v )0 of G v on itself defined by T C gv˙ ( f ) = g −1 f Adv˙ (g), f, g ∈ (G v )0 . In Section 4.1 we prove that this is the main part of the data which classifies double cosets of the dual Poisson-Lie group G + of G − in their classical double. Sections 4.2 and 4.3 realize the passage to symplectic leaves (dressing orbits) in G − . The set of symplectic leaves of G + can be described in a similar way, but the corresponding results are not stated. Section 5 deals with the general case of symplectic leaves on the full group G. The classification data is partly a “doubling” of the one for G − . One starts with a choice of a pair (v1 , v2 ) of minimal length representatives in W for cosets from W/W1 and W/W2 . The second part of the data is a twisted conjugation orbit on the maximal semisimple subgroup of L 1 whose Lie algebra is stable under Adv˙1 θ −1 Adv˙2 θ. As in the G − case, we show that this group is the derived subgroup of a standard Levi subgroup of G. The map θ : Lie(L 01 ) → Lie(L 02 ) is an isomorphism known as the Cayley transform of the r -matrix r. The first three sections contain both a review of known facts and some new results (see in particular Sections 1.3, 2.2, 2.3, and 3.3). For simplicity, we have chosen to state and prove fully the results for the PoissonLie subgroup G − of G. The results for G are stated, but their proofs are mostly sketched. Although they imply the ones for G − , their proofs compared to the G − case only contain additional complications. Our proofs are based on a certain inductive procedure that resembles (and in some sense should be considered as a Poisson-Lie analog of) the construction of M. Duflo and of C. Moeglin and R. Rentschler for computing the primitive spectrum of universal enveloping algebras of Lie algebras which are not necessarily solvable or semisimple (see [6], [25]; see also the reviews [17], [26]). The reason for this relation comes from the fact that the linearization of the Poisson structure on G at the origin and a quantization produce the universal enveloping algebra U (g∗ ) of the dual Lie bialgebra g∗ of g = Lie(G). The Lie algebra g∗ is, roughly speaking, isomorphic to
456
MILEN YAKIMOV
a sum of two standard parabolic subalgebras of g with identified Levi factors. It is solvable only in the cases of the simplest twistings of the standard structure. An explicit quantization of the Belavin-Drinfeld r -matrices was obtained recently by P. Etingof, T. Schedler, and O. Schiffmann [7]. An appropriate extension of the construction of [6], [25] is essential for the classification of the primitive ideals of the corresponding quantized algebras of functions. In a subsequent publication (see [32]), we will extend our methods to classify the symplectic leaves of all Poisson homogeneous spaces for the considered Poisson-Lie groups. This includes in particular the interesting case of their classical Heisenberg doubles. A word on terminology and notation: All double cosets are considered as orbits of actions of Lie groups and as such are equipped with the induced quotient topology. All results refer to this topology. Often there are formulas that mix direct sums of linear spaces and direct sums of Lie algebras. To distinguish them, the former is denoted by u and the latter by ⊕. 1. Lie bialgebras and Poisson-Lie groups All Lie algebras and groups are assumed to be finite-dimensional. 1.1. Lie bialgebras Definition 1.1 A Lie bialgebra is a Lie algebra (g, [., .]) equipped with a map δ : g → ∧2 g, called Lie cobracket, such that (1) the dual map to δ defines a Lie algebra structure on g∗ , and (2) δ([a, b]) = [δ(a), 1 ⊗ b + b ⊗ 1] + [1 ⊗ a + a ⊗ 1, δ(b)], ∀a, b ∈ g. The notion of Lie bialgebra is self dual. More precisely, the dual space g∗ is naturally equipped with a structure of Lie bialgebra with Lie bracket induced by δ and Lie cobracket induced by [., .]. The same Lie algebra equipped with the opposite cobracket is denoted by g∗ op . Definition 1.2 The space gug∗ admits a canonical structure of Lie bialgebra called a classical (Drinfeld) double of g and denoted by D(g). It is uniquely determined from the following requirements. (1) The natural embeddings i : g ,→ D(g) = g u g∗ and i ∗ : g∗ op ,→ D(g) = g u g∗ are morphisms of Lie bialgebras.
COMPLEX REDUCTIVE POISSON-LIE GROUPS
(2)
457
The inner product on g u g∗ , which is equal to the pairing between g and g∗ and is trivial otherwise, is invariant with respect to the adjoint action of D(g).
The following is an important construction for studying the structure of a classical double. Definition 1.3 A Manin triple is a triple of Lie algebras (g, a+ , a− ) for which g is equipped with an invariant nondegenerate inner product (., .) and (1) a+ and a− are isotropic Lie subalgebras of g, (2) g = a+ u a− . The double D(g) of any Lie bialgebra g gives rise to a Manin triple (D(g), g, g∗ ) with inner product on g u g∗ as defined in Definition 1.2. Conversely, assume that (g, a+ , a− ) is a Manin triple. Then a+ , a− , and g have natural structures of Lie bialgebras with costructures induced from the bilinear form on g by identifying a∗+ ∼ = a− , ∗ op a− ∼ = a+ and setting g = a+ ⊕ a− as Lie coalgebras. One finds that as bialgebras, ∗ op a− ∼ = a+ and g ∼ = D(a+ ). Definition 1.4 A Lie bialgebra (g, [., .], δ) is called coboundary if there exists an element r ∈ g ⊗ g such that δ(a) = [r, a ⊗ 1 + 1 ⊗ a], ∀a ∈ g. (1.1) The map δ from (1.1) defines a Lie bialgebra structure on g if and only if the following conditions hold for r : r + r21 ∈ (g⊗2 )g ,
(1.2)
[r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] ∈ (g⊗3 )g .
(1.3)
For the standard notation r12 , r13 , and so on, we refer to any book on quantum groups, for example, [2], [8], and [23]. In both formulas, (.)g means the g-invariant part with respect to the natural adjoint action of g on tensor powers of g. Definition 1.5 A coboundary Lie bialgebra is called quasitriangular if its r -matrix satisfies the YangBaxter equation [r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] = 0. (1.4)
458
MILEN YAKIMOV
The classical double D(g) of any Lie bialgebra g is a quasitriangular bialgebra with r -matrix r ∈ g ⊗ g∗ representing the graph of the trivial map id : g → g. For any quasitriangular Lie bialgebra g, one defines (see [28]) the maps r+ and r− : g∗ → g by r+ (x) = (x ⊗ id)r,
∀x ∈ g∗ ,
r− (x) = −(id ⊗ x)r,
∀x ∈ g∗ .
(1.5) (1.6)
1.6 The maps r+ , r− : g∗ → g are morphisms of Lie bialgebras. LEMMA
See [28], [27], and [8] for a proof of this lemma. Denote the images of r+ and r− by g+ and g− , respectively. Definition 1.7 A quasitriangular Lie bialgebra g is called factorizable (see [28]) if r + r21 ∈ S 2 g defines a nondegenerate inner product on g∗ . (Here S k V denotes the kth symmetric power of the vector space V.) For any quasitriangular Lie bialgebra g, the spaces g+ and g− are Lie sub-bialgebras ∗ op of g, and g− ∼ = g+ . Thus D(g+ ) as a vector space is isomorphic to g+ u g− , and one can define a linear map π : D(g+ ) → g (1.7) which, restricted to g+ and g− , coincides with the natural embeddings of these algebras in g. LEMMA 1.8 The map π defined in (1.7) is a morphism of Lie bialgebras.
Proofs of this lemma can be found in [28], [27], and [8]. 1.2. Symplectic leaves of Poisson-Lie groups Recall that a symplectic leaf of a Poisson manifold (M, π ) is a maximal connected symplectic submanifold. Any symplectic leaf S can also be described as a union of all piecewise smooth paths starting at a given point of S with segments that are integral curves of Hamiltonian vector fields on M. In particular, this implies that M is a disjoint union of its symplectic leaves.
COMPLEX REDUCTIVE POISSON-LIE GROUPS
459
A Poisson-Lie group G is a Lie group equipped with a Poisson bivector field for which the multiplication map G × G → G is a Poisson map. Its tangent Lie algebra g carries a natural Lie bialgebra structure, and conversely for any Lie bialgebra g, there exists a unique connected and simply connected Poisson-Lie group whose tangent Lie bialgebra is g (see [2]). Any Poisson-Lie group G with coboundary tangent Lie bialgebra g can be equipped with the Sklyanin Poisson-Lie structure πx = L x (r ) − Rx (r ),
x ∈ G,
(1.8)
where r ∈ g ⊗ g ∼ = Te G ⊗ Te G is an r -matrix for the Lie bialgebra structure on g and where L x , Rx denote the differentials of the right/left translations by x ∈ G. The bivector field π is skew-symmetric because of the Ad-invariance of r +r21 (see (1.2)). A double D(G) of a Poisson-Lie group G is a connected Lie group with Lie(D(G)) = D(g) for which the embedding i : g ,→ D(g) can be integrated to a (possibly not proper) embedding of Lie groups i : G → D(G). Since D(g) is a quasitriangular Lie bialgebra, any such group D(G) can be equipped with the Sklyanin Poisson structure, and then i becomes an embedding of PoissonLie groups. By G r we denote the (generally not closed) subgroup of D(G) with Lie(G r ) = i ∗ (g∗ ) ⊂ D(g). G r is equipped with the topology that has as a basis of open sets the connected components of intersections of open sets of D(G) with G r . Since i ∗ (g∗ op ) is a Lie sub-bialgebra of D(g), G r is a (nonproperly embedded) Poisson-Lie subgroup of D(G) with tangent Lie bialgebra i ∗ (g∗ op ) ∼ = g∗ op . It naturally plays the role of (opposite) dual of G. The intersection 6 := i(G) ∩ G r is a countable subgroup of D(G) because i(g) ∩ i ∗ (g∗ ) = 0. The right action of 6 on i(G) preserves the Poisson bivector field. By abuse of notation, the projections of the symplectic leaves of G on i(G)/6 are called symplectic leaves of i(G)/6. In the important case when i(G) and G r are closed subgroups of D(G), 6 is a discrete subgroup of i(G) and the latter are indeed the symplectic leaves of i(G)/6. In this case, i(G)/6 is also diffeomorphic as a Poisson manifold to the dense, open subset i(G)G r /G r of the Poisson homogeneous space D(G)/G r of D(G). The projection map G → i(G)/6 is denoted by p. In this setting, an algebraic way to describe the symplectic leaves of G was found by M. Semenov-Tian-Shansky [30] (see also [24]). In this formulation, the statement was taken from [4] and [10].
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MILEN YAKIMOV
1.9 The symplectic leaves of i(G)/6 are the connected components of the intersections of double cosets G r x G r in D(G) with i(G)G r projected into i(G)G r /G r ∼ = i(G)/6. The symplectic leaves of G are the connected components of the pullbacks of symplectic leaves of i(G)/6 under the projection p.
THEOREM
(1)
(2)
In the case when i(G) and G r are closed subgroups of D(G), the left (local) action of G r on i(G)G r /G r can be lifted to a (local) Poisson action of G r on G using the covering map p. In the general case such an action is obtained by integrating the socalled dressing vector fields (see [24]). This action is called the left dressing action of the (opposite) dual Poisson-Lie group G r of G, on G. The traditional way of stating the result of Theorem 1.9 is to say that the symplectic leaves of G are the orbits of the dressing action of G r . 1.3. The passage from double cosets to symplectic leaves According to Theorem 1.9, the main question in passing from G r double cosets in D(G) to symplectic leaves of i(G)/6 is to understand whether (and how) each such double coset intersects the dense, open subset i(G)G r of D(G). In this section we provide an answer to this question for an arbitrary Poisson-Lie group. 1.10 For any connected complex Poisson-Lie group G, the intersection of each double coset G r x G r in D(G) with i(G)G r is a dense, open, and connected subset of the coset G r x G r . THEOREM
Taking into account i(g) u i ∗ (g∗ ) = D(g), Theorem 1.10 can be obtained as a consequence of the following lemma. LEMMA 1.11 Let C be a connected complex Lie group, and let A, B be two Lie subgroups of C satisfying Lie(A) u Lie(B) = Lie(C). (1.9)
Then the intersection Ax A ∩ B A is a dense, open, and connected subset of Ax A for any x ∈ C. Proof Equation (1.9) implies that the products AB and B A are dense, open subsets of C. Recall that if U and V are two dense, open subsets of C, then U V = C (see, e.g.,
COMPLEX REDUCTIVE POISSON-LIE GROUPS
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[15, Lemma 7.4]). Applying this for U = AB and V = B A shows that AB A = C. Thus the intersection Ax A ∩ B A is a nonempty open subset of Ax A, for all x ∈ C. B A can be considered as the orbit of e ∈ C under the obvious action of A × B. Consider the general situation of an action of a Lie group F on a complex manifold X for which there is a dense, open orbit O of dimension dim F. We construct a global section of the anticanonical bundle K X∗ on X whose set of zeros is the complement N to O . Fix a linear basis { f k }k=1 of Lie(F), N = dim F. The infinitesimal action of N on X. The assumption on the action Lie(F) gives rise to a set of vector fields {vk }k=1 of F implies that the set of vectors {v1 (x), . . . , v N (x)} ⊂ Tx X is linearly independent for x ∈ O and linearly dependent otherwise. Thus the global section v1 ∧ · · · ∧ v N of the anticanonical bundle on X vanishes exactly on the complement to the orbit O in X. Applying this argument to the above-mentioned action of A × B on C shows that Ax A ∩ B A is a nonempty open subset of B A whose complement in B A is the zero set of a global section of K C∗ . This implies that it is a dense, open subset of B A which in addition is connected since the ground field is C. Note also that Lemma 1.11 without the connectedness part is still valid over R. 2. Lie algebra structure of certain doubles and duals This section is devoted to the explicit description of the Lie algebra structure of certain doubles and duals associated to a factorizable Lie bialgebra g. (In all statements the costructure of the involved bialgebras is not considered.) First we recall a result of N. Reshetikhin and Semenov-Tian-Shansky [28] on the Lie algebra structure of D(g) and the embeddings of g and g∗ in it (see also [21]). From it we deduce a similar result about the Lie structure of D(g+ ) and the embeddings of g+ and g− . A second derivation is also presented. Compared to the first one, the second has the advantage that it does not rely on an initial guess of what the structure is. The nondegenerate bilinear form on g induced by r + r21 ∈ S 2 g is used often and is denoted by (., .)r . 2.1. Description of D(g) and of the embeddings g, g∗ ⊂ D(g) The homomorphisms r± (see (1.5), (1.6)) can be combined to define an embedding of Lie algebras r : g∗ → g ⊕ g (2.1) by r (x) = r+ (x), r− (x) ,
x ∈ g∗ .
(2.2)
462
MILEN YAKIMOV
This map is injective due to the nondegeneracy of r + r21 . Its image is denoted by gr ∼ = g∗ . The Lie algebra g by itself can be embedded diagonally in g ⊕ g: d(g) = (g, g) ∈ g ⊕ g for g ∈ g,
(2.3)
so d(g) := Im d ∼ = g. 2.1 (Reshetikhin and Semenov-Tian-Shansky) The double D(g) of a factorizable Lie bialgebra g is isomorphic as a Lie algebra to g ⊕ g. In addition, g∗ and g are embedded in it by the maps r and d defined in (2.2) and (2.3). PROPOSITION
Proof Equip g ⊕ g with the bilinear form
(x1 , x2 ), (y1 , y2 ) = (x1 , y1 )r − (x2 , y2 )r .
(2.4)
One checks that d(g) and gr are isotropic subspaces of g. Since d(g) ∩ gr = 0 and dim d(g) + dim gr = 2 dim g, the triple (g ⊕ g, d(g), gr ) is a Manin triple, which implies the statement. The Lie algebra gr = r (g∗ ) can be described more explicitly in terms of the so-called Cayley transform of r. To formulate this result, we need to introduce the Lie algebras m+ = r+ (Ker r− ), m− = r− (Ker r+ ). Clearly, m± are Lie ideals of g± . The quotients g+ /m+ and g− /m− are isomorphic as Lie algebras, and the isomorphism is given by θ r+ (x) + m+ = r− (x) + m− , ∀x ∈ g∗ . (2.5) It is correctly defined since r+ (x) ∈ m+ implies x ∈ Ker r+ + Ker r− and thus r− (x) ∈ m− . It is also straightforward to check that θ is a Lie algebra homomorphism. The fact that it is an isomorphism is proved directly by constructing the inverse map θ −1 : g− /m− → g+ /m+ by θ r− (x) + m− = r+ (x) + m+ , ∀x ∈ g∗ . The map θ is the classical Cayley transform f /( f − 1) of the linear map f : g → g, defined by f = −r− ◦ j : g → g, (2.6)
COMPLEX REDUCTIVE POISSON-LIE GROUPS
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where j : g → g∗ is the linear isomorphism associated with the nondegenerate form (., .)r . In terms of it, r+ ◦ j = 1 − f. The algebras g± , m± , are g+ = Im( f − 1), g− = Im f, m+ = Ker f, m− = Ker( f − 1), as needed for the definition of the Cayley transform. Consider also the projections p± : g± → g± /m± .
(2.7)
The following result of Semenov-Tian-Shansky [29] describes gr in terms of θ. PROPOSITION 2.2 The subalgebra gr of g+ ⊕ g− ⊂ g ⊕ g consists of those (x+ , x− ) ∈ g+ ⊕ g− for which θ ◦ p+ (x+ ) = p− (x− ).
This proposition is obtained directly from the definition of θ. Note that m± are isotropic subalgebras of g± with respect to the bilinear form (., .)r . Indeed, for all x ∈ Ker r− ⊂ g∗ and y ∈ g∗ , we have r+ (x), r+ (y) r = x r+ (y) = (y ⊗ x)r = −y r− (x) = 0. Therefore one can equip g+ /m+ and g− /m− with inner products induced by (., .)r . In this setting the map θ is an isometry. 2.2. Description of D(g+ ) and of the embeddings g+ , g− → D(g+ ) Here we describe the Lie structure of D(g+ ) and the way g+ and g− are embedded in it. PROPOSITION 2.3 As a Lie algebra,
D(g+ ) ∼ = g ⊕ (g+ /m+ ).
(2.8)
The embeddings i ± : g± → D(g+ ) ∼ = g ⊕ (g+ /m+ ) are given by the formulas i + (x) = x, p+ (x) for x ∈ g+ , i − (y) = y, θ −1 ◦ p− (y) for y ∈ g− .
(2.9) (2.10)
Proof Equip the Lie algebra g ⊕ (g+ /m+ ) with the invariant bilinear form
(x1 , x2 ), (y1 , y2 ) = (x1 , y1 )r − (x2 , y2 )r , x1 , y1 ∈ g, x2 , y2 ∈ g+ /m+ . (2.11)
464
MILEN YAKIMOV
As was noticed, this is possible since m+ is an isotropic subspace of g+ with respect to the bilinear form (., .)r . Equations (2.9) and (2.10) define embeddings of g± in g ⊕ (g+ /m+ ). Their images i ± (g± ) are isotropic subspaces with trivial intersection. This follows from i + (g+ ) = (id × p+ )d(g+ ), i − (g− ) = (p+ × id) gr ∩ (g+ ⊕ g) , and the fact that d(g) and gr are isotropic subspaces of g ⊕ g with respect to the inner product (2.4). In the second equation the elements of g ⊕ (g+ /m+ ) in the right-hand side are written as pairs in the opposite order. Taking into account dim g+ +dim m+ = dim g, one gets that (g ⊕ (g+ /m+ ), i + (g+ ), i − (g− )) is a Manin triple, which implies the statement. 2.3. Another approach to the structure of D(g+ ) Here we present a second approach to the results from Section 2.2 on the Lie structure of D(g+ ). All results are formulated without proofs. The starting point is the Lie bialgebra morphism π : D(g+ ) → g (see (1.7)). Its kernel Cg := (x, −x) x ∈ g+ ∩ g− ⊂ g+ u g− (2.12) is a Lie bialgebra ideal of D(g). Define a map I : g → D(g+ )
(2.13)
I (x) = r+ ◦ j (x), −r− ◦ j (x) ∈ g+ u g− ∼ = D(g+ ),
(2.14)
by where, as before, j : g → g∗ is the identification map coming from the nondegenerate bilinear form (., .)r on g. 2.4 I is an injective homomorphism of Lie algebras such that r ◦ I = idg .
LEMMA
This proposition allows us to split D(g+ ) as a direct sum of Lie algebras D(g+ ) = Im I ⊕ Cg ∼ = g ⊕ Cg. To investigate the Lie structure of Cg, we define the maps F± : g± /m± → Cg ⊂ D(g+ )
(2.15)
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by F+ (x) = ( f x, − f x) ∈ Cg,
∀x ∈ Im( f − 1), F− (x) = ( f − 1)x, −( f − 1)x ∈ Cg, ∀x ∈ Im f. They are clearly well defined on the factors. LEMMA 2.5 The maps F+ and F− are Lie algebra isomorphisms, and the composition
(F− )−1 ◦ F+ : g+ /m+ → g− /m− is equal to the Cayley transform θ : g+ /m+ → g− /m− of f. Combining (2.15) with this proposition implies the isomorphism (2.8). The statement about the inclusions i ± of g± in D(g+ ) from Proposition 2.3 is obtained by computing the composition of the embeddings g± ,→ D(g+ ) with this isomorphism. Finally, note that when D(g+ ) is identified with g ⊕ (g+ /m+ ), the projection π : D(g+ ) → g (see (1.7)) is just the projection on the first component.
3. Factorizable reductive Lie bialgebras and Poisson-Lie groups 3.1. Belavin-Drinfeld classification Consider a complex reductive Lie algebra g. Let h ⊂ g be a Cartan subalgebra of g. Choose a set of positive roots 1+ , and denote by b± the corresponding positive and negative Borel subalgebras of g. Denote by 0 the set of simple roots. For a fixed nondegenerate invariant inner product (., .) on g, denote by {xβ |β ∈ ±1+ } a set of root vectors xβ ∈ gβ such that (xβ , x−β ) = 1,
∀β ∈ 1+ .
(3.1)
Definition 3.1 A Belavin-Drinfeld triple is a triple (01 , 02 , τ ), where 01 , 02 ⊂ 0 and τ : 01 → 02 is such that (1) (τ (α), τ (β)) = (α, β) for all simple roots α and β from the set 01 , and (2) for any γ ∈ 01 , there exists a positive integer n for which τ n (γ ) ∈ 02 \01 (nilpotency condition). Given a Belavin-Drinfeld triple (01 , 02 ⊂ 0), one defines a partial ordering on 1+ by α < β if α ∈ 01 , β ∈ 02 , and β = τ n (α) for an integer n.
466
MILEN YAKIMOV
THEOREM 3.2 (Belavin and Drinfeld) Any factorizable Lie bialgebra structure on a complex reductive Lie algebra g has an r -matrix X X r = r0 + x−α ∧ xβ (3.2) x−α ⊗ xα + α∈1+
α,β∈1+ ,α<β
for an appropriate choice of a Cartan subalgebra h, an invariant bilinear form (., .) on g, root vectors xβ ∈ gβ normalized by (3.1), and a Belavin-Drinfeld triple (01 , 02 , τ ). The component r0 ∈ h⊗2 of r has to be such that (τ α ⊗ 1)r0 + (1 ⊗ α)r0 = 0,
∀α ∈ 01 ,
r0 + r021 = 0 .
Here 0 is the component in h ⊗ h of the Casimir of g associated to the form (., .). For such an r -matrix, the bilinear form (., .)r on g induced by r + r21 (see Section 2) is equal to the form (., .). The original result of Belavin and Drinfeld dealt with the case of non-skewsymmetric r -matrices on complex simple Lie algebras. It was later observed (see [9]) that their proof can be easily extended to the above generality. It is also important to note that all Lie bialgebra structures on a complex semisimple algebra g are quasitriangular since H 2 (g, g) = 0. But this is not true for a reductive algebra. For example, when g is commutative, any Lie algebra structure on g∗ gives rise to a Lie bialgebra structure on g. Let us fix an r -matrix as in Theorem 3.2 associated to a Belavin-Drinfeld triple (01 , 02 , τ ). Denote by p+ the parabolic subalgebra of g containing b+ and the root spaces gα for all negative roots α that are linear combinations of simple roots from 01 . In a similar way, define a parabolic subalgebra p− containing b− and gβ for all positive roots β that are linear combinations of simple roots from 02 . Denote by n+ , n− the nilpotent radicals of p+ , p− and by l1 , l2 their Levi factors containing h. In the terminology from the introduction, li are standard Levi subalgebras of g. The Levi decompositions of p± read p± = l1,2 u n± .
(3.3)
The derived subalgebras of l1 , l2 are denoted by l01 , l02 and their centers by z1 , z2 , so li = li0 ⊕ zi ,
i = 1, 2.
(3.4)
The spans of first and second components of r0 contain l01 ∩ h and l02 ∩ h, respectively. Denote by h1 and h2 their intersections with z1 and z2 . Finally, define ⊥ hiort = hi ⊕ (li0 ∩ h) , i = 1, 2, (3.5)
COMPLEX REDUCTIVE POISSON-LIE GROUPS
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where the orthogonal complement is taken in h with respect to the form (., .). One proves (see [1], [8]) that m± are the orthogonal complements to g± , which implies hiort ⊂ hi ,
i = 1, 2.
Therefore g± /m± can be identified with l01,2 ⊕ (h1,2 /hort 1,2 ), and the Cayley transform θ can be considered as an orthogonal Lie algebra isomorphism between the last two algebras. One can choose complements ai for hiort in hi (i = 1, 2), for which the map θ can be lifted to a Lie algebra isomorphism θ : l01 ⊕ a1 → l02 ⊕ a2 ,
(3.6)
preserving the form (., .). PROPOSITION 3.3 For a Belavin-Drinfeld triple (01 , 02 , τ ), the algebras g± , m± are given by
m± = hort 1,2 u n± ,
(3.7)
(l01,2
(3.8)
g± =
⊕ a1,2 ) u m± ,
and thus the quotients g± /m± are g± /m± ∼ = l01,2 ⊕ a1,2 .
(3.9)
The Cayley transform θ : l01 ⊕ a1 → l02 ⊕ a2 maps l01 to l02 extending the map τ to root spaces and a1 to a2 preserving the form (., .) (where it is completely determined by the term r0 of the r -matrix r ). The projections p± : g± → l01,2 ⊕ a1,2 ∼ = g± /m± (recall (3.7)) are just the projections on the first components in (3.8). Example 3.4 Consider the special case when the r -matrix r is such that g+ ⊃ h.
(3.10)
This means that the Cartan subalgebra h sits entirely inside the subalgebra of g spanned by the first components of r. Equation (3.10) is equivalent to the condition hiort = 0. Now g± coincide with the parabolic subalgebras p± , g± = p± , and their ideals m± are equal to the nilpotent radicals n± of p± , m± = n± .
468
MILEN YAKIMOV
The quotient algebras g± /m± are isomorphic to the Levi factors l1,2 of p± : g± /m± ∼ = l1,2 . In this case, ai can be naturally taken as zi (i = 1, 2), and the Cayley transform θ : l1 → l2 is an orthogonal Lie algebra isomorphism between the full Levi subalgebras l1 and l2 . Example 3.5 The standard Lie bialgebra structure on a complex semisimple Lie algebra g comes from the r -matrix X X r= xi ⊗ xi + x−α ⊗ xα , x1
α∈1+
where {xi } is an orthonormal basis of h with respect to the Killing form on g. It is associated with the trivial Belavin-Drinfeld triple for which 01 = 02 = ∅. The algebras g± are equal to the Borel subalgebras b± , and m± are their nilpotent radicals n± . The quotients g± /m± are naturally isomorphic to the Cartan subalgebra h (li0 and hiort , i = 1, 2, are trivial). Under this identification, the map θ : h → h is θ = −idh . Example 3.6 Let g = sl(n + 1), let h be its Cartan subalgebra consisting of diagonal matrices, and let {α1 , . . . , αn } be the set of its simple roots. The Cremmer-Gervais (see [3], [9]) Lie bialgebra structure on it is associated to the Belavin-Drinfeld triple 01 = {α1 , . . . , αn−1 }, 02 = {α2 , . . . , αn }, τ (α j ) = α j+1 , j = 1, . . . , n − 1. The algebras hiort are trivial, and l1 = diag(A, − tr A) A ∈ gl(n) ∼ (3.11) = gl(n), ∼ l2 = diag(− tr A, A) A ∈ gl(n) = gl(n) (3.12) (in block notation). The r0 component of the r -matrix is chosen in such a way that its Cayley transform θ : l1 → l2 is θ diag(A, − tr A) = diag(− tr A, A). 3.2. Related Poisson-Lie groups For a complex reductive group R, we denote its derived group by R 0 and the identity component of its center by Z (R)◦ . Recall the standard fact that R = R 0 Z (R)◦ and that R 0 ∩ Z (R)◦ is a finite abelian group.
(3.13)
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In the rest of this paper, G denotes a connected complex reductive Lie group. We fix a factorizable Lie bialgebra structure on g := Lie(G) associated to a BelavinDrinfeld triple (01 , 02 , τ ) and equip G with the corresponding Sklyanin Poisson-Lie structure (see (1.8)). Let G + and G − denote the connected subgroups of G with tangent Lie bialgebras g+ and g− . They are Poisson-Lie subgroups of G since g± are Lie sub-bialgebras of g. Let M± be the connected subgroups of G + with Lie(M± ) = m± . Assuming the notation from Section 3.1 for Lie(G) = g, we give explicit formulas for M± and G ± that are similar to (3.7) and (3.8). The maximal torus of G corresponding to the fixed Cartan subalgebra h of g is denoted by H. Let B± be the Borel subgroups of G relative to H with Lie(B± ) = b± . Denote by P± the parabolic subgroups of G with Lie algebras p± . Let L 1,2 be their Levi factors containing H , and let N± be their unipotent radicals. The Levi decompositions of P± are P± = L 1,2 n N± . The connected subgroups of Z (L i )◦ with Lie algebras ai and hiort are denoted by Ai and Hiort . The group analogs of (3.7) and (3.8) are ort M± = H1,2 n N± ,
(3.14)
G ± = (L 01,2 A1,2 )M± .
(3.15)
The intersections L i0 ∩ Ai ⊂ Z (L i0 ) are finite groups, but they are nontrivial in general. The groups L 01,2 A1,2 normalize M± . The intersections ort (L 01,2 A1,2 ) ∩ M± = Z (L 01,2 )A1,2 ∩ H1,2 are discrete subgroups of Z (L 01,2 A1,2 ). Generally, they are also nontrivial. The factor groups G ± /M± are ort . G ± /M± ∼ = (L 01,2 A1,2 )/ Z (L 01,2 ) A1,2 ∩ H1,2
(3.16)
Unfortunately, the Cayley transform θ : l01 ⊕ a1 → l02 ⊕ a2 cannot be lifted in general to an isomorphism between the two groups in (3.16). One of the reasons for this is that it might not map the kernel of the exponential map for L 01 A1 to the one of the exponential map for L 02 A2 . Nevertheless, there exist two countable subgroups 3i of Z (L i0 Ai ) containing (L i0 Ai ) ∩ Hiort such that the factor groups (L i0 Ai )/3i are isomorphic (as abstract groups only). We consider the minimal such groups 3i . Explicitly, they are given by 0 ort 0 ort 3i = expi ◦ θ −i ◦ exp−1 , 3−i (Z (L 3−i )A3−i ∩ H3−i ) Z (L i )Ai ∩ Hi
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MILEN YAKIMOV
where i = (−1)i , i = 1, 2. The corresponding isomorphism is denoted by 2: 2 : (L 01 A1 )/31 → (L 02 A2 )/32 .
(3.17)
Clearly, the groups 3i consist of semisimple elements of L i0 Ai . Thus the restrictions of the projections L i0 Ai → (L i0 Ai )/3i to any unipotent subgroup of L i0 Ai are one to one. Remark 3.7 Note that the groups 3i may have in general some accumulation points and that then (L i0 Ai )/3i does not have the structure of differential manifolds. In Sections 3.2.1 and 4, we assume that the r -matrix that we start with is such that this does not happen, that is, that 3i are closed subgroups of L i0 Ai . In Section 5, full generality is assumed. It brings the inconvenience that G r is not a closed subgroup of G × G, but this is not essential for the proofs there. Let p± : G ± → (L 1,2 A1,2 )/31,2 denote the compositions of the projection maps ort ) → (L 01,2 A1,2 )/31,2 . G ± → G ± /M± ∼ = (L 01,2 A1,2 )/ (L 01,2 A1,2 ) ∩ H1,2 Their differentials at the origin are equal to the maps p± : g± → g± /m± introduced in Section 2. Remark 3.8 If the Poisson structure on G comes from an r -matrix r satisfying g+ ⊃ h (see Example 3.4), then the groups G ± are the full parabolic subgroups P± of G, M± are their ort are trivial.) The unipotent radicals N± , and L 01,2 A1,2 are their Levi factors L 1,2 . (H1,2 groups 31,2 are generally nontrivial even if P± = B± and L i = H. Example 3.9 The standard Poisson structure on a semisimple Lie group G is the integration of the standard Lie bialgebra structure on g = Lie(G) reviewed in Example 3.5. In this case, G ± = B± , M± = N± , where N± are the unipotent radicals of the Borel subgroups ort are trivial, and A B± . The groups L 01,2 , H1,2 1,2 = H. There is no need to introduce additional groups 3i , and the Cayley transform 2 : H → H acts as 2(h) = h −1 . Example 3.10 The groups L i for the Cremmer-Gervais Poisson structure on SL(n +1) (see Example
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471
3.6) are L 1 = diag(A, (det A)−1 ) A ∈ GL(n) ∼ = GL(n), −1 L 2 = diag((det A) , A) A ∈ GL(n) ∼ = GL(n). G ± are the parabolic subgroups of SL(n + 1) containing B± with Levi factors L 1,2 , respectively. The map θ : l1 → l2 lifts to 2 : L 1 → L 2 : 2 diag(A, (det A)−1 ) = diag (det A)−1 , A , so 3i are trivial. 3.2.1. The double D(G + ) Combining the results from Section 2.2 and (3.16), we see that we can choose as a double Poisson-Lie group D(G + ) of G + the Lie group D(G + ) := G × (L 01 A1 )/31 , (3.18) equipped with the Sklyanin Poisson structure (1.8). The groups G + and G − are embedded in it by i + (x) = x, p+ (x) ,
x ∈ G +, i − (y) = y, 2−1 ◦ p− (x) , y ∈ G −
(3.19) (3.20)
(cf. Proposition 2.3). The group (3.18) is also a double of G − , but the corresponding ∗ op Poisson bivector field is opposite to the one for D(G + ) since g− ∼ = g+ . 3.2.2. The double D(G) The results from Section 2.1 imply that the Lie group D(G) := G × G, equipped with the Poisson structure (1.8), can be chosen as a double for G. The group G is embedded in it diagonally: d : G → G × G,
d(g) = (g, g) for g ∈ G.
Its image is denoted by d(G). The connected subgroup of G × G with Lie algebra gr is denoted by G r (see Propositions 2.1 and 2.2). It plays the role of the opposite dual Poisson-Lie group of G (see Theorem 1.9) and is explicitly given by G r := (g+ , g− ) g+ ∈ G + , g− ∈ G − , and 2 ◦ p+ (g+ ) = p− (g− ) . (3.21) Because of Proposition 2.2, we need only prove that the group in (3.21) is conˇ i ⊂ L 0 Ai by nected. Consider its identity component (G r )◦ , and define 3 i ˇ i = x ∈ L i0 Ai (x, e) ∈ (G r )◦ , i = 1, 2. 3
472
MILEN YAKIMOV
ˇ 1 to (L 0 A2 )/3 ˇ 2: The following map is an isomorphism from (L 01 A1 )/3 2 ˇ 1 7 → l2 3 ˇ2 l1 3
if li ∈ L i0 Ai and (l1 , l2 ) ∈ (G r )◦ .
ˇ i ⊃ (L 0 Ai ) ∩ H ort because It is correctly defined since L 01 A1 is connected. Besides, 3 i i H1ort × e and e × H2ort are subgroups of (G r )◦ . From the minimality property of 3i , ˇ i , which implies G r = (G r )◦ . we get 3i = 3 3.3. Induction on factorizable complex reductive Lie bialgebras and Poisson-Lie groups As in Section 3.1, we assume that g is a complex reductive Lie bialgebra with an r matrix as in (3.2) associated to the Belavin-Drinfeld triple (01 , 02 , τ ). Recall that the nilpotency condition for the map τ states that for each γ ∈ 01 , there exists an integer n(γ ) such that τ n(γ ) ∈ 02 \01 . Denote ord(τ ) = max n(γ ). γ ∈01
We canonically construct a chain of complex reductive factorizable Lie bialgebras (k) (k) g(k) (k = 0, . . . , ord(τ )) associated to Belavin-Drinfeld triples (01 , 02 , τ (k) ) such that g(0) ≡ g as Lie bialgebras and ord(τ (k) ) = ord(τ ) − k. Recall that D(g+ ) ∼ = g ⊕ (l01 ⊕ a1 ) and g+ and g− are embedded in it by the maps i ± (2.9) and (2.10). Let d be the subalgebra of g ⊕ (l01 ⊕ a1 ) defined by d = l2 ⊕ (l01 ⊕ a1 ) ∩ l2 = l2 ⊕ (l01 ∩ l2 ) ⊕ a1 . (3.22) It is clear that the restriction of the standard bilinear form of D(g+ ) (see the proof of Proposition 2.3) to d is nondegenerate. Define (1) (1) g+ = (l01 ∩ l2 ) ⊕ a1 u (hort (3.23) 1 u n+ ) ⊂ g+ , (1) (1) g− = θ(l01 ∩ l2 ) ⊕ a2 u (hort (3.24) 2 u n− ) ⊂ g− , (1)
(1)
where n+ , n− are the nilpotent subalgebras of l02 spanned by the root spaces of positive/negative roots of l02 that are not roots of l01 ∩ l02 , θ(l01 ∩ l02 ), respectively. (1) (1) The images i ± (g± ) of g± under the inclusions i ± (see (2.9), (2.10)) are isotropic subalgebras of d with trivial intersection since the same is true for the full images (1) i ± (g± ) as subalgebras of D(g+ ). Taking into account dim d = 2 dim g± , we get that (1) (1) (1) (d, i + (g+ ), i − (g− )) is a Manin triple. This induces Lie bialgebra structures on g± (1) and d such that d ∼ = D(g± ). One easily checks that the second term of d in (3.22) is a
COMPLEX REDUCTIVE POISSON-LIE GROUPS
473
Lie bialgebra ideal of d. The factor of d by it is isomorphic to l2 as a Lie algebra that induces a Lie bialgebra structure on l2 . Set g(1) := l2 . (1)
(1)
Define the subsets 01 and 02 of 02 by (1)
01 = 01 ∩ 02 , (1)
02 = τ (01 ∩ 02 ), and let
(1)
(1)
τ (1) := τ |0 (1) : 01 → 02 .
(3.25)
1
The definition of the Lie bialgebra structure on g(1) as a quotient of the double d (see (3.22)) implies that it comes from the r -matrix X X r (1) = r0 + x−α ⊗ xα + x−α ∧ xβ , (3.26) α∈12+
α,β∈12+ ,α<β
where 12+ denotes the set of positive roots of l2 and < is the restriction of the initial partial ordering of 1+ to 12+ (see Theorem 3.2). The Belavin-Drinfeld triple (1) (1) associated to the r -matrix r (1) is (01 , 01 , τ (1) ). The images of the corresponding (1) (1) (1) homomorphisms r± : (g(1) )∗ → g(1) are exactly the Lie bialgebras g+ and g− . From (3.25) it is clear that ord(τ (1) ) = ord(τ ) − 1. Repeatedly applying this construction, one builds a sequence g(0) = g, g(1) , . . . , g(ord τ ) of reductive factorizable Lie bialgebras with the stated property. The Lie bialgebra g(ord τ ) obtained at the end is the Cartan subalgebra h of g with (ord τ ) trivial Lie cobracket (since it is an abelian coboundary bialgebra). Besides, g± and D(g(ord τ ) ) are given by (ord τ )
g±
(ord τ )
D(g
= g± ∩ h,
) = h ⊕ (l01 ⊕ a1 ) ∩ h = h ⊕ (l01 ∩ h) ⊕ a1 .
On the group level, one constructs a chain of connected complex reductive Poisson-Lie groups G (0) = G, G (1) , . . . , G (ord τ ) = H. As a Lie group, G (1) is (1) just L 2 , and G ± are given by (1)
(1)
(1)
(1)
ort G ± = (L 1,2 A1,2 )(H1,2 n N± ), (1)
(1)
(1)
(1)
(3.27)
where L 1 , L 2 , A1 , and A2 denote the connected subgroups of G with Lie alge(1) bras l01 ∩ l02 , θ(l01 ∩ l02 ), (l01 ∩ z2 ) ⊕ a1 , and θ(l01 ∩ z2 ) ⊕ a2 , respectively. N± are the (1) unipotent subgroups of G with Lie algebras n± .
474
MILEN YAKIMOV (1)
(1)
The corresponding groups M± ⊂ G ± (see Section 3.2) are the second factors in these formulas. (1) (1) On the Lie algebra level, the Cayley transform θ (1) : Lie(L 1 A1 ) → (1) (1) (1) (1) Lie(L 2 A2 ) is the restriction of θ to the subspace Lie(L 1 A1 ) = (l01 ⊕ a1 ) ∩ l2 of (1) (1) l01 ⊕ a1 . Since 3i are subgroups of Z (L i0 Ai ) ⊂ H , they sit inside L i Ai , and θ (1) can be lifted to an isomorphism (1)
(1)
(1)
(1)
2(1) : (L 1 A1 )/31 → (L 2 A2 )/32 . A double Poisson-Lie group of G 1± is the Lie group (1)
(1)
(1)
D(G ± ) := G (1) × (L 1 A1 /31 )
(3.28)
equipped with the Poisson structure (1.8). It is naturally a subgroup of G × (L 1 A1 /31 ), and the embeddings (1)
(1)
i ± : G ± → D(G (1) ) (1)
are simply the restrictions of the embeddings i ± (see (3.19), (3.20)) to G ± . The last object G (ord τ ) is G (ord τ ) = H, and
(ord τ )
G±
ort = (L 01,2 ∩ H ) A1,2 H1,2 .
At each step of the described procedure, there is a second canonical way of constructing a Lie bialgebra gk+1 from gk , having the stated property. Restricting to the first step (k = 0), define g(1) = l1 , (1) −1 (1) g+ = θ −1 (l02 ∩ l1 ) ⊕ a1 u hort 1 u θ (n1 ) , (1) −1 (1) g− = (l02 ∩ l1 ⊕ a2 ) u hort 2 u θ (n1 ) , and induce Lie bialgebra structure on g(1) using the subalgebra l1 ⊕ ((l02 ⊕ a2 ) ∩ l1 ) of D(g+ ). This structure comes from an r -matrix as in (3.26) with sums over the roots of l01 instead of l02 . The two choices for Lie bialgebras g(1) are isomorphic if hort ± are trivial, but seem to be slightly different in general. 4. Symplectic leaves in G − Section 4.1 is devoted to the description of the set of i + (G + ) double cosets of D(G + ), while Section 4.2 studies the discrete subgroup i − (G − ) ∩ i + (G + ) of D(G + ). In Section 4.3 these results are combined with Theorem 1.10 to describe the set of symplectic leaves of G − .
COMPLEX REDUCTIVE POISSON-LIE GROUPS
475
4.1. Double cosets of i + (G + ) in D(G + ) Denote the double coset of the element ˜ ∈ G × (L 01 A1 )/31 = D(G + ) (g, l) ˜ by [(g, l)]:
˜ = i + (G + )(g, l)i ˜ + (G + ). (g, l)
Since L 01 A1 ⊂ G + , for all g ∈ G, l˜ ∈ L 01 A1 , ˜ ˜ + (l) = (gl −1 , l) ˜ , g, p+ (l)l˜ = (l −1 g, l) and g, lp
∀l ∈ L 01 A1 . (4.1) The following proposition illustrates the problem of classifying double cosets.
PROPOSITION 4.1 The i + (G + ) double cosets of D(G + ) are in one-to-one correspondence with the conjugation orbits of G + in G/(31 M+ ).
Recall that M+ is a normal subgroup of G + and 31 ∈ Z (L 1 ), so the conjugation action of G + on G indeed can be pushed down to the homogeneous space G/31 M+ . Note also that this action can equivalently be described as the action of (L 01 A1 ) M+ on G/(31 M+ ), where the first factor acts by conjugation and the second one by left multiplication. Proof By a straightforward computation for any g ∈ G, we have [(g, e)] = (m 0l 0 gl 00 m 00 , p+ (l 0 )p+ (l 00 )) l 0 , l 00 ∈ L 01 A1 , m 0 , m 00 ∈ M+ = (m 0l 0 g(l 0 )−1 m 00 k, p+ (k)) l 0 , k ∈ L 01 A1 , m 0 , m 00 ∈ M+ .
(4.2)
The second expression is obtained from the first by letting l 0l 00 = k ∈ L 01 A1 and expressing l 00 in terms of l 0 and k. We also use the fact that M+ is a normal subgroup of G + . To the G + conjugation orbit of g(31 M+ ) we associate the double coset [(g, e)]. Equation (4.2) shows that this map is defined correctly. It is surjective since each double coset of D(G + ) is equal to a coset [(g, e)] for some g ∈ G because of (4.1). To show that it is injective, assume that [(g1 , e)] = [(g2 , e)]. If m 01l10 g1 (l10 )−1 m 001 k1 , p+ (k1 ) = m 02l20 g2 (l20 )−1 m 002 k2 , p+ (k2 ) is an element of this double coset (the notation is as in (4.2)), then k1 (k2 )−1 ∈ 31 . The first components give immediately that the G + orbits of g1 31 M+ and g2 31 M+ coincide.
476
MILEN YAKIMOV
In the special case of an r -matrix r for which g+ ⊃ h (see Remark 3.8), Proposition 4.1 concerns the conjugation action of the parabolic subgroup P+ of G on G/(31 N+ ), where N+ is its unipotent radical. 4.1.1. Relation to double cosets in G and the Bruhat lemma Let 5 : D(G + ) = G × (L 01 A1 )/31 → G
(4.3)
be the projection map on the first component of D(G + ). It is clear that ˜ = G + gG + , ∀g ∈ G, l˜ ∈ L 01 A1 , 5 [(g, l)]
(4.4)
which connects i + (G + ) double cosets in D(G + ) with G + double cosets in G. The group G + differs from the parabolic subgroup P+ of G just on a part of the maximal torus H (see (3.15)). This makes the description of the G + double cosets of G an easy consequence from the one for P+ double cosets. The latter is given by the Bruhat lemma. For any element w ∈ W , we denote by w˙ a representative of it in the normalizer N (H ) of H in G. The conjugation and the adjoint actions of w˙ on G and g are denoted by Adw˙ . LEMMA 4.2 (Bruhat) Let P1 and P2 be two parabolic subgroups of G, each of which contains one of the Borel subgroups B+ or B− . If W1,2 denote the Weyl groups of their Levi factors L 1,2 (naturally embedded in W ), then G G= P1 w˙ P2 , (4.5)
with w running over a set of representatives in W of all cosets from W1 \W/W2 . After commuting w˙ with the part Z (L 2 )◦ of the torus H ⊂ P2 , one gets G G= P1 w(L ˙ 02 n N2 ), where N2 is the unipotent radical of P2 and the disjoint union is over the same subset of W as in (4.5). The standard length function on W is denoted by l(.). In the following lemma we collect some facts for minimal length representatives of double cosets of the Weyl group W which are needed later. LEMMA 4.3 We use the notation of Lemma 4.2.
COMPLEX REDUCTIVE POISSON-LIE GROUPS
477
(1) Each double coset from W1 \W/W2 has a unique representative w ∈ W of minimal length. It can be equivalently described as the representative w ∈ W of the coset for which the following statement is true. For all positive roots α of Lie(L 1 ) and β of Lie(L 2 ), w−1 (α) and w(β) are positive roots of g.
(∗)
For any such w ∈ W, lw = Lie(L 1 ) ∩ w˙ −1 (Lie(L 2 )) is a standard Levi subalgebra of g (see the introduction). Its Weyl group is denoted by W w . (2) Any element u ∈ W can be uniquely represented as u = w1 ww2 ,
(4.6)
with w, w1 being minimal length representatives in W, W1 of cosets from W1 \W/W2 , W1 /W w , respectively, and w2 ∈ W2 . Its length is given by l(u) = l(w1 ) + l(w) + l(w2 ).
(4.7)
(3) Let P3 be a parabolic subgroup of G such that B± ⊂ P3 ⊂ P1 , and let W3 be the Weyl group of its Levi factor. Then the set of minimal length representatives in W for the cosets from W3 \W/W2 consists of the elements u ∈ W whose representation from part (2) has w2 = id and w1 —the minimal length representative in W of a coset from W3 \W1 /W w . Proof In the case of P1 = B± (i.e., W1 trivial) the lemma is proved in [16, Proposition 1.10]. We use this fact and refer to it as the single coset version of the lemma. Parts (1) and (2). Fix a coset from W1 \W/W2 . Choose a representative w of the coset in W having the property (∗). Any minimal length representative w0 of the double coset has this property since otherwise l(s1 w0 ) < l(w0 ) or l(w0 s2 ) < l(w0 ) for some simple reflections si ∈ Wi , i = 1, 2. It is clear that any element u ∈ W1 wW2 can be represented in the form (4.6) with wi ∈ Wi as in part (2). We show the uniqueness of this representation and prove (4.7). The latter implies the statement of part (1). Since w1 ∈ W1 is the minimal length representative of a coset from W1 /W w , it has the property that w1 (α) is a positive root of Lie(L 1 ) for any positive root α of lw . This combined with the definition of w implies that w1 w is a minimal length representative in W of the coset w1 wW2 . (Here we also use the fact that w1 maps positive roots of g that are not roots of Lie(L 1 ) into positive roots of g.) The single coset version of the lemma applied to the coset w1 wW2 implies that w2 is determined uniquely from u, and therefore so is w1 .
478
MILEN YAKIMOV
The single coset version of (4.7) for the cosets w1 wW2 ∈ W/W2 and w1 W w ∈ W1 /W w with minimal representatives w1 w and w1 gives l(w1 ww2 ) = l(w1 w) + l(w2 ), l(w1 w) = l(w1 ) + l(w), which together imply (4.7). Part (3). Part (1) implies that the elements of W listed in part (3) are minimal length representatives for cosets from W1 \W/W2 . In the opposite direction, any minimal representative u = w1 ww2 ∈ W written as in part (2) should have w2 = id, and w1 (α) should be a positive root of Lie(L 1 ) for any positive root α of lw . This implies the stated property of w2 . For the given r -matrix, denote by W1,2 the Weyl groups of the Levi factors L 1,2 of P± . The Bruhat lemma and (4.4) imply that any i + (G + ) double coset in D(G + ) is ˜ for some h ∈ H, l˜ ∈ (L 0 A1 )/31 , and w—the minimal equal to a coset [(h w, ˙ l)] 1 length representative in W of a double coset from W1 \W/W1 . Taking into account (4.1), we obtain the following lemma. LEMMA 4.4 Each G + double coset of D(G) coincides with one of the cosets [(l w, ˙ e)] for w—the minimal representative in W of a coset from W1 \W/W1 and l ∈ L 1 .
4.1.2. Main result Let V ⊂ W denote the set of minimal length representatives of the cosets from W/W1 . According to Lemma 4.3, each such coset has a unique representative in V. For an element v ∈ V , denote by gv the largest Lie subalgebra of l1 that is stable under Adv˙ . It does not depend on the choice of representative of v in N (H ) and can also be defined by gv = span h, gα for the roots α of l1 , such that v n (α) is a root of l1 , ∀ integer n . (4.8) The minimality condition on v guarantees that gv is a standard Levi subalgebra of g (i.e., a Levi subalgebra containing h of a parabolic subalgebra of g containing b+ ; see the introduction). This follows from the fact that for any positive root α of l1 , v(α) is a positive root of g (cf. Lemma 4.3). Denote by G v the reductive subgroup of G with Lie algebra gv and by W v its Weyl group. It is a standard Levi subgroup of G and is clearly stable under Adv˙ . The definition of gv combined with the minimality property of v implies that for any positive root α of gv , v(α) is again a positive root of gv , and thus so is v −1 (α).
COMPLEX REDUCTIVE POISSON-LIE GROUPS
479
From Lemma 4.3(1), it follows that any v ∈ V is the minimal length representative of the double coset W v vW1 . The fact that Adv˙ preserves G v implies that it also preserves its derived group (G v )0 and the identity component of its center Z (G v )◦ . Thus we can define an action of (G v )0 on itself by T C gv˙ ( f ) = g −1 f Adv˙ (g),
g, f ∈ (G v )0 ,
(4.9)
which is called twisted conjugation action. The fact that 31 is a subgroup of Z (L 01 A1 ) ⊂ H implies that 31 ⊂ G v . Let v 31 be the subgroup of (G v )0 × Z (G v )◦ which is the inverse image of 31 under the multiplication map (G v )0 × Z (G v )◦ → G v = (G v )0 Z (G v )◦ . Note that 3v1 ⊂ Z ((G v )0 × Z (G v )◦ ) and that each element of 31 has finitely many preimages in 3v1 (more precisely, #(G v )0 ∩ Z (G v )◦ ). Denote by L v the connected subgroup of G v with Lie algebra gv ∩ (l01 ⊕ a1 ). Let Congv (L v ) be the subgroup of the abelian group Z (G v )◦ defined by Congv (L v ) = h −1 Adv˙ (h) h ∈ Z (L v )◦ . The groups H1ort and Adv˙ (H1ort ) are also subgroups of Z (G v )◦ since by the definition of H1ort , H1ort ⊂ Z (L 1 )◦ . 4.5 The i + (G + ) double cosets of D(G + ) are classified by a minimal length representative v in W of a coset from W/W1 , an orbit of the twisted conjugation action (4.9) of (G v )0 on itself, and an element of the abelian group Z (G v )◦ / Congv (L v )H1ort Adv˙ (H1ort ) . (4.10) THEOREM
For a fixed v, this data has to be taken modulo the multiplication action of 3v1 ⊂ Z ((G v )0 × Z (G v )◦ ). The dimension of the coset corresponding to the orbit through f ∈ (G v )0 is 0 v dim g+ − dim hort 1 + dim(l1 ⊕ a1 ) − dim l + l(v) v˙ v v ort ort + dim T C(G v )0 ( f ) + dim Cong (L )H1 Adv˙ (H1 ).
(4.11)
(The described data corresponding to different choices of a representative in N (H ) of a fixed element v ∈ W can be canonically identified.)
480
MILEN YAKIMOV
Proof In Section 4.1.1 we showed that any i + (G + ) double coset of D(G + ) is equal to a coset [(l w, ˙ e)] for some l ∈ L 1 and w—the minimal length representative in W of a coset from W1 \W/W1 . Based on the elements l and w, ˙ we define a sequence of reductive Lie subalgebras gk of g, elements g k ∈ gk , and wk ∈ W (k ≥ 1). It starts with g1 = l1 and g 1 = l, w1 = w. The sequence has the property that gk and gk = Ad(w˙ k−1 ···w˙ 1 ) (gk )
(4.12) k
are standard Levi subalgebras of gk−1 ⊂ g. If W k and W are their Weyl groups considered as subgroups of W k−1 ⊂ W, wk has the property that it is the representative k of minimal length of a coset from W k \W k−1 /W . If Ad(w˙ k ...w˙ 1 ) does not preserve gk , then define gk+1 by gk+1 = gk ∩ Ad−1 (gk ). (w˙ k ···w˙ 1 )
(4.13)
The minimality property of w j implies that wk · · · w1 is the minimal length representative of a coset from W k \W/W 1 (see Lemma 4.3(3)), and therefore for any positive root α of gk , (w k · · · w1 )(α) and (wk · · · w1 )−1 (α) are positive roots of g. From this we get that both gk+1 and gk+1 = Ad(w˙ k ···w˙ 1 ) (gk+1 ) are standard Levi subalgebras of g. Let nk+1 and nk+1 be the two nilpotent subalgebras of gk spanned by the root + + spaces of positive roots of gk that are not roots of gk+1 and gk+1 , respectively. k+1 The connected subgroups of G with Lie algebras gk+1 , gk+1 , nk+1 + , and n+ are denoted by G k+1 , G
k+1
k+1
, N+k+1 , and N + . They give rise to two parabolic subgroups
k+1
k+1
G k+1 n N+k+1 and G n N + of G k . To define recursively g k+1 and wk+1 from g k and wk , we apply the Bruhat lemma for the above parabolic subgroups of G k . We get that g k ∈ G k can be represented as g k = n 0 g 0 w˙ k+1 g 00 n 00
(4.14)
for some wk+1 ∈ W k+1 that is the representative of minimal length of a coset from k+1 k+1 k+1 W k+1 \W k /W and some g 0 ∈ G k+1 , g 00 ∈ G , n 0 ∈ N+k+1 , n 00 ∈ N + . Equation (4.14) is the defining relation for wk+1 , and g k+1 is defined by 00 −1 0 g k+1 = Ad−1 (g ) g. (4.15) k 1 (w˙ ···w˙ ) Clearly, the described process continues for finitely many steps k = 1, . . . , K , and at the end Ad(w˙ K ···w˙ 1 ) (g K ) = g K = g K . (4.16)
COMPLEX REDUCTIVE POISSON-LIE GROUPS
481
Again, due to Lemma 4.3(3), the minimality properties of w1 , . . . , w K add up to v := (w K · · · w1 ) ∈ V
(4.17)
(see the beginning of this subsection for the definition of the set V ). Furthermore, we use the representative v˙ = w˙ K · · · w˙ 1 of v in N (H ). Observe that g K = gv . Indeed, in view of (4.16) it is sufficient to prove that gv ⊂ g K . One proves by induction that gv ⊂ gk for k = 1, . . . , K . The step from k to k + 1 follows from definition (4.13) of gk+1 and the fact that (w K · · · wk+2 ) ∈ W k+1 . Thus we have G K = G v = (G v )0 Z (G v )◦ , and we let gK = f K cK
(4.18)
for some f K ∈ (G v )0 and c K ∈ Z (G v )◦ , defined uniquely modulo multiplication of f K and division of c K by an element of the finite group (G v )0 ∩ Z (G v )◦ . The statement from Theorem 4.5 about the set of i + (G + ) double cosets follows from the next two lemmas. LEMMA 4.6 In the above notation, [(l w, ˙ e)] = [(g K v, ˙ e)] and two double cosets such as the one in the right-hand side coincide if and only if they are related to one and the same v ∈ V ; v˙ K ) that correspond to them and the images the twisted conjugation orbits T C(G v )0 ( f of their elements c K in the factor group (4.10) are equal modulo the multiplication action of 3v1 on (G v )0 × Z (G v )◦ .
For Theorem 4.5 we associate to the double coset [(l w, ˙ e)] = [(g K v, ˙ e)] the T C v˙ K K orbit of f and the image of c in the abelian factor group (4.10) (recall (4.18)). LEMMA 4.7 Any element of V can be obtained in a unique way as a product (w K · · · w1 ) through the described iterative procedure.
We need some more notation. Let nk− denote the nilpotent subalgebras of gk−1 spanned by root spaces of negative roots of gk−1 that are not roots of gk . For convenience we set g0 = g and also define nk± and nk+ for k = 1. Let lk = gk ∩ (l1 ⊕ a1 ), and let L k be the corresponding connected subgroup of G. Note that L K = L v .
482
MILEN YAKIMOV
Proof of Lemma 4.6 Our strategy in dealing with the double coset [(l w, ˙ e)] = [(g 1 w˙ 1 , e)] is to commute 1 the right factor i(G + ) with w˙ = w˙ , splitting it according to how the result is related to the left factor i + (G + ). Then we continue inductively, following the described procedure. Note that l1 = l01 ⊕ a1 and m+ admit the following decompositions as direct sums of linear spaces, each term of which is a Lie subalgebra: −1 1 −1 1 1 1 1 l1 = l1 ∩ Ad−1 (n ) u l ∩ Ad (g ) u l ∩ Ad (n ) , 1 1 + w˙ w˙ w˙ 1 − m+ = n1+ ∩ Ad−1 (n1 ) u m+ ∩ Ad−1 (g1 ) u n1+ ∩ Ad−1 (n1 ) . w˙ 1 + w˙ 1 w˙ 1 −
(4.19) (4.20)
These formulas imply that the product 1 L 1 ∩ Ad−1 (N+1 ) N + ∩ Ad−1 (N+1 ) L 1 ∩ Ad−1 (G 1 ) M+ ∩ Ad−1 (G 1 ) w˙ 1 w˙ 1 w˙ 1 w˙ 1 1 × L 1 ∩ Ad−1 (N−1 ) N + ∩ Ad−1 (N−1 ) (4.21) w˙ 1 w˙ 1 is a dense, open subset of G + because the linear span of the Lie algebras of all factors 1 is g+ . Although n1+ = n1+ (= n+ ) and N + = N+1 (= N+ ), we use both notations since k
this is how the groups N + 6= N+k appear at the next step k ≥ 2. From this it follows that the double coset [(l w, ˙ e)] = [(g 1 w˙ 1 , e)] of D(G + ) has a dense, open subset consisting of elements of the form 00 00 00 00 00 00 00 00 m 0l 0 g 1 w˙ 1l+ n +l∗ m ∗ l− n − , p+ (l 0 )p+ (l+ )p+ (l∗00 )p+ (l− ) (4.22) 00 , n 00 , l 00 , m 00 , l 00 , and n 00 of the groups of the product for some arbitrary elements l+ + ∗ ∗ − − 00 and n 00 with (4.21) (taken in the same order). Note that after commuting the terms l+ + w˙ 1 in the first component, they are absorbed by the term m 0 ∈ M+ . The group L 1 by itself has the following dense, open subset: L 1 ∩ Ad−1 (N−1 ) L 1 ∩ Ad−1 (G 1 ) L 1 ∩ Ad−1 (N+1 ) . (4.23) w˙ 1 w˙ 1 w˙ 1
If the term l 0 from (4.22) is taken in it, after some manipulations one shows that the coset [(g 1 w˙ 1 , e)] has the following dense, open subset: (N+1 , e) (id, p+ )(N−2 ) e, p+ (N+2 ) 2 × (H1ort N+2 , e) (id, p+ )(L 2 ) (g 1 , e) (Adw˙ 1 , p+ )(L 2 ) N + Adw˙ 1 (H1ort ), e 1 (4.24) × (id, p+ Ad−1 ) Adw˙ 1 (L 1 ) ∩ N−1 Adw˙ 1 (N + ) ∩ N−1 , e (w˙ 1 , e). w˙ 1 Here, for two homomorphisms σi : S → R, i = 1, 2 by (σ1 , σ2 )(S), we denote the subgroup of R × R which is the image of the diagonal embedding of S in S × S, composed with the homomorphism σ1 × σ2 .
COMPLEX REDUCTIVE POISSON-LIE GROUPS
483
An important point in the derivation of (4.24) is that H1ort commutes with L 1 = (see (3.5)) and thus with N−2 ⊂ L 1 . Later this is used for all groups N−k ⊂ L 1 , k = 2, . . . , K . Note that 31 ⊂ Z (L 01 A1 ) ⊂ H implies 31 ⊂ L k (∀k). The product on the L 01 A1
2
second line of (4.24) is the left (H1ort N+2 ) L 2 and right Adw˙ 1 (L 2 ) (N + Adw˙ 1 (H1ort )) double coset of (g 1 , e) in G 1 × (L 2 /31 ), where the two groups are embedded in G 1 × (L 2 /31 ) as it is shown in the formula. Using the representation (4.14) of g 1 and (4.1), one easily shows that [(g 1 w˙ 1 , e)] = [(g 2 w˙ 2 , e)] and proceeds by induction on k. This procedure proves that [(l w, ˙ e)] = [(g K v, ˙ e)] (recall (4.17)). To prove the rest of Lemma 4.6, let us rearrange the coset from the last step similarly to (4.2): K (H1ort N+K , e) (id, p+ )(L K ) (g K , e) (Adv˙ , p+ )(L K ) N + Adv˙ (H1ort ), e v˙ g = (N+K , e) (id, p+ )(L K ) T C L v ×H ort ×H ort (g K ), e , (4.25) 1
K
1
where we used the fact that N + = N+K normalizes G L v × H1ort × H1ort on G v is defined by
K
v˙ g = G K . The action T C of
v˙ g T C (l,h 0 ,h 00 ) (g) = h 0l −1 g Adv˙ (lh 00 ) for g ∈ G v , l ∈ L v , h 0 , h 00 ∈ H1ort .
(4.26)
In (4.25) the term i + (H1ort L K ) is replaced by i + (H1ort )i + (L K ) and (Adv˙ ×id) ·(i − (L K H1ort )) by (Adv˙ ×id)(i − (L K )i − (H1ort )). (The intersection H1ort ∩ L K = H1ort ∩ L 01 A1 is nontrivial in general.) ort are trivial, the action (4.26) gives a simpler classification In the case when H1,2 of the i + (G + ) double cosets (see Corollary 4.8). Using the decomposition (4.18) of g K , the middle set in (4.25) can be rewritten as v˙ v˙ K g T C L v ×H ort ×H ort (g K ) = T C(G ) c K Congv (L v )H1ort Adv˙ (H1ort ) . (4.27) v )0 ( f 1
1
The inductive argument shows that the coset [(l w, ˙ e)] has the following dense, open subset: K Y
(N+1 , e)
Ek
(id, p+ )(L K )
k=2
v˙ K K v v ort ort × T C(G v )0 (g ) c Cong (L )H1 Adv˙ (H1 ) , e ×
1 Y k=K −1
Fk (w˙ K . . . w˙ 1 , e),
(4.28)
484
MILEN YAKIMOV
where the subsets E k and F k of G are defined by E k = (id, p+ )(N−k ) N+k × p+ (N+k ) and k Fk = (Ad(w˙ K ···w˙ k ) , Ad(w˙ K ···w˙ k+1 ) p+ )(L k ∩ Ad−1 (N )) k − w˙ k × Ad(w K ···wk+1 ) (Adw˙ k (N + ) ∩ N−k ), e . The order of the factors in (4.28) is k = 2, . . . , K in the first product and k = K − 1, . . . , 1 in the second one. Note that Adw˙ 1 (n1+ ) ∩ n1− and Adw˙ 1 (l1 ) ∩ n1− are subalgebras of n1− and thus do not intersect n1+ and g1 . Using this, the fact that p± are one to one on any unipotent subgroup of L 01,2 A1,2 , and the Bruhat lemma, we see that if two subsets of G of the type (4.24) are equal, then they are related to one and the same w1 ∈ W and the corresponding double cosets of G 1 × (L 2 /31 ) (on line two) coincide. To finish the proof of Lemma 4.6, assume that two double cosets of the type [(g K v, ˙ e)] obtained from the described procedure coincide. Then their dense, open subsets from (4.24) have nontrivial intersection. Using the above argument repeatedly, we get that the two cosets are related to the same elements w K , . . . , w1 ∈ W (and thus the same v˙ g v ∈ V ) and that the corresponding orbits T C L v ×H ort ×H ort (g K ) in G v coincide. That 1
1
v˙ K ) corresponding to the two double cosets and the images of is, the orbits T C(G v )0 ( f c K in the factor group (4.10) are equal modulo the multiplication action of 3v1 on (G v1 )0 × Z (G v )◦ .
Proof of Lemma 4.7 Let v 0 be an arbitrary element of V. Decompose it as v 0 = v 1 w1 u 1 with v 1 , u 1 ∈ W1 and w1 —the minimal length representative in W of the double cosets W1 vW1 . Since v 0 is the minimal length representative of a coset from W/W1 , using Lemma 4.3(3) we get that u 1 is the identity element of W1 . Define two reductive Lie subalgebras g2 2 and g2 of g1 by (4.13) and (4.12) with k = 1, and let W 2 , W be the corresponding subgroups of W. The minimality property of v 0 implies that for any positive root α 1 of g1 , v(α 1 ) is a positive root of g. This means that for any positive root α 2 of g2 , v 1 (α 2 ) is a positive root of g1 . Therefore, v 1 is the minimal length representative 2 in W 1 of a coset in W 1 /W . Continuing by induction, we define wk , v k , gk , gk , k W k , W . At some step k = K + 1, we have g K +1 = g K +1 , which is the same as g K = Ad(w˙ K ...w˙ 1 ) (g K ). As in the proof of Lemma 4.6, set v = w K · · · w1 and then v 0 = v K v.
(4.29)
COMPLEX REDUCTIVE POISSON-LIE GROUPS
485
We have already proved that gv = g K . From (4.29) it follows that gv = g K , so both v and v 0 have the property that they are minimal length representatives of one and the same coset from W K \W/W 1 . The uniqueness of the minimal length representative implies v 0 = v (see Lemma 4.3(1)). Comparing this procedure with the one from the beginning of the proof of Theorem 4.5 completes the proof of the existence part of Lemma 4.6. The uniqueness part is obtained by repeated application of Lemma 4.3(1). 0
Proof of (4.11) The double coset [(g K v, ˙ e)] can be considered as the orbit of (g K v, ˙ e) under the natural multiplication action of the (2 dim g+ )-dimensional group G + × G + on D(G + ). At the kth step a group isomorphic to k
N + ∩ Ad−1 (N+k ) w˙ k acts trivially (k = 1, . . . , K − 1) (see the second group in (4.21)). Its dimension is k dim nk+ − dim nk+1 + − l(w ).
The proofs of Lemma 4.6 and Lemma 4.3(2) imply that these groups do not intersect, and, moreover, the dimension of their product is dim n+ −
K X
l(wk ) = dim n+ − l(v).
k=1
At the end one needs to subtract also the dimension of the stabilizer of g K of the v˙ g action T C of L v × (H ort )×2 . Passing to dimensions of orbits and using formula (4.27), we see that K v ort v˙ K dim StabTg ) v˙ (g ) = dim l + 2 dim h1 − dim T C (G v )0 ( f C v v ort ort − dim Cong (L ) H1 Adv˙ (H1 ) .
Formula (4.11) is obtained by combining the above results with the fact that two elements of the set (4.28) are equal if and only if they have equal terms in each factor of the product. Identification of the data of Theorem 4.5 corresponding to different choices of a representative of v ∈ V in N (H ). If v˙ and v¨ are two different representatives of v ∈ V in N (H ), then v¨ = h v˙ for some h ∈ H
486
MILEN YAKIMOV
since the centralizer of a maximal torus in a complex reductive group coincides with the torus. Let h = h v h ◦ for some h v ∈ (G v )0 and h ◦ ∈ Z (G v )◦ . A direct computation shows that the right multiplication by h v in (G v )0 maps T C v¨ orbits to T C v˙ orbits: T C v¨ (gh v ) = T C v˙ (g) h v
for all g ∈ G v .
Since 3v1 is in the center of (G v )0 × Z (G v )◦ , the right multiplication of h v on the first component of (G v )0 × Z (G v )◦ commutes with the regular action of 3v1 . This identifies the data in Theorem 4.5 corresponding to two different choices of representatives v˙ and v¨ of v in N (H ). 4.8 If the Poisson structure on G comes from an r -matrix r for which g+ ⊃ h, the i + (G + ) double cosets of D(G + ) are classified by a minimal length representative v of a coset in W/W1 and an orbit of the twisted conjugation action of G v on G v /31 defined by COROLLARY
v˙ g T C g ( f ) = g −1 f Adv˙ (g),
g ∈ G v , f ∈ G v /31 .
(4.30)
v˙
g The dimension of the coset corresponding to the T C orbit through f ∈ G v /31 is v˙
g dim g+ + dim l1 − dim lv + l(v) + dim T C G v ( f ).
(4.31)
The action (4.30) is the pushdown of the twisted conjugation action (4.26) of G v on itself to the factor group G v /31 . It is correctly defined since 31 ⊂ Z (G v ). Proof ort are trivial and L v = G v (see Remark 3.8). In the considered case, the groups H1,2 The statement follows directly from the proof of Theorem 4.5 (see (4.25)). Example 4.9 The simplest coset in W/W1 is W1 . Its minimal length representative in W is the identity element id of W. In this example we discuss the i + (G + ) double cosets corresponding to it. We assume that id is represented in N (H ) by the identity element e of G. The group L id is simply L 01 A1 , and for any coset of this type, the inductive procedure of the proof of Theorem 4.5 has only one step, that is, K = 1. In the product (4.21) all groups except the second, the third, and the fourth are trivial. Besides this, the full group G + is represented as G + = N+ (L 01 A1 )H1ort , not just a dense subset of it. (This is nothing but (3.15).) The expression (4.28) reduces to ( f c, e) = (N+ , e) (id, p+ )(L 01 A1 ) T C Le 0 ( f )(cH1ort ), e (4.32) 1
COMPLEX REDUCTIVE POISSON-LIE GROUPS
487
for all f ∈ L 01 and c ∈ Z (L 1 )◦ . The action T C e is the usual conjugation action of L 01 on itself: T Cle ( f ) = l −1 f l, l, f ∈ L 01 . Note also that Congid (L id )H1ort Ade (H1ort ) = H1ort . 0 0 ◦ The subgroup 3id 1 of L 1 × Z (L 1 ) is the pullback of 31 under the map L 1 × 0 ◦ ◦ Z (L 1 ) → L 1 Z (L 1 ) = L 1 . Thus the cosets corresponding to id ∈ V are classified by the conjugation orbits on L 01 and the elements of the abelian group Z (L 1 )◦ /H ort taken modulo the multipli0 ◦ cation action of 3id 1 on L 1 × Z (L 1 ) . The dimension of the coset (4.32) is dim(g+ ) + dim T C Le 0 ( f ) , 1
which follows from (4.11) after some cancellations. Example 4.10 Consider the case of the standard Poisson structure on a semisimple Lie group G (see Examples 3.5 and 3.9). The groups W1,2 are trivial, so W/W1 = W. Any i + (B+ ) double coset of D(B+ ) = G × H is equal to a coset of the type [(h w, ˙ e)], with w ∈ W, h ∈ H. For all w ∈ W , the procedure from the proof of Theorem 4.5 again has only one step and L w = H. All groups in the product (4.21) except the second, the third, and the sixth are trivial, and it reduces to −1 B+ = N+ ∩ Ad−1 w˙ (N+ ) H N+ ∩ Adw˙ (N− ) . Since the full group G + = B+ is represented in this way, using (4.28), we can get an explicit expression for any double coset of D(B+ ):
(h w, ˙ e) = (N+ , e)d(H ) h Congw (H ), e Adw˙ (N+ ) ∩ N− , e (w, ˙ e), ∀w ∈ W, h ∈ H, (4.33)
where
Congw (H ) = (h 0 )−1 Adw˙ (h 0 ) h 0 ∈ H
and d(H ) is the image of the diagonal embedding of H in G×H. The set h Congw (H ) w˙ g is the T C H orbit through h from Corollary 4.8. The general twisted conjugation orbits in Theorem 4.5 and Corollary 4.8 can be considered as nonabelian analogs of this action. The different cosets (4.33) are classified by the elements of the Weyl group W of g and the elements of the abelian group H/ Congw (H ) (or, equivalently, the orbits of
488
MILEN YAKIMOV w˙
g the T C action of H on itself). According to Theorem 1.10, this data classifies the symplectic leaves of i − (B− )/6− . Both (4.33) and (4.31) give dim (h w, ˙ e) = dim b+ + l(w) + dim Congw (H ) . The above description of i(B+ ) double cosets of D(B+ ) (and symplectic leaves of i − (B− )/6− ) for the standard Poisson-Lie structure on G agrees with the results in [4] and [14]. Example 4.11 The Cremmer-Gervais Poisson structure on SL(n + 1) is reviewed in Examples 3.6 and 3.10. In this case, W ∼ = S n+1 and Wi can be identified with the subgroups Sin+1 of S n+1 which permute the first and last n numbers. Denote by H the maximal torus of SL(n + 1) consisting of diagonal matrices. We assume that any element of S n+1 is represented in the normalizer of H by a multiple of the corresponding standard permutation matrix. Any minimal length representative σ in S n+1 of a coset from S n+1 /S1n should have the property that σ (1), . . . , σ (n) is an increasing sequence (see Lemma 4.3(1)). From this, one finds that the set of such representatives consists of j σ1 ∈ S n+1 , j = 0, . . . , n, defined by j
σ1 ( p) = p,
j
p = 1, . . . , j; j σ1 (n
σ1 ( p) = p + 1,
p = j + 1, . . . , n;
+ 1) = j + 1.
In a compact form, they are the permutations j σ1 = ( j + 1)( j + 2) · · · (n + 1) .
(4.34)
j
The groups G σ˙ 1 are the following standard Levi subgroups of SL(n + 1): j G σ˙ 1 = diag(A, a j+1 , . . . , an+1 ) A ∈ GL( j), a p ∈ C; det(A)a j+1 · · · an+1 = 1 . j
The twisted conjugation action of G σ1 on itself is given by j
σ˙ 1 g T C diag(A,a diag(B, b j+1 , . . . , bn+1 ) j+1 ,...,an+1 ) −1 = diag(A−1 B A, a −1 j+1 b j+1 aµ( j+1) , . . . , an+1 bn+1 aµ(n+1) ), (4.35) j
where µ = (σ1 )−1 . To each such orbit one can associate the conjugation orbit of GL( j) through B. The correspondence is bijective, and the difference in their dimenj sions is n − j. Note also that l(σ1 ) = n − j.
COMPLEX REDUCTIVE POISSON-LIE GROUPS
489
Thus in the case of the Cremmer-Gervais structure, the i + (G + ) double cosets of D(G + ), or, in other words, the symplectic leaves of i − (G − )/6− (recall Theorem 1.10), are classified by a choice of a number j = 0, . . . , n and a conjugation orbit on GL( j). The dimension of the corresponding double coset is n(n + 1) + (n − j)(n + j + 1) + dim Ad A (B) A ∈ GL( j) . 4.2. Structure of the group 6− = i + (G + ) ∩ i − (G − ) ⊂ D(G + ) Denote 6− = i + (G + ) ∩ i − (G − ) ⊂ D(G + ). It is a countable subgroup of D(G + ) since Lie(i + (G + )) ∩ Lie(i − (G − )) = 0. THEOREM 4.12 The group 6− is the intersection of the i ± images of the tori (L i0 ∩ H )Hiort Ai ⊂ G ± in the maximal torus H × ((L 01 ∩ H )A1 /31 ) of the reductive group G × (L 01 A1 /31 ) = D(G + ).
Proof Using the recursive procedure from Section 3.3, we show that the statement holds for G (k) , assuming its validity for G (k+1) . Note that it is trivial for G (ord τ ) since G (ord τ ) = H. Clearly, we can restrict ourselves to the step from G (1) to G = G (0) . Let li ∈ L i0 Ai , i = 1, 2, m ± ∈ M± be such that l1 m + , p+ (l1 ) = m −l2 , 2−1 ◦ p− (l2 ) ∈ i + (G + ) ∩ i − (G − ).
(4.36)
From the first components we get l1 m + = m −l2 ∈ (L 01 A1 )M+ ∩ (L 02 A2 )M− .
(4.37)
(1)
Let N1± denote the unipotent subgroups of L 01 generated by one-parameter subgroups (1)
of positive/negative roots of l01 that are not roots of l02 . Similarly, we define N2± as the unipotent subgroups of L 02 corresponding to roots of l02 which are not roots of l01 . Using the standard fact that the intersection of the two parabolic subgroups P+ and P− of G is connected, we get (1)
m + ∈ H1ort n N2+ , m− ∈
(1) N1−
o
H2ort ,
(4.38) (4.39)
490
MILEN YAKIMOV
and (1) l1 ∈ N1− o (L 01 ∩ L 2 ) A1 , (1) l2 ∈ (L 02 ∩ L 1 ) A2 n N2+ .
(4.40) (4.41)
(1)
Let m + = h 1 n 2+ and l1 = n 1−l1 be the decompositions of m + and l1 according to the semidirect products (4.38) and (4.40). Equation (4.37) implies that the decompositions of m − and l2 according to (4.39) and (4.41) should be m − = n 1− h 2 , (1) (1) l2 = k2 n 2+ for some h 2 ∈ H2ort and k2 ∈ (L 02 ∩ L 1 ) A2 such that (1)
(1)
l1 h 1 = h 2 k2 .
(4.42)
Observe that (1)
(1)
(1)
N2+ = N+ , (1) (1) θ Lie(N1− ) = Lie(N− ).
(L 01 ∩ L 2 )A1 = L 1 , (1) (1) θ Lie((L 01 ∩ L 2 )A1 ) = Lie(L 2 A2 ),
(4.43) (4.44)
(1)
Since N− is a unipotent subgroup of L 02 A2 and the kernel 32 of the projection p− : L 02 A2 → L 02 A2 /32 consists of semisimple elements, equations (4.44) imply (1) (1) (1) the existence of elements n − ∈ N− and l2 ∈ (L (1) )0 A(1) such that (1)
2 ◦ p+ (n 1− ) = p− (n − ), (1)
(4.45)
(1)
2 ◦ p+ (l1 ) = p− (l2 ).
(4.46) (1)
(1)
Comparing the second components of (4.36), we get 2 ◦ p+ (n 1−l1 ) = p− (k2 n 2+ ). Combining this equation with (4.45) and (4.46) gives (1) (1)
(1)
p− (n − l2 ) = p− (k2 n 2+ ). (1)
By modifying l2 ∈ (L (1) )0 A(1) with an element of 32 ⊂ (L (1) )0 A(1) , one can make it satisfy (1) (1) (1) n − l2 = k2 n 2+ (4.47) (1)
while keeping (4.46). Expressing k2 from (4.42) and substituting it in (4.47) gives (1) (1)
(1)
h 2 n − l2 = l1 h 1 n 2+ . (1)
(1)
(4.48) (1)
(1)
Because l1 ∈ (L 01 ∩ L 2 ) A2 = L 1 and n 2+ ∈ N2+ = N+ , equations (4.48) and (4.46) imply that (1) (1) l1 h 1 n 2+ , p+ (l1 ) (1) (1) (1) (1) (1) (1) (1) (1) = h 2 n − l2 , 2−1 ◦ p− (l2 ) ∈ i + (G + ) ∩ i − (G − ) ⊂ D(G + ).
COMPLEX REDUCTIVE POISSON-LIE GROUPS
491
Using the inductive assumption for validity of the statement of the lemma for (1) G (1) , we find that l1 ∈ H, n 2+ = e and therefore (l1 m + , l1 ) ∈ H × ((L 01 ∩ H ) × A1 ). As a consequence of Theorem 4.12, we also find that the 6− -groups associated to the Poisson-Lie groups G, G (1) , . . . , G (ord τ ) are naturally isomorphic. From Theorem 4.12 one easily derives an explicit formula for 6− in terms of the lattices of the kernels of the exponential maps for certain tori. We restrict ourselves to the case of an r -matrix for which hiort are trivial since in the general case the formula is too cumbersome. Denote by ker and ker0 the kernels of the exponential maps for the tori H and H/32 . 4.13 In the case when the r -matrix r is such that g+ ⊃ h, the group 6− is isomorphic to ker0 / ker0 ∩ (1 − θ) ker . (4.49) COROLLARY
If in addition the groups 3i are trivial, then 6− = (h, h) h ∈ H, 2(h) = h .
(4.50)
One easily deduces from the assumption g+ ⊃ h and the fact D(g+ ) = i + (g+ ) u i − (g− ) that 1 − θ is a linear automorphism of h. This implies that (1 − θ )−1 ker0 is a lattice in h. The quotient (4.49) of the two lattices is not a finite group in general. Proof According to Theorem 4.12, 6− is the subgroup of H × (H/31 ) consisting of the elements l1 , p+ (l1 ) = l2 , 2−1 ◦ p− (l2 ) (4.51) for some li ∈ H. Let l1 = exp(x) for some x ∈ h. The second component of (4.51) gives l2 = exp(θ x)λ2 for some λ2 ∈ 32 . From the first component we obtain exp (1 − θ)x = λ2 ∈ 32 and thus x ∈ (1 − θ )−1 ker0 . Taking the quotient (1 − θ)−1 ker0 / (1 − θ)−1 ker0 ∩ ker
(4.52)
cancels the elements x ∈ h for which exp(x) = e. The final answer is obtained after applying (1 − θ ) to (4.52). Equation (4.50) follows directly from (4.51).
492
MILEN YAKIMOV
Example 4.14 Consider the case of the standard Poisson structure on a complex semisimple Lie group G. Then 32 is trivial and ker0 = ker . Corollary 4.13, combined with the facts that the Cayley transform θ : h → h is −id|h and 32 is trivial, gives rank g 6− ∼ = ker /(2 ker) ∼ = Zrank g /(2Zrank g ) ∼ = Z2
(4.53)
6− = (h, h) h ∈ H, h 2 = e .
(4.54)
or, equivalently, This coincides with the results in [4] and [14]. Example 4.15 For the Cremmer-Gervais structure on SL(n + 1), the groups 3i are trivial. The group 6− is explicitly given by 6− = (h, h) h = diag(h 1 , . . . , h n+1 ); h i+1 = h i , i = 1, . . . , n = ( E, E) ∈ C, n+1 = 1 ∼ (4.55) = Zn+1 , where E denotes the identity matrix of size n + 1. This can also be proved directly from (4.49) using the facts that the kernel of the exponential map on h is ker = 2πiZα1 ⊕ · · · ⊕ 2πiZαn and the Cayley transform on Lie algebra level acts by θ(α j ) = α j+1 for j = 1, . . . , n − 1 and θ(αn ) = −α1 − · · · − αn . 4.3. Summary of the results on the symplectic leaves of G − In this subsection we summarize the results for the symplectic leaves of G − and i − (G − )/6− obtained by combining the results from Sections 4.1 and 4.2 and Theorems 1.9 and 1.10. 4.16 (1) The symplectic leaves of i − (G − )/6− are classified by the data from Theorem 4.5. An element v ∈ V (see Section 4.1.2), an orbit T C v˙ ( f ) in (G v )0 , and an element of the abelian factor group Z (G v )◦ /(Congv (L v )H1ort Adv˙ (H1ort )) are all taken modulo the multiplication action of 3v1 . The dimension of the leaf corresponding to this data is THEOREM
v˙ dim(l01 ⊕ a1 ) − dim lv − dim hort 1 + l(v) + dim T C (G v )0 ( f )
+ dim Congv (L v )H1ort Adv˙ (H1ort ). (4.56)
COMPLEX REDUCTIVE POISSON-LIE GROUPS
493
(2) The leaves of G − are coverings of the leaves of i − (G − )/6− under the projection map G − → i − (G − )/6− . The countable group 6− is characterized in Theorem 4.12. In a special case Corollary 4.8 gives a simpler description of the symplectic leaves of G −. COROLLARY 4.17 In the case when g+ ⊃ h, the symplectic leaves of i − (G − )/6− are in one-to-one v˙ g correspondence with the elements v ∈ V and the orbits of the action T C of G v on v˙ g G v /31 (see (4.30)). The dimension of the leaf corresponding to the orbit T C ( f ) is v˙ g dim l1 − dim gv + l(v) + dim T C G v ( f ).
The group 6− is explicitly described in Corollary 4.13. Note also that the results from Sections 4.1 and 4.2 give some information for the structure of the symplectic leaves of i − (G − )/6− . In particular, the leaf corresponding to the choice of the data specified there has a dense, open subset that is the intersection of the set (4.28) (for f K = f, c K = c) with i − (G − )i + (G + ) projected to i − (G − )i + (G + )/i + (G + ). 5. Symplectic leaves of G This section describes the set of symplectic leaves of the full group G. The proofs of all results are only sketched since they go along the lines of the proofs in Section 4. We restrict ourselves to the case when the derived subgroup G 0 of G is simply connected. This is not an essential restriction since any factorizable reductive Poisson-Lie group R has a covering that is a Poisson-Lie group G of the considered type. The leaves of R are simply projections of the ones of G. The above assumption implies in particular that L i0 are also simply connected and that the restriction of the map θ to l01 can be lifted to an isomorphism 20 : L 01 → L 02 .
(5.1)
It clearly satisfies 2 ◦ p+ (l) = p− ◦ 20 (l),
∀l ∈ L 01 .
5.1. G r double cosets of D(G) The double coset of an element (g1 , g2 ) ∈ G × G = D(G) is denoted by [(g1 , g2 )]: [(g1 , g2 )] := G r (g1 , g2 )G r ⊂ G × G.
494
MILEN YAKIMOV
Similarly to (4.1), one has (g1l, g2 ) = (g1 , g2 20 (l −1 )) ,
(lg1 , g2 ) = (g1 , 20 (l −1 )g2 ) ,
∀l ∈ L 01 . (5.2) As in Section 4.1.1, it can be shown that any G r double coset of G × G is equal to a coset of the form (l1 w˙ 1 , w˙ 2l2 ) , where li ∈ L i and wi are representatives in W of cosets from Wi \W/Wi , i = 1, 2. For simplicity of notation, it is more convenient to have the cosets written as (l1 w˙ 1 , w˙ 2−1l2 ) , where li and wi belong to the same sets as above. 5.1.1. Description of the set of double cosets The map τ from the Belavin-Drinfeld triple (01 , 02 , τ ) associated to the r -matrix r is assumed to be extended to a linear bijection between the set of roots of l1 and l2 . Define the set V as the subset of W ×W consisting of pairs of elements (v1 , v2 ) of W which are representatives of minimal length in W of cosets from W/Wi , i = 1, 2. For each such pair (v1 , v2 ), let gv11 ,v2 be the maximal subalgebra of l1 containing h whose set of roots is stable under v1 τ −1 v2 τ : gv11 ,v2 = span h, gα for the roots α of l1 such that (v1 τ −1 v2 τ )n (α) is well defined and is a root of l1 , ∀ integer n . (5.3) Note that v1 τ −1 v2 τ is not defined on all root spaces gα of l1 but only on those for which v2 τ (α) is a root of l2 . In this notation the maximal subalgebra of l2 containing h whose set of roots is preserved by v2 τ v1 τ −1 is gv21 ,v2 = Adv˙2 θ(gv11 ,v2 )0 + h.
(5.4)
The minimality property of vi implies that giv1 ,v2 are standard Levi subalgebras g (see the introduction). Denote by G iv1 ,v2 the connected subgroups of G with Lie algebras giv1 ,v2 and by Wiv1 ,v2 their Weyl groups, i = 1, 2. Similarly to Section 4.1.2, one shows that vi are minimal length representatives for the double cosets Wiv1 ,v2 vi Wi . Clearly, Adv˙1 (20 )−1 Adv˙2 20 is an automorphism of (G v11 ,v2 )0 . The connected subgroups of G iv1 ,v2 with Lie algebras giv1 ,v2 ∩(li0 ⊕ai ) and giv1 ,v2 ∩ v ,v Adv˙i (li0 ⊕ ai ) are denoted by L iv1 ,v2 and L i 1 2 . Note that v ,v Adv˙2 θ : Lie Z (L v11 ,v2 ) → Lie Z (L 21 2 )
COMPLEX REDUCTIVE POISSON-LIE GROUPS
495
and v ,v2
1 θ Ad−1 v˙1 : Lie Z (L 1
) → Lie Z (L v21 ,v2 )
are linear isomorphisms. For any (v1 , v2 ) ∈ V , we denote by Z iv1 ,v2 the following subgroups of the abelian group Z (G v11 ,v2 )◦ × Z (G v21 ,v2 )◦ : ◦ v ,v Z 1v1 ,v2 = (x, y) ∈ Z (L v11 ,v2 )◦ × Z (L 21 2 )◦ 2p+ (x) = p− Ad−1 v˙2 (y) , ◦ v ,v −1 Z 2v1 ,v2 = (x, y) ∈ Z (L 11 2 )◦ × Z (L v21 ,v2 )◦ p+ Ad−1 v˙1 (x) = 2 p− (y) . Define also the following subgroup of Z (G v11 ,v2 )◦ × Z (G v21 ,v2 )◦ : Z v1 ,v2 = Z 1v1 ,v2 Z 2v1 ,v2 H1ort Adv˙1 (H1ort ) × H2ort Adv˙2 (H2ort ) .
(5.5)
THEOREM 5.1 The G r double cosets of G × G are classified by pairs (v1 , v2 ) of minimal length representatives in W of cosets from W/Wi , an orbit of the twisted conjugation action T C v˙1 ,v˙2 of (G v11 ,v2 )0 on itself defined by
T C gv˙1 ,v˙2 ( f ) = g −1 f Adv˙1 (20 )−1 Adv˙2 20 (g),
g, f ∈ (G 1v1 ,v2 )0 ,
(5.6)
and an element of the abelian factor group Z (G v11 ,v2 )◦ × Z (G v21 ,v2 )◦ /Z v1 ,v2 . This data has to be taken modulo the action of the finite group (G v11 ,v2 )0 ∩ Z (G v11 ,v2 )◦ × (G v11 ,v2 )0 ∩ Z (G v21 ,v2 )◦
(5.7)
(5.8)
on (G v11 ,v2 )0 × Z (G v11 ,v2 )◦ × Z (G v21 ,v2 )◦ , where an element (q1 , q2 ) of (5.8) acts on the direct product by multiplication by (q1−1 Adv˙1 (20 )−1 (q2 ), q1 , q2 ). (The choice of representatives of vi in the normalizer of the maximal torus H of G is not essential.) An inductive procedure Let [(l1 w˙ 1 , w˙ 2−1l2 )] be an arbitrary G r double coset of G × G, where li ∈ L i and wi are minimal length representatives of cosets from Wi \W/Wi , i = 1, 2. We define two sequences of reductive Lie subalgebras gik of g, elements gik ∈ gik , and wik ∈ W (k ≥ 1, i = 1, 2). Set gi1 = li , gi1 = li , wi1 = wi . For convenience we also define gi0 = g and extend θ to a linear isomorphism between l1 and l2 . As in the proof of Theorem 4.5, the defined objects have the following two properties: (1) The Lie algebras gk1 and gk1 := Ad(w˙ k−1 ···w˙ 1 ) θ −1 (gk2 ) 1
1
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MILEN YAKIMOV
are standard Levi subalgebras of gk−1 ⊂ g. The subalgebras gk2 and 1 gk2 = Ad(w˙ k−1 ···w˙ 1 ) θ(gk1 ) 2
of
gk−1 2
2
have the same property. k
Let G ik , G i be the connected subgroups of G with Lie algebras gik , gik , and let k
Wik , W i be their Weyl groups. (2) The elements wik are representatives of minimal length in Wik−1 of cosets k
from Wik \Wik−1 /W i . If gk+1 6= gk1 or gk+1 6= gk2 , we define gk+1 and gk+1 by 1 2 1 2 gk+1 := gk1 ∩ θ −1 Ad−1k 1 (gk2 ) = θ −1 Ad−1k−1 1 gk2 ∩ Ad−1k (gk2 ) , (5.9) 1 (w˙ 2 ···w˙ 2 ) w˙ 2 (w˙ 2 ···w˙ 2 ) gk+1 := gk2 ∩ θ Ad−1k 1 (gk1 ) = θ Ad−1k−1 1 gk1 ∩ Ad−1k (gk1 ) . (5.10) 2 (w˙ 1 ···w˙ 1 )
(w˙ 1
w˙ 1
···w˙ 1 )
k+1
k+1 Denote by Ni± and N i± the unipotent subgroups of G ik generated by oneparameter subgroups of positive/negative roots of gik which are not roots of gik+1 and gik+1 . Set k = (−1)k . Let us apply the Bruhat lemma to g1k ∈ G k1 , g2k ∈ G k2 and to the k+1
k+1
k+1 k+1 pairs of parabolic subgroups N1 o G k+1 n N 1+ of G k1 and N 2− o G 2k+1 , 1 , G1 k+1 k+1 G k+1 n N2 of G k2 . We obtain 2 k
g1k = n 01 g10 w˙ 1k+1 g100 n 001 ,
(5.11)
g2k = n 02 g20 (w˙ 2k+1 )−1 g200 n 002 ,
(5.12)
k+1 0 ),
00 for some g10 ∈ G k+1 1 , g1 ∈ (G 1 k+1
k+1 0 ),
g20 ∈ (G 2
k+1 0 g200 ∈ G k+1 2 , n 1 ∈ N1k+1 ,
k+1
k+1 n 001 ∈ N 1+ , n 02 ∈ N 2− , and n 002 ∈ N2 . The elements wik+1 are minimal length k k+1
representatives in Wik of cosets from W i \W/Wik+1 . Equations (5.11) and (5.12) define them uniquely. The elements gik+1 ∈ G ik+1 are defined by g1k+1 = (20 )−1 Ad−1k 1 g20 g10 , (5.13) (w˙ 2 ···w˙ 2 ) g2k+1 = g200 20 Ad−1k 1 g100 (5.14) (w˙ 1 ···w˙ 1 )
(see (5.1) and (4.1)). The described procedure has finitely many steps k = 1, . . . , K . Denote vi = wiK · · · wi1 , i = 1, 2. For their representatives in N (H ), we set v˙i = w˙ iK · · · w˙ i1 . Similarly to the proof of Lemma 4.6, one shows that (v1 , v2 ) ∈ V and that any pair (v10 , v20 ) ∈ V can be obtained through the described procedure. At the last step of the procedure, giK = giv1 ,v2 .
COMPLEX REDUCTIVE POISSON-LIE GROUPS
497 k
We also need the connected subgroups of G ik and G i with Lie algebras gik ∩ (li0 ⊕ k
k,r ai ) and gik ∩ Ad(w˙ k−1 ···w˙ 1 ) (li0 ⊕ ai ). They are denoted by L ik and L i . Let G k,r 1 and G 2 i
i
denote the subgroups of G k−1 × G k−1 defined by 1 2 k −1 k G k,r 1 = (x, y) ∈ L 1 × L 2 2p+ (x) = p− Ad k−1 (w˙ 2
×
H1ort N1k−1
···w˙ 21 )
◦ (y)
k × Ad(w˙ k−1 ···w˙ 1 ) (H2ort )N 2− 2 2
(5.15)
and k −1 k G k,r 2 = (x, y) ∈ L 1 × L 2 p+ Ad k−1 (w˙ 1
×
k Ad(w˙ k−1 ···w˙ 1 ) (H1ort )N 1+ 2 2
···w˙ 11 )
◦ (x) = 2−1 p− (y)
× H2ort N2k .
(5.16)
Similarly to the proof of (3.21), using the fact that 3i ⊂ H, one shows that the groups G ik,r are connected. They play the role of G r (as left and right factors) at the kth step of the procedure. In particular, G i1,r = G r , i = 1, 2. Proof of Theorem 5.1 As in the proof of Theorem 4.5, we construct inductively a sequence of dense, open subsets of the double coset [(l1 w˙ 1 , (w˙ 2 )−1l2 )]. The set from the first step consists of the following elements of G × G: 00 −1 00 0 −1 00 0 0 0 0 0 0 m + ((20 )−1l+ )(p−1 ˙ 1 l+ n + l∗ m ∗ l− n − , + 2 p− (l2 ))((2 ) l− )l1 w
00 00 00 00 00 0 0 0 0 n 00+l+ m ∗ l∗ n +l+ (w˙ 2 )−1l2 (20l+ )(p−1 − 2p+ (l∗ ))(2 l− )m − , (5.17) 0 , n 0 , l 0 , m 0 , l 0 , n 0 belong to the factors of the following where the elements l+ + ∗ ∗ − − dense, open subset of G + :
1 1 1 L 11 ∩ Ad−11 (N1+ ) N1+ ∩ Ad−11 (N1+ ) L 11 ∩ Ad−11 (G 11 ) w˙ 1 w˙ 1 w˙ 1 1 1 1 ) . (5.18) ) N1+ ∩ Ad−11 (N1− × M+ ∩ Ad−11 (G 11 ) L 11 ∩ Ad−11 (N1− w˙ 1
w˙ 1
w˙ 1
00 , m 00 , l 00 , n 00 , l 00 belong to the factors of the following dense, open Similarly, n 00+ , l+ ∗ ∗ + + subset of G − :
1 1 1 N 2− ∩ Ad−11 (N2+ ) L 12 ∩ Ad−11 (N2+ ) M− ∩ Ad−11 (G 12 ) w˙ 2 w˙ 2 w˙ 2 1 −1 −1 1 1 1 1 × L 2 ∩ Ad 1 (G 2 ) N 2− ∩ Ad 1 (N2− ) L 12 ∩ Ad−11 (N2− ) . (5.19) w˙ 2
w˙ 2
w˙ 2
The sets (5.18) and (5.19) are constructed according to the way Adw˙ 1 (G + ) in1
1 G 1 N 1 of G and the way Ad (G ) intersects tersects the dense, open subset N1+ − w˙ 1 1 1− 2
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MILEN YAKIMOV
1 G 1 N . The set (5.17) is obtained by replacing the first component of the right N2+ 2 2− factor G r with (5.18) and the second component of the left factor G r with the set (5.19). Similarly to (4.24), one shows that the set of elements (5.17) can be represented as 1 1 2 1 1 e, (w˙ 21 )−1 N1+ × (Adw˙ 1 (N 2− ) ∩ N2+ ) ((20 )−1 Ad−11 , id)(Adw˙ 1 (L 1 ) ∩ N2+ ) w˙ 2
2
×
2
2,r 1 1 G 2,r 1 (g1 , g2 )G 2
1 1 1 1 1 1 ) (Adw˙ 1 (N 1+ ) ∩ N1− ) × N2− (w˙ 1 , e) × (id, 20 Ad−11 )(Adw˙ 1 L 1 ∩ N1− w˙ 1
1
1
(5.20) (recall (5.15), (5.16) for the definitions of G i2,r ). One proceeds with the left G 2,r 1 and 2,r 1 1 right G 2 double coset in G 1 × G 2 in the same way as in the proof of Theorem 4.5. k,r Unfortunately, the unipotent radicals of the projections of G k,r 1 and G 2 on the first factor (and also on the second one) correspond to roots of a different sign for even k k k k .) This k and to roots of the same sign for odd k. (Compare N1 , N 1+ and N 2− , N2 k k+1 does not allow one to obtain a closed formula for the dimensions of the cosets. The necessary modifications are made in the next section. At the last step of the procedure, we get the double coset G 1K +1,r (g1K , g2K )G 2K +1,r K +1 in G v11 ,v2 × G v21 ,v2 . The groups Ni± and N i±
K +1
(5.21)
are trivial, and
G 1K +1,r = H1ort × Adv˙2 (H2ort ) (id, Adv˙2 20 )(G v11 ,v2 )0 Z 1v1 ,v2 , G 2K +1,r = Adv˙1 (H1ort ) × H2ort (Adv˙1 (20 )−1 , id)(G v21 ,v2 )0 Z 2v1 ,v2 .
(5.22) (5.23)
This is a consequence of p± Z (L iv1 ,v2 )◦ = Z (L iv1 ,v2 /3i )◦ , which can be proved as follows. The group on the left is a connected subgroup of the one on the right, and the two have one and the same tangent Lie algebras, so they coincide. Decompose giK ∈ G iv1 ,v2 as giK = f iK ciK ,
f iK ∈ (G iv1 ,v2 )0 , ciK ∈ Z (G iv1 ,v2 )◦ .
Each pair ( f iK , ciK ) is only defined modulo multiplication of f iK and division of ciK by an element of the finite abelian group (G iv1 ,v2 )0 ∩ Z (G iv1 ,v2 )◦ , i = 1, 2. Using
COMPLEX REDUCTIVE POISSON-LIE GROUPS
499
(5.22) and (5.23), one rewrites (5.21) as v˙1 ,v˙2 v1 ,v2 0 K 0 −1 T C(G ( f 2K )−1 ), e (c1K , c2K )Z v1 ,v2 (id, 20 Ad−1 ) v1 ,v2 )0 ( f 1 Adv˙ 1 (2 ) v˙1 )(G 1 (5.24) (recall definition (5.5) of the group Z v1 ,v2 ). The final result of the procedure is that (l1 w˙ 1 , w˙ 2−1l2 ) = f 1K (Adv˙1 (20 )−1 ( f 2K )−1 )c1K v˙1 , v˙2−1 c2K (5.25) and that this coset has the dense, open subset K Y (e, v˙2−1 ) E k T C v˙1 ,vv˙12,v2 0 ( f 1K Adv˙1 (20 )−1 ( f 2K )−1 ), e
× ×
(G 1 ) k=1 v1 ,v2 0 ) (c1K , c2K )Z v1 ,v2 (id, 20 Ad−1 v˙1 )(G 1
1 Y
F k (v˙1 , e),
(5.26)
k=K
where the subsets E k and F k of G × G are defined by k k E k = N1 × Ad(w˙ K ···w˙ k+1 ) (Adw˙ k (N2− ) ∩ N2k ) k+1 2
× ((2 )
0 −1
2
2
k Ad−1k , Ad(w˙ K ···w˙ k+1 ) )(Adw˙ k (L 2 ) ∩ w˙ 2 2 2 2
k N2 ) k+1
and k k )) F k = (Ad(w˙ K ···w˙ k+1 ) , 20 Ad−1k )(Adw˙ k (L 1 ) ∩ (N1 k w ˙ 1 1 1 1 k k × Ad(w˙ K ···w˙ k+1 ) (Adw˙ k (N 1+ ∩ N1 ) × N ) . 2, k k 1
1
1
(Here, as before, k = (−1)k .) v˙1 ,v˙2 To the coset (5.25) one associates the pair (v1 , v2 ) ∈ V, the T C(G v1 ,v2 )0 orbit of K K 0 −1 v ,v 1 2 f 1 v1 (2 ) ( f 2 ) ∈ G , and the coset (c1K , c2K )Z v1 ,v2 ∈ Z (G v11 ,v2 )◦ × Z (G v21 ,v2 )◦ /Z v1 ,v2 . This is only defined modulo the action of the finite group (5.7) due to the freedom in the definition of f iK and ciK . Analogously to the proof of Theorem 4.5, one shows that two sets of the type (5.26) have a nontrivial intersection if and only if they correspond to one and the same pair (v1 , v2 ) ∈ V, orbit of T C v˙1 ,v˙2 , and cosets of the abelian factor group (5.7), modulo the action of the group (5.8). The equivalence relation respects the nonuniqueness in the definition of ciK and f iK .
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MILEN YAKIMOV
From this proof one also derives the following corollary. 5.2 If the Poisson structure on the group G comes from an r -matrix for which g± ⊃ h and the groups 3i are trivial, then the G r double cosets of G × G are classified by a pair of minimal length representatives (v1 , v2 ) in W of cosets from W/Wi and an orbit of the twisted conjugation action of G v11 ,v2 on itself defined by COROLLARY
v˙ ,v˙ g T C g1 2 ( f ) = g −1 f Adv˙1 2−1 Adv˙2 2(g) ,
f, g ∈ G 1v1 ,v2
(5.27)
(simple connectedness for G 0 is not assumed). 5.1.2. Another view on G r double cosets and a dimension formula The goal of this subsection is to prove a formula for the dimensions of the G r double cosets of G × G. The idea is to conjugate the group G r by some elements of N (H ) × N (H ) in such a way that the iterative procedure of the proof of Theorem 5.1 is still applicable to the cosets of the resulting subgroups of G × G. At the same time, the k,r corresponding groups G k,r 1 and G 2 from the iteration have unipotent radicals of their first and second components, of opposite sign (independent of the parity of k). Let wmax and wimax be the elements of maximal length in the Weyl groups W and Wi , i = 1, 2. Denote by u imax the representatives of minimal length in W of the cosets wmax Wi ∈ W/Wi . More explicitly, u imax = wmax (wimax )−1 .
(5.28)
Denote br1 = (id × Adu˙ max )G r ⊂ G × G, G 2 br2 = (Adu˙ max ×id)G r ⊂ G × G. G 1 br and right G br double cosets of G × G are called for simplicity G br − G br The left G 1 2 1 2 double cosets of G ×G. There is a simple connection between them and the G r double cosets of G × G: −1 max −1 br br max (5.29) (g1 , g2 ) = e, (u˙ max G 1 (g1 (u˙ max 2 ) 1 ) , u˙ 2 g2 )G 2 (u˙ 1 , e). THEOREM 5.3 br − G br (1) Any G 1 2
double coset of G × G is of the form r br1 f c1 v˙1 (u˙ max )−1 , (v˙2 (u˙ max )−1 )−1 c2 G b2 , G 1 2
(5.30)
where vi are minimal length representatives in W of cosets from W/Wi , f ∈ (G v11 ,v2 )0 , and ci ∈ Z (G iv1 ,v2 )◦ . All such cosets are classified by the same data as
COMPLEX REDUCTIVE POISSON-LIE GROUPS
501
in Theorem 5.1 and are related to the G r double cosets of G × G by r b1 ( f c1 v˙1 (u˙ max )−1 , u˙ max v˙ −1 c2 )G br2 (u˙ max , e). ( f c1 v˙1 , v˙ −1 c2 ) = e, (u˙ max )−1 G 2
2
1
2
1
2
(5.31) (2) The dimension of the cosets (5.30) and (5.31) is 2 dim g − 2 dim n+ − dim(gv11 ,v2 )0 + l(v1 ) + l(v2 ) + dim T C v˙1 ,vv˙12,v2 0 ( f ) − dim Z v1 ,v2 . (G 1
)
(5.32)
Proof The images of li , L i , Ai , m± , and M± under conjugation with u˙ imax are denoted by b bi , m b∓ . b∓ , and M li , b Li , A max max Since w , w are the elements of maximal length in W, Wi , they map (simple) positive roots of g, li into (simple) negative roots of the same algebras. Formula (5.28) for u imax implies that Adu˙ imax maps (1) (simple) positive roots of li into (simple) positive roots ofb li , and (2) positive roots of g which are not roots of li into negative roots of g which are not roots ofb li . From (1) we get thatb li are standard Levi subalgebras of g (see the introduction). b± are spanned by negative root spaces of Property (2) implies that the Lie algebras m g and (u imax )−1 (hiort ). The groups G ri can be more explicitly written as br1 = ( M b− , M+ ) (x, y) ∈ (b b1 ) × (L 02 A2 ) 2p+ Ad−1 G L 01 A (x) = y , u˙ max 1 br2 = (M− , M b+ ) (x, y) ∈ (L 01 A1 ) × (b b02 ) 2p+ (x) = p− Ad−1 G L 02 A u˙ max (y) . 2
The two groups are of the same type as G r . On the Lie algebra level, the role of the Cayley transformation θ : l01 ⊕ a1 → l02 ⊕ a2 is played by the map 0 b0 b θ Ad−1 u˙ max : l1 ⊕ a1 → l2 ⊕ a2 1
for the first group and by Adu˙ max θ : l01 ⊕ a1 → b l02 ⊕ b a2 2 for the second one. The procedure from the proof of Theorem 5.1 can be applied to describe the set b br double cosets of G × G. It starts with g1 = b of G r1 − G l1 , g11 = l1 , g12 = l2 , and 2 1 1 g2 = b l2 . The final result depends on a choice of a pair (b v1 ,b v2 ) of minimal length bi . Denote the set of such pairs by V b, and representatives in W of cosets from W/W
502
MILEN YAKIMOV
recall the definition of the set V from Section 5.1.1. Properties (1) and (2) of the elements u imax ∈ W imply −1 max −1 b. (v1 , v2 ) ∈ V if and only if v1 (u max ∈V 1 ) , v2 (u 2 ) max −1 of −1 For any (v1 , v2 ) ∈ V , we use the representatives v˙1 (u˙ max 1 ) , v˙ 2 (u˙ 2 ) max max −1 −1 br − G br v1 (u 1 ) , v2 (u 2 ) in N (H ). As in Theorem 5.1, the procedure for the G 1 2 double cosets of G × G gives that any such coset is equal to a coset of the form (5.30) max b for some pair (v1 u max 1 , v2 u 2 ) ∈ V (i.e., (v1 , v2 ) ∈ V ). The important point now is that at each step of the procedure the unipotent radk,r icals of the first and second components of the (new) groups G k,r 1 and G 2 have opposite signs. This gives the following closed formula for the dimension of the coset (5.30):
−1 −1 2 dim g − dim(gv11 ,v2 )0 − l v1 (u max − l v2 (u max 1 ) 2 ) + dim T C v˙1 ,vv˙12,v2 0 ( f ) − dim Z v1 ,v2 . (G 1
)
(The proof is identical to the one of the dimension formula (4.11).) Formula (5.32) follows from it and l(vi u imax ) = dim n+ − l(vi ). The last equation is a consequence of properties (1) and (2) of u imax and the fact that the length of w ∈ W is equal to the number of positive roots α of g for which w(α) is a negative root. 5.2. The group 6 = d(G) ∩ G r Let 6 denote the group d(G) ∩ G r . It is a countable subgroup of G × G since d(g) ∩ gr = 0. The group 6 is isomorphic to 6− = i + (G + ) ∩ i − (G − ) ⊂ D(G − ) for the Poisson structures on G considered in Section 4, that is, when the groups 3i are closed subgroups of L i0 Ai . Indeed, (x, y) ∈ 6 implies y = x and p+ (x) = p− (x). So (x, p+ (x)) ∈ 6− and one can define a map φ : 6 → 6− by φ(x, y) = x, p+ (x) (5.33) (recall the notation from Section 4.2). It is an isomorphism because of (3.21). PROPOSITION 5.4 If the Poisson structure on G is such that 3i are closed subgroups of L i0 Ai , then the groups 6 and 6− are canonically isomorphic by (5.33). For any factorizable Poisson structure on G, 6 = d(H ) ∩ G r . (5.34)
COMPLEX REDUCTIVE POISSON-LIE GROUPS
503
In the general case, (5.34) is proved in the same way as Theorem 4.12. In fact, one can prove that Theorem 4.12 is valid in the general case of an arbitrary Belavin-Drinfeld structure on G dealing with D(G + ) = G × ((L 01 A1 )/31 ) as an abstract group. The abstract groups 6 and 6− would then also be isomorphic. Note that all points of 6 are fixed under the dressing action of G r on G (i.e., they are zero-dimensional leaves of G). Thus (5.34) can be viewed as a dressing analog of the classical fact that the center of a reductive group R belongs to any maximal torus of R. The theorem deals with a fixed torus because the Poisson structure depends on the choice of the torus. 5.3. Summary of the results on the symplectic leaves of G As a direct consequence of the results of the previous subsections and Theorems 1.9 and 1.10, we obtain the following classification result for the set of symplectic leaves of the Poisson-Lie group G. THEOREM 5.5 Let G be a complex reductive Poisson-Lie group with simply connected derived subgroup G 0 . (1) The symplectic leaves in d(G)/6 are classified by the data of Theorem 5.1. The dimension of the leaf corresponding to the pair of minimal length representatives (v1 , v2 ) of cosets from W/Wi and the T C v˙1 ,v˙2 orbit through f ∈ (G 1v1 ,v2 )0 is v1 ,v2 0 dim(l01 ⊕ a1 ) + 2 dim hort ) + l(v1 ) + l(v2 ) 1 − dim(g1
+ dim T C v˙1 ,vv˙12,v2 0 ( f 1 ) − dim Z v1 ,v2 . (G 1
(2)
)
The symplectic leaves of G are coverings of the ones of d(G)/6 under the projection G → d(G)/6. The group 6 can be explicitly computed from Proposition 5.4.
In a particular (but important) case this theorem simplifies a lot (see Corollaries 5.2 and 4.13). 5.6 Let G be a factorizable complex reductive Poisson-Lie group for which the associated r -matrix satisfies g± ⊃ h and the groups 3i are trivial. Then the symplectic leaves of d(G)/6 are classified by a pair of minimal length representatives (v1 , v2 ) of cosets from W/Wi and an orbit of the twisted conjugation v˙ ,v˙ g action T C 1 2 of G v11 ,v2 on itself defined in (5.27). COROLLARY
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MILEN YAKIMOV
The dimension of the leaf that corresponds to this data is dim l1 − dim gv11 ,v2 + dim T C v˙1v,1v˙,v2 2 ( f ) + l(v1 ) + l(v2 ). G1
The group 6 is explicitly given by 6 = (h, h)|h ∈ H, 2(h) = h ∼ = ker /(1 − θ)(ker), where ker denotes the kernel of the exponential map for the maximal torus H. Remark 5.7 Consider the special choice of the data of Theorem 5.5 and Corollary 5.6 when v2 is equal to the identity element id of W and only a subgroup F1 of the abelian factor group (Z (G v11 ,id )◦ × Z (G v21 ,id )◦ )/Z v1 ,id is considered. It is defined as the image of the projection of the subtorus Z (G v11 ,id )◦ × e of Z (G v11 ,id )◦ × Z (G v21 ,id )◦ . The symplectic leaves of G corresponding to this special choice of the data are the symplectic leaves of the Poisson subgroup G − of G. The connection with Theorem 4.16 and Corollary 4.17 is as follows. For any minimal representative v1 in W of a coset from W/W1 , one has G v11 ,id = G v1 and the twisted conjugation actions T C v˙1 ,e and T C v˙1 are the same. Using ◦ Z 1v1 ,id = (x, y) ∈ Z (L v11 ,id )◦ × Z (L v21 ,id )◦ 2p+ (x) = p− (y) , one identifies the part of the data of Theorems 4.5 and 5.1 coming from the abelian factor groups F1 ∼ = Z (G v1 )◦ / Congv1 (L v1 )H1ort Adv˙1 (H1ort ) . The fact that the groups L i0 are simply connected implies that 3i ∈ Z (L i )◦ . From it one also gets 3v11 = d (G v1 )0 ∩ Z (G v1 )◦ (e, 31 ) ⊂ (G v1 )0 × Z (G v1 )◦ , where d(.) stays for the diagonal embedding. This gives the connection between the “modulo” parts of Theorems 4.5 and 5.1. The iterative procedures from the proofs of these theorems are also related. Let us start with the G r double coset of the element (l1 w˙ 1 , e) ∈ G × G and the i + (G + ) double coset of the same element but considered in G×((L 01 × A1 )/31 ). The reductive Lie algebras gk , gk and gk1 , gk1 from these proofs are related by k g2k 1 =g ,
g2k−1 = gk . 1 At odd (even) steps k, the algebras gk1 (resp., gk1 ) are kept unchanged.
COMPLEX REDUCTIVE POISSON-LIE GROUPS
505
Example 5.8 If the complex semisimple group G is equipped with the standard Poisson structure, then any G r double coset of G × G is equal to a coset of the form [(h 1 v˙1 , v˙2−1 h 2 )] for some h i ∈ H and vi ∈ W. The procedure from the proof of Theorem 5.1 has only one step, and it produces the following representation of the above coset (see also Corollary 5.2): v˙ ,v˙ g (v˙2−1 , e) N+ × (Adv˙2 (N− ) ∩ N+ ) T C H1 2 (h 1 Adv˙1 (h 2 )), e × (h −1 , Adv˙ (h)) h ∈ H (Adv˙ (N+ ) ∩ N− ) × N− (v˙1 , e), (5.35) 1
where
1
v˙ ,v˙ g T C h1 2 (h 0 ) = h −1 h 0 Adv˙1 v˙2 (h).
(5.36)
All such double cosets and thus all symplectic leaves of d(G)/6 are classified by a pair (v1 , v2 ) ∈ W and an orbit of the action (5.36). Both Corollary 5.6 and (5.35) imply that the dimension of the corresponding symplectic leaf is v˙ ,v˙2
g dim T C H1
(h) + l(v1 ) + l(v2 ).
The group 6 is isomorphic to the group 6− from Example 4.14 (see (4.53) and (4.54)). This example agrees with the results in [10] and [14]. Example 5.9 Here we discuss the set of symplectic leaves of the group SL(n + 1) equipped with Cremmer-Gervais Poisson structure. The first step is the classification of the SL(n +1)r double cosets of D(SL(n +1)). Recall from Example 4.11 that the minimal length representatives in W ∼ = S n+1 of j n n+1 the cosets from W/W1 ∼ = S /S1 are the permutations σ1 , j = 0, . . . , n defined in (4.34). In the same way, one finds that the minimal length representatives for the cosets from W/W2 ∼ = S n+1 /S2n are the permutations −1 σ2k = 1 · · · (n + 1 − k) ,
k = 0, . . . , n.
As in Example 4.11, we assume that all elements of the Weyl group of SL(n + 1) are represented in N (H ) by multiples of permutation matrices. j
σ˙ ,σ˙ 2k
Case 1: j + k ≥ n. The stable subgroup G 11 j
σ˙ ,σ˙ 2k
G 11
is given by
n = diag(a1 , . . . , an−k , A, a j+1 , . . . , an+1 ) a p ∈ C, A ∈ GL( j + k − n), det A
Y p
o ap = 1 .
506
MILEN YAKIMOV j
Under the twisted conjugation action T C σ˙ 1 ,σ˙ 2 of this group on itself, the element diag(a1 , . . . , an−k , A, a j+1 , . . . , an+1 ) acts as follows: k
diag(b1 , . . . , bn−k , B,b j+1 , . . . , bn+1 ) −1 7→ diag(a1−1 b1 aµ(1) , . . . , an−k bn−k aµ(n−k) , −1 A−1 B A, a −1 j+1 b j+1 aµ( j+1) , . . . , an+1 bn+1 aµ(n+1) ),
(5.37) where the permutation µ ∈ S n+1 is given by µ = ( j + 1) · · · (n + 1)(n − k) · · · 1 . j
j
σ˙ ,σ˙ k
There is a one-to-one correspondence between the set of T C σ˙ 1 ,σ˙ 2 orbits on G 11 2 and the conjugation orbits of GL( j + k − n). To the orbit of the element (5.37) of k
j
σ˙ ,σ˙ 2k
G 11
is associated the conjugation orbit of B ∈ GL( j + k − n). Therefore the SL(n + 1)r double cosets of SL(n + 1)×2 = D(SL(n + 1)) and j the symplectic leaves of SL(n + 1)/6 corresponding to the permutations σ1 , σ2k with j + k ≥ n are classified by the conjugation orbits of GL( j + k − n). The dimension of the leaf corresponding to the orbit through B ∈ GL( j + k − n) is (2n − j − k)( j + k + 1) + dim Ad A B A ∈ GL( j + k − n) . The group 6 coincides with the group 6− from Example 4.11 (see (4.55)), and the symplectic leaves in SL(n + 1) are coverings of the ones of SL(n + 1)/6 under the covering map SL(n + 1) → SL(n + 1)/6. j
σ˙ ,σ˙ 2k
Case 2: j + k ≤ n − 1. Then the stable subgroup G 11 n
is
diag(a1 , . . . , a j+1 , A, an−k+1 , . . . , an+1 ) a p ∈ C, A ∈ GL(n − j − k − 1), det A
Y
o ap = 1 .
p j
The T C σ˙ 1 ,σ˙ 2 action of the element diag(a1 , . . . , a j+1 , A, an−k+1 , . . . , an+1 ) on diag(b1 , . . . , b j+1 , B, bn−k+1 , . . . , bn+1 ) produces the element k
diag(a1−1 b1 aµ(1) , . . . , a −1 j+1 b j+1 aµ( j+1) , −1 −1 A−1 B A, an−k+1 bn−k+1 aµ(n−k+1) , . . . , an+1 bn+1 aµ(n+1) ),
where −1 µ = 1 · · · ( j + 1) (n − k + 1) · · · (n + 1) ∈ S n+1 .
COMPLEX REDUCTIVE POISSON-LIE GROUPS
507 j
The symplectic leaves of SL(n + 1)/6 corresponding to σ1 , σ2k in the case j + k ≤ n − 1 are classified by the conjugation orbits of GL(n − j − k − 1) on itself and a complex nonzero number. (It comes from the fact that the product of the entries a1 , . . . , a j+1 is invariant under the twisted conjugation action.) The dimension of the leaf corresponding to the orbit through B ∈ GL(n − j − k + 1) is (2n − j − k − 2)( j + k + 1) + 2n + dim Ad A B A ∈ GL(n − j − k − 1) . The group 6 is the same as in Case 1, and the symplectic leaves of SL(n + 1) are recovered in the same way. 6. Symplectic leaves of the dual Poisson-Lie group of G Let G be an arbitrary complex reductive factorizable Poisson-Lie group. The symplectic leaves of the opposite dual Poisson-Lie group G r of G (see Section 3.2.2) can be described in a much simpler way than the ones of G. The d(G) double cosets of D(G) = G × G are classified by the conjugation orbits in G. Indeed, every d(G) double coset of G × G is equal to a coset of an element (g, e) ∈ D(G). Associating to this coset the conjugation orbit in G through g gives the needed one-to-one correspondence. Theorem 1.10 with the roles of the groups G and G r exchanged now implies the following proposition. PROPOSITION 6.1 For an arbitrary complex reductive Poisson-Lie group G from the Belavin-Drinfeld class, the set of symplectic leaves of the dual Poisson-Lie group is described as follows: (1) The leaves of G r /6 are classified by the conjugation orbits in G, and the dimension of each leaf is exactly equal to the dimension of the related conjugation orbit. (2) The symplectic leaves of G r are coverings of the ones of G r /6 under the projection G r → G r /6. The group 6 can be explicitly described from Proposition 5.4.
Acknowledgments. I am grateful to Nicolai Reshetikhin for many stimulating discussions. I also benefited a lot from fruitful conversations with G. Ames, E. Frenkel, A. Givental, T. Hodges, T. Levasseur, A. Okounkov, A. Weinstein, and J. Wolf. I thank the referee for several remarks that helped me improve the exposition. This work is part of the author’s Ph.D. thesis for the University of California at Berkeley.
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MILEN YAKIMOV
Department of Mathematics, Cornell University, Ithaca, New York 14853-4201, USA;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 3,
KOSZUL DUALITY FOR MODULES OVER LIE ALGEBRAS TOMASZ MASZCZYK and ANDRZEJ WEBER
Abstract Let g be a reductive Lie algebra over a field of characteristic zero. Suppose that g acts on a complex of vector spaces M • by i λ and Lλ , which satisfy the same identities that contraction and Lie derivative do for differential forms. Out of this data one defines the cohomology of the invariants and the equivariant cohomology of M • . We establish Koszul duality between them. 1. Introduction Let G be a compact Lie group. Set 3• = H∗ (G) and S • = H ∗ (BG). The coefficients are in R or C. Suppose that G acts on a reasonable space X . In the paper [GKM], M. Goresky, R. Kottwitz, and R. MacPherson established a duality between the ordinary cohomology that is a module over 3• and the equivariant cohomology that is a module over S • . This duality is on the level of chains, not on the level of cohomology. Koszul duality says that there is an equivalence of derived categories of 3• -modules and S • -modules. One can lift the structure of an S • -module on HG∗ (X ) and the structure of a 3• -module on H ∗ (X ) to the level of chains in such a way that the obtained complexes correspond to each other under Koszul duality. Equivariant coefficients in the sense of [BL] are also allowed. Later, C. Allday and V. Puppe [AP] gave an explanation for this duality based on the minimal Brown-Hirsh model of the Borel construction. One should remark that Koszul duality is a reflection of a more general duality: the one described by D. Husemoller, J. Moore, and J. Stasheff in [HMS]. Our goal is to show that this duality phenomenon is a purely algebraic affair. We construct it without appealing to topology. We consider a reductive Lie algebra g and a complex of vector spaces M • on which g acts via two kinds of actions: i λ and Lλ . These actions satisfy the same identities that contraction and Lie derivative do in the case of the action on the differential forms of a G-manifold. Such differential gmodules were already described by H. Cartan [Ca] (see also [AM], [GS]). We do not assume that M • is finite-dimensional or semisimple. We also wish to correct a small DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 3, Received 9 January 2001. Revision received 26 April 2001. 2000 Mathematics Subject Classification. Primary 17B55, 17B20; Secondary 55N91. Weber’s work supported by Komitet Bada´n Naukowych grant number 2P03A 00218. 511
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MASZCZYK and WEBER
inaccuracy in [GKM, proof of Lem. 17.6]. The distinguished transgression plays a crucial role in our construction. This is a canonical identification between the space of primitive elements of (3• g∗ )g ' H ∗ (g) with certain generators of (Sg∗ )g . 2. Category Let g be a reductive Lie algebra over a field k of characteristic zero. We consider differential graded vector spaces M • over k equipped with linear operations i λ : M • → M •−1 of degree −1 for each λ ∈ g. We define Lλ = di λ + i λ d : M • → M • .
We assume that i λ is linear with respect to λ, and for each λ, µ ∈ g the following identities are satisfied: i λ i µ = −i µ i λ , [Lµ , i λ ] = i [µ,λ] . Then L is a representation of g in M • . The category of such objects with obvious morphisms is denoted by K (g). Example 2.1 Let G be a group with Lie algebra g, and let X be a G-manifold. Then the space of differential forms • (X ) equipped with the contractions with fundamental vector fields is an example of an object from K (g). Example 2.2 Another example of an object of K (g) is 3• g∗ , the exterior power of the dual of g. The generators of 3• g∗ are given the gradation 1. This is a differential graded algebra with a differential d3 induced by the Lie bracket. The operations i λ are the contractions with g. Example 2.3 For a representation V of g, define the invariant subspace V g = {v ∈ V : ∀λ ∈ g, Lλ v = 0}. Then V g with trivial differential and i λ ’s is an object of K (g). In particular, we take V = S • g∗ , the symmetric power of the dual of g. The generators of S • g∗ are given the gradation 2. Example 2.4 Suppose there are given two objects M • and N • of K (g). Then M • ⊗ N • with operations i λ defined by the Leibniz formula is again in K (g).
KOSZUL DUALITY FOR MODULES OVER LIE ALGEBRAS
513
Note that the objects of K (g) are the same as differential graded modules over a dg– Lie algebra Cg (the cone over g) with Cg0 = Cg−1 = g,
Cg6=−1,0 = 0.
The elements of Cg0 are denoted by Lλ , and the elements of Cg−1 are denoted by i λ . They satisfy the following identities: di λ = Lλ ,
[Lλ , Lµ ] = L[λ,µ] ,
[Lλ , i µ ] = i [λ,µ] ,
[i λ , i µ ] = 0.
The enveloping dg-algebra of Cg is the Chevaley-Eilenberg complex V (g) (see [Wei, p. 238]) which is a free U (g)-resolution of k. Thus the objects of K (g) are just the dg-modules over V (g). Example 2.5 Suppose that a Lie algebra g acts on a graded commutative k-algebra A by derivations. Let k A be the algebra of forms; it is generated by symbols a and da with a ∈ A. The cone of g acts by derivations. The action on generators is given by i λ a = 0,
Lλ a = λa,
i λ da = λa,
Lλ da = dλa.
Then k A is in K (g). Another point of view (as in [GS]) is that the objects of K (g) are the representations of the super Lie algebra b g = Cg ⊕ k[−1] (where k[−1] is generated by d) with relations [d, d] = 0, [d, x] = d x for x ∈ Cg. 3. The twist We describe a transformation that plays the role of the canonical map 8 : G × X → G × X, (g, x) 7 → (g, gx)
for topological G-spaces. We define a linear map i : 3• g∗ ⊗ M • → 3• g∗ ⊗ M • by the formula X i(ξ ⊗ m) = ξ ∧ λk ⊗ i λk m, k
where {λk } is a basis of g and where {λk } is the dual basis. It commutes with Lλ . The operation i is nilpotent. Define an automorphism T : 3• g∗ ⊗ M • → 3• g∗ ⊗ M •
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MASZCZYK and WEBER
(which is not in K (g)): T = exp(−i) =
∞ X (−1)n n=1
T(ξ ⊗ m) =
X
n!
in ,
(−1)n(n+1)/2 ξ ∧ λ I ⊗ i λ I m.
I ={i 1 <···
It satisfies (see [GHV, Prop. V, p. 286]; see also [AM]) i µ T(ξ ⊗ m) = T (i µ ξ ) ⊗ m , X d T(ξ ⊗ m) = T d(ξ ⊗ m) + λk ∧ ξ ⊗ Lλk m . k
Remark 3.1 Note that for the self-map 8 of G × X we have X d8∗ ω(1, x) = dω + p ∗ λk ∧ Lλk ω (1, x), k
where p : G × X → X is the projection and where x ∈ X . The twist on the level of the Weil algebra was used by Cartan [Ca] and later by V. Mathai and D. Quillen [MQ]. From another point of view, for a dg–vector space to be an object of K (g) is equivalent to having such a twist that satisfies certain axioms. We do not state them here. We just remark that understanding this twist allows us to develop a theory of actions of L∞ -algebras. 4. Weil algebra Following [GHV, Chap. VI, p. 223], we define the Weil algebra W (g) = 3• (Cg)∗ = S • g∗ ⊗ 3• g∗ . The generators of S • g∗ are given the gradation 2, whereas the generators of 3• g∗ are given the gradation 1. The differential in W (g) is the sum of three operations: X X dW (a ⊗ b) = a ⊗ d3 b + λk a ⊗ i λk b + adλ∗k a ⊗ λk ∧ b, k
k
where {λk } is a basis of g. The differential dW satisfies the Maurer-Cartan formula dW (1 ⊗ ξ ) − 1 ⊗ d3 ξ = ξ ⊗ 1 for ξ ∈ g∗ .
KOSZUL DUALITY FOR MODULES OVER LIE ALGEBRAS
515
The operations i λ are contractions with the second term. The resulting action Lλ = dW i λ + i λ dW is induced by the coadjoint action on S • g∗ ⊗ 3• g∗ (see [GHV, (6.5), p. 226]). With this structure, W (g) becomes an object of K (g). The cohomology of W (g) is trivial except in dimension zero, where it is k (see [GHV, Prop. I, p. 228]). There are given canonical maps in K (g): • inclusion (S • g∗ )g ' (S • g∗ )g ⊗ 1 ⊂ W (g), • restriction W (g) −→ → 3• g∗ , which sends all the positive symmetric powers to zero. The Weil algebra is a model of differential forms on E G, and the sequence of morphisms in K (g), (S • g∗ )g ,−→ W (g) −→ → 3• g∗ , is a model of • (BG) ,−→ • (E G) −→ → • (G). Remark 4.1 It is easy to see that W (g) = k 3• g∗ . Thus for any commutative dg-algebra A, Homdg-comm (3• g∗ , A) = Homdg-comm W (g), A . 5. The distinguished transgression The invariant algebra (3• g∗ )g is the exterior algebra spanned by the space of primitive elements P • , whereas (S • g∗ )g is the symmetric algebra spanned by some space e• . The point is that P e• can be canonically chosen. P PROPOSITION 5.1 ([GHV, Prop. VI, p. 239]) Suppose that ξ ∈ P • is a primitive element. Then there exists an element ω ∈ W (g)g such that
ω|3• g∗ = ξ, i λ ω = i λ (1 ⊗ ξ ) for all λ ∈ (3• g)g , dW (ω) = e ξ ⊗ 1. The element ω is not unique, but e ξ is. The set of e ξ for ξ ∈ P ∗ is the distinguished space of generators of (S • g∗ )g . 6. Example: su2 The algebra su2 is spanned by i, j, and k with relation [i, j] = 2k and its cyclic transposition. In 3• g∗ we have d3 i∗ = 2j∗ ∧ k∗ and cycl.
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In the Weil algebra we have dW (1 ⊗ i∗ ) = 1 ⊗ 2j∗ ∧ k∗ + i∗ ⊗ 1 and cycl., dW (i∗ ⊗ 1) = 2(k∗ ⊗ j∗ − j∗ ⊗ k∗ ) and cycl. The primitive elements in 3• g∗ are spanned by ξ = i∗ ∧ j∗ ∧ k∗ . As ω of Proposition 5.1 we take 1 ω = 1 ⊗ i∗ ∧ j∗ ∧ k∗ + (i∗ ⊗ i∗ + cycl.). 2 Then e ξ ⊗ 1 = dW (ω) =
1 ∗2 (i + j∗ 2 + k∗ 2 ) ⊗ 1, 2
whereas dW (1 ⊗ ξ ) = i∗ ⊗ j∗ ∧ k∗ + cycl. 7. Invariant cohomology and equivariant cohomology Denote (3• g)g by 3• . Let D(3• ) be the derived category of graded differential 3• modules. For M • ∈ K (g) the invariant submodule (M • )g is a differential module over 3• . We obtain an object in D(3• ). We call it the invariant cohomology of M • . Remark 7.1 Let X be a manifold on which a compact group G acts. Let g be the Lie algebra of G. Then • (X ) is a g-module. The invariants of 3• g act on • (X ), but this action does not commute with the differential in general. We have [d, i λ ]ω = Lλ ω. To obtain an action that commutes with the differential, one restricts it to (• (X ))g . Fortunately, the resulting cohomology does not change. We obtain a complex with an action of 3• = H∗ (G), which is quasi-isomorphic to • (X ). The cohomology is equal to H ∗ (X ). Denote (S • g∗ )g by S • . Let D(S • ) be the derived category of graded differential S • modules. Following [Ca], we define (M • )g := (S • g∗ ⊗ M • )g with differential d M,g (a ⊗ m) = a ⊗ d M m −
X
λk a ⊗ i λk m.
k
It is a differential S • -module. We obtain an object (M • )g in D(S • ). We call it the equivariant cohomology of M • .
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For an object N • of K (g), we define horizontal elements (N • )hor = {n ∈ N • : ∀λ ∈ g, i λ n = 0}. Then we define basic elements g
(N • )basic = (N • )hor = {n ∈ N • : ∀λ ∈ g, i λ n = 0, i λ dn = 0}, which form a complex. The following lemma can be found in [Ca], but we need to have an explicit form of the isomorphism, as in [AM, §4.1]. 7.2 ([Ca]) The map ψ0 is an isomorphism of differential graded S • -modules: ψ0 = 1 ⊗ T|1⊗M • : (M • )g → W (g) ⊗ M • basic . LEMMA
Proof The elements of (3• g∗ ⊗ M • )hor are of the form X X T(1 ⊗ m) = 1 ⊗ m − λk ⊗ i λk m − λk ∧ λl ⊗ i λk i λl m ± . . . ; k
k
thus they are determined by m. The conclusion follows since W (g) ⊗ M • hor = S • g∗ ⊗ (3• g∗ ⊗ M • )hor . Remark 7.3 From Lemma 7.2 we see that (M • )g is an analog of • (E G ×G X ). Let X and g be as in Remark 7.1. The construction of the equivariant cohomology presented here is the so-called Cartan model of • (E G ×G X ). We obtain a complex with an action of S • = H ∗ (BG). The cohomology is equal to HG∗ (X ). 8. Koszul duality + + By [GKM, §8.5], the functor h : D (S • ) → D (3• ) is an equivalence of categories: h(A• ) = Homk (3• , A• ) = (3• g∗ )g ⊗ A• with differential dh (ξ1 ∧ · · · ∧ ξn ) ⊗ a X = (−1) j+1 (ξ1 ∧ · · · ∨ j · · · ∧ ξn ) ⊗ e ξ j a + (−1)n (ξ1 ∧ · · · ∧ ξn ) ⊗ da, j
where ξ j ’s are primitive.
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THEOREM 8.1 (Koszul duality) + Let g be a reductive Lie algebra. Suppose that M • is an object of K (g); then in + D (3• ), h (M • )g ' (M • )g .
Proof The action of g on H ∗ (W (g)) is trivial, and since g is reductive, W (g) is semisimple. Thus, by [GHV, Th. V, p. 172], the inclusion g k ⊗ (M • )g = W (g)g ⊗ (M • )g ⊂ W (g) ⊗ M • is a quasi-isomorphism. We want to construct a quasi-isomorphism ψ from h (M • )g = (3• g∗ )g ⊗ (S • g∗ ⊗ M • )g to W (g) ⊗ M •
g
= (S • g∗ ⊗ 3• g∗ ⊗ M • )g .
First we choose a linear map ω : P ∗ → W (g) satisfying the conditions of Proposition 5.1. We construct ψ by the formula extending ψ0 of Lemma 7.2 with the help of the distinguished transgression of §5: ψ (ξ1 ∧ · · · ∧ ξn ) ⊗ m = ω(ξ1 ) . . . ω(ξn )ψ0 (m). It is well defined since (W (g) ⊗ M • )g is a W (g)g -module and ψ0 (m) ∈ W (g) ⊗ M • basic , ω(ξ1 ) ∈ W (g)g . The map ψ commutes with i λ since the image of ψ0 is horizontal. It commutes with the differential because dW (ω(ξ )) = e ξ ⊗ 1. This corrects an error in [GKM, proof of Lem. 17.6], where ψ does not commute with the differential unless g is abelian. We check that ψ is a quasi-isomorphism. Let us filter both sides by S ≥i (g∗ ). Then the corresponding quotient complexes are GriS ψ : (3• g∗ )g ⊗ (S i g∗ ⊗ M • )g −→ (S i g∗ ⊗ 3• g∗ ⊗ M • )g . The differential on the left-hand side is just ⊗ 1 ⊗ d M (where = (−1)degξ ), and the differential on the right-hand side is X 1 ⊗ d3 ⊗ 1 + adλ∗k ⊗ λk ∧ · ⊗ 1 + 1 ⊗ ⊗ d M . k
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The map GriS ψ equals 1 · (1 ⊗ T|1⊗M • ). When we untwist it (i.e., when we apply exp(i) = T−1 to the right-hand side), the differential takes the form X X 1 ⊗ d3 ⊗ 1 + adλ∗k ⊗ λk ∧ · ⊗ 1 + 1 ⊗ ⊗ d M + 1 ⊗ λk ∧ · ⊗ Lλk . k
k
Since we stay in the invariant subcomplex, the differential equals X 1 ⊗ d3 ⊗ 1 − 1 ⊗ λk ∧ adλ∗k ⊗ 1 + 1 ⊗ ⊗ d M . k
Moreover,
P
k
1 ⊗ λk ∧ adλ∗k = 2d3 ; thus the differential on the right-hand side is −1 ⊗ d3 ⊗ 1 + 1 ⊗ ⊗ d M .
The cohomology of the left-hand side is H ∗ (3• g∗ )g ⊗ (S i g∗ ⊗ M • )g = (3• g∗ )g ⊗ H ∗ (S i g∗ ⊗ M • )g , and the cohomology of the right-hand side is H ∗ (S i g∗ ⊗ 3• g∗ ⊗ M • )g = H ∗ (3• g∗ )g ⊗ H ∗ (S i g∗ ⊗ M • )g , again by [GHV, Th. V, p. 172] since the action on H ∗ (3• g∗ ) is trivial and 3• g∗ is semisimple. Thus cohomologies of the graded complexes are the same. The conclusion of Theorem 8.1 follows. Remark 8.2 Let X and g be as in Remark 7.1. Following [GKM], let us explain the meaning of Koszul duality. The invariant and the equivariant cohomology of X are defined on the level of derived categories. Theorem 8.1 gives a procedure to reconstruct the invariant cohomology from the equivariant cohomology. Since this duality is an isomorphism of categories, the invariant cohomology determines the equivariant cohomology as well. The corresponding statement on the level of graded modules over 3• and S • is not true, as an easy example in [GKM] shows. One cannot recover HG∗ (X ) from H ∗ (X ) with an action of H∗ (G) even in the case X = S 3 , G = S 1 . Acknowledgments. The results of the present paper were obtained with the help and assistance of Marcin Chałupnik. It is a part of joint work. Our aim is to describe Koszul duality in a much wider context. We thank the referee for careful corrections. References [AM]
A. ALEKSEEV and E. MEINRENKEN, The non-commutative Weil algebra, Invent.
Math. 139 (2000), 135 – 172. MR 2001j:17022 511, 514, 517
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[AP]
C. ALLDAY and V. PUPPE, On a conjecture of Goresky, Kottwitz and MacPherson,
[BL]
J. BERNSTEIN and V. LUNTS, Equivariant Sheaves and Functors, Lecture Notes in
[Ca]
H. CARTAN, “Notions d’alg`ebre diff´erentielle; application aux groupes de Lie et aux
Canad. J. Math. 51 (1999), 3 – 9. MR 2000e:18008 511 Math. 1578, Springer, Berlin, 1994. MR 95k:55012 511
[GKM]
[GHV]
[GS] [HMS]
[MQ] [Wei]
vari´et´es o`u op`ere un groupe de Lie” in Colloque de topologie (espaces fibr´es) (Bruxelles, 1950), Thone, Li`ege, 1951. MR 13:107e 511, 514, 516, 517 M. GORESKY, R. KOTTWITZ, and R. MACPHERSON, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), 25 – 83. MR 99c:55009 511, 512, 517, 518, 519 W. GREUB, S. HALPERIN, and R. VANSTONE, Connections, Curvature, and Cohomology, Vol. III: Cohomology of Principal Bundles and Homogeneous Spaces, Pure Appl. Math. 47, Academic Press, New York, 1976. MR 53:4110 514, 515, 518, 519 V. W. GUILLEMIN and S. STERNBERG, Supersymmetry and Equivariant de Rham Theory, Math. Past Present, Springer, Berlin, 1999. MR 2001i:53140 511, 513 D. HUSEMOLLER, J. C. MOORE, and J. STASHEFF, Differential homological algebra and homogeneous spaces, J. Pure Appl. Algebra 5 (1974), 113 – 185. MR 51:1823 511 V. MATHAI and D. QUILLEN, Superconnections, Thom classes, and equivariant differential forms, Topology 25 (1986), 85 – 110. MR 87k:58006 514 C. A. WEIBEL, An Introduction to Homological Algebra, Cambridge Stud. Adv. Math. 38, Cambridge Univ. Press, Cambridge, 1994. MR 95f:18001 513
Maszczyk Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland;
[email protected] Weber Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 3,
GALOIS REPRESENTATIONS WITH CONJECTURAL CONNECTIONS TO ARITHMETIC COHOMOLOGY AVNER ASH, DARRIN DOUD, and DAVID POLLACK
Abstract In this paper we extend a conjecture of A. Ash and W. Sinnott relating niveau 1 Galois representations to the mod p cohomology of congruence subgroups of SLn (Z) to include Galois representations of higher niveau. We then present computational evidence for our conjecture in the case n = 3 in the form of three-dimensional Galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture. Our examples include Galois representations with nontrivial weight and level, as well as irreducible three-dimensional representations that are in no obvious way related to lower-dimensional representations. In addition, we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture. 1. Introduction In [22], J.-P. Serre published his conjecture (which had existed in some form since 1973) relating continuous odd absolutely irreducible Galois representations ρ : G Q → GL2 (F¯ p ) to the mod p reductions of modular forms. He not only conjectured that a relationship existed but also gave precise formulae describing where to find the predicted modular forms. In [4], Ash and Sinnott presented a conjecture giving a relationship between odd niveau 1 Galois representations of arbitrary dimension n and certain cohomology groups of congruence subgroups of GLn (Z). In the two-dimensional case, this conjecture is closely related to Serre’s conjecture. Ash and Sinnott presented computational evidence for their conjecture in certain three-dimensional cases, primarily in the case of three-dimensional level 1 reducible representations. In this paper we present additional computational evidence for the conjecture, including cases with nontrivial DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 3, Received 26 January 2001. Revision received 13 June 2001. 2000 Mathematics Subject Classification. Primary 11F75, 11F80. Ash’s work supported by National Science Foundation grant number DMS-9531675 and National Security Agency grant number MDA 904-00-1-0046. This manuscript is submitted for publication with the understanding that the United States government is authorized to produce and distribute reprints. Doud’s work supported by a National Science Foundation postdoctoral research fellowship. 521
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weight, level, and nebentype. We also expand the conjecture to include representations of higher niveau and present computational evidence for this generalization. The representations in Section 7.2 are particularly interesting—they are the first examples in which a cohomology eigenclass seems to correspond to a native three-dimensional Galois representation (i.e., a Galois representation that is in no obvious way related to a one-dimensional or two-dimensional Galois representation). There is no problem in finding many Galois representations to which the conjecture that we make applies. The challenge is in finding Galois representations for which the predicted weights and levels allow computation of the associated cohomology classes and their Hecke eigenvalues. Our verifications of the conjecture involve finding representations that have fairly small weight and level and computing the predicted cohomology groups and the action of Hecke operators on these groups for primes up to 47. We then compare the Hecke eigenvalues with the coefficients of the characteristic polynomials of the images of Frobenius, and if they match, we claim to have evidence for the conjecture. We present many examples of Galois representations with weight and level small enough for us to work with, resulting in over 200 predictions (counting each weight associated to a Galois representation by Conjecture 3.1 separately). These examples are summarized in Tables 1 through 10, in which we describe Galois representations and give predicted weights, levels, and characters. For all the examples listed in the tables, we have computed the homology groups (which are naturally dual to the cohomology groups), and in all cases an eigenclass with the correct eigenvalues up to ` = 47 (` = 3 in Table 1) did exist in the predicted weight, level, and character. Our examples include cases with niveau 1, 2, and 3, as well as wildly ramified niveau 1 representations. We also call attention to Theorem 4.1 and the examples that follow it, in which the theory of symmetric squares is used to prove a prediction of Conjecture 3.1 for certain irreducible three-dimensional Galois representations. 2. Definitions Let p be a prime number, and let F¯ p be an algebraic closure of the finite field F p with p elements. By a Galois representation we mean a continuous representation of the absolute Galois group G Q of Q to a matrix group GLn (F¯ p ). The representations with which we work in this paper will always, in addition, be semisimple. We say that a Galois representation is odd if the image of complex conjugation is a nonscalar matrix and that it is even if the image of complex conjugation is a scalar. For a given prime q, we denote a decomposition group at q in G Q by G q . This decomposition group then has a filtration by ramification subgroups G q,i , with the whole inertia group above q equal to G q,0 . We often denote the inertia group G p,0 at p by I p . We fix a Frobenius element Frobq for each q, and we fix a complex
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conjugation Frob∞ . We denote the fundamental characters of niveau n in characteristic p (see [19]) by ψn,d , d = 1, . . . , n, and we note that they are all Galois conjugates (over F p ) of ψn,1 . In many cases we are interested in working with fundamental characters of niveau 2 and 3, so for brevity we let ψ = ψ2,1 , ψ 0 = ψ2,2 , and we let θ = ψ3,1 , θ 0 = ψ3,2 , and θ 00 = ψ3,3 . Note that the cyclotomic character ω is equal to ψ1,1 . 2.1. Hecke operators Let 00 (N ) be the subgroup of matrices in SLn (Z) whose first row is congruent to (∗, 0, . . . , 0) modulo N . Define S N to be the subsemigroup of integral matrices in GLn (Q) satisfying the same congruence condition and having positive determinant relatively prime to N . Let H (N ) denote the F¯ p -algebra of double cosets 00 (N )\S N / 00 (N ). Then H (N ) is a commutative algebra that acts on the cohomology and homology of 00 (N ) with coefficients in any F¯ p [S N ]-module. When a double coset is acting on cohomology or homology, we call it a Hecke operator. Clearly, H (N ) contains all double cosets of the form 00 (N )D(`, k)00 (N ), where ` is a prime not dividing N , 0 ≤ k ≤ n, and 1 .. . 1 D(`, k) = ` .. . ` is the diagonal matrix with the first n − k diagonal entries equal to 1 and the last k diagonal entries equal to `. When we consider the double coset generated by D(`, k) as a Hecke operator, we call it T (`, k). Definition 2.1 Let V be an H ( pN )-module, and suppose that v ∈ V is a simultaneous eigenvector for all T (`, k) and that T (`, k)v = a(`, k)v with a(`, k) ∈ F¯ p for all ` 6 | pN prime and all 0 ≤ k ≤ n. If ρ : G Q → GLn (F¯ p ) is a representation unramified outside pN and n X
(−1)k `k(k−1)/2 a(`, k)X k = det I − ρ(Frob` )X
k=0
for all `6 | pN , then we say that ρ is attached to v or that v corresponds to ρ.
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2.2. Level and nebentype Let ρ : G Q → GLn (F¯ p ) be a continuous representation. We define a level and nebentype associated to ρ exactly as Serre does in [22]. For a fixed prime q 6= p, and i ≥ 0, let gi = |ρ(G q,i )|. Note that this is a finite integer since by continuity the image of ρ must be finite. Let M = F¯ np be acted on by G Q via ρ in the natural way, and define nq =
∞ X gi dim M/M G q,i . g0 i=0
The sum defining n q is then a finite sum since eventually the G q,i are trivial. Definition 2.2 With ρ as above, define the level N (ρ) =
Y
q nq .
q6 = p
Note that this product is also finite since ρ is ramified at only finitely many primes, and n q is zero if ρ is unramified at q. In order to define the nebentype character, we again proceed exactly as Serre does in [22]. We factor det ρ = ωk , where ω is the cyclotomic character modulo p, and is a character G Q → F¯ p unramified at p. By class field theory, we may then consider as a Dirichlet character × : Z/N (ρ)Z → F¯ × p. We then pull back the definition of to S N by defining to be the composite character × S N → Z/N (ρ)Z → F¯ × p, where the first map takes a matrix in S N to its (1, 1) entry, and we define F to be the one-dimensional space F¯ p with the action of S N given by . For a GLn (F p )-module V , we now define V () = V ⊗ F . Letting 00 (N ) act on V by reduction modulo p, we see that V () is a 00 (N )-module. In addition, since S pN acts on F , we see that V () is also an S pN -module. In specifying the nebentype, we often refer to the unique quadratic character modulo p ramified only at a prime q > 3, and we denote this character by q : G Q → F× p.
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We also refer to the character 4 , which is ramified only at 2 and cuts out the field √ Q( −1). 2.3. Irreducible GLn (F p )-modules The natural generalization of the weight in Serre’s conjecture is an irreducible GLn (F p )-module. To see this, we note that the Eichler-Shimura theorem (see [23]) relates the space of modular forms of weight k to cohomology with coefficients in Symg (C2 ) with g = k − 2. Hence, an eigenform f of level N , nebentype , and weight k gives rise to a collection of Hecke eigenvalues which, when reduced modulo p, also occurs in H 1 00 (N ), Vg () , where Vg ∼ = Symg (F¯ 2p ) is the space of two-variable homogeneous polynomials of degree g over F¯ p with the natural action of SL2 (F¯ p ). Ash and G. Stevens have shown in [5] that any system of Hecke eigenvalues occurring in the cohomology of 00 (N ) with coefficients in some GLn (F p )-module also occurs in the cohomology with coefficients in at least one irreducible GLn (F p )-module occurring in a composition series of the original module. Hence, there is some irreducible GLn (F p )-module W such that the system of eigenvalues coming from f also occurs in H 1 (00 (N ), W ()). Given this fact, it is natural to ask which irreducible modules give rise to the system of eigenvalues. We may parameterize the complete set of irreducible GLn (F p )-modules as in [10]. Definition 2.3 We say that an n-tuple of integers (b1 , . . . , bn ) is p-restricted if 0 ≤ bi − bi+1 ≤ p − 1,
1 ≤ i ≤ n − 1,
and 0 ≤ bn < p − 1. PROPOSITION 2.4 The set of irreducible GLn (F p )-modules is in one-to-one correspondence with the collection of all p-restricted n-tuples.
The one-to-one correspondence in this proposition may be described explicitly as follows: the module F(b1 , . . . , bn ) corresponding to the p-restricted n-tuple
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(b1 , . . . , bn ) is the unique simple submodule of the dual Weyl module W (b1 , . . . , bn ) with highest weight (b1 , . . . , bn ). Theorem 8.1 gives an explicit model for the module F(b1 , b2 , b3 ) in the case n = 3, but for larger n no general computational models are known to the authors. In dealing with Galois representations, it often becomes necessary to associate an irreducible module to an n-tuple that is not p-restricted. We do this via the following definition. Definition 2.5 Let (a1 , . . . , an ) be any n-tuple of integers. Define F(a1 , . . . , an )0 = F(b1 , . . . , bn ), where (b1 , . . . , bn ) is a p-restricted n-tuple for which ai ≡ bi
(mod p − 1).
We note that in certain cases (namely, when some ai ≡ ai+1 (mod p − 1)) the module F(a1 , . . . , an )0 may not be well defined. In this case we interpret any statement concerning F(a1 , . . . , an )0 to mean that the statement is true for some choice of F(b1 , . . . , bn ) as in the definition. For example, if p = 5, a statement concerning F(1, 0, 0)0 is true if the statement is true for either F(1, 0, 0) or F(5, 4, 0) (or both). When dealing with modules defined by the prime notation, we say that a module F(a1 , . . . , an )0 is determined unambiguously if there is a unique p-restricted sequence congruent to (a1 , . . . , an ) modulo p − 1. 2.4. The strict parity condition We modify slightly the statement of the strict parity condition in [4] for ease of exposition, but our formulation is logically equivalent to that in [4]. Definition 2.6 Let V = F¯ np be an n-dimensional space with the standard action of GLn (F¯ p ). A Levi subgroup L of GLn (F¯ p ) is the simultaneous stabilizer of a collection D1 , . . . , Dk of L subspaces such that V = i Di . If each Di has a basis consisting of standard basis vectors for V , then L is called a standard Levi subgroup. Example 2.7 The standard Levi subgroups of GL2 (F¯ p ) are the subgroup of diagonal matrices and the whole of GL2 (F¯ p ).
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Example 2.8 The standard Levi subgroups of GL3 (F¯ p ) are the subgroup of diagonal matrices, the whole of GL3 (F¯ p ), and the three subgroups ∗ 0 0 0 ∗ ∗ , 0 ∗ ∗
∗ 0 ∗ 0 ∗ 0 , ∗ 0 ∗
∗ ∗ 0 ∗ ∗ 0 . 0 0 ∗
Definition 2.9 Let ρ : G Q → GLn (F¯ p ) be a continuous representation. A standard Levi subgroup L of GLn (F¯ p ) is said to be ρ-minimal if L is minimal among all standard Levi subgroups that contain some conjugate of the image of ρ. Definition 2.10 A semisimple continuous representation ρ : G Q → GLn (F¯ p ) satisfies the strict parity condition with Levi subgroup L if it has the following properties: (1) L is ρ-minimal; (2) the image of complex conjugation is conjugate inside L to a matrix 1 ± −1 .. . with strictly alternating signs on the diagonal. Example 2.11 Any odd irreducible two-dimensional (resp., three-dimensional) representation satisfies strict parity, with L = GL2 (F¯ p ) (resp., L = GL3 (F¯ p )). Example 2.12 Let ρ be the direct sum of a two-dimensional odd irreducible representation and a one-dimensional representation, with image contained inside ∗ 0 0 L = 0 ∗ ∗ 0 ∗ ∗
or
∗ ∗ 0 L = ∗ ∗ 0 . 0 0 ∗
Then ρ satisfies the strict parity condition, with Levi subgroup L. Example 2.13 Let ρ be the direct sum of a two-dimensional even irreducible representation and a
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one-dimensional representation, with the image of ρ contained inside ∗ ∗ L = ∗ . ∗ ∗ Then ρ satisfies strict parity with this Levi subgroup exactly when ρ is odd. Remark 2.14 Note that any odd three-dimensional Galois representation is conjugate to a representation that satisfies the strict parity condition for some standard Levi subgroup L. More generally, if ρ is an n-dimensional representation where the number of +1 eigenvalues and the number of −1 eigenvalues of complex conjugation differ by at most one, then ρ satisfies the strict parity condition for some standard Levi subgroup L. Definition 2.15 If ρ : G Q → GLn (F¯ p ) lands inside a Levi subgroup L, and σ : G Q → GLn (F¯ p ) is another representation of G Q , we say that ρ ∼L σ if there is a matrix M ∈ L such that Mρ(g)M −1 = σ (g) for all g ∈ G Q . If L = GLn (F¯ p ), then we may write ρ ∼ σ. 2.5. Weights We now begin to predict the weights (or irreducible modules) for which we expect to find cohomology eigenclasses with ρ attached. Following the example of Serre’s conjecture, we expect these weights to be determined by the restriction of ρ to a decomposition group at p, so we are interested in studying representations of the decomposition group G p . For convenience we denote the inertia group G p,0 by I p and the wild ramification group G p,1 by Iw . We begin by considering simple representations of G p . LEMMA 2.16 Let V be a simple n-dimensional F¯ p [G p ]-module, with the action of G p given by a
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representation ρ : G p → GL(V ). Then we may choose a basis for V such that ϕ1 .. ρ| I p = , . ϕn
m , . . . , ψ m for some with the characters ϕ1 , . . . , ϕn equal to some permutation of ψn,1 n,n m ∈ Z.
Proof This proof is almost identical to the proof in [13] for two-dimensional representations. We first note that ρ has finite image, so that we may actually realize it over a finite extension of F p . Hence, we may find an F pm [G p ]-module V 0 such that V = V 0 ⊗ F¯ p . We note that Iw must act trivially on V 0 since the invariants V 0 Iw are a nontrivial G p submodule of the simple module V 0 (since the image of Iw under ρ is a p-group). Hence, we may diagonalize ρ| I p . Since the Frobenius acts on the tame inertia as pth powers, we see that the set of diagonal characters must be stable under taking pth powers. Finally, since V is simple, the Frobenius must permute the diagonal characters transitively, resulting in the characterization given above. Remark 2.17 Note that for a given V , Lemma 2.16 yields n distinct values of m modulo ( p n − 1). If m 0 is one of them, the others are congruent to pm 0 , p 2 m 0 , . . . , p n−1 m 0 modulo ( p n − 1). Definition 2.18 Let V be a simple G p -module, diagonalized as in Lemma 2.16 with some choice of exponent m. If possible, write m as m = a1 + a2 p + · · · + an p n−1 , with 0 ≤ ai − an ≤ p − 1 for all i. Suppose that (b1 , . . . , bn ) satisfies bi ≥ bi+1 for all i < n and is obtained by permuting the entries of (a1 , . . . , an ). Then we say that (b1 , . . . , bn ) is derived from V . If the action of G p on V is given by a representation ρ, we say that the n-tuple is derived from ρ. Remark 2.19 Note that not all values of m have an expansion of the form given here. For example, if p = 5, n = 3, m = 30, there is no expansion satisfying the above properties. It is a simple exercise to see that every simple module has at least one derived n-tuple and that a given value of m yields a unique n-tuple if it yields any. Hence, a simple
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n-dimensional G p -module may have at most n n-tuples derived from it, but it can have fewer. Now let V be any n-dimensional G p -module, with the action of G p given by ρ : G p → GL(V ). We may find a composition series {0} = V0 ⊂ V1 ⊂ · · · ⊂ Vk = V. Let each composition factor Vi /Vi−1 have dimension di , and set d0 = 0. By diagonalizing ρ on each simple composition factor, we may find a basis (e1 , . . . , en ) of V such that ρ is upper triangular, with diagonal characters (ϕ1,1 , . . . , ϕ1,d1 , ϕ2,1 , . . . , ϕ2,d2 , . . . , ϕk,1 , . . . , ϕk,dk ), where the first d1 characters come from the action on V1 /V0 , the next d2 from the action on V2 /V1 , and so on. For each composition factor, choose m i such that for some i j, ψdmi ,1 = ϕi, j , and such that m i yields a di -tuple derived from Vi /Vi−1 . Concatenating these di -tuples gives us an n-tuple (a1 , . . . , an ). We wish to preserve the order of the integers in our n-tuple which come from an individual composition factor, so we make the following definition. Definition 2.20 A permutation σ of the integers {1, . . . , n} is compatible with the filtration 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vk = V given above if for 0 ≤ s < k and a, b ∈ [1 + we have σ (a) < σ (b).
Ps
j=0 d j , ds+1
+
Ps
j=0 d j ]
with a < b,
Definition 2.21 Let V be an n-dimensional G p -module with chosen basis {e1 , . . . , en } with respect to which the action of G p is upper triangularized, and let (a1 , . . . , an ) be an ntuple obtained as above. If σ is a permutation of the integers {1, . . . , n} compatible with the filtration above and such that the action of G p with respect to the ordered basis {eσ (1) , . . . , eσ (n) } remains upper triangular, then we say that the n-tuple (aσ (1) , . . . , aσ (n) ) is derived from V . Remark 2.22 Note that there is at least one (and possibly more) n-tuple derived from V , namely, the original n-tuple (a1 , . . . , an ). In addition, even the choice of this original n-tuple is not unique, so that there usually are many n-tuples derived from a given V .
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Definition 2.23 Let ρ : G Q → GLn (F¯ p ) be a semisimple continuous representation, conjugated to land in a ρ-minimal standard Levi subgroup L. Let D1 , . . . , Dk be the subspaces of F¯ np given in the definition of L. Then we have representations ρi : G Q → GL(Di ), which make each Di into a G Q -module. Let G p be a decomposition group above p, and consider each Di as a G p -module. Let di = dim Di , and let (a1 , . . . , adi ) be a di -tuple derived from Di as above. If the standard basis elements of F¯ np which span Di are e jr , 1 ≤ r ≤ di , with jr < js for r < s, then set b jr = ar for r = 1, . . . , di . Doing this for each Di produces an n-tuple (b1 , . . . , bn ). Such an n-tuple is said to be derived from ρ, with Levi subgroup L. Remark 2.24 Note that the above discussion may (in many cases) be summarized more informally as follows. Given a representation ρ : G Q → GLn (F¯ p ) which lands inside a ρminimal standard Levi subgroup L, we may upper triangularize its restriction to inertia by conjugating by an element of L. This gives a sequence of characters of the tame inertia group on the diagonal. Group these characters together into niveau d collections. (A niveau d collection is set of d characters, each a power of a different fundamental character of niveau d with the same exponent m and all appearing in the same “Levi block”.) For a given niveau d collection, write the exponent m as a1 + a2 p + · · · + ad p d−1 , with 0 ≤ ai − ad ≤ p − 1 for all i, and let (b1 , . . . , bd ) be the ordered (decreasing) d-tuple with the same components as (a1 , . . . , ad ). Then construct an n-tuple (c1 , . . . , cn ) as follows: if the ith character in the niveau d collection is in the kth diagonal position in the image of ρ, set ck = bi . (Note that the order of the bi should be preserved in the n-tuple.) This procedure gives the same derived n-tuples as above, except when there is a combination of wild ramification and multiple niveau d collections containing the same characters, in which case the more complicated procedure described above is needed.
3. Conjecture CONJECTURE 3.1 Let ρ : G Q → GLn (F¯ p ) be a continuous semisimple Galois representation. Suppose that ρ satisfies the strict parity condition with Levi subgroup L. Let (a1 , . . . , an ) be an n-tuple derived from ρ with the Levi subgroup L, and let V = F(a1 −(n −1), a2 − (n − 2), . . . , an − 0)0 . Further, let N = N (ρ) be the level of ρ, and let = (ρ) be
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the nebentype character of ρ. Then ρ is attached to a cohomology eigenclass in H ∗ 00 (N ), V () . Remark 3.2 We note that in the case of two-dimensional Galois representations, we may take ∗ to be 0 or 1, and in fact, for irreducible two-dimensional representations, we may take ∗ to be 1. In the case of three-dimensional Galois representations, we may take ∗ to be at most 3, and for irreducible Galois representations (or sums of an even twodimensional representation with a one-dimensional representation) we may take ∗ to be equal to 3, as explained in [4]. As mentioned previously, any odd two-dimensional or three-dimensional representation is conjugate to a representation that satisfies strict parity for some standard Levi subgroup L. In our computations we test the conjecture for three-dimensional representations by computing H 3 . In cases where ρ is the sum of three characters or the sum of an odd two-dimensional representation and a character, we are thus actually testing a stronger assertion than Conjecture 3.1, namely, that the cohomology class exists in H 3 (see, e.g., Tables 4 and 9). We did not test any ρ that are sums of three characters in this paper, but several examples of such may be found in [3] and [1]. In addition, we do not present computational examples for p = 2 as this would involve rewriting portions of our computer programs. In addition, for p = 2 and p = 3, our computational techniques (based on those in [1]) do not always compute the whole of H 3 . Nevertheless, we have no reason to doubt our conjecture for these primes. In particular, problems with the weight and nebentype that occur when p = 2 or p = 3 for Serre’s original conjecture involving classical modular forms modulo p should not occur for our conjecture, which involves mod p cohomology. Remark 3.3 Note that Conjecture 3.1 applies to Galois representations of arbitrary dimension, but that we have no computational evidence for dimension higher than 3. Forthcoming work of Ash with P. Gunnells and M. McConnell touches on the case of certain fourdimensional representations. Remark 3.4 Note that the conjecture makes no claim of predicting all possible weights that yield an eigenclass with ρ attached. In fact, we have three types of computational examples in which additional weights (not predicted by the conjecture) do yield eigenclasses that appear to have ρ attached. The first type of additional weight occurs if ρ is attached to a quasi-cuspidal
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eigenclass (e.g., if ρ is either irreducible or reducible as a sum of an even twodimensional representation and a character). In this case, for certain weights, we may define an extra weight as follows. Definition 3.5 Let F(a, b, c) be an irreducible GLn (F p )-module, with a − c < p − 2. Then we may define ( F p − 2 + c, b, a − ( p − 2) if a ≥ p − 2, M = F(d, e, f ) = F 2( p − 2) + c + 1, b + ( p − 1), a + 1 if a < p − 2. Then we say that M is the extra weight associated to F(a, b, c). Applying [10, Proposition 2.11], it is easy to see that if F(d, e, f ) is the extra weight associated to F(a, b, c), there is an exact sequence 0 → F(d, e, f ) → W (d, e, f ) → F(a, b, c) → 0. Now, suppose that ρ is attached to a quasi-cuspidal homology eigenclass in weight F(a, b, c). Examining the long exact homology sequence associated to this short exact sequence, we find that a quasi-cuspidal eigenclass α in H3 (00 (N ), F(a, b, c)()) (in particular, any eigenclass corresponding to an irreducible Galois representation) either is the image of an eigenclass in H3 (00 (N ), W (d, e, f )()) or has nonzero image β in H2 (00 (N ), F(d, e, f )()). In the second case, β is an eigenclass, and using Theorem 3.10 and Lefschetz duality, we find that there is an eigenclass γ in H3 (00 (N ), F(d, e, f )()) which has the same eigenvalues as α. Hence, for each quasi-cuspidal eigenclass in an appropriate weight there are two possibilities: either the eigenclass lifts to the dual Weyl module, or the eigenclass gives rise to another eigenclass with the same eigenvalues in the extra weight. Our experimental evidence supports the hypothesis that in all such cases a quasi-cuspidal eigenclass gives rise to another eigenclass with the same eigenvalues in the extra weight. The second class of additional weights which we have observed consists of certain weights which would be predicted by our conjecture if we eliminated the strict parity condition. These additional weights have been observed only for representations ρ that are either the sum of three characters or the sum of an odd twodimensional representation and a character. These additional weights seem to occur fairly rarely and sporadically and may be related to the occurrence of eigenclasses in H 2 which have ρ attached. A full investigation of them would require new computational techniques, beyond those developed in this paper. The third class of additional weights consists of extra weights associated to weights that would be predicted by our conjecture but for the strict parity condi-
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tion. As in the second case, these additional weights occur only rarely, and only for reducible ρ. Before beginning to present computational evidence for Conjecture 3.1, we begin by proving several facts about the conjecture. THEOREM 3.6 If Conjecture 3.1 is true for a representation ρ, then it is true for the representation ρ ⊗ ωs , where ω is the cyclotomic character modulo p.
Proof First, note that twisting by ωs does not affect the predicted level or nebentype in any way. Denote the level of ρ by N and the nebentype of ρ by . If ρ has niveau 1, then this is just [4, Proposition 2.6]. For higher niveau representations, we note that twisting by ωs changes the value of m coming from a niveau d character by s(1 + p + · · · + p d−1 ); hence, it changes all the values of ai arising from m by s. Following this change through the permutations involved in deriving an n-tuple, we find that twisting a representation ρ by ωs adds s to each element of a derived n-tuple. This change is then reflected in the predicted weight, and we have that the set of predicted weights for ρ ⊗ ωs is precisely the set of twists by dets of the predicted weights of ρ. Finally, if an eigenclass v shows up in weight V and has ρ attached, then we may consider v as lying in cohomology with weight V ⊗ dets , and we see easily (as in [4]) that in this new cohomology group v has ρ ⊗ ωs attached. Hence, if ρ is attached to a cohomology class in each of the weights predicted by Conjecture 3.1, then ρ ⊗ ωs satisfies the conjecture as well. We now note that there is a correspondence between systems of Hecke eigenvalues arising from modular forms and systems of eigenvalues arising from arithmetic cohomology in characteristic p, similar to that given by the Eichler-Shimura isomorphism in characteristic zero. In particular, we note that by [6, Proposition 2.5], for p > 3, any system of Hecke eigenvalues comes from the mod p reduction of an eigenform of level N , nebentype , and weight k = g + 2 if and only if it comes from a Hecke eigenclass in H 1 (00 (N ), Vg (F¯ p )()), where Vg (F¯ p ) is the gth symmetric power of the standard representation of GL2 (F p ). THEOREM 3.7 If p > 3, Serre’s conjecture implies Conjecture 3.1 for n = 2.
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Proof For a complete description of Serre’s conjecture, including Serre’s prediction of the weight, see [22] or [13]. There are two cases: where ρ is niveau 1 and where ρ is niveau 2. In either case we note that the level and nebentype predicted by Serre’s conjecture are identical to those predicted by Conjecture 3.1, so that we need only deal with the weight. Suppose that ρ : G Q → GL2 (F¯ p ) is odd, semisimple, and has niveau 1. If ρ is reducible, Conjecture 3.1 is true (see [4, Proposition 2.7]), so we may assume that ρ is irreducible. If ρ is tamely ramified, we have a ω 1 , ρ| I p ∼ ω a2 with 0 ≤ a1 , a2 ≤ p − 2. Conjecture 3.1 predicts a weight of F(a1 − 1, a2 )0 . If a2 < a1 , then a −a ω1 2 ρ ⊗ ω−a2 | I p ∼ , ω0 and Serre’s conjecture claims that ρ ⊗ ω−a2 corresponds to a modular form of weight 1 + a1 − a2 or (via [6, Proposition 2.5]) that ρ ⊗ ω−a2 corresponds to a cohomology class with coefficients in F(a1 − a2 − 1, 0). Twisting by ωa2 (which corresponds to twisting the weight by deta2 ), we find that ρ corresponds to a cohomology class with coefficients in F(a1 − 1, a2 ), exactly as predicted by Conjecture 3.1. If a2 ≥ a1 , then p−1+a −a 1 2 ω p−1−a2 ρ⊗ω |I p ∼ , ω0 and by Serre’s conjecture (together with [6, Proposition 2.5]), ρ ⊗ ω p−1−a2 corresponds to a cohomology class with coefficients in F( p − 2 + a1 − a2 , 0). Twisting by ωa2 as before, we find that ρ has weight F(a1 − 1 + ( p − 1), a2 ), exactly as predicted. Now if ρ is wildly ramified at p, then β ω ∗ ρ| I p ∼ , ωα with 0 ≤ α ≤ p − 2 and 1 ≤ β ≤ p − 1, and Conjecture 3.1 predicts a weight of F(β − 1, α)0 . Before applying Serre’s conjecture, we twist ρ by ω−α to obtain β−α ω ∗ −α ρ⊗ω ∼ . ω0 Applying Serre’s conjecture to this representation, we find that it has weight
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1 + (β − α) (i.e., F(β − α − 1, 0)) if β > α + 1, 2 (i.e., F(0, 0)) if β = α + 1 and ρ ⊗ ω−α is peu ramifi´ee, p + 1 (i.e., F( p − 1, 0)) if β = α + 1 and ρ ⊗ ω−α is tr`es ramifi´ee, 1 + (β − α) + ( p − 1) (i.e., F(β − α + ( p − 1) − 1, 0) if β ≤ α. Twisting each of these weights by detα , we find that ρ corresponds to a cohomology class in weight F(β − 1, α)0 in every case. (Note that when β − 1 ≤ α we may add p − 1 to β − 1 to obtain a p-restricted pair.) This proves the theorem in the case when ρ has niveau 1. Suppose that m ψ ρ|I p ∼ , ψ 0m (1) (2) (3) (4)
where m = a + bp, and 0 < a − b ≤ p − 1. (Note that if a = b, we are really in niveau 1.) For simplicity we use the fact that ψ and ψ 0 have order p 2 − 1 to reduce to the case where 0 < m < p 2 − 1, so that b < p − 1. The weight predicted by Conjecture 3.1 is then F(a − 1, b)0 . Now, ! ψ a−b −b ρ ⊗ ω ∼L , ψ 0 a−b so that by Serre’s conjecture ρ ⊗ ω−b corresponds to a cohomology class with coefficients in F(a − b − 1, 0). Twisting by ωb , we see that ρ then corresponds to a cohomology class with coefficients in F(a − 1, b), exactly as predicted. Hence, Serre’s conjecture implies Conjecture 3.1 for n = 2. We now prove a partial converse to Theorem 3.7, which shows that in certain cases Conjecture 3.1 is actually equivalent to Serre’s conjecture. THEOREM 3.8 Assume Conjecture 3.1. Let p > 3, and let ρ : G Q → GL2 (F¯ p ) be a semisimple continuous odd Galois representation. If each weight predicted by Conjecture 3.1 is defined unambiguously, then Serre’s conjecture is true for ρ.
Proof We may clearly assume that ρ : G Q → GL2 (F¯ p ) is irreducible since Serre’s conjecture says nothing about reducible representations. First, note that ρ cannot be attached to any class in H 0 since, according to [2, Theorem 4.1.4], any class in H 0 is a twist of a punctual class, and a punctual class corresponds to a reducible representation by [2, Lemma 4.1.2]. Conjecture 3.1 implies that ρ is attached to an eigenclass in H 1 (00 (N ), V ()), where N , V , and are as predicted in the conjecture. We note that the level and
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nebentype predicted by Conjecture 3.1 are exactly the same as those predicted by Serre’s conjecture. If ρ is tamely ramified and has niveau 1, then we have a ω ρ| I p ∼ , ωb and we may further conjugate ρ so that 0 ≤ b ≤ a < p − 1. The weights predicted by Conjecture 3.1 are then F(a − 1, b)0 and (permuting the diagonal characters) F(b − 1, a)0 . These are defined unambiguously exactly when a 6= b + 1. For a > b + 1, we have that F(a − 1, b) embeds in Va−1+bp , so Conjecture 3.1 predicts that ρ is attached to a cohomology eigenclass in weight Va−1+bp since any system of eigenvalues occurring in a submodule occurs in the containing module (see [4]). This implies (by [6, Proposition 2.5]) that ρ is attached to an eigenform of weight 1+a+bp, which is exactly the weight predicted by Serre’s conjecture. For a = b = 0, the predicted weights for Conjecture 3.1 and Serre’s conjecture are both F( p − 2, 0). For a = b 6= 0, Conjecture 3.1 predicts a weight of F(a − 1 + p − 1, b), while Serre’s conjecture predicts a weight of Vb−1+ pa . Using [10, Table 1] (specifically, the last line, as b 6 = 0), we see that F(a − 1 + p − 1, b) is a subquotient of Vb−1+ pa . Hence, we are finished if we can show that the system of eigenvalues corresponding to ρ in weight F(a − 1 + p − 1, b) also shows up in weight Vb−1+ pa . Lemma 3.9 shows that for GL2 , systems of eigenvalues of eigenclasses that are not twists of punctual classes are inherited from subquotients, so that we are finished. For a tamely ramified niveau 2 representation, the proof is essentially identical— one of the weights predicted in Conjecture 3.1 embeds in the module corresponding to the weight predicted by Serre’s conjecture. If ρ is wildly ramified, then we have α ω ∗ ρ= . ωβ Conjecture 3.1 then predicts a weight of F(α−1, β)0 , which is unambiguously defined as long as α 6 ≡ β + 1 (mod p − 1). In order to apply Serre’s conjecture, we normalize so that 1 ≤ α ≤ p − 1 and 0 ≤ β ≤ p − 2. If α > β and α 6 = β + 1, then Serre’s conjecture predicts a weight of Vα−1+βp , which contains F(α − 1, β) as a submodule; hence we are finished, as before. If α ≤ β, then Serre’s conjecture predicts a weight of 1 + β + pα, and we have F(α − 1, β)0 = F(α − 1 + p − 1, β). Using [10, Table 1] as before, we find that F(α−1+ p−1, β) is a subquotient of Vβ−1+ pα , which is the module corresponding to the weight predicted by Serre’s conjecture. Hence, by Lemma 3.9, we are finished.
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LEMMA 3.9 If α is an eigenclass in H 1 (00 (N ), A), where A is a subquotient of a GL2 (F)-module B, α is not a twist of a punctual eigenclass, and p > 3, then there is an eigenclass in H 1 (00 (N ), B) with the same eigenvalues as α.
Proof Let T ⊂ S ⊂ B, with S/T ∼ = A, and examine the long exact cohomology sequence arising from the short exact sequence 0 → T → S → A → 0. Note that since p > 3, H 2 (00 (N ), T ) = 0, so that the eigenclass α must come from a class σ in H 1 (00 (N ), S). By [5], we may replace σ by an eigenclass having the same eigenvalues as α (calling the new class σ again). The long exact cohomology sequence arising from the short exact sequence 0 → S → B → B/S → 0 then shows that σ goes to a nonzero class β in H 1 (00 (N ), B) since it cannot come from H 0 (00 (N ), B/S) (as it is not a twist of a punctual class). Clearly, β has the same eigenvalues as σ . THEOREM 3.10 Assume that ρ : G Q → GLn (F¯ p ) is attached to an eigenclass α in H i (00 (N ), V ()), where N , , and V are the level, the nebentype, and a weight predicted for ρ. Then ρ ∨ = t ρ −1 is attached to a cohomology class β in H i (00 (N ), W ( −1 )), where W = V ∗ ⊗ det−(n−1) is a twist of the contragredient V ∗ of V . Further, the level, the nebentype, and a weight predicted for ρ ∨ are N , −1 , and W .
Proof The proof that there is a β in the indicated cohomology group with ρ ∨ attached is exactly the same as the proof of [4, Proposition 2.8]. For ρ of niveau 1, Ash and Sinnott also prove that the invariants of ρ ∨ are as above. The level and nebentype computations remain the same regardless of the niveau of the representation, so we need only show that W is a predicted weight for ρ ∨ . We show that if (b1 , . . . , bn ) is a derived n-tuple for ρ, then (−bn , . . . , −b1 ) is a derived n-tuple for ρ ∨ . Then, since (F(α1 , . . . , αn )0 )∗ = F(−αn , . . . , −α1 )0 , it follows that if V is a predicted weight for ρ, then W is a predicted weight for ρ ∨ . It is an easy exercise to reduce the question to simple representations of G p . Suppose that ρ is a simple representation of G p , with the n-tuple (b1 , . . . , bn ) derived
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from it. Then there must be some exponent m such that ϕ1 .. ρ| I p = , . ϕn
m , . . . , ψ m . Then −m is an exponent where (ϕ1 , . . . , ϕn ) is some permutation of ψn,1 n,n ∨ associated to ρ in the same way, as is any multiple of −m by a power of p. Now m = a1 + a2 p + · · · + an p n−1 , where the ai are some permutation of the decreasing n-tuple (b1 , . . . , bn ), with 0 ≤ ai − an ≤ p − 1. Let ak be the largest of the ai , which is equal to b1 . Then − p n−1−k m is congruent (modulo ( p n − 1)) to
−ak+1 − · · · − an p n−2−k − a1 p n−1−k − · · · − ak p n−1 , with 0 ≤ ai − ak ≤ p − 1, so that (−bn , . . . , −b1 ) is easily seen to be an n-tuple associated with ρ ∨ . 3.1. Heuristic for the niveau n case For the most part, we have derived our conjecture using Serre’s conjecture as a model. We can provide a suggestive heuristic for one feature of our conjecture: the weight of a niveau n representation into GLn (F¯ p ). Let ρ : G Q → GLn (F¯ p ) be given such that ρ| I p ∼
ϕ1
..
. ϕn
,
where the ϕi are powers of a fundamental character of niveau n and are conjugate to each other. Let us suppose that ρ lifts to a p-adic representation 2 unramified at almost all primes. Further, suppose that 2 comes from a motive M with good reduction at p, which would conjecturally be the case were 2 attached to an automorphic representation π of cohomological type of level N prime to p (cf. [9]). Then 2 is crystalline. So by analogy it is reasonable to assume that ρ is “crystalline” in the sense of [14], that is, that it corresponds to a filtered Frobenius module for F p . n−1 Now write ϕ1 = ψ a1 +a2 p+···+an p , with 0 ≤ ai ≤ p − 1. By [14, Theorem 0.8], there is indeed a unique filtered Frobenius module 8 over F p which corresponds to a representation of G Q p into GLn (F p ) whose restriction to I p is equivalent to ρ| I p . This is our motivation for choosing the ai in the given range. Assuming again that 2 and M exist, the Hodge numbers of M would be the same as the Hodge-Tate numbers of 2|G Q p , and these in turn would be the same as
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the jumps in the filtration of the filtered Frobenius module associated to 2|G Q p . If we take the latter to be the same as the jumps in 8, they are a1 , . . . , an . Now, suppose we are in the generic case; that is, suppose that |ai − a j | > 1 for i 6 = j. Let {b1 , . . . , bn } = {a1 , . . . , an }, with b1 > b2 > · · · > bn . Assuming the general picture of L. Clozel (following R. Langlands) of the relationship between automorphic representations and motives, as found in [9], especially Chapters 3 and 4, the motive M predicts the existence of an automorphic representation π attached to M such that π∞ ⊗ W has (g, K )-cohomology, where W is the irreducible representation of GLn (C) with highest weight (b1 − (n − 1), b2 − (n − 2), . . . , bn ). By analogy, we conjecture that ρ will be attached to a cohomology class with weight V = F(b1 − (n − 1), b2 − (n − 2), . . . , bn ). After all, ρ is the reduction of 2 modulo p, and W mod p (or, more precisely, the reduction modulo p of a model for W over Z p ) has V as a composition factor. If we now require our conjecture to be closed under twisting by powers of ω, a simple exercise yields the weights predicted by Conjecture 3.1 for niveau n, dimension n, in the generic case. By “continuity” we extend the heuristic to the nongeneric case. 4. Symmetric squares Using work of Ash and P. Tiep [7], who proved that certain Galois representations are in fact attached to cohomology eigenclasses, we are able to verify certain special cases of Conjecture 3.1. THEOREM 4.1 Let σ : G Q → GL2 (F p ) be a continuous irreducible odd Galois representation ramified only at p. Assume that Serre’s conjecture is true for σ , and let k be the weight predicted by Serre’s conjecture. Then if 2 < k < ( p+3)/2 and Sym2 σ is irreducible, Sym2 σ is attached to a cohomology eigenclass in weight F(2(k − 2), k − 2, 0), and this weight is predicted by Conjecture 3.1.
Proof By [6, Proposition 2.5], we see that σ is attached to a cohomology eigenclass in H 1 (SL2 (Z), Uh (F¯ p )), where h = k − 2 and Uh (F¯ p ) = Symh (F¯ 2p ), with the standard action of GL2 (F¯ p ) on F¯ 2p . Then, by [7, Corollary 5.3], Sym2 σ is attached to a cohomology eigenclass in H 3 (SL3 (Z), F(2h, h, 0)). Hence, we need only show that the weight F(2h, h, 0) is predicted by Conjecture 3.1. If σ has niveau 1, this is trivial since we must have k−1 ω ∗ σ |I p ∼ , 1
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2(k−1) ω ∼
∗ ∗ . 1
∗ ωk−1
If σ has niveau 2, then we must have σ |I p ∼
!
ψ k−1
with 1 ≤ k − 1 ≤ ( p − 1)/2, so that ψ 2(k−1) 2 Sym σ | I p ∼
ψ 0 k−1
,
ωk−1 ψ 0 2(k−1)
,
with 2 ≤ 2(k − 1) ≤ p − 1. Clearly, a predicted weight for this representation is F(2(k − 2), k − 2, 0). Example 4.2 Let K be a totally complex S4 -extension of Q, such that the quartic subfield of K has discriminant p 3 , where p is a prime congruent to 5 mod 8 (for examples of such fields, see [11]). The unique three-dimensional irreducible unimodular mod p representation of S4 gives rise to an irreducible unimodular representation ρ : G Q → GL3 (F p ) which is ramified only at p. This representation is (up to a twist by a power of the cyclotomic character) the symmetric square of a two-dimensional irreducible representation σ : G Q → GL2 (F¯ p ) with projective image isomorphic to S4 and image of order 96 (see [20]). Serre’s conjecture is true for σ since σ has a lift to a twodimensional irreducible complex Galois representation with solvable image to which we apply the theorem of Langlands and J. Tunnell [24]. Hence, σ is modular and so, by the -conjecture, Serre’s conjecture holds for σ (see [11] for more details). One easily checks that σ has niveau 1 and that the weight predicted by Serre’s conjecture for σ is ( p + 3)/4, so that Theorem 4.1 applies. Hence, at least one of the weights predicted for ρ by Conjecture 3.1 yields an eigenclass with ρ attached. In fact, this weight is F(( p − 5)/2, ( p − 5)/4, 0) ⊗ det3( p−1)/4 . Example 4.3 Let K be a totally complex S4 -extension of Q such that the quartic subfield of K has discriminant − p, where p is a prime congruent to 3 mod 8. Let ρ be the unimodular irreducible three-dimensional Galois representation associated to K as above. Again, there is a two-dimensional irreducible representation σ : G Q → GL2 (F p ), with projective image isomorphic to S4 , such that σ is ramified only at p (see [20]; this time
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the image of σ has order 48) and (again up to a twist by a power of the cyclotomic character) ρ is the symmetric square of σ . (Note that up to twisting, the symmetric square depends only on the projectivization of a representation.) One checks easily that Serre’s conjecture predicts a weight of ( p + 1)/2 for σ (again σ has niveau 1) and that (just as above) Serre’s conjecture is true for σ . Hence, one of the weights predicted by Conjecture 3.1 does in fact contain an eigenclass with ρ attached. In this case, the weight is F( p − 3, ( p − 3)/2, 0) ⊗ det( p−1)/2 . Example 4.4 Let K be a complex S4 -extension of Q with K ramified at only one prime p, with p congruent to 3 modulo 8, and with ramification index at p equal to 4 (for examples of such extensions, see [11]). Let ρ be the unique unimodular irreducible three-dimensional mod p Galois representation with image isomorphic to S4 and such that the fixed field of the kernel of ρ is K . Then, up to twisting, ρ is the symmetric square of a representation σ : G Q → GL2 (F p ) with image isomorphic to S˜4 (i.e., isomorphic to GL2 (F3 )). In this case, σ has niveau 2, Serre’s conjecture is true for σ and its twists, and a twist of σ has weight ( p + 5)/4 (see [11]), so that Theorem 4.1 applies. Hence, one of the weights predicted for ρ gives a cohomology group that contains an eigenclass predicted for ρ. In this case, the weight that works is F(( p − 3)/2, ( p − 3)/4, 0) ⊗ det(3 p−5)/4 .
5. Niveau 1 representations 5.1. Reducible representations in level 1 In [4] Ash and Sinnott dealt extensively with reducible representations ramified at only one prime. Each of their examples was a direct sum of an even two-dimensional representation with a one-dimensional representation, and they included cases where the two-dimensional representation had image isomorphic to a dihedral group or projective image isomorphic to A4 . They did not give examples in which the projective image was isomorphic to S4 or A5 . We recall their construction from [4]. Let σ : G Q → GL2 (F¯ p ) be an irreducible representation with the following properties: (1) σ is unramified outside p; (2) the image of σ has order relatively prime to p; (3) σ (Frob∞ ) is central, where Frob∞ is a complex conjugation in G Q ; (4) σ (G p,0 ) has order dividing p − 1.
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Then, choosing integers j and k appropriately, we find that the representation ρ = (σ ⊗ ω j ) ⊕ ωk is three-dimensional and odd, and by adjusting j and k, we may adjust the predicted weight of ρ to some extent. In particular, we need to choose j and k to have opposite parity if σ (Frob∞ ) = 1 and the same parity if σ (Frob∞ ) = −1. In addition, we choose j and k to give the simplest possible weight. The reducibility of these representations makes it possible to reduce the weight to calculable levels; however, in the examples that we consider here the weight is still quite high. Hence, rather than being able to calculate many Hecke eigenvalues, we found it impractical to calculate more eigenvalues than those at 2 and 3, due to time constraints. We begin by specifying the fixed field of the kernel of the projective image of σ , which is a totally real number field. 5.1.1. Representations of type A4 In [4] Ash and Sinnott presented several examples of reducible Galois representations that are sums of one-dimensional characters with even two-dimensional representations having projective image isomorphic to A4 . Using the same computational techniques as in [1], we have been able to find other A4 -extensions for which we can compute the predicted quasi-cuspidal homology classes. These examples are given in Table 1. We begin with a quartic polynomial f that has four real roots and whose splitting field K is an A4 -extension of Q ramified only at one prime p. We know (by [4, Lemma 4.1]) that K sits inside an Aˆ 4 -extension Kˆ of Q, with Kˆ /Q ramified only at p. In fact, there are two possibilities for Kˆ ; following [4], we take Kˆ to be the one that has ramification index 3 at p. Let K 4 be the quartic extension of Q defined by f . We note that K 4 must be contained in an octic subextension K 8 of Kˆ , with K 8 /K 4 unramified at all finite primes. Since K 8 has Kˆ as its Galois closure, we may determine whether Kˆ is totally real or totally complex by comparing the two-ranks of the class group and the narrow class group of K 4 . For instance, when p = 1009, the class number of K 4 is two, and the narrow class group is cyclic of order four. Thus, the two class groups have the same two-rank, so Kˆ must be real (since K 8 and all its conjugates must be real). If Kˆ is totally real, we write its sign as 1; otherwise its sign is −1. Now Aˆ 4 has a unique two-dimensional irreducible unimodular mod p representation σ : G Q → GL2 (F p ). We see easily that σ | I p = ωd ⊕ ω−d with d = ( p − 1)/3. We now take ρ = (σ ⊗ ω j ) ⊕ ωk with j = 2d and k = 1 if the sign of Kˆ is 1, and k = 2 otherwise. We note that ρ satisfies the conditions of the construction of [4] and has a predicted weight of F(( p − 1)/3 − 2, 0, 0) if k = 1 and F(( p − 1)/3 − 2, 1, 0) if k = 2. For each of the examples in Table 1, we have calculated the interior homology of
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Table 1. Reducible representations of type A4
Polynomial x 4 − 2x 3 − 13x 2 − 9x + 4 x 4 − x 3 − 16x 2 + 3x + 1 x 4 − x 3 − 10x 2 + 3x + 20 x 4 − x 3 − 13x 2 + 12x + 16 x 4 − 2x 3 − 19x 2 + 29x + 1 x 4 − 2x 3 − 31x 2 − 51x − 4 x 4 − 2x 3 − 39x 2 + x + 125 x 4 − 2x 3 − 51x 2 + 100x + 83 x 4 − 2x 3 − 51x 2 + 32x + 192 x 4 − 2x 3 − 37x 2 + 10x + 29 x 4 − 2x 3 − 43x 2 + 127x − 55
Sign −1 1 −1 −1 −1 1 1 1 1 1 1
p 163 277 349 397 547 607 1009 1399 1699 1777 1951
k 2 1 2 2 2 1 1 1 1 1 1
Weight F(52, 1, 0) F(90, 0, 0) F(114, 1, 0) F(130, 1, 0) F(180, 1, 0) F(200, 0, 0) F(334, 0, 0) F(464, 0, 0) F(564, 0, 0) F(590, 0, 0) F(648, 0, 0)
SL3 (Z) in the given weight using the techniques described in [1] and found it to be one-dimensional. We have also calculated the Hecke eigenvalues at 2 and 3 and found that they exactly match the values predicted from the characteristic polynomial of the image of Frobenius under ρ by Conjecture 3.1. 5.1.2. Representations of type S4 Totally real S4 -extensions ramified at only one prime can have two types of ramification; either the ramification index is 2, or the ramification index is 4. For our purposes, the extensions with ramification index 4 are better (since they yield lower weights), although they are more difficult to find. They can, however, be found by application of explicit class field theory, and many such examples are known. Only the two below yield predicted weights that are feasible for computation. Example 5.1 Let K be the splitting field of the polynomial x 4 −x 3 −1017x 2 +9665x +60608. Then K is a totally real S4 -extension of Q, ramified only at p = 2713, with ramification index e = 4. Let S˜4 be the central extension of S4 by Z/2Z which is isomorphic to GL2 (F3 ). Then K embeds in an S˜4 -extension K˜ of Q (by [4, Lemma 4.1]), and K˜ /K must further ramify at p (as described in [11]), so that in K˜ , p = 2713 has e = 8. We need to determine whether K˜ is totally real or totally complex. To do this, we note that S˜4 has three conjugacy classes of subgroups of order 6 and that each subgroup of order 12 contains exactly one subgroup of order 6 from each conjugacy class. In terms of field extensions then, each subfield of K˜ of degree 4 has exactly three
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quadratic extensions lying in K˜ . Hence, if K˜ is totally real, the degree 4 subfield K 4 of K must have a Klein four extension contained inside K˜ , hence ramified only at p (in particular, not ramified at infinity). Such an extension would lie inside the ray class field of K 4 modulo pm (where p is the unique prime of K 4 lying over p). However, the two-part of the ray class group of K 4 modulo pm is cyclic for every m (see [17]). Hence, K˜ must be totally complex. Now, we let σ be the two-dimensional representation σ : G Q → GL2 (F¯ p ), with image isomorphic to S˜4 and kernel equal to G K˜ , chosen such that σ |I p =
3( p−1)/8 ω
ω( p−1)/8
=
3(339) ω
ω339
.
Taking j = −339, k = 1 (with the same parity since σ (Frob∞ ) = −1), we see that ρ = (σ ⊗ ω j ) ⊕ ωk has 678 ω , ρ| I p ∼ L ω1 0 ω where we take ∗ ∗ L = ∗ . ∗ ∗ Then the weight predicted by Conjecture 3.1 is F(678 − 2, 1 − 1, 0)0 = F(676, 0, 0), the level is 1, and the nebentype is trivial. Computations using the techniques of [1] show that the interior cohomology is in fact one-dimensional. The Hecke eigenvalues at 2 and 3 correspond exactly to σ , as predicted by Conjecture 3.1. A similar construction can be performed with the splitting field K of the polynomial x 4 − 6668x 3 + 16598046x 2 − 18278822428x + 7514424150025, which is a totally real S4 -extension of Q ramified only at p = 3137. In this case, K˜ is totally real, and the predicted weight is F(782, 0, 0). Again, the homology is one-dimensional and the eigenvalues at 2 match ρ. The image of the Frobenius at 3 is of order 8, however, and presents some difficulty. We have determined σ (and hence also ρ) by a local condition at p, namely, its restriction to inertia at I p . Determining the Frobenius at 3 is a local condition at 3, and combining these two determinations in order to determine Tr(ρ(Frob3 )) is a global problem that involves calculations in a large number field. We thus have not distinguished between two possibilities for eigenvalues at 3 which would correspond to ρ. One of these possibilities does in fact occur in the predicted cohomology.
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5.1.3. Representations of type A5 Ash and Sinnott’s construction works best with A5 -extensions if the ramification index of the unique ramified prime is as large as possible. However, totally real A5 extensions of Q ramified at only one prime with ramification index 5 are quite difficult to find. D. Doud thanks S. Harding for showing him the second example below with p = 3821. Example 5.2 Let K be the splitting field of the polynomial f = x 5 − 7402x 3 − 3701x 2 + 14804x + 11103. Then K is a totally real Galois extension of Q, with Galois group A5 , ramified only at p = 3701. K must lie inside an extension Kˆ of Q with Galois group Aˆ 5 (the unique nonsplit central extension of A5 by Z/2Z). In fact, K lies inside two such extensions, one in which primes above p ramify further, and one in which primes above p do not ramify further. Let Kˆ be an Aˆ 5 -extension of Q containing K , in which p has ramification index 5. Let H be a subgroup of Aˆ 5 of order 20. Using the computer algebra system Magma, one can see that H has a quotient group that is cyclic of order 4. Hence, the degree 6 subextension of K must have a cyclic quartic extension contained in Kˆ which is unramified at all finite primes. A defining polynomial for the degree 6 subextension of K may be found as the minimal polynomial of the element α1 α2 + α2 α3 + α3 α4 + α4 α5 + α5 α1 , where αi , 1 ≤ i ≤ 5, are the roots of f . Using PARI/GP (see [18]) to compute the ideal class group and the narrow class group, we find that both are cyclic of order 4. Hence, the only possible cyclic quartic extension of the degree 6 subfield of K which is unramified at all finite primes is also unramified at infinity, so that Kˆ is totally real. Now Aˆ 5 has two two-dimensional mod p representations. Call them σ and σ 0 . On inertia at p = 3701, we may choose σ and σ 0 such that 3( p−1)/5 ( p−1)/5 ω ω 0 σ |I p ∼ and σ |I p ∼ . ω2( p−1)/5 ω−( p−1)/5 If we let ρ = (σ ⊗ω−2( p−1)/5 )⊕ω, then ρ is an odd three-dimensional representation, and if it is conjugated to land inside ∗ ∗ L = ∗ , ∗ ∗
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then it satisfies strict parity. We then have ( p−1)/5 ω ρ| I p ∼ L
547
,
ω 1
giving a predicted weight of F(( p − 1)/5 − 2, 0, 0) = F(738, 0, 0). We may calculate the quasi-cuspidal homology in this weight and find that it is one-dimensional and has the appropriate eigenvalues at 2 and 3 to correspond to ρ. In this case, there is an ambiguity similar to that in the preceding example, in that we have not determined which of the two conjugacy classes of order 5 contains the Frobenius at 2. The computed eigenvalues at 2 are in fact one of the two possible pairs of values. A similar calculation may be carried out for the A5 -extension defined by the polynomial x 5 − 3821x 3 − 3821x 2 + 3821x + 3821 and ramified only at p = 3821. In this case Kˆ is again totally real. Hence, as above, we get a predicted weight for ρ of F(( p − 1)/5 − 2, 0, 0) = F(762, 0, 0). Calculating the quasi-cuspidal homology in this weight yields a one-dimensional space, which has appropriate eigenvalues at 2 and 3 to correspond to ρ, with the ambiguity that we have not determined the conjugacy class of elements of order 10 (resp., 5) containing the Frobenius at 2 (resp., 3), just as in the previous examples. 5.2. Reducible representations in higher level With the introduction of levels higher than one, we gain immensely in reducing the weight of the representations that we can find. In particular, we find that we can actually compute “companion forms,” or classes with different weights, attached to the same representation. These offer important examples of Conjecture 3.1. We work out one interesting example in full detail and describe others in a table format. Example 5.3 Let K be the S3 -extension of Q given as the splitting field of the polynomial x 3 − x 2 − 3x + 1. Then K is ramified only at p = 37 (with ramification index 2) and at q = 2 (with ramification index 3). Since S3 has a two-dimensional mod 37 representation, we obtain a two-dimensional Galois representation σ : G Q → GL2 (F37 ), with the fixed field of the kernel of σ equal to K . Let ω be the cyclotomic character modulo 37, and let ρ = σ ⊕ ω. We note that σ is an even representation since K is totally real, so we want to conjugate ρ to land inside the Levi subgroup ∗ ∗ L = ∗ . ∗ ∗
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Table 2 Eigenvalues a(`, 1) a(`, 2)
2 * *
3 2 24
5 5 22
7 6 15
11 10 26
13 13 17
17 17 13
19 19 35
23 23 8
29 29 29
31 31 31
37 1 1
41 3 27
43 6 6
47 9 25
Now inside L, the image of complex conjugation is conjugate to the matrix 1 −1
, 1
so that ρ satisfies strict parity. One sees easily that the level of ρ is equal to the level of σ , which is 22 (since the ramification at 2 is tame and the image of inertia at 2 under σ does not fix a subspace). The nebentype of ρ is trivial since the determinant is just ω19 . Finally, if we examine the restriction of ρ to inertia at 37, we find that 18 ω . ρ| I37 ∼ L ω1 ω0 Thus, the weight predicted by Conjecture 3.1 is F(18 − 2, 1 − 1, 0)0 = F(16, 0, 0). When we compute the cohomology in this weight, in level 4 with trivial nebentype, we obtain a fifteen-dimensional space containing a one-dimensional eigenspace with eigenvalues given by Table 2. We now compute the trace Tr(ρ(Frob` )) and T2 (ρ(Frob` )) (the sum of products of pairs of eigenvalues) for ` between 2 and 47. To do this, we note that the characteristic polynomial of ρ(Frob` ) is det I − xρ(Frob` ) = det I − xσ (Frob` ) 1 − xω(Frob` ) = 1 − Tr(σ (Frob` ))x + det(σ (Frob` ))x 2 (1 − `x) = 1 − Tr(σ (Frob` )) + ` x + det(σ (Frob` )) + Tr(σ (Frob` ))` x 2 − det σ (Frob` ) `x 3 ,
so that the trace of ρ(Frob` ) is Tr(σ (Frob` ))+` and T2 (ρ(Frob` )) = det(σ (Frob` ))+ Tr(σ (Frob` ))`. Using PARI/GP, we may calculate these two values for ` from 2 to 47 (excluding the ramified primes 2 and 37), and we find that they exactly match the values of a(`, 1) and `a(`, 2) calculated above.
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Other reducible examples are easily computed just as above. In each row of Table 3, we give a polynomial whose splitting field K is a totally real Galois extension of Q with Galois group G, such that G has a unique two-dimensional representation σ modulo p. We also give the predicted weight(s), level, and nebentype of the cohomology classes corresponding to ρ = σ ⊕ ω. Several examples have more than one predicted weight, coming from multiple orderings of the diagonal characters. Such predictions actually occur in all of these examples, but most are too large for us to calculate and hence do not appear in this table. For all of the examples in this table, all Hecke eigenvalues up to ` = 47 coincide exactly with the coefficients of the characteristic polynomial of the image of Frobenius, as predicted by Conjecture 3.1. We may also apply Conjecture 3.1 to reducible representations that are the sum of an odd two-dimensional representation and a character. In order to satisfy strict parity, such a representation must land inside a Levi subgroup of the form ∗ ∗ ∗ L = ∗ ∗ or L = ∗ ∗ . ∗ ∗ ∗ For each such three-dimensional ρ, we thus have four predicted weights, two from each choice of Levi subgroup. In Table 4, for each example we give a polynomial f that has Galois group G = S3 or D4 , together with a prime p and the ramification index of p in the splitting field K of f . If we let σ be the unique two-dimensional mod p Galois representation arising from K , and ρ = σ ⊕ ω0 , we also give the level N and nebentype associated to ρ, and the set of predicted weights arising from Conjecture 3.1. In this case we are able to compute with all the predicted weights, and we find that in every case an eigenclass with the correct eigenvalues (up to ` = 47) appears in every predicted weight. The last examples in the table, in which σ has image isomorphic to D4 (the dihedral group with 8 elements), are interesting in that fewer than four weights are predicted. In these cases the four predicted weights are not distinct, so that the total number of weights in which we expect to find eigenvalues with ρ attached is less than four. For instance, in the last example in Table 4, in which p = 5, the image of inertia at 5 is contained in the center of D4 , so that the restriction of σ to inertia at 5 has diagonal characters ω2 and ω2 . The coincidence of these diagonal characters results in the fact that only two distinct weights are predicted. 5.3. Irreducible representations in higher level In order to find irreducible three-dimensional Galois representations, it is necessary to find Galois groups that have irreducible three-dimensional mod p representations. For p larger than 3 this is easily done: the groups A4 , S4 , and A5 all have threedimensional irreducible mod p representations. We thus concentrate primarily on rep-
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Table 3. Reducible higher-level niveau 1 examples (even two-dimensional plus ω1 )
Polynomial x 3 − x 2 − 3x + 1 x 3 − x 2 − 4x + 2 x 3 − x 2 − 5x − 1 x 3 − x 2 − 4x + 1 x 3 − x 2 − 5x + 4 x 3 − 5x − 1 x 3 − 7x − 5
G S3 S3 S3 S3 S3 S3 S3 S3
x 3 − x 2 − 6x + 5 x 3 − 7x − 1 x 3 − x 2 − 9x + 8 x 4 − x 3 − 3x 2 + x + 1
S3 S3 S3 D4
x 4 − x 3 − 5x 2 + 2x + 4
D4
x 4 − 2x 3 − 4x 2 + 5x + 5
D4
x 4 − x 3 − 7x 2 + 3x + 9
D4
x 4 − 2x 3 − 4x 2 + 5x + 2
D4
x 4 − x 3 − 6x 2 + 8x − 1
D4
x 4 − x 3 − 5x 2 + x + 1
D4
p 37 79 101 107 67 43 11 41 17 5 5 7 5 29 5 89 5 101 5 181 17 53 13 61 13 53
Weight(s) F(16, 0, 0) F(37, 0, 0) F(48, 0, 0) F(51, 0, 0) F(31, 0, 0) F(19, 0, 0) F(3, 0, 0) F(18, 0, 0) F(6, 0, 0) F(0, 0, 0) F(0, 0, 0) F(8, 6, 2), F(6, 6, 4) F(0, 0, 0), F(6, 4, 2) F(12, 0, 0) F(0, 0, 0), F(6, 4, 2) F(42, 0, 0) F(0, 0, 0), F(6, 4, 2) F(48, 0, 0) F(0, 0, 0) F(88, 0, 0) F(6, 0, 0) F(24, 0, 0) F(4, 0, 0) F(28, 0, 0) F(4, 0, 0) F(24, 0, 0)
Level 4 4 4 3 7 11 43 17 41 157 269 53 29 5 89 5 101 5 181 5 53 17 61 13 53 13
1 4 1 3 7 11 43 17 41 157 269 53 29 5 89 5 101 5 181 5 53 17 61 13 53 13
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Table 4. Reducible higher-level niveau 1 examples (odd two-dimensional plus ω0 )
Galois representation p = 7, e = 3 N = 19, = 19 G = S3 x 3 − x 2 + 5x − 6 p = 7, e = 3 N = 47, = 47 G = S3 x 3 − x 2 − 2x − 27 p = 7, e = 3 N = 59, = 59 G = S3 x 3 − x 2 + 5x + 8 p = 7, e = 3 N = 59, = 59 G = S3 x 3 − x 2 − 9x + 36 p = 7, e = 3 N = 59, = 59 G = S3 x 3 − x 2 − 2x − 20 p = 19, e = 3 N = 3, = 3 G = S3 x 3 − x 2 − 6x − 12 p = 13, e = 3 N = 43, = 43 G = S3 x 3 − x 2 − 17x + 38 p = 3, e = 2 N = 13, = 13 G = D4 x4 + x2 − 3 p = 3, e = 2 N = 37, = 37 G = D4 x 4 + 5x 2 − 3 p = 3, e = 2 N = 61, = 61 G = D4 x 4 − 7x 2 − 3 p = 3, e = 2 N = 73, = 73 G = D4 x 4 + 34x 2 − 3 p = 5, e = 2 N = 39, = 3 13 G = D4 x 4 − x 3 − 8x − 1
Weights F(2, 1, 0), F(4, 3, 2) F(6, 3, 0), F(10, 7, 4) F(2, 1, 0), F(4, 3, 2) F(6, 3, 0), F(10, 7, 4) F(2, 1, 0), F(4, 3, 2) F(6, 3, 0), F(10, 7, 4) F(2, 1, 0), F(4, 3, 2) F(6, 3, 0), F(10, 7, 4) F(2, 1, 0), F(4, 3, 2) F(6, 3, 0), F(10, 7, 4) F(10, 5, 0), F(16, 11, 6) F(22, 11, 0), F(34, 23, 12) F(6, 3, 0), F(10, 7, 4) F(14, 7, 0), F(22, 15, 8) F(2, 1, 1), F(1, 1, 0) F(0, 0, 0) F(2, 1, 1), F(1, 1, 0) F(0, 0, 0) F(2, 1, 1), F(1, 1, 0) F(0, 0, 0) F(2, 1, 1), F(1, 1, 0) F(0, 0, 0) F(6, 5, 2) F(4, 1, 0)
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Table 5
Triple (2( p − 1)/3, ( p − 1)/3, 0) (( p − 1)/3, 0, 2( p − 1)/3) (0, 2( p − 1)/3, ( p − 1)/3) (( p − 1)/3, 2( p − 1)/3, 0) (2( p − 1)/3, 0, ( p − 1)/3) (0, ( p − 1)/3, 2( p − 1)/3)
Weight F(2( p − 4)/3, ( p − 4)/3, 0) F(2( p − 4)/3, ( p − 4)/3, 0) ⊗ det2( p−1)/3 F(2( p − 4)/3, ( p − 4)/3, 0) ⊗ det( p−1)/3 F(2(2 p − 5)/3, (2 p − 5)/3, 0) F(2(2 p − 5)/3, (2 p − 5)/3, 0) ⊗ det( p−1)/3 F(2(2 p − 5)/3, (2 p − 5)/3, 0) ⊗ det2( p−1)/3
resentations (up to a twist) whose images are isomorphic to one of these groups. Of course, we deal only with odd representations. For all the irreducible niveau 1 representations presented in this section, the three-dimensional Galois representation is a symmetric square of an odd two-dimensional representation; hence the correspondences presented here are not native three-dimensional phenomena. 5.3.1. Representations of type A4 Suppose that p is a prime congruent to 1 mod 3 and that K is a totally complex A4 -extension ramified at p, with ramification index 3. There may be other ramified primes, which would then contribute to the level. Since A4 has an irreducible 3-dimensional mod p representation, we obtain an irreducible three-dimensional representation ρ : G Q → GL3 (F p ). We observe that the restriction of ρ to inertia at p is a ω , ρ| I p = ωb c ω where (a, b, c) is some permutation of (2( p − 1)/3, ( p − 1)/3, 0). The six permutations of (2( p − 1)/3, ( p − 1)/3, 0) then give six predicted weights for ρ. The six weights are displayed in Table 5. Hence, we expect to find three cohomology eigenclasses, each with one of ρ, ρ ⊗ ω( p−1)/3 , and ρ ⊗ ω2( p−1)/3 attached, in each of the two weights F(2( p − 4)/3, ( p − 4)/3, 0) and F(2(2 p − 5)/3, (2 p − 5)/3, 0). In fact, however, since ρ ⊗ ω( p−1)/3 ∼ ρ, the three eigenclasses may coincide, and there may actually be only one such eigenclass in each weight. In practice, in order to compute the cohomology associated to a representation as above, we often have to twist by a character that is unramified at p in order to reduce the level. We illustrate with an example. Example 5.4 Let K be the splitting field of the polynomial x 4 −x 3 +5x 2 −4x +3, which is ramified
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Table 6. Irreducible higher level niveau 1 examples Galois representation p = 7, e = 3 N = 13, = 13 G = A4 , χ = 13 x 4 − x 3 + 5x 2 − 4x + 3 p = 7, e = 3 N = 29, = 29 G = A4 , χ = 29 x 4 − x 3 + 5x 2 − 6x + 7 p = 7, e = 3 N = 26 , = 1 G = A4 , χ = 1 x 4 − 2x 3 + 2x 2 + 2 p = 13, e = 3 N = 5, = 5 G = A4 , χ = 5 x 4 − x 3 − 3x + 4 p = 13, e = 3 N = 52 , = 1 G = A4 , χ = 1 x 4 − x 3 − 3x + 4 p = 19, e = 3 N = 7, = 7 G = A4 , χ = 7 x 4 + 3x 2 − 7x + 4 p = 19, e = 3 N = 11, = 11 G = A4 , χ = 11 x 4 + 15x 2 − 11x + 81 p = 7, e = 3 N = 53, = 53 G = S4 , χ = 1 x 4 − x 3 + 4x 2 + 1 p = 13, e = 4 N = 19, = 19 G = S4 , χ = 1 x 4 − x 3 + 2x 2 + 4x − 88 p = 7, e = 3 N = 73, = 73 G = A5 , χ = 73 x 5 − 5x 3 − x 2 + 9x + 7
Predicted weights F(2, 1, 0), F(4, 3, 2), F(6, 5, 4) F(6, 3, 0), F(8, 5, 2), F(10, 7, 4) F(2, 1, 0), F(4, 3, 2), F(6, 5, 4) F(6, 3, 0), F(8, 5, 2), F(10, 7, 4) F(2, 1, 0), F(4, 3, 2), F(6, 5, 4) F(6, 3, 0), F(8, 5, 2), F(10, 7, 4) F(6, 3, 0), F(10, 7, 4), F(14, 11, 8) F(14, 7, 0), F(18, 11, 4), F(22, 15, 8) F(6, 3, 0), F(10, 7, 4), F(14, 11, 8) F(14, 7, 0), F(18, 11, 4), F(22, 15, 8) F(10, 5, 0), F(16, 11, 6), F(22, 17, 12) F(22, 11, 0), F(28, 17, 6), F(34, 23, 12) F(10, 5, 0), F(16, 11, 6), F(22, 17, 12) F(22, 11, 0), F(28, 17, 6), F(34, 23, 12) F(2, 1, 0), F(4, 3, 2), F(6, 5, 4) F(6, 3, 0), F(8, 5, 2), F(10, 7, 4) F(7, 5, 3), F(13, 8, 6), F(16, 14, 9) F(16, 8, 3), F(19, 14, 6), F(25, 17, 9) F(2, 1, 0), F(4, 3, 2), F(6, 5, 4) F(6, 3, 0), F(8, 5, 2), F(10, 7, 4)
at p = 7 (with e = 3) and at 13 (with e = 2). The predicted weights are F(2, 1, 0) and F(6, 3, 0). The level of ρ is 132 , and the nebentype is trivial. Unfortunately, this level is too large for us to use in computations. However, ρ ⊗13 is easily seen to have level 13 and nebentype 13 . Thus, we predict the existence of cohomology eigenclasses in weights F(2, 1, 0) and F(6, 3, 0), level 13, and nebentype 13 , which are attached to ρ ⊗ . Direct computation shows that these eigenclasses do in fact exist and that the eigenvalues match, at least up to ` = 47. Other A4 -extensions that give rise to computable cohomology classes are shown in Table 6. Each example in this table gives a polynomial f with Galois group G. The prime p is given, together with its ramification index e in the splitting field of f . When G equals A4 , ρ is the twist by the character χ of the unique irreducible threedimensional mod p representation of G Q cutting out the splitting field of f . The level N , nebentype , and predicted weights for ρ are indicated in the table. In all cases, we have computationally verified the existence of an eigenclass in the predicted weight, level, and character, with the correct eigenvalues (up to ` = 47) to have ρ attached.
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5.3.2. Representations of type S4 For p > 3, S4 has two absolutely irreducible three-dimensional representations defined over F p . Hence, by finding extensions K /Q with Galois group S4 , we may easily construct irreducible three-dimensional Galois representations that have image isomorphic to S4 . Two such examples are given in Table 6. Here the format is as in the A4 case, except that we take ρ to be the unique irreducible three-dimensional representation of G Q cutting out the splitting field of f and taking transpositions to elements of trace 1. In both of these cases, the twisting character χ is trivial. 5.3.3. Representations of type A5 The group A5 has two three-dimensional irreducible representations defined over F¯ p , for each p > 5. By composing these representations with the projection G Q → Gal(K /Q), where K is a field with Galois group A5 , we obtain irreducible threedimensional Galois representations with image isomorphic to A5 . We give one example in Table 6, which we now explain in detail. Example 5.5 Let K be the splitting field of the polynomial x 5 −5x 3 − x 2 +9x +7. Then Gal(K /Q) is isomorphic to A5 , and K is ramified only at p = 7 (with ramification index 3) and at 73 (with ramification index 2). Let ρ1 and ρ10 be the two characteristic 7 Galois representations alluded to above. Then it is easy to see that ρ1 and ρ10 are Galois conjugates of each other over the field F7 . The trace of both ρ1 and ρ10 on a generator of inertia at 73 is −1, so that both representations have level 732 and trivial nebentype. This level is too large for us to work with, so we twist both representations by the character χ = 73 to obtain ρ = ρ1 ⊗ 73 and ρ 0 = ρ10 ⊗ 73 . Now ρ and ρ 0 have level 73 and nebentype 73 . Just as in Example 5.4, the restriction of ρ (and of ρ 0 ) to inertia at 7 has diagonal characters ω0 , ω2 , and ω4 . Hence, the predicted weights are the same as in those examples, namely, F(2, 1, 0) ⊗ deta and F(6, 3, 0) ⊗ deta with a = 0, 2, 4. Computing the cohomology in each of these six weights with level 73 and nebentype 73 , we find that there is a unique eigenspace with the correct eigenvalues to correspond to ρ, and a unique eigenspace with the correct eigenvalues to correspond to ρ 0 (at least up to ` = 47). As expected, these eigenspaces are defined over F72 rather than over F7 , and they are Galois conjugates of each other over F7 . 5.3.4. Wildly ramified representations In addition to the preceding representations, we are able to calculate cohomology classes corresponding to irreducible three-dimensional representations ρ : G Q → GL3 (F¯ p ) which are wildly ramified at p. We have two types of examples of such
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representations, those having image A5 , which are wildly ramified at 5, and those having image PSL2 (F7 ), which are wildly ramified at 7. We begin our study of the type A5 representations by noting that there is a unique (up to isomorphism) injective homomorphism from A5 to GL3 (F5 ), with image generated by the three matrices 1 1 0 4 2 2 4 1 4 1 1 , 1 2 , 0 4 1 , 1 4 2 4 2 of orders 5, 2, and 3, respectively. The fields from which we obtain our Galois representations have inertia group at 5 of order 5 or 10. In the case of representations with inertia group of order 10, we choose our representation so that the image of inertia is generated by the first two matrices above. With this choice of Galois representation, it is clear that we have 2 ω ∗ ∗ ρ| I p ∼ ω0 ∗ . ω2 Hence, we obtain a triple of (2, 0, 2) yielding a predicted weight of F(0, −1, 2)0 = F(4, 3, 2) = F(2, 1, 0) ⊗ det2 . In order to keep the level to a manageable size, we work with a twist of ρ by a character unramified at p (so that the weight is not affected). Let be the product of the characters q , where q runs through the set of primes at which ρ is ramified with ramification index 2. Then each prime q at which ρ has ramification index 2 contributes a factor of q to the level of ρ ⊗ , and each prime q at which ρ has ramification index 3 contributes a factor of q 2 to the level of ρ ⊗ . The nebentype of ρ ⊗ is easily seen to be . We have one example in which the inertia group has order 5. In this case we choose the representation so that the image of inertia is generated by the first matrix above. It is then clear that 1 ∗ ∗ ρ| I p ∼ 1 ∗ , 1 yielding a predicted weight of F(−2, −1, 0)0 = F(6, 3, 0). The level of this representation is 34 (note that 3 is wildly ramified), and the nebentype is trivial. For all of these examples, we have found that the predicted eigenclass does exist in the given weight, level, and character and has the correct eigenvalues (at least up to ` = 47) to have ρ attached.
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Table 7. Wildly ramified Galois representations in niveau 1 Polynomial x 5 + 5x 3 − 10x 2 − 45 x 5 + 5x 3 − 10x 2 − 1 x 5 + 5x 3 − 10x 2 + 5 x 5 + 5x 3 − 10x 2 + 9 x 5 + 5x 3 − 10x 2 + 20 x 5 + 25x 2 − 75 x 7 − 7x 5 − 7x 4 − 7x 3 − 7x 2 − 7 x 7 + 14x 6 + 14x 5 − 14x 4 + 35 x 7 − 21x 3 + 7x − 27 x 7 − 7x 5 − 28x 2 + 7x + 4 x 7 − 7x 5 − 21x 4 − 49x 3 − 21x 2 + 1 x 7 − 14x 4 + 42x 2 − 21x − 9 x 7 + 7x 5 − 7x 4 − 49x 3 − 98x − 107
G A5 A5 A5 A5 A5 A5 PSL2 (F7 ) PSL2 (F7 ) PSL2 (F7 ) PSL2 (F7 ) PSL2 (F7 ) PSL2 (F7 ) PSL2 (F7 )
p 5 5 5 5 5 5 7 7 7 7 7 7 7
Weight F(4, 3, 2) F(4, 3, 2) F(4, 3, 2) F(4, 3, 2) F(4, 3, 2) F(6, 3, 0) F(6, 5, 4) F(8, 5, 2) F(6, 5, 4) F(8, 5, 2) F(8, 5, 2) F(6, 5, 4) F(6, 5, 4)
Level 13 31 37 41 22 · 13 34 17 52 47 26 26 34 112
13 31 37 41 13 1 17 1 47 1 1 1 1
We have also found Galois representations with image isomorphic to PSL2 (F7 ) which have low enough level that we can compute the predicted cohomology classes. The image of the representation is generated by the three matrices 1 1 0 2 3 3 1 0 0 0 1 1 , 0 1 3 , 2 6 0 , 0 0 1 0 0 4 4 0 6 of orders 7, 3, and 2, respectively. The image of inertia at 7 under the representations that we examine always has order 21 and may be chosen to be the subgroup generated by the first two matrices above. Hence, on inertia, we have 2 4 ω ∗ ∗ ω ∗ ∗ ρ| I7 ∼ or ρ| I7 ∼ ω0 ∗ ω0 ∗ . 4 ω ω2 In order to distinguish between these two possibilities, we use the action of tame ramification on wild ramification. Let K be our PSL2 (F7 )-extension, and let K p be its localization at a prime above 7. Then K p /Q p is a degree 21 extension, which is totally ramified. Denote its Galois (inertia) group by G 0 and its higher ramification subgroups by G 1 , G 2 , . . . . Clearly, there is a unique i ≥ 1 such that G i /G i+1 is nontrivial, since G 1 is simple. By [21], for a ∈ G 0 /G 1 and b ∈ G j /G j+1 , we have θ j (aba −1 ) = θ0 (a) j θ j (t), ( j)
( j+1)
where θ j : G j /G j+1 → U K /U K is the injective homomorphism sending σ to σ (π)/π, where π is a uniformizer of K p .
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2 If we consider θ0 as a map into F× p , then we see (by [19]) that θ0 = ω . We ∼ identify G i /G i+1 = G i with its image under θi . Then we have
sts −1 = t ω
2i (s)
.
However, we see easily (by matrix multiplication) that sts −1 = t 2 , so that ω2i (s) = 2, and we have 2i ω ∗ ∗ ρ| I7 ∼ 1 ∗ . ω4i Finally, analysis of the ramification groups shows that if the discriminant of the degree 7 subfield of K is exactly divisible by 78 , then i = 1, and if it is exactly divisible by 710 , then i = 2. Clearly, if i = 1, we get a predicted weight of F(0, −1, 4)0 = F(6, 5, 4), and if i = 2, we get a predicted weight of F(2, −1, 2)0 = F(8, 5, 2). The level and nebentype are easily calculated, and in the case of odd primes q that have inertia group of order 2, we twist by q to lower the level from q 2 to q. Table 7 contains information on the wildly ramified Galois representations we have studied. Each line of the table gives a polynomial whose Galois closure is a G-extension of Q (where G = A5 or PSL2 (F7 )), as well as the weight, level, and nebentype predicted by Conjecture 3.1 for the appropriate twist of ρ. In each case the representation for which the invariants were computed is actually ρ ⊗, where is the indicated nebentype (as described above, this lowers the level to a manageable size). In every example an eigenclass with the correct eigenvalues (up to ` = 47) occurs in the predicted cohomology group.
6. Niveau 2 representations 6.1. Reducible representations in higher level Each line of Table 8 gives a polynomial with splitting field a totally real S3 -extension K of Q. In each case we define σ to be the unique two-dimensional Galois representation σ : G Q → GL2 (F¯ p ) which cuts out K , and we note that σ is niveau 2. Letting ρ = σ ⊕ ωk , where k is indicated in the table, Conjecture 3.1 predicts two possible weights corresponding to ρ, as indicated in the table. We have checked that for each row of Table 8, the cohomology in the given weights, level, and nebentype does contain an eigenclass with the correct eigenvalues to correspond to ρ, at least up to ` = 47. Table 9 contains examples of Galois representations, each of which is the sum of an odd irreducible two-dimensional Galois representation and the trivial character.
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Table 8. Reducible niveau 2 representations σ ⊕ ωk with σ even Fixed field of ker(σ ) x 3 − x 2 − 8x + 7
G S3
x 3 − x 2 − 7x + 2
S3
x 3 − 11x − 11
S3
k 1 3 5 7 9 11 13 5 7 9 11 13
p 5 5 11 11 11 11 11 11 11 11 11 11
Weights F(5, 4, 1) F(4, 4, 2) F(4, 2, 2) F(5, 2, 1) F(5, 4, 3) F(22, 14, 6) F(15, 6, 3) F(12, 6, 6) F(15, 8, 3) F(12, 8, 6) F(15, 10, 3) F(12, 10, 6) F(15, 12, 3) F(12, 12, 6) F(5, 4, 3) F(22, 14, 6) F(15, 6, 3) F(12, 6, 6) F(15, 8, 3) F(12, 8, 6) F(15, 10, 3) F(12, 10, 6) F(15, 12, 3) F(12, 12, 6)
Level 73 73 13 13 13 13 13 17 17 17 17 17
73 73 13 13 13 13 13 17 17 17 17 17
In each example a polynomial is given that has Galois group G. For all but the last two examples, we let σ be the unique two-dimensional mod p representation of G, and in all cases we take ρ to be σ ⊕ 1. The ramification index e of p, and the level N and nebentype of ρ, are indicated. For each such representation, Conjecture 3.1 predicts four weights (two of the predicted weights are the same in the p = 3 cases), as indicated in the table, and in all cases we have been able to check that the predicted eigenvalues exist in the cohomology in all of the predicted weights. We explain the last two examples in Table 9 in detail in Example 6.1. Example 6.1 Let K be the splitting field of the polynomial f = x 5 − 19x 2 + 38x − 95. Then K is a totally complex D5 -extension of Q, ramified only at 7 (with ramification index 2 and residual degree 1) and 19 (with ramification index 5 and residual degree 2). The group D5 has two irreducible two-dimensional representations over F19 — we denote the corresponding Galois representations by σ and σ 0 . Let ρ (resp., ρ 0 ) be the direct sum of σ (resp., σ 0 ) with the trivial character. Then both ρ and ρ 0 are easily seen to have level 7 and nebentype 7 . We may conjugate each of ρ and ρ 0 to land in either of the standard Levi subgroups ∗ ∗ ∗ L = ∗ ∗ or L 0 = ∗ ∗ , ∗ ∗ ∗ and each representation satisfies strict parity with Levi subgroup L (or L 0 ), as σ and σ 0 are both odd.
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Table 9. Reducible niveau 2 representations σ ⊕ ω0 with σ odd
Galois representation p = 5, e = 3 N = 7, = 3 G = S3 x 3 − x 2 + 2x − 3 p = 5, e = 3 N = 43, = 3 G = S3 x 3 − x 2 + 2x + 12 p = 5, e = 3 N = 47, = 3 G = S3 x 3 + 5x − 5 p = 5, e = 3 N = 67, = 3 G = S3 x 3 − x 2 + 7x + 2 p = 5, e = 3 N = 83, = 3 G = S3 x 3 − 10x − 15 p = 5, e = 3 N = 83, = 3 G = S3 x 3 − x 2 + 7x − 8 p = 5, e = 3 N = 83, = 3 G = S3 x 3 − x 2 − 3x − 8 p = 17, e = 3 N = 3, = 3 G = S3 x 3 − x 2 + 6x − 12 p = 17, e = 3 N = 7, = 7 G = S3 x 3 − x 2 + 6x + 5 p = 3, e = 4 N = 7, = 7 G = D4 x 4 − 3x 2 − 3 p = 3, e = 4 N = 19, = 19 G = D4 x 4 − 30x 2 − 3 p = 3, e = 4 N = 31, = 31 G = D4 x 4 + 9x 2 − 3 p = 3, e = 4 N = 43, = 43 G = D4 x 4 − 318x 2 − 3 p = 7, e = 4 N = 11, = 11 G = D4 x 4 − 7x 2 − 7 p = 19, e = 5 N = 7, = 7 G = D5 x 5 − 19x 2 + 38x − 95 p = 19, e = 5 N = 7, = 7 G = D5 x 5 − 19x 2 + 38x − 95
Predicted weights F(1, 0, 0), F(4, 1, 0) F(2, 2, 1), F(6, 5, 2) F(1, 0, 0), F(4, 1, 0) F(2, 2, 1), F(6, 5, 2) F(1, 0, 0), F(4, 1, 0) F(2, 2, 1), F(6, 5, 2) F(1, 0, 0), F(4, 1, 0) F(2, 2, 1), F(6, 5, 2) F(1, 0, 0), F(4, 1, 0) F(2, 2, 1), F(6, 5, 2) F(1, 0, 0), F(4, 1, 0) F(2, 2, 1), F(6, 5, 2) F(1, 0, 0), F(4, 1, 0) F(2, 2, 1), F(6, 5, 2) F(9, 4, 0), F(20, 9, 0) F(14, 10, 5), F(30, 21, 10) F(9, 4, 0), F(20, 9, 0) F(14, 10, 5), F(30, 21, 10) F(1, 0, 0), F(2, 2, 1) F(2, 1, 0) F(1, 0, 0), F(2, 2, 1) F(2, 1, 0) F(1, 0, 0), F(2, 2, 1) F(2, 1, 0) F(1, 0, 0), F(2, 2, 1) F(2, 1, 0) F(3, 0, 0), F(6, 3, 0) F(4, 4, 1), F(10, 7, 4) F(13, 2, 0), F(20, 13, 0) F(16, 14, 3), F(34, 21, 14) F(9, 6, 0), F(24, 9, 0) F(16, 10, 7), F(34, 25, 10)
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Both ρ and ρ 0 have niveau 2, but they differ on restriction to inertia at 19. We choose ρ (possibly swapping σ and σ 0 ) such that ψ 72 ρ| I19 ∼ L ψ 0 72 , 0 ω while ρ 0 has diagonal characters ψ 144 , ψ 0 144 , ω0 . Since 72 = 15 + 3 ∗ 19, we get a predicted weight of F(15 − 2, 3 − 1, 0)0 = F(13, 2, 0) for ρ. In addition, we may also conjugate ρ inside L, so that the diagonal characters on inertia are ψ 288 , ψ 0 288 , and ω0 . Since 288 = 22+14∗19, we also predict a weight of F(20, 13, 0) for ρ. Finally, we may conjugate ρ to land inside L 0 , yielding predicted weights of F(16, 14, 3) and F(34, 21, 14). In a similar fashion, we predict four weights for ρ 0 , namely, F(9, 6, 0), F(16, 10, 7), F(24, 9, 0), and F(34, 25, 10). In order to test whether the representations ρ and ρ 0 are attached to Hecke eigenclasses with these weights, we need to compute the characteristic polynomials of the images of Frobenius elements under ρ and ρ 0 . There is a subtlety introduced here by the fact that D5 has two conjugacy classes of order 5. On one of these classes, ρ has trace 5 and ρ 0 has trace 15, while on the other class these traces are reversed. We must determine which class contains each Frobenius element of order 5. Suppose τ ∈ G Q restricts to an element of order 5 in Gal(K /Q). Then there is some element η ∈ I19 such that τ ≡ η modulo G K . So 72 Tr ρ(τ ) = Tr ρ(η) = ψ 72 (η) + ψ 0 (η) + 1. Let P be the unique prime of K lying over p = 19, and let π be a uniformizer of P. Then ζ = ψ 72 (η) ≡ η(π )/π (mod P) is a fifth root of unity in the residue field F of P (which has order 192 ). Note that there are two possible images of ζ in F, depending on our choice of fundamental character ψ. We may, however, compute ζ + ζ p , which is in the prime field and is independent of this choice. These calculations are easily performed using PARI/GP since K is only of degree 10 over Q. A convenient uniformizer to use is a root α of the polynomial f defined above. We find, for instance, that Tr(ρ(Frob2 )) = 5, giving predicted Hecke eigenvalues a(2, 1) = 5 and a(2, 2) = 12 for the classes attached to ρ, and eigenvalues a(2, 1) = 15 and a(2, 2) = 17 for the classes attached to ρ 0 . We have computed the Hecke eigenvalues (for l ≤ 47) for cohomology classes with each of the 8 weights that arose above, and in each case we have found that the eigenvalues are exactly as predicted. 6.2. Irreducible representations in higher level In niveau 2 we again obtain several irreducible representations that are symmetric squares of odd two-dimensional representations, but we also obtain one set of exam-
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ples that are not. We begin by describing an example of the former type of representation. Example 6.2 Let p = 5, and let K be the splitting field of the polynomial f = x 4 + x 2 − x + 2. Then K has Galois group S4 and is ramified only at 5 (with ramification index 3) and at 73 (with ramification index 2). In fact, since the discriminant of f is 52 73, the quadratic subfield of K is ramified at 73, so the inertia group at 73 must be generated by a transposition. If we let ρ be the unique irreducible three-dimensional mod 5 representation of G Q cutting out K and taking transpositions to elements of trace 1, we easily determine that the level of ρ is 73 and that its nebentype is 73 . The weights predicted for ρ by Conjecture 3.1 are calculated by noting that ψ8 ρ| I5 ∼ ψ 08 , 0 ω with 8 = 3 + 1 ∗ 5, so that we have a predicted weight of F(3 − 2, 1 − 1, 0)0 = F(1, 0, 0)0 . We may also write ψ8 ω0 ω0 ψ8 ρ| I5 ∼ or ρ| I5 ∼ , 8 8 0 0 ψ ψ yielding predicted weights of F(3−2, 0−1, 1)0 = F(5, 3, 1) and F(0−2, 3−1, 1)0 = F(2, 2, 1)0 . In addition, we note that ψ 0 8 = ψ 16 , so we can permute the two characters of niveau 2 and write ψ 16 ρ| I5 ∼ ψ 0 16 , ω0 with 16 = 1 + 3 ∗ 5 = 6 + 2 ∗ 5, so that we have a predicted weight of F(6 − 2, 2 − 1, 0)0 = F(4, 1, 0). We may also write ω0 ψ 16 ω0 ψ 16 or ρ| I5 ∼ ρ| I5 ∼ , ψ 0 16 ψ 0 16 yielding predicted weights of F(6−2, 0−1, 2)0 = F(4, 3, 2) and F(0−2, 6−1, 2)0 = F(6, 5, 2). Calculating the cohomology in all six of these weights, we find eigenclasses with the correct Hecke eigenvalues to correspond to ρ (at least for primes up to 47). This
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Table 10. Irreducible niveau 2 representations Galois representation p = 5, e = 3 N = 73, = 73 G = S4 , χ = 1 x4 + x2 − x + 2 p = 5, e = 3 N = 144, = 1 G = S4 , χ = 1 x 4 − 2x 3 − 8x + 4 p = 7, e = 4 N = 67, = 67 G = S4 , χ = 1 x 4 − 56x + 112 p = 11, e = 3 N = 17, = 17 G = S4 , χ = 1 x 4 − x 3 + 3x + 2 p = 5, e = 3 N = 89, = 89 G = A5 , χ = 89 x 5 − 2x 3 − x 2 − 6x − 11 p = 5, e = 3 N = 151, = 151 G = A5 , χ = 151 x 5 − 3x 3 − x 2 + x − 3 p = 5, e = 3 N = 157, = 157 G = A5 , χ = 157 x 5 + 7x 3 − x 2 − 9x + 7
Predicted weights F(1, 0, 0), F(5, 3, 1), F(2, 2, 1) F(4, 1, 0), F(4, 3, 2), F(6, 5, 2) F(1, 0, 0), F(5, 3, 1), F(2, 2, 1) F(4, 1, 0), F(4, 3, 2), F(6, 5, 2) F(6, 3, 3), F(12, 8, 4), F(7, 7, 4) F(9, 6, 3), F(3, 2, 1), F(7, 4, 1) F(5, 2, 0), F(15, 9, 3), F(8, 6, 3) F(12, 5, 0), F(12, 9, 6), F(18, 13, 6) F(1, 0, 0), F(5, 3, 1), F(2, 2, 1) F(4, 1, 0), F(4, 3, 2), F(6, 5, 2) F(1, 0, 0), F(5, 3, 1), F(2, 2, 1) F(4, 1, 0), F(4, 3, 2), F(6, 5, 2) F(1, 0, 0), F(5, 3, 1), F(2, 2, 1) F(4, 1, 0), F(4, 3, 2), F(6, 5, 2)
yields a family of six “companion forms” of different weights, all of which seem to correspond to ρ. Other examples of irreducible niveau 2 representations with image isomorphic to S4 , as well as examples with image isomorphic to A5 (where the representation is the twist by χ of the unique irreducible three-dimensional mod 5 representation having image A5 and cutting out the splitting field of f ), are given in Table 10, in the same format as the examples in Table 6. In addition, examples with image of order 54 are given in [12]. These last examples have p = 5, level N = 83, with quadratic nebentype, and cannot be the symmetric square of any two-dimensional representation. 7. Niveau 3 representations We have two examples of odd niveau 3 representations, both of which support Conjecture 3.1. It is easy to see that a niveau 3 representation must be irreducible and that it cannot be the symmetric square of a two-dimensional representation. Our first example is induced from a one-dimensional representation of a subgroup of index 3 in G Q , and the second has image isomorphic to PSL2 (F7 ) in GL3 (F¯ 11 ) but is in no obvious way related to any representation of dimension less than 3. 7.1. An induced representation Let f = x 3 + 2x − 1. The Galois group of f is S3 . Let K be the splitting field of f , and let K 3 = Q(α), where α is a root of f . Then K 3 is ramified only at 59. Using
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PARI/GP, we may calculate the ray class group of K 3 modulo 7 and find that it is cyclic of order 9. If we let L be the ray class field of K 3 modulo 7, then the existence of L implies the existence of a character χ : G K 3 → µ9 ⊂ F73 of order 9, ramified only at primes above 7. If we now set G
ρ = IndG QK χ, 3
then ρ : G Q → GL3 (F¯ 7 ) must be irreducible since it has niveau 3 (as the ramification index at 7 is divisible by 9). Note that there are six choices of χ since there are six primitive ninth roots of unity in µ9 . Until we make a choice, everything we state is true for any choice of χ and hence for any ρ induced from χ . If we let M be the Galois closure of L, we see that M contains the composite field K L, which is abelian of degree 9 over K , and in fact, M is generated by the conjugates of K L over Q. We see from this that no element of Gal(M/K ) has order more than 9. Note that ρ factors through G = Gal(M/Q), so in particular, the image of inertia at 7 under ρ must be of order 9 (since inertia fixes K ). In fact, it is easy to see that the factorization of ρ through G is a faithful representation of G. Now let 2 [ GQ = gi G K 3 , i=0
where the gi are coset representatives of G K 3 in G Q , and for g ∈ G Q , note that 2 X Tr ρ(g) = χ 0 (g gi ), i=0
where χ (x) = 0
( 0
if x 6 ∈ G K 3 ,
χ(x) if x ∈ G K 3 .
Using this description of ρ, we may calculate values of Tr(ρ(g)) in terms of χ for various g, given that we know the order of π(g), where π : G Q → S3 is the natural projection onto the Galois group of K . Let g 0 ∈ G K 3 be a conjugate by some gi of g if such a conjugate exists; for π(g) of order 2, ρ(g) has trace χ (g 0 ), and for π(g) of order 3, ρ(g) has trace zero (since no conjugate of g is in G K 3 ). In fact, we may go even further and compute the values of χ (g 0 ) using class field theory. Class field theory shows the existence of an isomorphism between the ray class group J of K 3 modulo 7 and the group µ9 of ninth roots of unity. We fix this isomorphism by setting the image of the ideal p above 2 in K 3 with inertial degree 1 to have image χ(Frobp ) = ζ9 , where Frobp is a Frobenius above p (note that p has order 9 in the ray class group). Given any ideal of K 3 , we may then compute its image in J
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Table 11 p O(Frob p ) O(π(Frob p )) χ (Frob0p ) Tr(ρ(Frob p ))
2 18 2 ζ9 ζ9
3 9 3 * 0
5 9 3 * 0
7 * * * *
11 18 2 ζ95 ζ95
13 6 2 ζ96 ζ96
17 9 1 ζ9 0
19 9 3 * 0
23 18 2 ζ9 ζ9
29 3 3 * 0
31 18 2 ζ92 ζ92
37 18 2 ζ97 ζ97
41 3 3 * 0
43 2 2 1 1
47 18 2 ζ9 ζ9
in terms of the image of the ideal above 2 and hence find the image of any Frobenius element under χ . The ray class computations are easily done using PARI/GP since the degree of K 3 is only 3. Using these techniques, we find the values in Table 11. The only value that has not yet been explained is the trace of Frobenius at 17. This trace is zero since 17 splits completely in K (hence also in K 3 ). Hence, there are g three distinct conjugates Frob17i of Frob17 , all in G K 3 , and their images under χ are ζ9 , ζ94 , and ζ97 , so that the trace of ρ(Frob17 ) is zero. Direct computation in the ray class group shows that if p ≤ 47 is a rational prime with π(Frob p ) having order 2, and g is any Frobenius element for p, then χ (g 2 ) = χ(g 0 )2 . Since this is true for any conjugate of g 2 , we have Tr(ρ(g 2 )) = 3χ (g 0 )2 = 3 Tr(ρ(g))2 . Using this fact, a simple computation (using Magma) shows that the eigenvalues of ρ(g) must be ξ , ξ , and −ξ , where ξ = Tr(ρ(g)). Hence, the characteristic polynomial det(1 − ρ(g)X ) is equal to 1 − ξ X − ξ 2 X 2 + ξ 3 X 3. In particular, we use the fact that det ρ(g) = −ξ 3 = −(Tr ρ(g))3 . We now compute the level and character of ρ. The prime 59 has ramification index 2 in the fixed field of ρ, and if g is a generator of inertia at 59, then π(g) has order 2 (since 59 has ramification index 2 in K ). In addition, χ(g 0 ) must be simultaneously a ninth root and a square root of 1, hence equal to 1. Then Tr(ρ(g)) = χ (g 0 ) = 1, so the three eigenvalues of g are 1, 1, and −1, and the level of ρ must be 59, with nebentype 59 . Finally, we calculate the predicted weights for ρ. These weights in fact depend on the choice of χ . We recall that the fundamental characters of niveau 3 are denoted by θ , θ 0 , and θ 00 . Since 7 has ramification index 9 in M, we know that ρ must have niveau 3. In fact, we have that either θ 76 θ 38 or ρ| I7 ∼ ρ| I7 ∼ θ 0 76 θ 0 38 . 76 38 00 00 θ θ
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Note that det ρ = ω38 59 = ω2 59 in the first case, while det ρ = ω76 59 = ω4 59 in the second. Thus, in order to obtain the first case, we choose χ (and hence ζ9 = χ (Frob2 )) such that 3 −ζ93 = − Tr ρ(Frob2 ) = det ρ(Frob2 ) = ω2 (Frob2 )59 (Frob2 ) = −4, and in order to get the second, we choose χ (and hence ζ9 ) such that −ζ93 = −2. Note that each of the two possibilities comes from three choices of χ . Hence, we should expect three eigenclasses in each predicted weight—one for each choice of χ . Considering the first case, m = 38 = 3 + 5 ∗ 7 + 0 ∗ 72 , so we get a triple (a, b, c) = (5, 3, 0), yielding predicted weight F(5 − 2, 3 − 1, 0) = F(3, 2, 0). We may also permute the characters on the diagonal, which has the effect of multiplying m by 7 or 72 modulo 73 − 1, yielding predicted triples and weights as follows. For 7 ∗ m = 266 = 7 + 9 ∗ 7 + 4 ∗ 72 , we get predicted weight F(9 − 2, 7 − 1, 4) = F(3, 2, 0) ⊗ det4 . For 49 ∗ m ≡ 152 = 5 + 7 ∗ 7 + 2 ∗ 72 , we get predicted weight F(7 − 2, 5 − 1, 2) = F(3, 2, 0) ⊗ det2 . We may similarly calculate weights for the second possibility and find the following predicted weights: F(3, 1, 0) ⊗ det1 ,
F(3, 1, 0) ⊗ det3 ,
and
F(3, 1, 0) ⊗ det5 .
Computations show that cohomology classes with the correct eigenvalues (up to ` = 47) exist in all of these weights. In each weight there is a triple of eigenclasses, defined over F73 and conjugate over F7 , each corresponding to a choice of χ as above. 7.2. A representation with image PSL2 (F7 ) We begin by noting that the irreducible polynomial f 1 = x 7 − 11x 5 − 22x 4 + 33x 2 + 33x + 11 has Galois group PSL2 (F7 ), as reported by both PARI/GP and Magma. If L = Q(α), where α is a root of f , we find that the discriminant of L is 116 312 . Since 11 is tamely ramified, we may conclude that the ramification index of 11 in the
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Table 12. Character table of PSL2 (F7 )
Class Size Order χ1 χ2 χ3 χ4 χ5 χ6
1 1 1 1 3 3 6 7 8
2 21 2 1 −1 −1 2 −1 0
3 56 3 1 0 0 0 1 −1
4 42 4 1 1 1 0 −1 0
5 24 7 1 α α¯ −1 0 1
6 24 7 1 α¯ α −1 0 1
splitting field K of f is e = 7. Using the main result of [8], we see easily that the ramification index of 31 in K is 2. √ The character table of PSL2 (F7 ) is given in Table 12, where α = (−1 + −7)/2 √ and α¯ = (−1 − −7)/2. Over F¯ 11 , we have that α and α¯ are equal to 4 and 6, with √ the order depending on our choice of −7. The existence of the PSL2 (F7 )-extension K gives rise to two irreducible threedimensional Galois representations defined over F¯ 11 . The image of inertia at 11 under both representations has order 7, so they are both niveau 3. We choose σ to be the representation which, when restricted to inertia at 11, has diagonal characters θ 190 , θ 0 190 , and θ 00 190 , and we choose σ 0 to be the other (with diagonal characters on inertia equal to θ 570 , θ 0 570 , θ 00 570 ). We note that the level of σ (and of σ 0 ) is 312 since the elements of order 2 are mapped to matrices of trace −1. This level is too large for convenient calculation, so we investigate ρ = σ ⊗ 31 and ρ 0 = σ 0 ⊗ 31 , which are easily seen to have level 31 and nebentype 31 . In order to calculate the predicted eigenvalues of the image of a Frobenius element of order 7 under ρ, we need to distinguish between the two conjugacy classes of order 7 in PSL2 (F7 ). In order to do this, we use a method similar to that used in Example 6.1. In this case, the method needs to be modified slightly since we are dealing with much larger fields. We begin by using Magma to determine the Galois group G ∼ = PSL2 (F7 ) of f as a permutation group acting on the roots of f . We note that each root of f is a uniformizer for all primes lying above 11 in K (since 11 is tamely ramified, and all the ramification occurs in L/Q). Let α be a root of f , and let τ be an element of order 7 in G. Then we easily compute a complex approximation to β = τ (α)/α. If P is the prime of K lying over 11 and having inertia group generated by τ , then the
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image of β in the residue field of P is a Galois conjugate of the primitive seventh root of unity θ 190 (τ ). Hence, the trace of σ (τ ) is equal to β + β 11 + β 121 mod P. We actually compute a complex approximation to γ = β + β 2 + β 4 , which is equal to this trace modulo P. Knowing that this trace is congruent to either 4 or 6 modulo P, we compute δ1 = γ − 4, and δ2 = γ − 6. Exactly one of δ1 and δ2 should lie in P. We note that if K 8 is the unique degree 8 subfield of K fixed by hτ i (so that K 8 is the decomposition field of P), then there is a unique degree 1 prime p in K 8 , and P lies over p. Hence, we may determine whether δi lies in P by determining whether the norm of δi (from K to K 8 ) lies in p. We compute a complex approximation to this norm (and all of its Galois conjugates) and then easily find a complex approximation to the minimal polynomial of this norm. This polynomial should have rational integer coefficients, so after examining the polynomial to see that this is true to many decimal places, we round off. We then calculate the valuation of the norm of δi at the unique degree 1 prime in K 8 (using PARI/GP). For our choice of τ , we find that δ1 ∈ / p, while δ2 ∈ p. Hence, Tr(σ (τ )) = 6. Then, using similar techniques, we determine that τ is a Frobenius element for the prime 7, but not for the primes 2, 13, or 23. Hence, for example, we predict that Tr ρ(Frob2 ) = Tr σ (Frob2 ) 31 (Frob2 ) = 4 · (1) = 4 and Tr ρ(Frob13 ) = Tr σ (Frob13 ) 31 (Frob13 ) = 4 · (−1) = 7. Returning to our study of ρ, we have θ 190 ρ| I p ∼ θ 0 190
θ 00 190
.
Note that m = 190 = 3 + 6 ∗ 11 + 1 ∗ 112 . Hence, one weight predicted by the conjecture for ρ is F(6 − 2, 3 − 1, 1)0 = F(4, 2, 1) = F(3, 1, 0) ⊗ det1 . We may also take m = 11·190 or m = 112 ·190, which yield predicted weights of F(6, 6, 0)⊗det5 and F(8, 3, 0) ⊗ det2 . Computing the cohomology in weight F(3, 1, 0) ⊗ det1 , we find a one-dimensional eigenspace with the eigenvalues indicated in Table 13. These eigenvalues are exactly what Conjecture 3.1 predicts, in order for ρ to be attached to this eigenclass. The same system of eigenvalues (up to ` = 47) also occurs in the other two weights predicted for ρ. Similarly, the predicted weights for ρ 0 are F(6, 0, 0) ⊗ det7 , F(3, 2, 0) ⊗ det6 , and F(8, 5, 0) ⊗ det8 . Each of these weights yields an eigenclass with the correct eigenvalues to have ρ 0 attached (at least for ` ≤ 47).
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Table 13. Orders of ρ(Frob` ) and eigenvalues in weight F(4, 2, 1) ` O(ρ(Fr` )) a(`, 1) a(`, 2)
2 7 4 3
3 3 0 0
5 4 1 9
7 7 6 10
11 * 0 0
13 7 7 3
17 4 10 2
19 4 1 7
23 7 7 6
29 3 0 0
31 * * *
37 2 1 8
41 3 0 0
43 3 0 0
47 3 0 0
8. Computational techniques We now give an overview of our methods for computing the various Hecke eigenclasses on which we have reported in this paper. We begin by noting that we do not, in fact, do any direct calculations of cohomology. Instead we compute with homology, exploiting the natural duality, as in [7, Section 3]. Let p and N be positive integers with p prime, and let V be a representation of the semigroup generated by S pN and 00 (N ). Then we wish to calculate H3 (00 (N ), V ) along with the action of various Hecke operators. The groups H3 are easier to calculate than H1 or H2 since the virtual homological dimension of SL3 (Z) is 3 (see [1]). In addition, one can show that for many classes of three-dimensional Galois representations, if the representation is attached to any homology class, then it is attached to a class in H3 (cf. [4]). By Shapiro’s lemma, SL3 (Z) H3 00 (N ), V ∼ = H3 SL3 (Z), Ind00 (N ) V , and by [5, Lemma 1.1.4] this isomorphism is compatible with the action of the Hecke operators away from pN . This reduces our problem to computing the homology of the full group SL3 (Z) as long as we are willing to consider sufficiently general weights. The broad outline of our calculations follows that of [1]. In particular, we first use a slight modification of their [1, Theorem 1] to identify H3 (SL3 (Z), V ) with the subspace of all v ∈ V such that (1) v · d = v for all diagonal (but not necessarily scalar) matrices d ∈ SL3 (Z); (2) v · z = −v for all monomial matrices of order 2 in SL3 (Z); (3) v + v · h + v · (h 2 ) = 0, where 0 −1 0 h = 1 −1 0 . 0 0 1 We refer to conditions 1 and 2 as the semi-invariant condition and to condition 3 as the h-condition. Given a sufficiently concrete realization of V , computing the subspace satisfying these conditions is simply an exercise in linear algebra. In Section 8.2.2 we discuss some optimizations we have employed in carrying out the calculation. Once
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we have this subspace in hand, we then use [1, Lemma 3] to compute the actions of various Hecke operators with respect to a basis of this space. The main difference between our calculations and those in [1] is our use of more general coefficient modules. We describe below our construction of the modSL3 (Z) ules Ind00 (N ) F(a, b, c) for a p-reduced triple (a, b, c). Another significant difference is a sharp increase in efficiency and hence in the complexity of the calculations we can tackle. This increase is due partly to better algorithms (as described below) and partly to having the entire calculation done using C++ code rather than relying on Mathematica. 8.1. Models for weights We have performed our calculations with a variety of weight modules. Our basic strategy has been to build more complicated weights up from simpler ones. In this subsection we describe the GL3 (F p )-modules with which we have worked, giving in particular a model for F(a, b, c) for a general p-reduced triple (a, b, c). Details of the implementation of these representations and of the process of inducing from 00 (N ) are left to Section 8.2. To begin, we view F¯ p3 as the standard 3-dimensional (right) F¯ p [GL3 (F p )]-module on which S pN acts via reduction modulo p. Then Symg (F¯ p3 ) is the space of homogeneous polynomials over F¯ p of total degree g in three variables x, y, z. An element m of GL3 (F p ) acts on f ∈ Symg (F¯ p3 ) by f (x) · m = f (mx), where x is the column vector t (x, y, z). Note that for a ≤ p − 1, the representation Syma (F¯ p3 ) is irreducible and is in fact isomorphic to F(a, 0, 0) = W (a, 0, 0). Note also that this action is the contragredient of the standard action used in the statement of the conjecture, as required by the duality between homology and cohomology. Next we look at the module F(a, b, 0) for p-restricted (a, b, 0). THEOREM 8.1 Let (a, b, 0) be a p-restricted triple. Then the GL3 (F p )-submodule of Syma (F¯ p3 ) ⊗ Symb (F¯ p3 ) generated by
v=
b X i=0
(−1)i
b i
(x a−i y i ⊗ x i y b−i )
is isomorphic to F(a, b, 0). Proof Recall that for any nonincreasing triple (α, β, γ ) of integers, both W (α, β, γ ) and
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F(α, β, γ ) are modules over GL3 (F¯ p ) and not just over GL3 (F p ). We prove that the GL3 (F¯ p )-module generated by v is isomorphic to F(a, b, 0). Since (a, b, 0) is assumed to be p-restricted, F(a, b, 0) remains irreducible when viewed as a representation of GL3 (F p ). We may then conclude that the GL3 (F p )-module generated by v is isomorphic to F(a, b, 0). Since we are now looking at representations of GL3 of an algebraically closed field, we may employ the theory of highest weights in representations of algebraic groups (cf. [15, Section 31]). In particular, if we work with respect to the standard diagonal torus and the upper triangular Borel, we note that the nonincreasing triples (n 1 , n 2 , n 3 ) correspond to the dominant weights t1 0 0 0 t2 0 7 → t n 1 t n 2 t n 3 . 1 2 3 0 0 t3 Then F(n 1 , n 2 , n 3 ) is the unique irreducible representation of GL3 (F¯ p ) with highest weight (n 1 , n 2 , n 3 ). Now, Young’s rule (see [16, p. 129]) gives us that W (a, 0, 0) ⊗ W (b, 0, 0) has a filtration W0 ⊃ W1 ⊃ · · · ⊃ Wb+1 = 0, with the quotients Wi /Wi+1 isomorphic to the modules W (a + b, 0, 0), . . . , W (a + b − i, i, 0), . . . , W (a, b, 0) in the given order (so that W (a +b, 0, 0) is a quotient and W (a, b, 0) is a submodule). Since (a, b, 0) is p-restricted, each W (a +b−i, i, 0) is irreducible if a +b−i ≤ p −2 or a + b − 2i = p − 1, and otherwise has F(a + b − i, i, 0) and F( p − 2, i, a + b − i − p + 2) as composition factors (see [10, Proposition 2.11]). We see then that F(a, b, 0) appears only once as a composition factor of W (a, 0, 0) ⊗ W (b, 0, 0) and that it appears as a submodule and not just a subquotient. It follows that W (a, 0, 0) ⊗ W (b, 0, 0) has a unique highest weight vector w of weight (a, b, 0) and that the GL3 (F¯ p )-module generated by this vector is isomorphic to F(a, b, 0). The lemma below shows that v is such a vector, and hence the GL3 (F¯ p )module generated by v is isomorphic to F(a, b, 0). LEMMA 8.2 The vector
v=
b X b (−1)i (x a−i y i ⊗ x i y b−i ) i i=0
in Sym is a highest weight vector of weight (a, b, 0). Here “highest” refers to the usual lexicographic ordering of the weights. a
(F¯ p3 ) ⊗ Symb (F¯ p3 )
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Proof It is clear that v is a weight vector of weight images of v under the operators 1 0 0 1 0 g1 = 1 1 0 , g2 = 0 1 0 0 1 0 1
571
(a, b, 0). We need only show that the 0 0 , 1
1 0 0 g3 = 0 1 0 1 0 1
are all equal to v plus something in the span S of vectors of weight strictly less than (a, b, 0). Clearly, v · g2 and v · g3 are both equal to v modulo S. For v · g1 , we calculate v · g1 =
=
b X i=0 b X
(−1)
i
(−1)
i
b i
b i
X b−i i X
i=0
=
a X
b X
x a−i (x + y)i ⊗ x i (x + y)(b−i)
k=0 j=0
X v
u=a−b v=a−u i=a−u u a−u v b−v
·x y
⊗x y
(−1)
i
i k
b i
b−i j
x a−i+k y i−k ⊗ x i+ j y b−i− j
i u−a+i
b−i v−i
.
Setting α = i − (a − u), expanding the binomial coefficients, and canceling equal terms, the inner sum becomes ±
u+v−a X
(−1)α
α=0
b! (b − v)!(u + v − a − α)!α!(a − u)!
u+v−a X b! 1 (−1)α (b − v)!(a − u)! α!(u + v − a − α)! α=0 u+v−a X b! u+v−a (−1)α , =± α (b − v)!(a − u)!(u + v − a)!
=±
α=0
which is zero if u + v > a. Thus, the only terms x u y a−u ⊗ x v y b−v that appear in v · g1 with nonzero coefficient have u + v = a. It is now easy to see that v · g1 is in fact exactly equal to v. For arbitrary p-restricted (a, b, c), we note that F(a, b, c) ∼ = F(a −c, b−c, 0)⊗detc . In practice, we did all of our calculations with F(a − c, b − c, 0) and simply scaled by detc at the end. We have also made use of representations of the form Syma (F¯ p3 ) ⊗ Symb (F¯ p3 ), Syma (F¯ p3 ) ⊗ Symb (F¯ p3 )∗ and subquotients of Syma (F¯ p3 ) for a larger than p − 1. By
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keeping track of the irreducible constituents of these representations, we were sometimes able to show that certain systems of Hecke eigenvalues come from a specific irreducible module (see [12] for more details). 8.2. Implementation The implementation of our algorithms has two very distinct parts. On the one hand, we need to do calculations involving various GL3 (Z/ pN Z)-modules V . This includes the basic vector space operations as well as multiplying an element in V by an element of GL3 (Z/ pN Z). Further, we need to identify a basis of V and be able to decompose elements of V with respect to that basis. For efficiency reasons it is also important to be able to determine the coefficient of a given basis element in some product v · g without computing all of v · g. On the other hand, we need to carry out various higher-level computations, such as finding the solutions to the h-condition above in order to compute homology with coefficients in V . These calculations can be described in terms of the basic operations of the previous paragraph without any specific knowledge about the module V . We have made use of object-oriented programming techniques to keep these two computational issues strictly separated. This allows us to switch from computing with one module to computing with another without having to rewrite any of the code describing the higher-level algorithms. 8.2.1. Coefficient modules We now look at a few of the implementation details behind some of our coefficient modules. As we stated above, the basic building block for all of our representations is Symg (F¯ p3 ), the space of homogeneous polynomials of degree g in three variables. The monomials form a natural basis of this space, and it is a simple matter to compute the coefficient of any given monomial in a product v · g. We have optimized this code to work especially well when many of g’s entries are zero. This is the case for the element h above as well as for many of the matrices arising in our Hecke operator calculations. The representations Syma (F¯ p3 ) ⊗ Symb (F¯ p3 ) again have natural bases coming from the monomial bases of Syma (F¯ p3 ) and Symb (F¯ p3 ), and all operations on the tensor product can be carried out in terms of those on each factor. We denote by Bab = {wi } this basis of Syma (F¯ p3 ) ⊗ Symb (F¯ p3 ), and we let h·, ·i be the bilinear form with hwi , w j i = δi j . The subspace F(a, b, 0) of Syma ⊗ Symb does not come equipped with a canonical basis. For ease of computation we choose a basis in which each basis vector has a distinguished leading term. In other words, we choose a basis {vi } such that for each i there is an element wi ∈ Bab with hwi , vi i = 1 and hwi , v j i = 0 for j 6= i. We then
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let h·, ·i F be the bilinear form with hvi , v j i F = δi j . Then for v ∈ F(a, b, 0), we have hv j , vi F = hw j , vi, and so we can compute coordinates with respect to this basis of F(a, b, 0) in terms of those with respect to the basis Bab . The final step in obtaining our general weights is to induce a representation W from 00 (N ) to SL3 (Z). The W we use are of the form F(a, b, 0) ⊗ for some a SL3 (Z) character of (Z/N Z)× . We view V = Ind00 (N ) W as the space of functions V = f : SL3 (Z) → W : f (xg) = f (x) · g for g ∈ 00 (N ) with SL3 (Z) acting by left translation. We let {ri } be a set of representatives for SL3 (Z)/ 00 (N ), and we let {wa } be a basis for W . We again choose a bilinear form h, i on W with hwa , wb i = δab . Then we define φri ,wa : SL3 (Z) → W by ( wa · ri−1 x if x ∈ ri 00 (N ), φri ,wa (x) = 0 otherwise. It is clear that the functions φri ,wa comprise a basis of V . In order to express the action of SL3 (Z) on V with respect to this basis, we need to introduce a bit of notation. For x ∈ SL3 (Z), let {x} be the unique representative ri in x00 (N ). Then (φri ,wa g)(x) = φri ,wa (gx) ( wa · ri−1 gx if gx ∈ ri 00 (N ), = 0 otherwise ( wa · ri−1 g{g −1ri }{g −1ri }−1 x if x ∈ g −1ri 00 (N ), = 0 otherwise X
wa · ri−1 g{g −1ri }, wb φ{g−1 ri },wb (x). = b
Note that in order to compute the actions of Hecke operators on H3 (SL3 (Z), V ), we also need to know how elements of S = m ∈ M3 (Z) : det(m) is positive and prime to pN act on V . Let 6 be the semigroup generated by 00 (N ) and S pN . Then S = SL3 (Z)6 and 00 (N ) = SL3 (Z) ∩ 6. (This is part of what it means for the Hecke pair (00 (N ), 6) to be compatible with (SL3 (Z), S).) Thus, if m ∈ S, we have m = ns
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for some n ∈ SL3 (Z) and s ∈ 6. Moreover, n is determined modulo 00 (N ) and so the coset representative {n} depends only on m. If we extend our notation to write {m} = {n}, the formula above for the action of g on φri ,wa makes sense for any g ∈ S. This action of S on V described by the formula induces the correct action of H ( pN ) on H3 (SL3 (Z), V ) (i.e., the one compatible with the action on H3 (00 (N ), W )). The ri may be chosen so that each is congruent to the identity modulo p, which greatly speeds up some of the calculations. Note that SL3 (Z)/ 00 (N ) ∼ = P2 (Z/N ) and so is easy to work with. Also, note that our formula shows at once how to compute the coefficients of a basis element φr j ,wb in v · g for v ∈ V and g ∈ S. 8.2.2. Finding homology Now we move on to the general algorithms we have used to compute the homology of SL3 (Z) with coefficients in some representation V . While this is a simple exercise in linear algebra, we have found it useful to tailor certain optimizations to our situation to allow us to work with larger examples. A typical instance of finding the solutions to the h-condition, for example, involves finding the kernel of a 700000 × 30000 matrix. These optimizations have been largely heuristic. We make no claim of having optimal algorithms. Let V be a 6-module of dimension d, with basis {vi }, and let h·, ·i be the bilinear form with hvi , v j i = δi j . Let K be the 24-element group of monomial matrices in SL3 (Z). Then for p > 3, the space of semi-invariants in V is the image of the operator X P= (g)g, g∈K
where (g) is the sign of the permutation on three letters induced by g. Our computations do not include examples for which p = 2, and for p = 3 only a minor adjustment is needed. Computing the action of P on each vi is not computationally intensive since we have specially optimized all of our coefficient modules with regard to the operation of monomial matrices. We then use column reduction to find a basis for V · P. We note that the dimension dsemi of V · P is approximately d/24. The more serious stage of the calculation is finding the solutions of the hcondition on V · P. We describe our algorithm for finding the solutions of the h-condition on any c-dimensional subspace W of V · P with basis {bi }. We are looking for the nullspace of the (d × c)-matrix M = (m i j ), where m i j = hvi , b j · (1 + h + h 2 )i is the coordinate of vi in b j · (1 + h + h 2 ). Simply computing this matrix and performing Gaussian elimination would theoretically allow us to find the nullspace but is hopelessly inefficient in both space and time. Although the matrix M is quite sparse, it becomes much denser as the elimination proceeds. Since we work with very large d (d on the order of 7 × 105 is not uncommon), we would
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rapidly run out of memory. We touch on four optimizations we have made to speed up the calculation and to reduce the memory requirements. First, we note that the rows of M are highly redundant as there are at most about 1/24th as many columns as rows. We exploit this by computing the rows of M one at a time and only storing those that yield new information about the kernel. Recall that we have set up our coefficient modules so that we can individually compute the entries hvi , b j · (1 + h + h 2 )i in the ith row of M without having to compute all of b j · (1 + h + h 2 ). We discuss below another optimization that makes this separate computation especially efficient. As we find a new row R, we continue our elimination process by subtracting from R the appropriate multiples of all the previously stored rows. If we are left with a nonzero row, we append it to our stored matrix, which remains in row-echelon form. If we are left with the zero row, then R did not add any constraints on the kernel of M and we may discard it and move on to the next row. This guarantees that we never waste space by storing redundant rows and caps the maximum number of rows we will ever store at c ≤ dsemi ≈ d/24. We denote by E the matrix that we are building up row by row in this process. Our second optimization is motivated by the fact that most of the information about the kernel of M can be obtained from M’s early rows. At each stage in our calculation, we clearly have ker M ⊂ ker E. Since E is in row-echelon form, we can immediately read off the dimension of ker E. In practice, we find that the dimension of ker E drops below 1 or 2 percent of dsemi after we run through as few as one fifth or so of the rows of M. Once this happens, we pause our calculation and compute (a basis for) the kernel of E, which is relatively easy to do since E is already in rowechelon form. We have now reduced our problem to finding the kernel of 1 + h + h 2 not on W but on the much smaller space ker E. We then start the algorithm over, replacing W by ker E. Our new choice of W guarantees that the initial rows of the new matrix M will all be zero, and so we can resume our calculation with the row at which we had paused. It is crucial here that we have not computed M all at once and thus do not have to make any time-consuming adjustments to account for our new basis. Indeed, it is now much easier to compute the new M, as it has far fewer columns. In practice the calculation very rapidly proceeds through the remaining rows of M and then computes the kernel of the new E, which is equal to the kernel of M. Our choice of a cutoff on the dimension of ker E is entirely heuristic, and we adjust it based on the size of V . Both of the optimizations above rely on the efficiency of the calculation of the coefficients hvi , v · (1 + h + h 2 )i of each vi in v · (1 + h + h 2 ) for v ∈ V . Although our modules allow for the calculation of hvi , v · gi for any v and g without computing all of v · g, there is still a great deal of work duplicated if we separately perform this calculation for all of the vi . For our calculations of the Hecke operators (see below)
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this is not necessary, but as described above, we must do this in the cases g = h and g = h 2 . We have optimized for this by storing some of the common pieces of SL3 (Z) these calculations. For example, when V = Ind00 (N ) W , we begin by computing and storing the entire matrices describing the actions of h and h 2 on W , and also the permutations induced by h and h 2 on P2 (Z/N ). Since the dimension of W is small compared to the dimension of V (even when N = 2, the dimension of V is 7 times that of W ), this calculation is not terribly costly in space or time. These stored tables can then be used to compute the action of h and h 2 on elements of V very quickly. We have implemented similar strategies when V is not induced but is the tensor product of two smaller representations. Finally, we have increased our available memory by making use of disk space and swapping pieces of our matrix in and out of memory. This requires minor modifications to the reduction algorithm described above in order to reduce the number of disk swaps. In particular, we carry out our row reduction on several (1000) new rows at once. In the end, this does not have a dramatic effect on run time, but it slashes the amount of RAM required. 8.2.3. Computing the Hecke action Our computation of the action of the Hecke operators closely mirrors that in [1], and we refer the reader to [1, Sections 3 and 8] for a discussion of modular symbols and a description of the action of Hecke operators on homology. We just summarize by noting that for v ∈ V satisfying the semi-invariant condition and the h-condition, we have X T (l, k)v = v · Mi j Bi , i, j
where 00 (N )D(l, k)00 (N ) =
a
00 (N )Bi ,
i
P the Mi j are unimodular, and the modular symbol [Bi−1 ] is homologous to j [Mi j ]. We have not recomputed the matrices Mi j but have used the files generated in the course of the calculations in [1]. If { fl } is a basis for the semi-invariants in V satisfying the h-condition, then we P P know a priori that i j f k · Mi j Bi is a linear combination l akl fl of the fl . We wish to obtain the numbers akl . To do this efficiently, we use the same trick we employed in our choice of basis for F(a, b, 0) and adjust our basis { fl } such that for each l there is a basis vector vl of V such that hvl , f m i = δlm . Then akl is the sum over i and j of hvl , f k · Mi j Bi i. As we have discussed, we are able to compute these coefficients directly. This is vastly superior to computing all of f k · Mi j Bi since the dimension of the homology space is only a tiny fraction of the dimension of V . This technique
GALOIS REPRESENTATIONS AND COHOMOLOGY
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was used in [1], although it could not be implemented as efficiently there due to the reliance on Mathematica’s multivariate polynomial routines. A final optimization uses the fact that the Hecke operators we are dealing with all commute and so preserve each other’s eigenspaces. The ultimate goal of our calculation is to identify simultaneous eigenvectors v of the T (l, k) attached to given Galois representations, that is, with T (l, k)v = α(l, k) for some prescribed α(l, k). If we compute the entire matrix for the action of T (2, 1) (which is very easy since T (2, 1) involves only 13 Mi j terms, whereas T (47, 1) involves 55923 such terms) and find a single eigenvector v with eigenvalue α(2, 1), we need only compute the image of the other T (l, k) on v and not on the whole homology space. Moreover, we know that v is an eigenvector of each T (l, k), and so we only need to compute a single coefficient hv` , T (l, k)i in order to determine the eigenvalue. This gives an extraordinary reduction in the time required to make the calculation. For example, we find that the dimension of the homology space at level 00 (11), weight F(22, 11)(11 ), and p = 19 is 31. We are interested in a single eigenvector in this space. In order to compute the entire matrix of a Hecke operator, we would need to find 312 = 961 coefficients of basis vectors. Instead, we reduce this to a single coefficient, giving nearly a thousandfold increase in performance. We point out that this technique was not needed in [1] as the homology spaces dealt with there were much smaller. 8.2.4. Reliability Whenever relying on a large amount of computer calculation, one hopes for a number of consistency checks on the data. Our first check is that two entirely independent programs were written to carry out the calculations on several different computers by two different authors and both programs yielded identical data where compared. The programming was done in C and C++ and compiled with gcc running on a Sparc Ultra 5, a Pentium II under Linux, and a Pentium III under Linux. We also compared our data to some of the data obtained in [1] and [4] and found everything to be consistent. Other checks include the fact that, whenever tested, the operators T (l, k) on a given homology space all commuted and that (again when tested) the Hecke operators all did preserve the space of semi-invariants in V satisfying the h-condition. Perhaps more compelling is the fact that our data meshes exactly with the Galois representations we have studied. While the correspondence is only conjectural, the agreement we observed very strongly suggests the validity of our calculations. Acknowledgments. The authors thank Warren Sinnott and Richard Taylor for many helpful conversations in the course of this work. We thank the maintainers of the Bordeaux database of number fields (available at ftp://megrez.math.u-bordeaux.fr), from which we have obtained defining polynomials for many of the fields de-
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scribed in this paper. We also thank John Jones and David Roberts, whose tables of number fields ramified only at small primes (available on the web at http://math.la.asu.edu/∼jj/numberfields/) also provided many of our defining polynomials. Finally, we thank the creators of PARI/GP (see [18]) and Magma for their software, which was used for many calculations throughout this work. References [1]
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G. ALLISON, A. ASH, and E. CONRAD, Galois representations, Hecke operators, and
the mod- p cohomology of GL(3, Z) with twisted coefficients, Experiment. Math. 7 (1998), 361 – 390. MR 2000d:11068 532, 543, 544, 545, 568, 569, 576, 577 A. ASH, Galois representations attached to mod p cohomology of GL(n, Z), Duke Math. J. 65 (1992), 235 – 255. MR 93c:11036 536 A. ASH and M. MCCONNELL, Experimental indications of three-dimensional Galois representations from the cohomology of SL(3, Z), Experiment. Math. 1 (1992), 209 – 223. MR 94b:11045 532 A. ASH and W. SINNOTT, An analogue of Serre’s conjecture for Galois representations and Hecke eigenclasses in the mod p cohomology of GL(n, Z), Duke Math. J. 105 (2000), 1 – 24. MR 2001g:11081 521, 526, 532, 534, 535, 537, 538, 542, 543, 544, 568, 577 A. ASH and G. STEVENS, Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues, J. Reine Angew. Math. 365 (1986), 192 – 220. MR 87i:11069 525, 538, 568 , Modular forms in characteristic ` and special values of their L-functions, Duke Math. J. 53 (1986), 849 – 868. MR 88h:11036 534, 535, 537, 540 A. ASH and P. H. TIEP, Modular representations of GL(3, F p ), symmetric squares, and mod- p cohomology of GL(3, Z), J. Algebra 222 (1999), 376 – 399. MR 2001d:11058 540, 568 S. BECKMANN, On finding elements in inertia groups by reduction modulo p, J. Algebra 164 (1994), 415 – 429. MR 95h:12003 566 L. CLOZEL, “Motifs et formes automorphes: Applications du principe de fonctorialit´e” in Automorphic Forms, Shimura Varieties, and L-Functions (Ann Arbor, Mich., 1988), Perspect. Math. 10, Academic Press, Boston, 1990, 77 – 159. MR 91k:11042 539, 540 S. R. DOTY and G. WALKER, The composition factors of F p [x 1 , x 2 , x 3 ] as a GL(3, p)-module, J. Algebra 147 (1992), 411 – 441. MR 93h:20015 525, 533, 537, 570 D. DOUD, S4 and e S4 extensions of Q ramified at only one prime, J. Number Theory 75 (1999), 185 – 197. MR 2000b:11060 541, 542, 544 , “Three-dimensional Galois representations with conjectural connections to arithmetic cohomology” to appear in Number Theory for the Millennium (Urbana-Champaign, Ill., 2000), ed. B. Berndt et al., Peters, Boston, 2002. 562, 572
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(Boston, 1995), Springer, New York, 1997, 209 – 242. CMP 1 638 480 529, 535
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Ash Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467-3806, USA;
[email protected] Doud Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138, USA; current: Department of Mathematics, 292 TMCB, Brigham Young University, Provo, Utah 84602, USA;
[email protected] Pollack Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, Ohio 43210, USA;
[email protected]
DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 3,
THE TRANSCENDENTAL PART OF THE REGULATOR MAP FOR K 1 ON A MIRROR FAMILY OF K 3-SURFACES ¨ PEDRO LUIS DEL ANGEL and STEFAN J. MULLER-STACH
Abstract We compute the transcendental part of the normal function corresponding to the Deligne class of a cycle in K 1 of a mirror family of quartic K 3 surfaces. The resulting multivalued function does not satisfy the hypergeometric differential equation of the periods, and we conclude that the cycle is indecomposable for most points in the mirror family. The occurring inhomogenous Picard-Fuchs equations are related to Painlev´e VI–type differential equations. 1. The regulator map and Picard-Fuchs equations In this paper we study the first nonclassical higher K -group K 1 (X ) for a smooth complex projective surface X . It was conjectured by H. Esnault around 1995 that certain elements in this group can be detected in the transcendental part of the Deligne cohomology group HD3 (X, Z(2)) via the regulator (Chern class) map. The transcendental part of the regulator map is defined as an Abel-Jacobi-type integral of holomorphic 2-forms over nonclosed real 2-dimensional chains in X associated to these elements. At that time it was only known that one could detect such classes in the complementary (1, 1)-part of Deligne cohomology (see, e.g., [16]). The goal of our paper is to show that Esnault’s conjecture is true by looking at the differential equations that are satisfied by the normal functions arising from such classes in a family of surfaces. It turns out that the resulting equations for Abel-Jacobi-type integrals with parameters are strongly connected to a generalization of Painlev´e VI–type differential equations. The higher K -groups K 1 (X ), K 2 (X ), . . . of an algebraic variety X were defined around 1970 by D. Quillen [19]. Later S. Bloch [2] showed that on smooth quasiprojective varieties all their graded pieces with respect to the γ -filtration may be computed as grγp K n (X )Q ∼ = CH p (X, n)Q , DUKE MATHEMATICAL JOURNAL c 2002 Vol. 112, No. 3, Received 4 April 2001. Revision received 23 May 2001. 2000 Mathematics Subject Classification. Primary 14C25, 19E15; Secondary 34M55, 32Q25. Del Angel’s work supported by Consejo Nacional de Ciencia y Tecnolog´ıa grant number 00359. M¨uller-Stach’s work supported by a Deutsche Forschungsgemeinschaft Heisenberg fellowship. 581
¨ DEL ANGEL and MULLER-STACH
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where CH p (X, n) are Bloch’s higher Chow groups in [2]. This isomorphism gives an explicit presentation of higher K -groups modulo torsion via algebraic cycles. Let us look more closely at the particular case of K 1 (X ) for a smooth complex projective surface X . There it is known that CH1 (X, 1) = C× and CH p (X, 1) = 0 for p ≥ 4. The remaining interesting parts of K 1 are therefore CH2 (X, 1) and CH3 (X, 1). The last group consists of zero cycles on X × P1 in good position, and therefore the map τ : CH2 (X ) ⊗ Z C× → CH3 (X, 1), x ⊗ a 7 → (x, a) is surjective. Therefore the complexity of CH3 (X, 1) is governed by the complexity of CH2 (X ), which is fairly understood by Mumford’s theorem, respectively, Bloch’s conjecture. We say that CH3 (X, 1) is decomposable. For CH2 (X, 1) the situation is quite different, and the complex geometry of X plays an essential role in the understanding of it. The natural map τ:
CH1 (X ) ⊗ Z C× → CH2 (X, 1),
D ⊗ a 7 → D × {a}
is neither surjective nor injective in general. In the literature there are several examples where the cokernel of τ is nontrivial modulo torsion and even infinite-dimensional (see [4], [11], [16], [24]). The kernel of τ is related but not equal to Pic0 (X ) ⊗ C× even modulo torsion by [20, Th. 5.2]. Note that the cokernel of τ is a birational invariant (by localization) and hence vanishes for rational surfaces and, in fact, for all surfaces with geometric genus pg (X ) = 0 and Kodaira dimension less than or equal to 1. Bloch’s conjecture would imply that it also vanishes for all surfaces of general type which satisfy pg (X ) = 0. One way to study CH2 (X, 1) is to look at the Chern class maps c2,1 :
CH2 (X, 1) → HD3 X, Z(2) =
H 2 (X, C) . F 2 H 2 (X, C)
H 2 (X, Z) +
(1.1)
The decomposable cycles (the image of τ ) are mapped to the subgroup NS(X ) ⊗Z C× ⊆
H 2 (X, C) H 2 (X, Z) + F 2 H 2 (X, C)
generated by the N´eron-Severi group NS(X ) ⊂ H 2 (X, Z) of all divisors in X . It is known (see [16]) that the image of c2,1 is at most countable modulo this subgroup, so that the image of coker(τ ) in Deligne cohomology modulo NS(X ) ⊗Z C× is at most countable. One conjectures that even coker(τ ) itself is countable. P The Chern class maps c2,1 are defined as follows: let Z = a j Z j ∈ CH2 (X, 1) be a cycle. Each Z j is an integral curve and inherits a rational map f j : Z j → P1 from the projection map X × P1 → P1 . Let γ0 be a path on P1 connecting 0 with
THE TRANSCENDENTAL K 1 OF A K 3-SURFACE
583
S S −1 ∞ along the real axis; then γ := γ j := f j (γ0 ) is a closed homological 1cycle, Poincar´e dual to a cohomology class in F 2 H 3 (X, Z) and therefore torsion (see [16]). If we assume that γ = 0 (e.g., if b1 (X ) = 0), then we write γ = ∂0 for a real piecewise smooth 2-chain 0. As a current, that is, as a linear functional on differentiable complex-valued 2-forms on X , the defining property of c2,1 (Z ) is Z Z 1 X c2,1 (α) = log( f j )α + α. (1.2) 2πi Z j −γ j 0 j
Now let X be a projective K 3-surface. Then pg (X ) = 1, b2 (X ) = 22, and b1 (X ) = 0. The intersection form on H 2 (X, Z) is known to be the unimodular form 2E 8 ⊕ 3H , where H is the 2-dimensional standard hyperbolic form. The N´eron-Severi lattice NS(X ) ⊂ H 2 (X, Z) has an orthogonal complement T (X ) ⊂ H 2 (X, Z). In particular, there is a well-defined morphism Coker(τ ) →
T (X ) ⊗ C× . F2
If we have an arbitrary smooth family f : X → B of complex algebraic surfaces over a quasiprojective complex variety B, and an algebraic family of cycles Z b ∈ CH2 (X b , 1) for all b ∈ B, that is, Z b = Z |X b for some given cycle Z ∈ CH2 (X, 1), then we may define the normal function ν(b) := c2,1 (Z b ) ∈
T (X b ) ⊗ C× . F2
One can easily show that ν is a holomorphic section of the corresponding family of generalized tori T (X b ) ⊗ C× /F 2 . Coming back to the case of K 3-surfaces, there the canonical bundle ω X is trivial; hence the group H 0,2 (X ) = H 0 (X, 2X )∗ = C is 1-dimensional and generated by the dual of ω X . In a smooth algebraic family X b of K 3-surfaces, the composition of the regulator with the projection onto H 0,2 (X b ) Im H2 (X b , Z) produces a multivalued holomorphic function on B, denoted by ν¯ (b), which has poles at all b where the family degenerates (see proof of Lem. 3.1). It is given by the formula Z ν¯ (b) = ωXb 0b
since the integral of ω X over any effective divisor vanishes. If DPF denotes the PicardFuchs differential operator of the Gauss-Manin connection associated to the family X b of K 3-surfaces, then DPF annihilates all periods of the family. Therefore we obtain the following result.
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LEMMA 1.1 Let B ⊂ B¯ be a smooth compactification of B. Then with the notation above, DPF (¯ν ) extends to a single-valued meromorphic function on B¯ with poles only along degeneracies of X b , and therefore it satisfies a differential equation DPF ν¯ (b) = g(b), (1.3)
¯ where g is a rational function in b ∈ B. The proof is given in the appendix. Altogether, we have obtained a map families of cycles in CH2 (X b , 1) −→ {differential equations/rational functions}. (1.4) For each such family of K 3-surfaces, it sends a family of cycles to the equation DPF ν¯ = g, respectively, the rational function g, which is the same information on a given family. One should view the resulting solutions ν¯ (b) as new transcendental functions arising from the family of K -theoretic cycles in CH2 (X b , 1). If g is a nontrivial function, then ν¯ , and hence ν, is a nonflat section of the family of Deligne cohomology groups of X b . In [16] the relationship between the infinitesimal behavior of such normal functions and the mixed Hodge structure of the total space X has already been investigated. This situation is very reminiscent of a method developed by R. Fuchs [10] in the case of the Legendre family y 2 = x(x − 1)(x − t) of elliptic curves and investigated further in the work of Yu. Manin [14, p. 134]. In particular, there is a strong connection with differential equations of a generalized form of type Painlev´e VI (see [14]). There exists a formula to derive g: there is a so-called inhomogeneous PicardFuchs equation DPF ω X = drel β (1.5) before integration over 0, where β is a section of the vector bundle of (meromorphic) 1-forms in the fibers of the family f : X → B. We say that 0 does not depend on b if it can be defined as a real semialgebraic subset via flat coordinates, that is, via coordinate functions that are horizontal with respect to the Gauss-Manin connection, and such that the defining inequalities of 0 are polynomials not depending on b. This shows on the one hand that for closed 0 the periods satisfy the Picard-Fuchs equation, and on the other hand that for nonclosed 0 (not depending on b) with ∂0 = γ , we get Z Z Z g(b) = DPF ω X = drel β = β. (1.6) 0
0
γ
The last equality uses a version of Stokes’s theorem for currents since some of the differential forms involved, in general, have integrable singularities. Hence Stokes’s
THE TRANSCENDENTAL K 1 OF A K 3-SURFACE
585
theorem for currents (see [12, Chap. 3]) also implies that β is integrable over γ . In general, 0 depends also on b, and then there is an additional contribution from the derivatives of the boundaries of the integral. In the case of the Legendre family y 2 = x(x − 1)(x − t) of elliptic curves, β is a meromorphic function (zero-form) after a double covering y 2(x − t)2 by [10, p. 310] and [14, p. 76]. Manin has put these equations into a more formal context (so-called µ-equations), so that one can understand the sections and operators in a coordinate-free way in terms of certain locally free sheaves on B. This also plays a role in his work on the functional Mordell conjecture. Furthermore, after uniformizing the elliptic curves, the inhomogeneous Picard-Fuchs equation is equivalent to a version involving the Weierstrass p-function (see [14, p. 137]): 3 Tj d2z 1 X = α p z + ,τ , j z 2 dτ 2 (2πi)2 j=0
where α j are constants parametrizing the family of differential equations and where (T0 , . . . , T3 ) = (0, 1, τ, 1 + τ ) are the vertices of the fundamental parallelogram. In this way the transcendental aspect of the solutions and also the connection to integrable systems become apparent (see [14, p. 139]). In the future we hope to investigate further the transcendental properties of our solutions (using again uniformization) and to study the attached integrable systems. The rest of this article is devoted to a particular solution of the inhomogeneous Picard-Fuchs equation for a certain family of K 3-surfaces introduced in Section 2. In Section 3 we deduce Esnault’s conjecture from the nonvanishing of the DPF (¯ν ) in √ the special case of b = −1. In Section 4 we study a certain Shioda-Inose model of X b which has isomorphic transcendental cohomology. This leads to an explicit computation of β in this case. 2. An example: A mirror family of K 3-surfaces We study the 1-dimensional family of K 3-surfaces given by the quartic equations X b := (x, y, z, w) ∈ P3 f (x, y, z, w) = x yz(x + y + z + bw) + w4 = 0 (2.1) with b ∈ P1 . Note that this surface, for general b, is not smooth but has six singular points defining a rational singularity of type A3 . The six points are (see [17, Sec. 4]) P1 = (0, 1, −1, 0),
P2 = (1, −1, 0, 0),
P3 = (1, 0, −1, 0),
P4 = (1, 0, 0, 0),
P5 = (0, 1, 0, 0),
P6 = (0, 0, 1, 0).
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The minimal resolution of the singularities defines a generically smooth family of K 3-surfaces. In [17] the following theorem was shown. THEOREM 2.1 (N. Narumiya and H. Shiga) The family X b has the following properties. (1) It arises as a mirror family from the dual of the simplest polytope P of dimension three. The dual mirror family is the family of all quartic K 3-surfaces. (2) The rank of Pic(X ) is greater than or equal to 19 for all b ∈ P1 \ {0, ±4, ∞} and equal to 19 for very general b (see [17, Sec. 4]). (3) T (X b ) has signature (2, 1) for b ∈ P1 \ {0, ±4, ∞}. (4) The periods of X b satisfy the Picard-Fuchs equation
(5)
(6)
3 11 3 (1 − u)23 − u22 − u2 − u = 0 2 16 32 (where 2 = u(d/du)) of the generalized Thomae hypergeometric function (see [22]) 1 2 3 F3,2 , , , 1, 1; u 4 4 4 4 and where we set u := (4/b) . In other words, the Picard-Fuchs equation is given by 3 51 3 (1 − u)u 2 8000 + 3u 1 − u 800 + 1 − u 80 − 8 = 0. (2.2) 2 16 32 The mirror map of the family X b is given by the arithmetic Thompson series T (q) of type 2A in the classification of J. Conway and S. Norton [5]: T (q) =
1 + 8 + 4372q + 96256q 2 + 124002q 3 + 10698752q 4 + . . . . q
Proof We refer to [17] for more details, but we sketch the proof of (4) and (5) since this is crucial; (1) – (3) follow from the construction. In particular, six A3 -singularities give rise to 18 independent cohomology classes of type (1,1), so that the Picard number is greater than or equal to 19. Since (5) is an easy corollary of (4), we prove (4). In [17] the periods are computed as power series in 1/b, and the differential equation in (4) follows from [22]. As in [17], we consider the new affine equation f (x, y, z) = x yz(x + y + z + 1) +
1 =0 b4
obtained by substituting w0 := bw and setting w0 = 1. The periods are integrals of the form Z 1 d x dy dz I (b) = . 2πi |x|=|y|=|z|=1/4 x yz(x + y + z + 1) + 1/b4
THE TRANSCENDENTAL K 1 OF A K 3-SURFACE
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On the other hand, one has a geometric series expansion at b = ∞: ∞
X 1 (−1)n b−4n = . 4 n+1 x yz(x + y + z + 1) + 1/b (x yz) (x + y + z + 1)n+1 n=0
Changing the order of summation and integration, we obtain 1 I (b) = 2πi =
1 2πi
∞ X
Z
|x|=|y|=|z|=1/4 n=0 ∞ Z X n=0 |x|=|y|=|z|=1/4
(−1)n b−4n d x dy dz (x yz)n+1 (x + y + z + 1)n+1 (−1)n b−4n d x dy dz . (x yz)n+1 (x + y + z + 1)n+1
Now one can apply three times the residue theorem and get I (b) = (2πi)
2
∞ X (4n)! n=0
(n!)4
b−4n .
Observing the identity involving Pochhammer symbols (4n)! (1/4)n (2/4)n (3/4)n 4 n = (4 ) , (1)n (1)n (1)n (n!)4 we have shown that I (b) = (2πi) F3,2 2
4 4 1 2 3 , , , 1, 1; . 4 4 4 b
Substituting u := (4/b)4 , one gets a multiple of the functions F3,2 (1/4, 2/4, 3/4, 1, 1; u) which satisfies a differential equation of order three precisely of the type described in (4), respectively, (5), by [22]. COROLLARY 2.2 In b-coordinates, the Picard-Fuchs equation can be written as b 4 b 4 b 3 000 3 b 2 1 b b 4 3 −1 8 + 1+ 800 + − 6 80 + 8 = 0. 4 4 4 4 4 16 4 4 32 (2.3)
Proof Use the chain rule. To make the following computations easier, we follow [17] and perform the following birational coordinate change (written in affine coordinates): bx y 1 + x yz X = x y, Y =i + , Z = yz. 2 z
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Then X, Y, Z are affine coordinates and define the family of surfaces Sb in Weierstrass form b2 1 +1 Sb : Y 2 = X X 2 + X Z + − Z 4 as an elliptic fibration over P1 in the Z -coordinate. The inverse transformation is given by x = −2
i X (1 + Z X ) , (−2Y + ibX )Z
y=
1 i Z (−2Y + ibX ) , 2 1+ ZX
z = −2
i(1 + Z X ) −2Y + ibX
in affine coordinates. The following is taken from [17] (with a slight correction). 2.3 The surfaces Sb are ramified coverings of P1 × P1 (in X, Z coordinates). In X, Y, Z coordinates, the canonical holomorphic 2-form on Sb is given up to a nonzero constant by dX dZ d X dY = 2 ω= , (2.4) YZ X (Z − 1/Z ) LEMMA
where X, Z are flat coordinates and where s p 1 b2 2 Y = Y (b) = P(X, Z ) = X X + X Z + − +1 . Z 4 Proof In the x, y, z coordinate system, the holomorphic 2-form is given up to a constant by (d x dy)/ f z in affine (x, y, z)-coordinates with w = 1. Then, using the coordinate transformations above, one computes that ω=
√ d x dy d X dY dX dZ −1 · = 2 = fz YZ X (Z − 1/Z )
since 2Y dY = X 2 (1 − 1/Z 2 ) d Z + (∂ P/∂ X ) d X . Now, if we apply the Picard-Fuchs operator DPF from Corollary 2.2 to (d X d Z )/Y Z , we get an expression of the form DPF
d Xd Z d Xd Z = K (X, Z ) · 7 , YZ Y Z
where K (X, Z ) is a polynomial function in X, Z .
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3. The normal function and the Picard-Fuchs equation On any elliptic surface, the easiest way to find cycles in K 1 is to use fibers. However, sometimes configurations coming from N´eron fibers (degenerate into a union of P1 ’s) do have trivial class in K 1 , as has already been observed by A. Be˘ılinson in [1]. But one can use one smooth fiber together with a bunch of sections (rational) and rational curves in degenerate fibers. In our example, let us take the following cycles: denote by Sb the surface defined by the equation b2 2 2 2 X+Z . (3.1) Sb : Z Y = X X Z + Z + 1 − Z 4 Let Cb be the smooth elliptic fiber over Z = 1 of this surface. Hence its defining equation is b2 X +1 . Cb : Y 2 = X X 2 + 2 − 4 The quadratic term X 2 + (2 − b2 /4)X + 1 in the right-hand side has two negative √ real roots if b is purely imaginary, for example, if b = −1. The points X = 0 and X = ∞ are rationally equivalent on Cb after taking a multiple of two since they are ramification points. The real line from 0 to ∞ does not hit the other ramification points by this observation. The surface X b in this birational model has 0 and X = ∞ as sections. The fiber over Z = 0 on Sb decomposes into three rational curves with multiplicity counted. Hence one can construct a cycle in CH2 (Sb , 1) for general b by using Cb , the two sections, and the degenerate fibers and appropriate rational functions on all curves. In X, Z coordinates the region 0 is given by the real square √ 0 ≤ Z ≤ 1, 0 ≤ X ≤ ∞. For b = −1 we make the following observation. LEMMA 3.1 √ For b = −1, all coefficients occurring in DPF ((d X d Z )/Y Z ) = K (X, Z ) · ((d X d Z )/Y 7 Z ) are positive integers, that is, all coefficients of K (X, Z ) and all coefficients of Y 2 = X (X 2 + X (Z + 1/Z − b2 /4) + 1).
Proof Here the rules of differentiating are ∂ X /∂b = ∂ Z /∂b = 0 and ∂Y /∂b = (b/4) · (1/Y 3 ). This implies that odd derivatives of 1/Y get multiplied by even powers of b. Now if we look at the coefficients of (2.3), we see that the coefficients at 9 000 and 9 0 become positive since (b/4)4 − 1 and (b/4)4 − 6 are negative rational numbers and get multiplied with (b/4)6 , respectively, (b/4)2 , which are both also negative rational numbers. The coefficients at 9 and 9 00 already involve 4th powers of b and hence are positive. Consequently, all coefficients occurring are positive. Using any computer
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algebra program, this can be verified, and indeed one has d Xd Z d Xd Z DPF = YZ 8192 × (349951 X 3 Z 3 + 85952 X Z 3 + 85952 X 5 Z 3 + 171904 X 4 Z 4 + 171904 X 4 Z 2 + 85952 X 3 Z 5 + 171904 X 2 Z 4 + 294912 X 2 Z + 294912 X Z 2 + 98304 Z 3 + 98304 X 3 + 294912 X 4 Z + 909352 X 2 Z 3 + 909352 X 3 Z 2 + 98304 X 6 Z 3 + 294912 X 5 Z 4 + 294912 X 5 Z 2 + 909352 X 4 Z 3 + 294912 X 4 Z 5 + 909352 X 3 Z 4 + 294912 X Z 4 + 98304 X 3 Z 6 + 294912 X 2 Z 5 + 85952 X 3 Z + 171904 X 2 Z 2 ) √ /(4 X 2 Z + 4 X Z 2 + 4 X + X Z + 4 Z )7/2 X Z . This completes the proof. In particular, if we integrate over the positive region 0, we get a positive and nonzero integral. Since the boundary of 0 is defined as the rectangle 0 ≤ Z ≤ 1, 0 ≤ X ≤ ∞ and since X, Z are flat with respect to the connection, we say that 0 does not depend on b (see introduction), and this suffices to show that the normal function is nontrivial. So we have proved Esnault’s conjecture (see [16]). COROLLARY 3.2 The projected normal function ν¯ (b) does not satisfy the Picard-Fuchs equation b 4 b 4 b 3 000 3 b 2 1 b b 4 3 00 −1 8 + 1+ 8 + − 6 80 + 8 = 0. 4 4 4 4 4 16 4 4 32 (3.2) In particular, it is not a rational multiple of a period for all but a countable number of values b. For those b, the corresponding cycle Z b has no integer multiple that is decomposable modulo Pic(X b ) ⊗ C∗ .
The main open problem remains to find a 1-form β such that dβ = DPF ((d X d Z )/Y Z ). We compute such a β for the Kummer-type model of these K 3-surfaces in the following section. 4. The solution of D ω = dβ In [17] we can find the description of a 2:1 map π : Sb → Sb0 onto a Kummer surface Sb0 that has a birational model with the equation ν + 1 2 t (t − 1)(t − ν 2 ), (4.1) u 2 = s(s − 1) s − ν−1
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where ν and b are related via the algebraic equation b2 = −4 ·
(ν 2 + 1)2 . ν(ν 2 − 1)
We prefer to use this equation for a computation of the solution of D ω = dβ since it is slightly easier, but we do not lose essential information. In this description we see that the transcendental part of H 2 (Sb ) also denoted by T (Sb ) has an Inose-Shioda structure (in the sense of [15]) and is therefore related to a variation of a family of elliptic curves. In fact, there are two isogenous elliptic curves E 1 (ν) and E 2 (ν) with equations ν + 1 2 E 1 (ν) : u 21 = s(s−1)(s−ν 2 ), E 2 (ν) : u 22 = t (t −1) t − (4.2) ν−1 together with a Nikulin involution (see [23]) on the abelian surface A = E 1 × E 2 such that the associated Kummer surface is Sb0 , and one has an isomorphism T (Sb ) ∼ = T (Sb0 ) under π∗ . In addition, this explains why the periods of Sb are squares of other hypergeometric functions related to the family E 1 (ν), respectively, E 2 (ν). More details about the birational map can be found in [17]. Further instances where InoseShioda structures and modular forms arise can be found in [8]. Let us now compute the Picard-Fuchs equation of the family E 1 (ν). If we let λ = ν 2 , we have √ ∂ν/∂λ = 1/(2ν) = 1/(2 λ), and therefore for any function 8 we have the transformation rules ∂8 ∂ν ∂8 1 ∂8 ∂8 1 = · = √ = ∂λ ∂ν ∂λ ∂ν 2 λ ∂ν 2ν and, for the second derivative, ∂ 28 1 ∂ 28 1 ∂8 . = − 3 2 2 2 ∂λ 4ν ∂ν 4ν ∂ν Plugging this into the standard hypergeometric Picard-Fuchs equation 1 λ(1 − λ)800 (λ) + (1 − 2λ)80 (λ) − 8(λ) = 0, 4 we get the new equation 1 − 3ν 2 0 8 (ν) − 8(ν) = 0 ν and the inhomogeneous variant (equality of 1-forms) p 2 1 − 3ν 2 ∂ s(s − 1)(s − ν 2 ) 2 ∂ (1 − ν ) 2 ω(s) + ω(s) − ω(s) = 2drel ν ∂ν ∂ν (s − ν 2 )2 s 2 (s − 1)2 = 2drel p , 3 s(s − 1)(s − ν 2 ) (1 − ν 2 )800 (ν) +
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where ω(s) = p
ds s(s − 1)(s − ν 2 )
.
The rational normalized version of this equation is p 1 − 3ν 2 ∂ 1 2 s(s − 1)(s − ν 2 ) ∂2 ω(s) + ω(s) = drel . ω(s) − 2 2 2 2 ∂ν ν(1 − ν ) ∂ν 1−ν 1−ν (s − ν 2 )2 (4.3) In a similar way, we use the substitution λ = ((ν + 1)/(ν − 1))2 and get the formula ∂ν/∂λ = −(ν − 1)3 /(4(ν + 1)) and hence ∂8 ∂8 ∂ν (ν − 1)3 ∂8 = · =− , ∂λ ∂ν ∂λ 4(ν + 1) ∂ν ∂ 28 (ν − 1)6 ∂ 2 8 (ν − 1)5 (ν + 2) ∂8 = + . ∂ν ∂λ2 16(ν + 1)2 ∂ν 2 8(ν + 1)3 Combining all this, we get the equation ∂2 ν 2 − 2ν − 1 ∂ 1 ω(t) + ω(t) ω(t) + 2 2 ∂ν ν(ν − 1) ∂ν ν(ν − 1)2 p 2 t (t − 1)(t − ((ν + 1)/(ν − 1)))2 =− drel 2 ν(ν − 1) (t − ((ν + 1)/(ν − 1))2 )2 for ω(t) = p
dt t (t − 1)(t − ((ν + 1)/(ν − 1))2 )
(4.4)
.
We have to compute a sort of convolution product of these two equations in the following sense: set ω = ω(s) ∧ ω(t) = p
ds dt s(s − 1)(s
− ν 2 )t (t
− 1)(t − ((ν + 1)/(ν − 1))2 )
,
and notice that we have the product formula ∂3 ∂3 ∂2 ∂ ω(t) ω = 3 ω(s) ∧ ω(t) + 3 2 ω(s) ∧ 3 ∂ν ∂ν ∂ν ∂ν ∂2 ∂3 ∂ + 3 ω(s) ∧ 2 ω(t) + ω(s) ∧ 3 ω(t). ∂ν ∂ν ∂ν Similar formulas hold for lower derivatives. Note that s, t are flat coordinates, so that differentiating a differential form with respect to the coefficients is a well-defined procedure. Such formulas can be used to compute (∂ 3 /∂ν 3 )ω and to obtain a PicardFuchs differential operator D for ω together with a solution β of D ω = drel β.
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LEMMA 4.1 One has the following inhomogeneous Picard-Fuchs equation involving 2-forms:
∂ 3ω 2ν + 1 ∂ 2 ω 7ν 4 − 6ν 3 − 4ν 2 + 6ν + 1 ∂ω + 3 + ν(ν + 1) ∂ν 2 ∂ν ∂ν 3 (ν − 1)2 (ν + 1)2 ν 2 ν 4 − 2ν 3 − 2ν − 1 ω = drel β, (4.5) + (ν − 1)3 (ν + 1)2 ν 2 where s(s − 1)(2ν 4 + 3ν 3 − ν 2 − 3νs + sν 2 − 2s) p ω(t) ν(s − ν 2 )2 (ν 2 − 1)2 s(s − 1)(s − ν 2 ) p 6 s(s − 1)(s − ν 2 ) 0 + ω (t) 1 − v2 (s − ν 2 )2
β = −2
t (t − 1)(2tν 4 − 7tν 3 + 7tν 2 − tν − t − 2ν 4 − 7ν 3 − 7ν 2 − ν + 1) p ω(s) ν 2 (ν − 1)4 (t − ((ν + 1)/(ν − 1))2 )2 (ν 2 − 1) t (t − 1)(t − ((ν + 1)/(ν − 1))2 ) p t (t − 1)(t − ((ν + 1)/(ν − 1))2 ) 0 +6 ω (s). (4.6) ν(ν − 1)2 (t − ((ν + 1)/(ν − 1))2 )2
+2
Proof We denote the derivative of a function (or a form in flat coordinates) f of ν by f 0 . Let us carry out the computation in a more general setting. Assume that we have two Picard-Fuchs equations, ω00 (s) − As ω0 (s) − Bs ω(s) = dβs , ω00 (t) − At ω0 (t) − Bt ω(t) = dβt , with functions βs , βt and As , Bs , At , Bt depending on ν. Now first note that ω000 (s) = As ω00 (s) + (A0s + Bs )ω0 (s) + Bs0 ω(s) +
d dβs dν
and furthermore that
d dβs = dβs0 dν by symmetry of mixed derivatives. A similar relation holds for t. By using the product formulas ω000 = ω000 (s) ∧ ω(t) + 3ω00 (s) ∧ ω0 (t) + 3ω0 (s) ∧ ω00 (t) + ω(s) ∧ ω000 (t), ω00 = ω00 (s) ∧ ω(t) + 2ω0 (s) ∧ ω0 (t) + ω(s) ∧ ω00 (t),
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we obtain that h i 3 1 3 ω000 − (As + At )ω00 = A0s + Bs + 3Bt − A2s − As At ω0 (s) ∧ ω(t) 2 2 2 h i 1 3 + A0t + Bt + 3Bs − A2t − As At ω(s) ∧ ω0 (t) 2 2 i h 3 + As Bs + At Bt − (Bs + Bt )(As + At ) + Bs0 + Bt0 ω 2 1 3 0 + drel βs − As + At βs ω(t) + 3βs ω0 (t) 2 2 1 3 0 0 − βt − At + As βt ω(s) − 3βt ω (s) 2 2 does not involve any more terms of the form ω0 (s) ∧ ω0 (t). Now let 3 1 3 (As + At ), B := − A2s − As At + A0s + Bs + 3Bt , 2 2 2 1 3 1 C := − As Bs − (As Bt + At Bs ) − At Bt + Bs0 + Bt0 . 2 2 2
A :=
Then, assuming that we have the equality 1 3 1 3 − A2s − As At + A0s + Bs + 3Bt = − A2t − As At + A0t + Bt + 3Bs 2 2 2 2 (this condition is equivalent to the fact that the elliptic curves E 1 (ν) and E 2 (ν) are isogenous), we have the inhomogeneous Picard-Fuchs equation ω000 − Aω00 − Bω0 − Cω = drel β, where β is the 1-form 1 3 0 β := βs − As + At βs ω(t) + 3βs ω0 (t) 2 2 1 3 0 + βt − At + As βt ω(s) + 3βt ω0 (s). 2 2 In our case, As = −(1 − 3ν 2 )/(ν(1 − ν 2 )), Bs = 1/(1 − ν 2 ), At = −(ν 2 − 2ν − 1) /(ν(ν 2 − 1)), and Bt = −1/(ν(ν − 1)2 ). Therefore we get A = −3
7ν 4 − 6ν 3 − 4ν 2 + 6ν + 1 , (ν − 1)2 (ν + 1)2 ν 2 ν 4 − 2ν 3 − 2ν − 1 C =− , (ν − 1)3 (ν + 1)2 ν 2
2ν + 1 , ν(ν + 1)
B=−
and, for β, the expression above. This finishes the proof.
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Appendix In this section we give the proof of the following lemma from Section 1. LEMMA 1.1 In the situation of Section 1, DPF (¯ν ) is a single-valued meromorphic function on B with poles only along degeneracies of X b , and therefore it satisfies a differential equation DPF (¯ν ) = g,
where g is a rational function in b ∈ B. Proof Assume that we have a family f : X¯ → B¯ of projective surfaces over a compact Rie¯ Let 6 ⊆ B¯ be the finite subset over which there are singular fibers. mann surface B. Let h : X → B be the smooth part of f . We may assume that the family is semistable (semistable reduction) and that there is a cycle Z ∈ CH2 (X, 1) such that the restriction of Z to all fibers induces the family of cycles in CH2 (X b , 1). (Both reductions perhaps require a finite cover of B, which does not, however, change the assertion.) The cycle Z has a class c3,2 (Z ) ∈ HD3 (X, Z(2)) in Deligne cohomology. By semistability, 1 := f −1 6 is a divisor with strict normal crossings and its Deligne cohomology can be computed via the logarithmic de Rham complex. Let V 2 be the sheaf of transcendental cohomology classes in R 2 h ∗ C, a local system of rank 22 − ρ(X b ) for b general. The Deligne class vanishes in F 3 ∩ H 3 (X b , Z), since b3 (X b ) = 0 in our case, and therefore induces a holomorphic normal function ν ∈ H 0 (B, V 2 ⊗ O B /F 2 ) over B. However, since the family is semistable, there is a canonical extension of ν to a holomorphic section of the sheaf R 2 f ∗ ∗X¯ / B¯ (log 1). This can be seen as follows. Let ZD ,X (2) = Cone R j∗ Z(2) → ∗X¯ (log 1)/F 2 [−1] j
be the Be˘ılinson-Deligne complex (see [9]) of X using the inclusion X ,→ X¯ , and let ZD ,h (2) = Cone R j∗ Z(2) → ∗X¯ / B¯ (log 1)/F 2 [−1] be the relative Be˘ılinson-Deligne complex of h. There is a natural surjection of complexes ZD ,X (2) → ZD ,h (2) which induces a morphism ¯ R 3 f ∗ ZD , f (2) . HD3 X, Z(2) → H 0 B, Since all fibers of h satisfy b3 (X b ) = 0, we conclude that the image of ¯ R 3 f ∗ R j∗ Z(2)) vanishes. Therefore the image of c3,2 (Z ) this element in H 0 ( B, 0 3 ¯ in H ( B, R f ∗ ZD , f (2)) is coming (at least locally because of monodromy) ¯ R 2 f ∗ ∗ (log 1)/F 2 ) and is thus an extension of ν ∈ from a class in H 0 ( B, X¯ / B¯
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¯ By construction, it is meromorphic along 6 but still H 0 (B, R 2 f ∗ ∗X/B /F 2 ) to B. multivalued with indeterminacies in the local system of integral cohomology classes. Now we apply the Picard-Fuchs operator. This makes g(b) a single-valued complex function on B. DPF has meromorphic (rational) coefficients in b since they are the coefficients of the characteristic polynomial of the Gauss-Manin connection, which has regular singular points along 6 by P. Deligne [6]. Therefore the resulting function g(b) is holomorphic outside 6, but it can have higher order poles along 6. By Chow’s theorem, any meromorphic function on B¯ is rational. Acknowledgments. We are grateful to S. Bloch, H. Esnault, Yu. Manin, and J. Nagel for several valuable discussions in which they urged us to find the solution of the inhomogeneous equation. We also thank N. Narumiya, H. Shiga, and N. Yui for explaining their results and M. Saito for explaining his conjectures about the structure of CH2 (X, 1) and his independent work on the same problem. We thank the referee for suggestions to improve the paper. Finally, we are grateful to Universidad Autonoma de Mexico and Universit¨at Essen for supporting this project. References [1]
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del Angel Centro de Investigaci´on en Matem´aticas, A.P. 402, Guanajuato, Guanajuato C.P. 36000, Mexico;
[email protected] M¨uller-Stach Fachbereich 6 Mathematik/Informatik, Universit¨at Essen, D-45117 Essen, Germany;
[email protected]
