Algebraic and Analytic Methods in Representation Theory
PERSPECTIVES IN MATHEMATICS, Vol. 17 S. Helgason, editor
Algebraic and Analytic Methods in Representation Theory Edited by Bent Orsted Department of Mathematics and Computer Science Odense University Odense, Denmark
Henrik Schlichtkrull Department of Mathematics and Physics The Royal Veterinary and Agricultural University Frederiksberg, Denmark
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Library of Congress Cataloging-in-Publication Data Algebraic and analytic methods in representation theory / edited by Bent ¢trsted, Henrik Schlichtkrull. p. cm. - - (Perspectives in mathematics; vol. 17) Main lectures from the European School of Group Theory held Aug. 15-26, 1994, in SCnderborg, Denmark. Includes bibliographical references and index. ISBN 0-12-625440-0 (alk. paper) 1. Lie groups. 2. Representations of groups. I. Orsted, Bent. II. Schlichtkrull, Henrik, 1954- . III. European School of Group Theory (1994: SCnderburg, Denmark) IV. Series. QA387.A44 1996 512' .55--dc20 96-22985 CIP
Printed in the United States of America 96 97 98 99 00 EB 9 8 7 6 5 4 3 2 1
Contents Contributors Preface
vii ix
Modular Representations of Algebraic Groups and Relations to Quantum Groups HENNING HAAHR ANDERSEN Orbital Varieties, Goldie Rank Polynomials, and Unitary Highest Weight Modules 53 ANTHONY JOSEPH Discontinuous Groups and Clifford-Klein Forms of PseudoRiemannian Homogeneous Manifolds TOSHIYUKI KOBAYASHI
99
The Method of Stationary Phase and Applications to Geometry and Analysis on Lie Groups 167 V. S. VARADARAJAN The Orbit Method and Unitary Representations for Reductive Lie Groups DAVID A. VOGAN, JR.
243
Index
341
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Contributors HENNING HAAHR ANDERSEN, Department of Mathematics, Aarhus Universitet, 8000 Arhus C, Denmark ANTHONY JOSEPH, Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel TOSHIYUKI KOBAYASHI, Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153, Japan V. S. VARADARAJAN, Department of Mathematics, UCLA, Los Angeles, California 90095-1555, USA
DAVID A. VOGAN, JR., Department of Mathematics, MIT, Cambridge, Massachusetts 02139-4307, USA
vii
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Preface The European School of Group Theory for the summer of 1994 was held in S0nderborg, Denmark, from August 15 to August 26 at the Sandbjerg Estate. About 50 advanced students participated, with interests quite wide but centered around algebraic and analytic aspects of Lie groups and their representations. The main lectures were delivered by five outstanding researchers, covering some of the deepest areas of the field today, yet giving introductions that were a pleasure to follow. It was a summer school with a high level of participation and interest on the part of the students, and it is our hope that the present volume will convey a similar spirit to its readers. The theory of reductive groups is at present as active and important as ever. Building on classical tools and structure theory as well as ideas from classical and quantum physics, many problems have been solved, and new research avenues have opened up. The subject has always required knowledge in several branches of mathematics, but even more so today, where an expanding amount of algebraic, geometric, and analytic techniques are combined. Since the written material by the lecturers at the summer school went far beyond usual lecture notes and the different topics formed a unifled view, albeit from perspectives quite apart, it was decided to publish the five contributions together. Chapter 1 by Henning Haahr Andersen gives an introduction to algebraic groups and their representation theory with a clear aim at reaching recent results about modular representations and quantum groups. Indeed, one of the lessons of the past few years of quantized enveloping algebras is that one should develop a good part of the theory parallel to that of algebraic groups. For this point of view, these lectures are very valuable. In Chapter 2, Anthony Joseph gives a full account of the classification of unitary highest weight modules for semisimple Lie algebras, simplifying and expanding previously published work as well as proving results about associated varieties.
In addition, several recent results are given on the
algebra of differential operators on orbital varieties. Toshiyuki Kobayashi ix
x
Preface
treats, in Chapter 3, discontinuous actions of discrete groups on homogeneous spaces and the corresponding problem of constructing Clifford-Klein forms. A number of classification results are given, opening up promising new areas of research on analogs of automorphic functions and locally symmetric spaces. Chapter 4 by V. S. Varadarajan develops the method of stationary phase from its very beginnings to advanced applications in geometry and in analysis on Lie groups. This fundamental technique is present in much of modern physics; in these lectures emphasis is on applications to the Plancherel formula, asymptotics of eigenvalues on locally symmetric spaces, character formulae, and related questions. David Vogan constructs, in Chapter 5, unitary representations of reductive Lie groups by the orbit method, in analogy with the quantization procedures leading from classical mechanics to quantum mechanics. Starting with the basic concepts and some motivating examples, these lectures patiently develop Lie theory, symplectic geometry, the theory of ideals in universal enveloping algebras, Dixmier algebras, and the induction techniques that enter in the orbit method. Emphasizing the distinction between semisimple and nilpotent orbits, many deep results are obtained and intriguing examples as well as conjectures are discussed. It is a pleasure to thank the five authors for their contributions to this volume as well as for their enthusiastic lectures at the summer school. Also, we thank the participating students for their dedication to mathematics and other sports. For financial support, we acknowledge Thomas B. Thriges Fond, Odense University, Aarhus University, The Danish Natural Science Research Council, and a grant from the EC, Euroconferences in Group Theory under the program Human Capital and Mobility. Finally, we are grateful to the staff at Sandbjerg for adding to the pleasant memories of this summer school. Djursholm, October 1995 Bent Orsted, Henrik Schlichtkrull
Chapter 1 Modular Representations of Algebraic Groups and Relations to Quantum Groups HENNING HAAHR ANDERSEN
University of Aarhus
CONTENTS
.
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Introduction Algebraic groups and their representations Reductive algebraic groups Classification of simple G-modules Derived functors of induction Representations of SL2 (k) The Borel-Weil-Bott theorem Serre duality and complete reducibility Frobenius kernels and their representations Injective GrT-modules Kempf's vanishing theorem Quantum groups Tilting modules References
Introduction
The (very informal) notes in this chapter present some highlights in the "classical" representation theory for reductive algebraic groups over algebraically closed fields. The theory is developed in such a way that almost everything carries over to quantum groups. In particular, we have emphasized the similarities between the modular representation theory (the case
2
H . H . Andersen
where the underlying field for the algebraic group has positive characteristic) and the representation theory for q u a n t u m groups at roots of unity. We start out by giving the basic general definitions concerning algebraic groups and their representations.
One of the most important tools
for us when studying representations will be the induction functor. So already in the first section we introduce this concept and give some of its first properties. In Section 2 we specialize to the case of reductive groups. Here we need some of the key results about the relations between these groups and their associated root systems. Most of what we need is collected in Facts 2.6. Our first real theorem comes in Section 3, where we present the classification of finite dimensional simple modules for reductive groups. To obtain this we rely heavily on induction from a Borel subgroup.
In the
following sections we explore also the higher derived functors of this induction. In particular, we prove the Borel-Weil-Bott theorem (in Section 6) and Serre duality (in Section 7). From these two results we easily deduce the complete reducibility of all finite dimensional modules for algebraic groups in characteristic zero. Then we turn to characteristic p > 0. Here we shall explore the socalled Frobenius subgroups.
To handle these properly, we need a more general
definition of algebraic groups than the one we have been working with so far. So we begin Section 8 by introducing this more general concept (of algebraic group functors), and we have some remarks on how to generalize the previous results. Then we study (in the rest of Section 8 and in Section 9) the Frobenius subgroups and their representations in a wav quite parallel to our presentation of representations of reductive groups.
As a reward,
we show in Section 10 how we can now very easily deduce the i m p o r t a n t K e m p f vanishing theorem. In Section 11 we sketch how the representation theory of q u a n t u m groups behaves very similarly to the representation theory for reductive algebraic groups. In fact, we leave most of the quantization to the reader, giving only some indications in a few key places. We conclude by a section on tilting modules that makes sense both for algebraic groups and q u a n t u m groups. The results here have applications to the recent work on invariants of 3-manifolds. The material in these notes formed the major parts of my lectures at the Summer School on Group Theory at Sandbjerg in August 1994. It is
Chapter 1
Modular Representations
3
a pleasure for me to thank both the organizers and all the participants for their patience and enthusiasm. 1
Algebraic groups and their representations
Throughout, k will denote an algebraically closed field. For n C N consider the general linear group GLn(k) with coefficients in k, i.e., GLn(k) = {(aij)i,j=l,2
.....
n
l aij C k, det(aij) 7£ 0}
Clearly, this is an open subset (in the Zariski topology) of k n2 (being the complement of the zeroes of det), or alternatively, it is a closed subset of k n2+1 (being identified with the zeroes of the polynomial T det(Xij) - 1 C kiT, (X~j)]). Definition 1.1 By an algebraic group (over k) we understand a Zariski closed subgroup of GLn(k) for some n E N. E x a m p l e s 1.2 i) GL~(k), n E N. ii) SLn(k)-- {A e G L n ( k ) i d e t ( A ) = 1}, n e N. iii) The 1-dimensional additive group (k, +). This is a subgroup of GL2(k) viaxH
(ix) 0
1
'
xCk
iv) The 1-dimensional multiplicative group (k x, .). Here k x = k\{0}, i.e., (kx, ") = G L I ( k ) . v) Tn -- Tn (k) -
_( [
dimensional torus.
/ tl 0
"..
0/
} tC i
kx
o
This is called the n-
tn
vi) Bn - B n ( k ) - {(aij) e GLn(k) I aij = 0 for i > j}, the subgroup consisting of the upper triangular matrices in G L n ( k ) . vii) Un = U~(k) = {(aij) e Bn] aii - 1 for i - 1,2,...,n}. Note that U~ is a normal subgroup of Bn and that Bn = Tn U~. If V is an n-dimensional vector space over k we can (and shall) identify the group of invertible k-linear endomorphisms of V, GL(V) with the algebraic group GLn(k).
4
H.H. Andersen
D e f i n i t i o n 1.3
The coordinate ring k[G] of an algebraic group G is k[G] = {f: G ---, k l f is regular}.
(Here regular refers to the notion of "regular function on a variety," cf. [Ha]. In the following, all maps between varieties are tacitly assumed to be regular. In the case at hand, k is considered as the l-dimensional affine space. More generally, we always consider an n-dimensional vector space as the affine n-space.) T h e group structure on G gives rise to a Hopf algebra structure, cf. [Sw], on k[G].
The comultiplication A: k[G] ~
k[G] ® k[G] comes from
multiplication G x G ~ G, the counit e: k[G] ~ k from the unit 1 -~ G, and the antipode S: k[G] ~ k[G] from the inverse map G ~ G. Show t h a t we have the following identities:
E x e r c i s e 1.4
k[(k,-~-)]- k[X], k[Tn ] --
Definition
1.5
]~[(]gx,.)]_ k[X,X-1],
k[Xl,... ,xn,xll,...,xnl].
A homomorphism f:G1 --~ G2 between two algebraic
groups is a map of varieties t h a t is also a group homomorphism.
A representation of an algebraic group G on a finite dimensional vector space V is a h o m o m o r p h i s m p: G ~ GL(V). We say in short t h a t V is a representation of G (or a G-module), and we often write gv instead of
p(g)(v), when g e G and v e V. If V and V t are two G-modules, then a G-homomorphism from V to V t is a linear map t h a t commutes with the G-actions. The set of these is d e n o t e d H o m a ( V , V'). We shall need to extend the preceding definition of representations to a class of infinite dimensional ones. Namely, we say t h a t a group homom o r p h i s m p: G ~ GL(V) from an algebraic group G into the general linear group on some vector space V is a a rational representation of G if V is a union of finite dimensional representations as just defined. In the following, when we write G-module we u n d e r s t a n d such a rational representation of G.
Chapter 1
Modular Representations
5
E x a m p l e s 1.6 i) The trivial representation of an algebraic group G is the representation on k,
G~GL(k)=k
×
that takes any g C G into I C k. More generally, the trivial representation of G on any vector space V is given by the trivial homomorphism G ~ GL(V). ii) A homomorphism G ~ GLn(k) defines a representation of G on k n (via the identification GLn(k) - GL(k~)). In particular, the identity GLn(k) --, GL~(k) defines the natural representation of GLn(k) on ]¢n. iii) A rank one representation of G is a homomorphism G -~ GL(k) = k x. These representations are called the characters of G and they constitute the set X(G). We have for instance X ( ( k , +)) = 0 and
x(kx)_ z.
iv)
The coordinate ring k[G] of G may be equipped with several representations: a) (the right regular representation) Pr: G ---, GL(k[G]) is defined by
Pr(g) f : X ~ f (xg),
g, x e G, f e k[G].
b) (the left regular representation) Pl: G ~ GL(k[G]) is defined by
pl(g)f: x H f ( g - l x ) ,
g,x e G, f e k[G].
c) (the conjugation representation) p: G--~ GL(k[G])is defined by
p(g) f : x H f (g-lxg),
g, x e G ,
fek[G].
Using the correspondence between the algebraic group G and the Hopf algebra k[G], we see that a representation of G corresponds to a comodule for k[G]: If V is a representation of G, then we get a k-linear map Av: V --. V ® k[G] satisfying the appropriate comodule axioms, cf. [Sw, p. 30].
6
H . H . Andersen
D e f i n i t i o n 1.7 Let V be a representation of G. Then the fixed points of G in V are defined by
V G={vCVIgv=vforallgEG}.
Note that
V a "~ H o m a ( k , V).
D e f i n i t i o n 1.8
Let H be a closed subgroup of the algebraic group G and
let V denote a representation of H. The induced representation IndaH V is defined by IndaH V -- { f ' G ~ V l f ( g h ) -
h - i f ( g ) , g e G, h E H}
with G-action given by
(xf)(g) - f ( x - l g ) ,
x,g e G, f e IndGHV.
In the case where V is infinite dimensional, we require that the elements f C IndaHV are maps of varieties in the sense that f(G) is contained in a finite dimensional G-invariant subspace W of V such that f: G --~ W is a map of varieties. The map Ev" IndaH V --, V that takes f C IndaHV into f ( 1 ) i s called the evaluation map. It is clearly an H-homomorphism. Examples
1.9
i) Consider the trivial subgroup H = 1, and let k denote
the l-dimensional trivial representation of H. Then I n d l a k - k[G]. The action is the left regular representation of G on k[G], cf. Example 1.6(iv). ii) The other extreme is the case H = G. In that case, we have
Chapter 1
Modular Representations
7
Indaa V "~ V for all G-modules V. iii) Let G - SL2(k) and take H - T2 gl G. Then IndaHk ~ S ( V ) , the symmetric algebra on the natural 2-dimensional representation of G. (To see this takes a little work, cf. also Section 5.) P r o p o s i t i o n 1.10 (Frobenius reciprocity) Let H be a closed subgroup of G. Suppose V is an H-module and M is a G-module. Then the map
H o m a ( M , IndaHV) ---, HomH(M, V) that takes ¢ to Ev o ¢ is an isomorphism. Proof: The inverse map takes ¢ C HomH(M, V) into the homomorphism that maps m C M to g ~-, ~p(g-lm). [-7 Proposition 1.10 has several consequences. We mention a few of the more immediate ones: C o r o l l a r y 1.11 Let H denote a closed subgroup of G. i) The functor Ind/a/ is left exact (being the right adjoint of restriction). ii) Induction is transitive; i.e., if K is a closed subgroup of H, then IndaK ,-~ IndaH o Ind H. iii) IndaH takes injective H-modules into injective G-modules. (An injective G-module is by definition a G-module that is injective in the category of all rational representations of G.) By taking H = 1 in (iii) we get for any G-module V an embedding of V into IndlaV. The latter is injective by (iii), and hence
iv) The category of rational representations of G has enough injectives. We shall also need the following useful result (the proof of which takes a little more work, see [Ja, Proposition 1.3.6])
Let V be a G-module and W an H-module. Then there is a natural isomorphism of G-modules
P r o p o s i t i o n 1.12 (The tensor identity)
I n d ~ ( V ® W) "" V ® IndaH W.
8 2
H.H. Andersen Reductive algebraic groups
From here on, we shall concentrate on reductive algebraic groups. In this section, we give the appropriate definitions, prove a key result on representations of tori, and list some facts about reductive groups that we shall need in developing their representation theory. D e f i n i t i o n 2.1 i) An algebraic group T is called a torus if there exists n C N such that T __ Tn (cf. Example 1.2(v)). ii) An algebraic group G over k is called reductive (resp., sernisimple) if the maximal connected solvable normal closed subgroup in G is a torus (resp., trivial). E x a m p l e s 2.2
i) A torus is clearly reductive.
ii) GL~(k) is reductive: This is a consequence of Lie-Kolchin's theorem, [Hu2, 17.6], which says that a connected solvable closed subgroup N of GLn(k) is conjugate to an upper triangular subgroup. If N is also normal, it follows that N is upper triangular. By symmetry, N is therefore diagonal. But if Eij denotes the n x n matrix with 1 at the ( i , j ) - t h entry and 0 elsewhere, then
(1 + Eij)
/
"'. 0
0/
(1 + Eij) -1 -
tn
/
".. 0
0/
+ (tj - ti)Eij
tn
for all t l , . . . , tn E k. Hence, we see that
N<__{
I t °I "..
0
tEk×}-Z(GL~(k))
t
and the claim follows (the only connected subgroups of k x are 1 and
k×).
iii) SL~ (k) is semisimple: This follows from the same arguments as in (ii). Let now T be a torus. If T ~_ Tn, then X ( T ) ~_ Z n (in the following we shall identify ( A I , . . . , A n ) E Z n with the character A:Tn --- k x that
Chapter 1 Modular Representations takes
I
t1
•
0I
..
into
t h1 i ' ' '
t n)~n , t l , . .
"~
tn
9
C k X ). As T is abelian, we
0 tn know t h a t all irreducible representations are 1-dimensional, i.e., belong to X ( T ) . The following result says that all finite dimensional representations split into a direct sum of 1-dimensional representations (i.e., are completely reducible). P r o p o s i t i o n 2.3
Let V be a finite dimensional representation of the torts
T. Then V-
~V~,
ACX(T)
where for each A E X(T), v~ - {v e v l t v - A(t)v, t e T}.
Proof: Choose first arbitrarily a basis { V l , . . . , Vr} of V. Then we can write
tvi - E fij(t)vj, J
t C T,
for some fij E k[T].
Identifying T with Tn and recalling that k[Tn] = k [ X 1 , X ~ I , . . . , X n , X n 1] (cf. Exercise 1.4)), we have
fij
I
t1
"'.
0
0I
-- E
AEX(T)
t~
rij'xt~ll""t~n"
If we denote our representation of T on V by p: T ---, GL(V), then the preceding tells us that we may write
p(t)- Z
AEX(T)
where p~ E Endk(V) is the endomorphism that in the basis
Vl,...,
Vn
is
given by the matrix (rij~)i,j=l .....n. Now X ( T ) is a linearly independent subset of k[T]. This together with the fact that p(tt') = p(t)p(t'), t, t' E T, i.e.,
x(t)x(t')p XCX(T)
-
Z ~,,~X(T)
for
t,t'
T,
10
H. H. Andersen
means first that
ik(t')pa -
~ #(t')pap, ,~x(r)
and then
pxp,
if A - # ,
[ 0,
p
"'. 0
1
Z pa" a~x(r)
From (2.1) and ( 2 . 2 ) i t follows clearly that V -
p~ ( v ) -
(2.1)
otherwise.
/ 1 °/
Note t h a t we have also
Id-
_ fp~,
for all t' C T,
(~p~(V)
(2.2)
and t h a t
v~. D
Terminology
2.4
Let V be as in Proposition 2.3. If A E X ( T ) , then
we call Va the A-weight space in V, and we say that ~ is a weight in V if
v~, # o. Let G be a connected reductive algebraic group, and fix a maximal torus T <_ G (i.e., T is a torus, T _< G and T is maximal with these properties). If g E G, the conjugation action x H gzg -1, x C G, gives rise to an endomorphism Ad(g)" g + g, where t~ is the Lie algebra of G. In this way we obtain a representation g ~-~ Ad(g) of G on t~, i.e., Ad" G ~ GL(g) is a homomorphism of algebraic groups, called the adjoint representation of G. W h e n we restrict Ad to T we have a finite dimensional representation of T, and by Proposition 2.3 it splits into a sum of weight spaces. D e f i n i t i o n 2.5
Let G and T be as before. The nonzero weights of T in
g are called the roots of G (with respect to T). The set of roots will be denoted by R. We collect some basic facts about the correspondence between R and G; see, e.g., [Hu2].
Chapter 1
Modular Representations
11
Facts 2.6 Let G be a connected reductive algebraic group with a maximal torus T < G. The set of roots R of G w.r.t. T is a root system in the euclidian space span•R C_ X ( T ) ®z R. ii) For each a E R, there exists a (unique up to scalar) homomorphism x~: (k, +) ---, G such that t x ~ ( z ) t -1 = x~(c~(t)z), t ~ T, z C k. The image of x~ is called the root subgroup in G associated with. cx. It is denoted Us. iii) Choose a set R+ C R of positive roots, and let U (resp., U ~) denote the subgroup of G generated by {U~ !c~ E - R + }
(resp., {&~ t o~ C R~_}).
Then U - I - I ~ R+ V_~ (resp., U' = I-I~c R+ V,~ ), with ~ denoting direct product. Moreover, U M T = 1 = T A U ~, and if we set B = T U then U~B is an open subset of G.
iv)
W = N G ( T ) / T is a finite group, which via its natural action on X ( T ) becomes identified with the Weyl group for R.
E x e r c i s e 2.7
3
Check the facts (i)-(iv) for G = GL~(k).
Classification of s i m p l e G - m o d u l e s In this and the next sections we consider a connected reductive al-
gebraic group G over k with a fixed maximal torus T.
We will use the
n o t a t i o n X ( T ) , R, R+, B, U, U I, Us, and W from Section 2.
L e m m a 3.1 Let V be a finite dimensional representation of G. Then the weights in V (w.r.t. T ) form a W-stable subset of X ( T ) . Proof." Let w = n T C W , n E N a ( T ) .
Take v E V~\{0}. T h e n t(nv) =
n ( n - l t n ) v = n ( A ( n - l t n ) v ) = (wA)(t)nv, for all t C T; i.e., nv E Vw~\{0}. D
L e m m a 3.2
Let E be a finite dimensional representation of B. Suppose
e C E a \ { 0 } for some A C X ( T ) .
Then, for all ct E R+, we have
U_~e C_ e + ~ n>O
Ea_n(~.
H. H. Andersen
12
Proof:
Since by Proposition 2.3 we have E -- ~ # E X ( T ) E , ,
we may for
each c~ E R+ write z e k,
ttEX(T)
where e. E E . and f~ e k[(k, +)] - k[X]. If a.,n e k are determined by
f . ( z ) - }--~n>Oa.,n zn, z E k, then by Fact 2.6(ii)we get tx_a(z)t-le
-
-
x_a(ol(t)-lz)e.
The left hand side is equal to l ( t ) -1 ~--~, f , ( z ) p ( t ) e , ,
whereas the right
hand side is }--~ f~(c~(t)-lz)e~. This means
A(t)-lA(z)p(t)-
A(c~(t)-lz),
for all #, t, z,
i.e.,
A(t)-lp(t)a~,n -- a~,nOl(t) -n, Hence, f~ - 0 unless p - I -
for all #, t, n.
n a for some n _> 0 and f~-n~(Z) - an Zn
for some an E k. Using also that x_~(0) - 1, we see that fx - 1 and the lemma is proved. [3 The choice of R+ induces an ordering on X ( T ) , namely I _< # if P-
I-
~ER+
n~c~ for some n~ E N.
P r o p o s i t i o n 3.3
Let E be a finite dimensional B-module. Then we have,
for any )~ E X ( T ) and e E E~\{0}, Ue C_ e + E
E,.
tt
In particular, E~ C_ E U if I is minimal among the weights in E. P r o o f : Immediate from Lemma 3.2 and Fact 2.6(iii). W1 Recall that by Facts 2.6 we have U,~B and B / U ~_ T. Each I E X ( T ) therefore gives a character (also denoted by A) of B, namely,
A(tu) - l(t),
t E T, u E U.
Chapter 1
Modular Representations
13
Now X ( k , +) = 0, cf. E x a m p l e 1.6(iii), and hence also X ( U ~ ) = 0 for all a C R. It follows t h a t X ( U ) = 0. In other words, all characters of B come from T (as before) or
X ( B ) ": X ( T ) .
(3.1)
We m a y therefore consider (in a unique way) every ~ e X ( T )
as a 1-
dimensional representation of B.
Corollary 3.4 tration
Any finite dimensional representation E of B has a ill0-
F r+l
C F r C '''
C
F°
-
E
by B-invariant subspaces F i with F i / F i+1 ~ )~i, i - O, 1 , . . . , r , where Moreover, the filtration may be chosen such that )~i < Aj only if i > j.
{ A 0 , . . . , At} is the set of weights in E.
N o t a t i o n 3.5
If E is a representation of B, then we write H ° ( E )
-
IndaB E. A weight ~ C X ( T ) is called dominant if (,~, c~v) _> 0 for all c~ E R+. (Here, c~v denotes the coroot associated with a.
Recall t h a t s~A -
~-
(~, c~V)c~ defines the reflection s~ in W associated to c~ E R+.) Clearly, is d o m i n a n t iff s~A _< ~ for all c~ E R+. T h e set of d o m i n a n t weights is denoted X ( T ) +. Theorem
3.6
Let )~ e X ( T ) and suppose H°()~) # O. Then
i) ~ e X(T)+. ii) The restriction map ¢: H°()~) --~ k[U'], which takes f e g ° ( A ) into f l u ' , is injective. iii) If we let T act on k[U'] via t . h" u H )~(t)h(t-lut),
t~T, u~U' h~k[U']
then ¢ becomes a T-homomorphism. iv) The weight spaces H °()~)~ are all finite dimensional; and if H °(~), O, then # < A. v) dim H°(A)~ - 1. vi) dim H ° (~) < c~.
H. H. Andersen
14
Proof: Consider first (ii): If f e H°(A) restricts to 0 on U', then f(ub) = A(b)-lf(u) = 0 for all u E U', b E B; i.e., f is also zero on the open set U'B (cf. Fact 2.6(iii)). Now a connected group is an irreducible variety, and hence U'B is dense in G. It follows that f = 0. Next we deal with (iii): Take f E H°(~), t E T. Then ¢(tf)(u) = f ( t - l u ) = f ( t - l u t t -1) = A(t)f(t-lut) = t¢(f)(u) for all t e T, u E U'. Let us then look at (iv): Suppose h E k[U'],\{0} for some p E X(T) (we consider the action of T on k[U'] defined in (iii)). Then we have A(t)h(t-lut) - p(t)h(u), By Fact 2.6(iii) we have u write
(3.2)
I-[aER+ x~(za) for some z~ E k, and we may
h(u)- Ear_ r
for some at_ E k, r -
(ra)~eR+ (using t - l u t -- I-Is x~(a(t)-lz~))
a
~(~)
t E T, u e U'.
H
za
hER+
E N IR+I • Inserting this in (3.2) we get
zar~) -- #(t) Er
a r _ ( ~ zar~).
We conclude that # - ~ - ~--~R+ r ~ a for some r E NIR+I and (iv) now follows from (ii) and (iii). In fact, we get d i m g ° ( A ) , _< # { r E N IR+I ] - ~ -
~c~+
r~}.
To prove (v) we see by the preceding that we only have left to check that H °(~)~ # 0. But i f H °(A)~ = 0, then E v ~ : H °(~) +
~ would be
the zero map, which, by the universality (Proposition 1.10) of the induced module, would imply H ° (~) = 0. Now we have only left to observe that (i) follows from (iv) and (v) (by Lemma 3.1, we see that s~A is a weight of H°(A) for all a E R+) and that (vi) is a consequence of (iv) (it is well known that any weight is W-conjugate to a dominant weight and that there are only finitely many
p E X(T) + with # _< ~). [::]
Chapter 1
Modular Representations
15
C o r o l l a r y 3.7 If E is a finite dimensional representation of B, then N ° ( E ) is a finite dimensional representation of G.
Proof." Apply H ° to a filtration of E as in Corollary 3.4 and use Theorem 3.6(vi). (Recall that H ° is left exact). D It turns out that the converse of the implication H°(A) ¢- 0 :=~ A C X ( T ) + (see Theorem 3.6(i)) is also true, i.e., Let A E X ( T ) . Then H°(A) -y: 0 iff A E X ( T ) +
(3.3)
(This involves a little algebraic geometry; see, e.g., [Hu2, 31.4].) Theorem
3.8
Let ~ E X ( T ) +
i) There is a unique irreducible G-invariant subspace L(A) in, H°(A). ii) dim L(A)a - 1. P r o o f : We claim
U'-
(3.4)
Note that the containment 2 follows from Proposition 3.3 (with U replaced by U'), since H ° ( A ) t , -
0 for all # > A. To get the other inclusion, take f E H°()~) U' nonzero and note that f(u'b) - A(b)-I.W(1) for all u'b C U'B. Since U'B is dense in G (Fact 2.6(iii)) we conclude that E v a f f(1) is nonzero. But this means that EVaiHo(X)u, is injective, and hence d i m H ° ( A ) u ' < 1. Let us now prove (i). Suppose S and S I are two different irreducible G-invariant subspaces of H°(A). Then S N S I -- 0. On the other hand, using Proposition 3.3 we have 0 =/= S U' C H°(A) U' - H°(A)~; and since the latter space is 1-dimensional (Theorem 3.6(v)), we conclude S U' - H°(A)~. The same is, however, true for S ~, and we get a contradiction. The same reasoning proves L()~) v' - H°(A)~; i.e., L(A)~ ~- 0 and (ii) follows. IS] 3.9 If S is a finite dimensional irreducible representation of G, then there exists a unique ~ C X ( T ) + with S ~ L(/~).
Theorem
P r o o f : The uniqueness we get from Theorem 3.8(ii). To get the existence, By
we pick A E X ( T ) such that A is maximal among all weights of S.
Corollary 3.4, we can then find a B-invariant subspace F C_ S with S / F ~_
H. H. Andersen
16
A. In particular, 0 :/: HomB(S, A) -~ Homa(S,H°()~)); see Proposition 1.10. But this means that A E X ( T ) + (by (3.3)) and that S may be identified with a G-invariant subspace of H°()~). The conclusion follows by Theorem 3.8(i). K] E x a m p l e 3.10 i) 1 for all t C T. to ambiguities representation
Let ,~0 E X ( T ) denote the trivial character, i.e., ,~0(t) = (Sometimes we write 0 for this character, but this leads in the present context.) Then H°(,~0) = k, the trivial of G. (By Theorem 3.6, we see that A0 is the only
weight of H ° (,~0).) ii) Let co E X ( T ) +, and suppose co is minimal among dominant weights (i.e., no ,~ e X ( T ) + satisfies ,~ < co). Then H°(co) = L(co). (One checks that the only weights of H°(co) are {w(co) [ w e W}). Let V be a finite dimensional representation of T. Then we set (cf. Proposition 2.5) ch V -
E(dim
V;~)e~ C Z[X(T)],
and we call this the (formal) character of V. Clearly, ch is additive on short exact sequences.
(3.5)
Also, we see from
L e m m a 3.1 that If V is a representation of G, then ch V e Z[X(T)] W.
(3.6)
Here, W acts on Z[X(T)] via w(}-~ a~e~) - ~-~,~a~e w~, w E W. Problem
3.11
Determine ch L(A) for all A e X ( T ) +.
This problem is in some sense the very first problem one encounters when studying finite dimensional representations of reductive groups. W h e n char(k) = p > 0, it is still open in general. (We will have more to say about it later on; see Problems 8.9, 9.7, and 11.7 and Proposition 11.8.)
Chapter 1 4
Modular Representations
17
D e r i v e d f u n c t o r s of i n d u c t i o n We still keep the general assumptions from Section 3. Consider two
closed s u b g r o u p s H I _
Recall that IndH~ is a left exact functor from the category of HImodules to that of H2-modules and that these categories have enough injectives (Corollary 1.11(iv)). Hence, we may consider the right derived functors RiIndH~ of IndH~. If V is an Hi-module, then RiIndH~V is obtained by choosing an injective resolution O---> V----+ Io---+ I1 ---~ I 2 - - + ' " --H2 and applying IndH~. Then R i T lnCtH1V is the ith cohomology of the resulting complex, IndH~ Io.
In the case where Hi = B and/-/2 = G, we write H i-
RiIndCB .
(4.1)
(Note that this is consistent with the use of H ° in the previous section, see Notation 3.5). In the general case, we sometimes write H i ( H 2 / H 1 , - ) 1H2 instead of R i-lnOH1. If Hi -- 1, then Ind H2 (E) - k[H2] ® E, for any k-vector space E (this follows, e.g., from the tensor identity of Proposition 1.12). In particular, Ind H2 is an exact functor, and hence RiInd H2 - 0 for i > 0. Similarly, Ind H2 is exact for all /-/2 >_ T (because all T-modules are injective by Proposition 2.3). Note in particular that
IndBE - k[U] ® E for all T-representations E.
(4.2)
Here, U acts on k[U] via the left regular
representation and T acts by t f: u ~ f ( t - l u t ) , t e T, u e U, f e k[U]. P r o p o s i t i o n 4.1 Let E be an Hi-module and V an H2-module. we have for all i an H2-isomorphism,
Then
--H2 (V @ E ) ~ V ® R i IndH1 H2 E . R i-INCH1
Proof." For i = 0, this is the tensor identity from Proposition 1.12. It follows for i > 0 by noting that if E ~ I. is an injective Hi-resolution of
18
H. H. Andersen
E, then V ® E ~ V ® I. is an injective Hi-resolution of V ® E (injectivity is preserved when tensoring by an arbitrary representation). D The preceding derived functors of induction relate to sheaf cohomology of bundles on the homogeneous space
H2/H1 as follows:
Let V be an Hi-module. Then V induces a bundle £ ( V ) on the sections of which over an open subset U C F ( U , £ ( V ) ) - { f ' T r - l ( u ) --~ (here,
H2/H1,
H2/H1 are
V l.f(xb) - b - i f ( z ) , w E 7 r - l ( u ) , b C B}
7r:H2 ~ H2/H1 denotes the canonical map). F(H2/HI,£(V))- IndH~V. We claim that in fact
Note that
Hi(H2/H1 £ ( V ) ) ~ RiIndH2V •
~
,
HI
(4.3/
for all i. Since we already observed (4.3) for i = 0, it follows for all i by standard dimension shift once we check that an injective H i - m o d u l e I induces a bundle
£(I) that is acyclic for the global section functor. This in turn
reduces to the standard fact t h a t on the affine variety/-/2 any bundle has vanishing higher cohomology, see [Ja, 1.5.11]. Via (4.3) we are now able to carry over properties of sheaf cohomology to our derived functors of induction. Although we could make much more heavy use of this (e.g., in Theorem 3.6(vi) or later in Theorem 7.1), we shall prefer to give pure representation theoretic arguments whenever possible. However, we shall need the following vanishing theorem (see [Ha, Theorem III.2.7]):
Hi(H2/HI,.T ") = 0 for all sheaves 5c on
for a l l / >
dimH2/H1,
(4.4)
H2/H1.
Let us use this opportunity to also introduce Ext-groups as derived functors: If H _< G is a closed subgroup and if E is a representation of H we may consider the right derived functors of the left exact functor H o m H ( E , - ) from H-modules into k-vector spaces. As usual, we write Ext'(E,-)-
RiHomH(E,-).
Chapter 1
5
Modular Representations
19
R e p r e s e n t a t i o n s of SL2(k) In this section, we set G = SL2(k) and t
U-{(0
0
tEk×},
1 u 1)
~ E k}, t
t- 1
t E k x ~1,E k
It is then straightforward to check that G-BUU(0
\ 1
-1'~ B 0 /
(disjoint union).
Set V = k[X,Y], and let G act on V by
b)
a
d
c forall ( ac
f(X,Y)-
db ) E G, f ( X , Y )
f ( a X + cY, bX + dY),
E k[X,Y]. Then V is a rational G-module.
In fact, V = t~,>0 V,., where V~ = span k { x i Y j I i ÷ j = n}. Note that V,, is G-stable, and that we have
(o (o (o t
t -0 1 ) x i y j
__ t i _ J x i y j
1
1
(5.2)
1
m--0
-1)0 x~YJ - ( - 1 ) J x J Y i "
By (5.1), these formulas tell everything. In this case, X ( T ) = Z. For n E Z, let A,:T --, k × denote the character
0
t -1
~
'
"
H. H. Andersen
20
Proposition 5.1
For all n >_ O, there is a natural G-isomorphism Cn" Vn "~ H°()in).
Proof:
The projection V~ --~ k onto the basis element y n E Vn is a B-
homomorphism if we let B act on k via )in.
By Frobenius reciprocity
(Proposition 1.10), we get a G-homomorphism ¢'Vn ~
H°()in).
One
checks that this is an isomorphism. [] C o r o l l a r y 5.2
Let n >_ O. Then H°()in) has a basis { v 0 , . . . , v ~ } such
that, for all i - 0 , . . . , n , i) tvi -- )i2i_~(t)vi, ii)
(10 u) 1
~ ( im)U'--'~Vm, " Vi -- Em=O
Proof: Set v i -
Remark
t C T. u e k.
C n ( X n - i y i) and compare with (5.2). D
It is easy from this corollary to check that in characteristic zero
the H°()in) are all irreducible. This is a special case of a general fact; see Theorem 7.6. We now want to compute the higher cohomology groups H / ()in), i > 0, n E Z. Here we use the exact sequence 0 ~
)in ""+ ]g[U] (~))i n ~
Q ~ 0,
(5.3)
where the first homomorphism is the natural B-homomorphism )in --* IndB)i~ (see (4.2)) and Q is the quotient. A careful examination of the long exact sequences arising from (5.3) for various n C Z gives (cf. [Do, 12.2])"
Proposition 5.3
Let n C Z.
i) Hi(An) - 0 for i > 1. ii) If n >_ - 1 , then HI(An)- O. iii) If n <_ --1, then H°(An) - O. Moreover, H 1 ()~n) has a basis of elements wo, w l , . . . , W-n-2 such that twi --/k-n-2-2i(t)wi,
t E T,
-n-2
( 10
U 1 )wi_
E j=0
( ~ ) uY-~wJ"
Chapter 1
Modular Representations
21
If we compare the result in Proposition 5.3(iii) with Corollary 5.2 we obtain
Corollary 5.4
Let n >_ O. Then
k, O,
H°mG(HI(Am)'H°(An))-
if m -- - n - 2, otherwise.
A nonzero h o m o m o r p h i s m (~n" H 1 ( A - n - 2 ) ~ H ° (An) i8 given by
{~n(Wi)- (ni)Yn_i,
Corollary 5.5
i--O,...,n.
L e t n e N. If(~ e H o m G ( H I ( / ~ _ n _ 2 ) , H ° ( A n ) ) is nonzero,
then I m (1) - -
Remark
L(An).
Similar computations show that we have for each n _> 0 a G-
homomorphism ~n" N ° ( A n ) + H I ( A - n - 2 ) given by
~n(Vi) -- (n -- i)! i! Wn_ i. The composite (in either order) of/I/n and
6
On
is multiplication by n!.
The Borel-Weil-Bott theorem Let c~ be a simple root. Then we define a subgroup P~ _< G by P~ - B U B s ~ B ,
(6.1)
where (by abuse of notation) we have written s~ also for a representative for s~ E W in N o ( T ) (note that s ~ B is independent of which representative we choose). Now Ind P~ behaves exactly as IndB C in the case G - SL2(k) considered in the previous section. Writing H~ instead of RiInd g~ , we have thus
H. H. Andersen
22 Proposition
Let )~ E X ( T ) .
6.1
Then
i) If i > 1. then H~ ( A ) - O. ii) If (A,(~ v} - n _> - 1 , then H~(A) elements v0, V l , . . . , v. with t~ - (~-
(~-
O, and H(°()~) has a basis of
i)~)(t)~,:,
tET, zEk.
j=O
iii) If (£,c~ v) -
n < -1,
then H ° ( £ ) -
O, and H i ( A ) has a basis of
elements wo, wl . . . , W - n - 2 with twi - ()~- (n + i + 1)a)(t)wi, X-c~(Z)Wi
Notation
6.2
Set p -
--
.
.
t E T,
zJ-iwj.
~1 ~-~ER+ c~. We have then (p,c~ v) -- 1 for all
simple roots a, and we assume p E X ( T ) .
T h e n the dot action of W on
X ( T ) is defined by ~ . ~ - ~ ( ~ + p) - p,
wEW,
)~ E X ( T ) .
If w E W, then we let l(w) denote the length of w, i.e., the length of a reduced expression for w in terms of simple reflections. The aim of this section is to prove the following result, cf. [Bo], [De], or JAn1]. Theorem
6.3
Suppose c h a r ( k ) -
0, and let )~ E X ( T ) .
If w E iV is
chosen such that w(~ + p) E X ( T ) +, then we have G-isomorphisms
{ HO(~.~), m(~)~ O,
if i -
l(~),
otherwise.
Chapter 1
Modular Representations
23
Remark We say that A is singular if there exists c~ E R w i t h (A-+-p, ozv} -0. In this case, w.A ¢_ X ( T ) + and Theorem 6.3 therefore says H i ( A ) - 0 for all i. On the other hand, if A is regular (i.e., nonsingular), then the
E X ( T ) + is unique and w.A C X ( T ) +. So in this
w E Wwithw(A+p)
case there is a unique nonvanishing cohomology group, namely the one in degree l(w). To prove Theorem 6.3 we need two lemmas.
Let A C X ( T ) , and suppose (A,c~v) >_ - 1 for some simple root ct. Then we have a G-isomorphism
Lemma
6.4
Hi(A) ~ H~(G/p~, H°(A)) for all i > O. Proof:
Let 0 --~ A --~ I. be an injective B-resolution of A. Then, since
H ~ ( A ) - 0 for i > 0 (Proposition 6.1), we have that 0 ~ H°(A)--~ H ° ( I . ) is an injective P~-resolution. The injectivity comes from Corollary 1.11(iii). Hence, the right hand side in the lemma is the ith cohomology of the complex H ° ( G / P ~ , H ° ( I . ) ) .
By transitivity (Corollary 1.11(ii)), this complex
is isomorphic to H ° ( I . ) , which has H~(A) as its ith cohomology. [Z]
Let A C X ( T ) , and suppose (A,o~v} _< - 1 for some simple root (~. Then we have G-isomorphisms
Lemma
6.5
Hi(A) ~ H i - I ( G / P ~ , H ~ ( A ) ) for all i >_ 1. Pro@
Let again 0 --~ A ~ I. be an injective B-resolution. By Proposition
6.1 the complex 0 ~ H°(Io) ~ H°(I1) - . . . .
is exact except in degree 1,
where its cohomology is H~ (A). If we set Q - I0/,~ ~ Ker(I1 ~ means that we have exact sequences 0
and
no(z )
/2), this
H. H. Andersen
24
The long exact sequence arising from the first of these gives
Hi(G/P~,HI(A)) ~ Hi(G/P~,H°(Q)) for i _> 1. Arguing as in Lemma 6.4, we get
Hi(G/P~, H°(Q)) ~ Hi+I(A) for all i >_ 0, and we are done. E5
P r o o f of Theorem 6.3: From Proposition 6.1 it follows that if A E X ( T ) has (A,(~ v) >__ - 1 for some simple root a, then H°(A) ~ U~(s~.A) (cf. Corollary 5.4 it is at this point that we use char(k) - 0; otherwise, the binomial coefficients (n) are not necessarily all nonzero, i.e., the ~n are not necessarily isomorphisms). By Lemmas 6.4 and 6.5, we get from this Hi(A) Hi+l(sc~.A ) for all i. Using this repeatedly, we get (with w as ~
stated)
Hi(A) ~ Hi-l(W)(w.A)
for all i.
If i < l(w), we see that Hi(A) - 0. A symmetric argument gives
Hi(A) ~_ Hi+N-l(~)(WoW.A), where N = ]R+I and w0 E W is the maximal element in W (of length l(wo) - N). Hence if i > l(w) we get Hi(A) = 0 by appealing to (4.3) (note that dim G / B - N). E] Remarks i) Theorem 6.3 remains true when char(k) - p > 0 for those A which satisfy (A + p, av) < p for all a E R. The proof goes through unchanged because for such A the relevant binomial coefficients are all prime to p. If p >_ h (the Coxeter number of R), we see in particular that
dim H i ( w . O ) - 5i,z(~) ii) Suppose again char(k) - p
for all i >_ 0, w C W.
(6.2)
>_ h, and let r > 0. Then we have, for all
wEW, { H°((p ~ - 1)p),
Hi(p~w(P) - P) ~-
O,
if/=/(w), otherwise.
(6.3)
Again the proof of Theorem 6.3 goes through (this time the point is that (ap~-l) ~ 0 (mod p) for a l l / - 0, 1 , . . . , ap ~ - 1, a - 1 , . . . p). iii) Actually (6.2) and (6.3) hold also for p < h, cf. [An2, An3]
Chapter 1 Modular Representations 7
25
Serre d u a l i t y a n d c o m p l e t e r e d u c i b i l i t y In this section, we shall prove
Theorem
Let A C X ( T ) .
7.1
Then, for each i > O, there is a G-
isomorphism Hi(A) ~_ H N - i ( - A -
2p)*.
This theorem is a consequence of the Serre duality theorem for locally free sheaves on projective varieties [Ha, III.7]. However, we shall deduce Theorem 7.1 directly from the results of the previous sections. Combining it with Theorem 6.3 we obtain the complete reducibility of all finite dimensional G-modules in characteristic zero. First, let us make sure we agree about dual representations: If V is a representation of G, then G acts on V* = Homk(V, k) via
gf" v ~ f (g-lv),
H N(-2p)
L e m m a 7.2
g E G , v E E f E V*.
~ k.
Proof: This is the w - w0 case of (6.2). [:] Let E and F be two B-modules. Then we have natural Ghomomorphisms
L e m m a 7.3
Hi(E) ® H i (F) ~ H i+j (E ® F) for all i, j > O. Proof: We go by induction on i + j . For i - j - 0, we have the map g ° ( E ) ® H°(F) ~ H ° ( E ® F) given by fl Q f2 ~-+ (g ~-+ f l ( g ) Q f2(g), g e G),
fl e H°(E), f2 e H°(F). Suppose j > 0, and embed F ~ I with I an injective B-module. Then
0 ~ H°(F)~
H°(I)~
H°(I/F)~
H~(F)-~ 0
(7.1)
H. H. Andersen
26
is exact, and for j > 1 we have
Hi(F) ~ HJ-I(I/F).
The last s t a t e m e n t
clearly gives the lemma for j > 1 via the induction hypothesis. To handle the j - 1 case, tensor (7.1) by
Hi(E)
to obtain the commutative d i a g r a m
0
Hi(E) ® H°(F)
,
Hi(E ® F)
H ~(E) ® H ° (I)
,
H ~(E ~ I)
H~(E) ® H ° ( I / r )
, H~(E 0 I/F)
re(E) 0 Hi(r)
m÷l(E ® r)
0 where the columns are exact and the horizontal maps exist by induction hypothesis. We get the desired map
Hi(E)®
H I ( F ) --~ H i + I ( E @ F ) . 7]
Let E be a finite dimensional B-module. The n a t u r a l map E ® E* ~ k t h a t takes e ® f into
f(e)
is clearly a B - h o m o m o r p h i s m . By the previous
two lemmas, we get G - h o m o m o r p h i s m s
OiE"Hi(E)® HN-i(E * ®-2p)
~
HN(--2p)"~ k
for all i _> O. This gives us a G - h o m o m o r p h i s m ~iE" H i (E) ~ H N-i (E* @
Now T h e o r e m 7.1 is a special case of
-2p)*,
(7.2)
Chapter 1 C l a i m 7.4
Proof:
Modular Representations
27
9 ~ is an isomorphism for each i > O.
Consider first the case i -
0 and E -
A C X ( T ) + • Set R -
K e r ( H ° ( 1 ) ~ - ~ l ) . If we tensor the diagram 0-+
RNL(1)~
L(1)
n
n
I
H°(t)~
1
0---+ by - I -
R
--+
-+
1
~
0
~
0
2p and take cohomology, we get
L(1) ® H N ( - l -
2p)
-+
n H°(1) ® H N ( - l -
H N(-2p) ~ k
-+
II 2p)
~
H N ( - 2 p ) _~ k
~
0
where we have used Proposition 4.1 to rewrite the terms on the left and (4.4) to get the 0's on the right. This and a little weight argument (note t h a t all weights of the quotient H ° ( 1 ) / L ( 1 ) are strictly less t h a n I ) prove t h a t 9 o is nonzero when restricted to L(1).
Hence, 9 o is injective (because by T h e o r e m 3.8 we
know t h a t L(1) is the unique irreducible G-invariant subspace of H ° ( 1 ) ) . It is now an easy induction on dim E to check t h a t ~ o is injective for all E. We prove next t h a t 9 ~ is surjective. For t - ( p r - 1 ) p , this follows from (6.3). For arbitrary A E X ( T ) +, we choose r E N such t h a t (p'r-1)p+wo,k E X ( T ) + Note t h a t wo((p ~ - l ) p + w o t ) =-(S-1)p+I is the minimal weight in H°((p " - 1)p + woA), so t h a t we have an exact B-sequence
0 ~ A ~ H°((p r - 1)p + woA)® (p~ - 1)p -+ Q -~ 0. This gives us the diagram 0
1 ,
;
2p) *
+
H°((p " - 1)p + w0A) ®H°((p ~ - 1)p)
'
H°(Q)
,
H°((p r - 1)p + w0A) ®Hg((--p r-
H x (Q* ® - 2 p ) *
1)p)*
H. H. Andersen
28
As we have observed, the middle horizontal map 1 ® ~ is an isomorphism. Since we also know that bottom horizontal map ~ is injective, it follows that ~ is surjective. Again, an easy induction gives that ~ o is surjective for all E. Hence, ~ o is an isomorphism. Standard degree shift arguments show that so are all ~ , i >_ 0. [3 C o r o l l a r y 7.5
Let A, # E X ( T ) +. Any short exact sequence of G-modules 0--~ H°(A)--~ E ~ H°(#) * --~ 0
splits, i.e., ExtlG(H°(p)*, H°(A)) = 0
for all A,# E X ( T ) +.
Proof: Suppose first that A ~ -w0(#). Then A is maximal among the weights of E, and this implies that we have a B-homomorphism E --+ A, see Corollary 3.4. By Frobenius reciprocity (Proposition 1.10) this is the same as a G-homomorphism E ~ H°(A); i.e., we have found a retraction H ° (A) --+ E of the given embedding. If A < - w 0 ( # ) , we dualize the sequence and repeat the previous argument. [] Combining Serre duality and the Borel-Weil-Bott theorem, we obtain the following classical result. T h e o r e m 7.6
Suppose char(k) - 0.
i) H°(A) is an irreducible representation of G for all A C X ( T ) +. ii) Any finite dimensional representation of G is completely reducible. Proof: i) By Theorems 6.3 and 7.1, we have H°(A) ~- H N (wo.£) ~- g ° ( - w o A ) *. Now we know already that H°(A) has a unique irreducible G-invariant submodule L(A) (Theorem 3.8(i)). The displayed isomorphism shows that L(A) is also the only irreducible G-quotient of H°(A). It follows that H°(A) L(A). =
ii) follows from (i) and Corollary 7.5. [--]
Chapter 1 Modular Representations Remark
29
The results in Theorem 7.6 remain true in char(k) = p > 0
if we require (A + p , a v) _< p for all a E R+ in (i), resp., if the same condition holds for all weights in the representation in (ii). On the other hand, the results fail without this bound: For SL2(k) we have, for instance,
L()~p) =/=H°(Ap), p = char(k). 8
Frobenius kernels and their representations In the previous sections we have been working with the "pedestrian"
definition of an algebraic group given in Definition 1.1.
When dealing
with algebraic groups over more general fields and rings (e.g., Z) or when considering infinitesimal groups (like the ones we are going to introduce shortly), we need a better definition: D e f i n i t i o n 8.1
Let R be a commutative ring.
By an
algebraic group
over R, we understand a functor G from the category of R-algebras into the category of groups such that there exists an R-algebra type over R (i.e.,
R[G] of finite
R[G] is a quotient of a polynomial ring in finitely many
variable over R) for which we have
G(A) = HomR_alg(R[G], A) for all R-algebras A. We shall call R[G] the coordinate ring of G. Then the definition says that an algebraic group is a representable functor from the category of Ralgebras into the category of groups (with a coordinate ring of finite type). The coordinate ring is a Hopf algebra over R.
It clearly determines G
uniquely. When R = k is an algebraically closed field, an algebraic group over
G(k) = Homk_alg(k[G],k). In fact, k[G] G(k) is nothing but (the closed points in) the affine variety defined by k[G]. In the k is determined by its k-points,
may be identified with the set of regular functions on G(k), since previous sections we have just been working with these k-points.
When S is an R-algebra and G is an algebraic group over R, we can define an algebraic group G s over S by extensions of scalars as follows: Gs(A) = G ( A )
H. H. Andersen
30
for all S-algebras A. Note that S[Gs] = RIG] ®R S. On the other hand we say that an algebraic group H over S is defined
over R if there exists an algebraic group H0 over R with H = H0s. Examples
8.2
i) The additive group ~a
R is defined by
over
Ga(A):(A,+) for any R-algebra A. Note that R[G] = R[T], where T is an indeterminate. ii) The multiplicative group Gm over R is defined by Gm(A)=(A×,
.)
for any R-algebra A. (Here A × denotes the set of units in A.) We have R[Gm] = R[T, T-~]. iii) The general linear group over R is given by GLn(A)-
{(aij)i,j=l .....n E Mn(A) I det(aij) 6 A ×}
for any R-algebra A.
For this algebraic group, we have R[GLn] =
R[Xij, T]/(T det(Xij) - 1). We leave it as an exercise for the reader to generalize the other algebraic groups we have encountered in the previous sections and to compute their coordinate rings (as well as to verify that the notation k[G] in Definition 1.3 is consistent with the conventions just given). Likewise, the various definitions and constructions found in the previous sections also generalize.
Let us give the details in the case of the
construction of induced modules: Let G be an algebraic group over k, and suppose V is a vector space over k. Then a representation of G on V is a morphism of functors between the two algebraic groups G and GL(V).
Here the latter is the algebraic
group whose value on a k-algebra A is the set of A-automorphisms of V®k A. If now H is a subgroup of G (i.e., H is itself an algebraic group, and
H(A) <_ G(A) for all k-algebras A), we shall construct a functor IndCH • {Representations of H} ~ {Representations of G},
Chapter 1 Modular Representations
31
which is a right adjoint of the restriction functor. Suppose M is a representation of H. Let Ma denote the functor
]Via: { k-algebras } --~ { k-modules }, A~MQA. We then define I n d C H ( M ) - {f e Mor(G, Ma) ] f A ( g h ) -
h-lfA(g),
for all g E G(A), h G H(A), and all k-algebras A}.(8.1) Here, fA: G(A) --~ Ma(A) denotes the A-component of f. Note that IndCH(M) is a representation of G via x e G(R), g e G(A), f E Mor(G, Ma)® A,
(gf)R: x ~ fR(g-lx),
for any k-algebra A and any A-algebra R (we identify Mor(G, M a ) ® A with
Mor(GA,MQA)). We have M ® k[G] -- Ma(k[G]) -~ Mor(G, Ma), and using this we get Ind/~(M) - (M ® k[G]) H,
(8.2)
where H acts on M ® k[G] via the given action on M and the right regular representation on k[G], see Example 1.6(iv)(a) (properly generalized). Our purpose in this and the next two sections is to explore the representations of Frobenius kernels. These are certain subgroup functors of G whose only point over the field k is 1: We continue to assume that k is an algebraically closed field and that G is a connected reductive algebraic group over k, but now this is considered as an algebraic group in the new sense (whose k-points form a connected reductive algebraic group in the old sense).
From now on, we assume
p = char(k) to be positive.
i Suppose A is a k-algebra. Let k ~ A be the structure map. Then, for
each r C Z, we define a new k-algebra A (r) by setting A (r) = A as a ring
H. H. Andersen
32
and choosing the composite k~-2fki-~A as the new structure map. Here, F is the Frobenius homomorphism on k, i.e., F(x) - x p, x C k. If X is a functor from {k-algebras} into {Sets}, then we denote by X (r) the functor given by
X(~)(A) - X(A(-r)),
A e {k-algebras}.
The r t h Frobenius morphism on X is the morphism x
x
given by
F~:(A)" X ( A ) X ~ ) X ( A where "yr"A ~
(-~))
A (-r) is the k-algebra homomorphism that takes a into
a p~, a C A . W h e n we apply this to G, we obtain in particular the Frobenius hom o m o r p h i s m F~" G --~ G (r). This is clearly a homomorphism of algebraic groups. Definition
8.3
Gr - Ker(F~). Similarly, we have Frobenius subgroups Tr, Br, Ur, etc., of the subgroups T, B, U of G. From now on, we shall assume that all our groups are defined over Fp C k. Since F is the identity on Fp, this assumption implies that G (r) = G for all r. We thus have an exact sequence of algebraic groups: 1 ~ Gr ~ G ~
G ~ 1.
(8.3)
It turns out that the representation theory for Gr is quite similar to the one for G. W i t h o u t worrying too much about details and proofs (the reader is referred to [Ja] for these), we now give some of the highlights.
Chapter 1 Modular Representations Definition
8.4
33
For any B~-module E, we set
Z~(E) - IndCB~E. One big difference between the induction functor Z~ and the "global" version H ° (induction from B to G) is that (8.4)
Zr is an exact functor.
The reason is that G~/B~ is affine, namely G~/B~ ~_ U¢, so that all higher sheaf cohomology of bundles on G~/Br vavishes. This fact also gives
(8.5)
dim Z~ (E) = pNr dim E, because dim k[U¢] = pN~ (cf. Fact 2.6(iii)). In analogy with Theorems 3.8 and 3.9, we have Theorem
8.5
Let A E X(T).
i) Z~()~) contains a unique irreducible Gr-submodule, which we denote Lr()~). ii) Any finite dimensional simple G~-module is isomorphic to some such L~()~). iii) If also # E X ( T ) , then L~()~) ~_ Lr(#) iff A - # (mod p X ( T ) ) . Definition
8.6
Set
Xp~(T) = {A E X(T) I 0 _< (A, av> < p" for all simple roots a}. The elements in this set are called the p~-restricted weights. Note that any A E X ( T ) can be written uniquely
)~ = )~0 + pr)~l
with )~0 E Xpr(T),
)k 1 E
X(T).
(8.6)
Note that prA1 is trivial as a Gr-module. Hence Z~(A) _~ Z~()~°) and L~(A) _~ L~(A°). This explains the if-part of Theorem 8.5(iii), which can be rephrased The finite dimensional simple Gr-modules are parametrized by the set of p~-restricted weights. Now comes the surprise:
(8.7)
H. H. Andersen
34 T h e o r e m 8.7
If )~ e Xp~(T), then L~()~) ~_ L(~X)Ia..
For r = 1 this result goes back to Curtis [Cu]. It may be proved by closely examining the natural G~-homomorphism N°(A) -~ Z~(~) (this idea is due to G. Kempf [Ke2]). Alternatively, see [Ja, II.3.15]. For many purposes it is convenient to develop an analogous theory for the subgroup schemes G~T and G~B. Note that
G~T = (F~)-I (T) and
G~B = ( r h ) - ~ ( B ) . We shall use the notation 2~ (resp., Z~) for the induction functor from {B~T-modules} to {G~T-modules} (resp., from {B-modules} to {a B-modul s}). Th n we T h e o r e m 8.8
Let A E X ( T ) .
i) Z~(A) (resp., Z~(A)) contains a unique simple G~T-submodule (resp., G,.B-submodule), which, we denote L~ (~) (resp., L~ (~)). ii) Any finite dimensional simple G~T-module (resp., G~B-module) is isomorphic to some such (unique!) L~(.k) (resp., L~ (~)). iii) L~(A)IG ~ ~_ L~(A) ~_ L~(A)IG,. One advantage of G~T-modules over G~-modules is that we may consider their (formal) characters. In particular, we can therefore pose the P r o b l e m 8.9
Determine ch Lr(&) for ~ e X ( T ) , r ¢ N.
Note that by the preceding, ~,r(A) _~ L(/~ °) ®
pr)tl,
and hence this
problem is equivalent to Problem 3.11. It turns out that the crucial case is the case r = 1; see [Ja].
9
Injective GrT-modules We keep the assumptions from Section 8.
Chapter 1 Modular Representations P r o p o s i t i o n 9.1
35
The category of finite dimensional G~T-modules has
enough injectives. Proof." Let M be a finite dimensional G~T-module. We have to find a finite dimensional injective GrT-module I containing M. Recall that M is injective as a T-module (this follows immediately from Proposition 2.3). Hence I - IndTa~TM is an injective G rT-module, see Corollary 1.11(iii). Moreover, I is finite dimensional (the dimension being given by a formula analogous to (8.5)). By Frobenius reciprocity, the identity M ~ M induces a G~T-injection M --~ I. [:] D e f i n i t i o n 9.2
Let A E X(T). Then we set Q~(A)equal to the injective
hull of L~(A). Note that by Proposition 9.1 (or rather its proof) we have (~(A) C Indra~TL, r(A). Hence, by injectivity, Q~ (A) is a direct summand of IndTa'TL~ (A).
(9.1)
D e f i n i t i o n 9.3 A G
c F.~ = M
in such a way that Fi/Fi-1 ~ 2~(Ai) for certain A~: E X(T). When M has a 2~-filtration we denote by []VI" Zr(#)] the number of times 2~(#) occurs in such a filtration; i.e., in the preceding notation
[M- 2<(p)]- #{i I A{ = p}. E x a m p l e 9.4 BrT-filtration
Let V be a finite dimensional BrT-module. Then V has a 0--
Vo C V 1 C
"'" C Vt -
V
with V i / ~ - i -~ Ai for certain Ai C X ( T ) (this is the same argument as in Corollary 3.4). Applying the exact functor Z~, we obtain a Z~-filtration of Z~V.
In other words, any G~T-module that is induced from a finite
dimensional B~T-module has a Zr-filtration. Moreover, [ 2 r V - 2~(#)] - dim V,.
H. H. Andersen
36
P r o p o s i t i o n 9.5 Let A e X(T). Then Q~(A) has a Z~-filtration, and for each # E X(T) we have
[Q~(~). 2~(~)1- [2~(~). L~(~)] (the latter symbol denoting composition factor multiplicity). Proof." Note that IndTG~T -- 2r o Ind B~T. Hence the first statement is a consequence of (9.1) and Example 9.4. To prove the second half, we first need some preparation: Let B' denote the opposite Borel subgroup; i.e., B' = woBwo, where w0 is (a representative in G of) the longest element in W. Set 2 ' - I n d ~ T T. Then, clearly, Z~ has properties completely analogous to Z~. Elementary weight considerations show ch 2~(A) - ch 2'~(A- 2 ( / -
1)p)
(9.2)
for all A E X(T). We now have
Let A,# E X(T). Then k, i f # - A + 2 ( p r - 1 ) p , i) Homa~T(2'~(A),2~(tt)) ~ O, otherwise.
L e m m a 9.6
ii) Ext~T(2'~(A), 2~(#)) -- 0. Proof: (ii) is proved just as Corollary 7.5 noticing that the (contravariant) dual of Z~(A) is isomorphic to Zr(A + 2 ( / - 1)p). This also implies (i) via Frobenius reciprocity and some easy weight considerations. [3
Proof of Proposition 9.5 (continued): From Lemma 9.6, it follows that [M" 2 ~ ( , ) ] - dimHomG~T(Z'r(#- 2 ( / -
1)p), M)
for any G~T-module M that has a Z~-filtration. Combining this with the equation
IV. L~(A)] - dimHomc~T(V, Q~(A)), valid for any finite dimensional G~T-module V, we are done because of (9.2). I-1 The equality in Proposition 9.5 is known as the Brauer-Humphreys reciprocity law. It shows that Problem 8.9 (or 3.11) is equivalent to the following:
Chapter 1 Modular Representations P r o b l e m 9.7 C o r o l l a r y 9.8
37
Determine ch Or(A) for all )~ e X(T). L((p r - 1 ) p ) - 2~((p ~ - 1)p) - Q ~ ( ( p ~ - 1)p).
Proof: By Lemma 9.6(i) we have a nontrivial homomorphism from the module Zr~(-(p r - 1)p)* to the module Zr((p r - 1)p). But the first (resp., second) module has L((p r - 1)p) as its unique simple quotient (resp., submodule), and so the first equality follows. The last equality is then a consequence of Proposition 9.5 (combined with the fact that all Zr(A) have the same dimension). Z] The irreducible G-module L((p r - 1)p) is known as the r t h Steinberg module and also denoted Str. By Corollary 9.8, it is an irreducible injective and induced (from BrT) GrT-module. The standard argument shows that injectivity is preserved when tensoring by an arbitrary finite dimensional module. It is then easy to see that for any )~ C Xr(T) we have an isomorphism of GrT-modules,
Str @ L((pr - 1 ) p + wo)~) ~- (~r(A) ® (~> (~r(#) "~)
(9.3)
for certain m~ > 0. In other words, (~r(A) appears as a distinguished GrTsummand of this tensor product. For p > 2 ( h - 1) it is known also to be a G-summand, whereas for smaller p the question of a G-structure on (~r(A) is still open; see [Ja, 11.11]. 10
Kempf's vanishing theorem
The Frobenius morphisms F~ and the Steinberg modules Str, r >_ 0, from the last sections are used here to prove the following. Theorem
10.1 ([Kel])
Let )~ E X(T) +. Then H~(~)-0
for/>0.
Note that in characteristic zero this result is a special case of the Borel-Weil-Bott theorem 6.3. So, as in the previous two sections, we shall assume char(k) - p > 0.
H. H. Andersen
38 D e f i n i t i o n 10.2
Let V be a representation of G. Then we define V (r)
to be the representation of G on the same vector space but with G-action given by g v -
F~(g)v, g E G, v E V.
We shall derive Theorem 10.1 from
Let E be a B-module. Then, for any r > O, we have an isomorphism of G-modules
Theorem
10.3
Hi(E(r) ® ( p r _ 1 ) p ) ~ Hi(E)(r) ® Str. Proof: Since H ° - IndaB -- IndaaB o Zr and since 2r is an exact functor (8.2), which takes injective B-modules into acyclic modules for IndaaB (in fact, into injective GrB-modules, see Corollary 1.11(iii)), we have
H ~ - H~(G/GrB, 2 r ( - ) ) . Now 2 r ( E ( r ) ® ( p r - 1 ) p ) - E ( r ) ® 2 r ( ( p r - 1 ) p ) - E ( r ) ® S t r , the first equality being the tensor identity for Zr and the second coming from Corollary 9.8. Hence, the tensor identity for H i ( G / G r B , - )
gives
Hi(E(r) @ ( p r _ 1 ) p ) ~ Hi(G/GrB, E (r)) ® Str, and we are done if we show that
H i ( G / G r B , E (r)) ~ Hi(E) (r).
(10.1)
One checks this easily for i - 0. To do the obvious dimension shift., we need to check that H i ( G / G r B , I (r)) - 0 for i > 0 whenever I is an injective B-module. In fact, it is enough to consider I -
k[B], and since k[B] (r) -~
Indaa;B(k) we have H i ( O / O r B , k[B] (r)) ~ Hi(G/Or, k). Induction from Gr to G is exact (because G/Gr ~ G is affine), and we are done. [3
P r o o f of T h e o r e m 10.1" By Theorem 10.3 we have, for all ,~ C X ( T ) , Hi(A) (~) ® St~ __ Hi(p~(A + p) - p), and for r >> 0, A E X ( T ) + the right hand side vanishes whenever i > 0. This is so because the line bundle £ ( ~ + p) on G / B is ample when A E X ( T ) + ; see [Ha, III.5.2]. [-1
Chapter 1 Modular Representations
39
Let I E X ( T ) +. Then
C o r o l l a r y 10.4 (Weyl's character formula)
ch H°(1) = J(e~+P)/J(eP),
where J is the Z-endomorphism on J(e~) - E
Z[X(T)] given
by
p E X(T).
(-1)l(W)eW(~)'
wEW
Proof: By Theorem 10.1 we have ch H ° (A) = ~--~i( - 1)ich Hi(1). The rest is standard; see, e.g., [Ja, II.5.10]. [3
11
Q u a n t u m groups
In this section we introduce quantum groups and briefly sketch how many of the preceding results in the representation theory for algebraic groups have quantized counterparts. Let (aij)i,j=l .....n be a Cartan matrix, and choose a set of relatively prime integers d l , . . . , dn such that (diaij) is symmetric. Fix also an indeterminate v, and set vi = v d~, i = 1 , . . . , n . If m C N we write [m] = (v m - v - m ) / ( V - - V - I ) and [m]! = [ m ] [ m - 1]... [2][1]. For t C N, we set [t] - [m][m - 1]-.. [m - t + 1]/[t]!. We write [m]i, [m]i!, and [~]i for the corresponding expressions with v replaced by vi. D e f i n i t i o n 11.1 The quantum group attached to (aij) is the Q(r)-algebra U with generators Ei, Fi, Ki, K/--I, i = 1 , . . . , n, and relations i) K i K ( -1 = 1 = K ~ I K i and KiKj = K j K i ii) K i E j K ( -1 - v ia i j Ej, iii) EiFj - FjEi
iv/
z r+s=l--aij
v)
E
K i F j K i -1
gi-gi -
~ij
-- v-iaiJFj
-1
vi -
-1 v i
8
i
(-1)s [1-aiJ 1 F[FjF~-O,
r+s--1--aij
8
i#j
i
It turns out that U is a Hopf algebra with comultiplication A given by A ( K i ) - Ki®Ki,
A ( E ~ ) - Ei®I+K~®Ei,
A(Fi) - Fi®K~ - l + l ® F i ,
H. H. Andersen
40 coidentity e given by
e(Ki) = 1,
=
and antipode S given by
S ( K i ) -- K[ -1,
S(Ei)-m--K[-1Ei,
Set now A = Z[v, v - l ] , a n d let generated by Ki,K~ 1, E} m) , F (m) , i Em/[rn]i! and similarly for F (m).
S(Fi) - - - F i K i.
UA denote the A-subalgebra of U 1 n, m C N, where E} m) , ' ' ' ,
If q is a nonzero element in some field F, then F becomes a A-algebra by specializing v to q, and we define
UF, q = UA @A F. Sometimes we write just
UF or Uq instead of gF, q. By the preceding, UF is
a Hopf algebra. It is called the q u a n t u m group over F associated with (aij). Of course, this construction makes sense also for any commutative ring F and q E F a unit. However, we shall here always assume that F is a field, and in fact for simplicity we only consider fields of characteristic zero. For instance, we may take F = C and let q be a nonzero complex number. By abuse of notation, we shall write F/(m), etc., also for the elements in Uq corresponding to
F.(,m), etc. E UA.
We have
- v: vov:,
where Uq (resp., U +) is the subalgebra of E}m)), i -
1,...,n,
Uq generated by F.(,m) (resp.,
rn E N, and U ° is the subalgebra generated by
Ki, KC 1, [K~;c], i - - 1 , . . . , n, c, t E N. Here, [K~.c] is the element
[K~;c]- l~i Ki vc+l-j Ki-lv-i -c-l+j j=l Vi _ Vi 2 Let R be the root system associated with the set of simple roots.
(aij), and let
O ~ 1 , . . . ,OZn
be
Set X equal to the set of integral weights, i.e.,
X - - Z n where A C X is an n-tuple A = (/~1,...,/~n) with Ai = (A, ai}, i =
Chapter 1
41
Modular Representations
1 , . . . , n. Such a weight gives rise to a character of U° (and hence of U°) via
If V is a U°-module and • E tU/ °, we set L/
v~ - {~ e v l ~ -
~(~)~, ~ e Uq°}.
It turns out (but this is a nontrivial fact; see, e.g., [APW, Section 9]) that any finite dimensional Uq-module V (of type 1) is a sum of weight spaces (note that replacing vi by - v i in the foregoing formulas defines an equally good character )~ and type 1 simply means that we work with all signs positive). We shall consider a slightly more general category of Uqmodules. Namely, define fq to be the category consisting of all Uq-modules V that satisfy i) V = ( ~ e z V~. ii) For every v C V, we have F(m)v - E}m)v - 0 for all m >> 0, i 1,...,n. Note that if V is arbitrary, then F(V) =
{v E e
V,x r.,i
v-
v-O,
m >> O, i - 1 , . . . , n
,~EX
}
is a Uq-submodule of V. By definition, F ( V ) E fq. Let f q (resp., fro) be the category obtained in the same way when we replace Uq by Uq U° (resp., U°). D e f i n i t i o n 11.2
The induction functor H°.¢~ - ~ Cq
is defined by Hqo (V) = F ( H o m G- uo (Uq, V)),
v~¢;.
Here, Uq is considered a UqU°-module via left multiplication, and the Uq-structure on H ° ( V ) comes via right multiplication on Uq" u . f : x ~ f(xu),
x , u 6 Uq, f C H ° ( V ) .
H. H. Andersen
42
It is straightforward to check that H ° satisfies Frobenius reciprocity, etc. We leave to the reader the task of verifying that the results from Section 4 on remain true when we add index q. One result, which is not clear how to quantize, is (4.4). We shall now indicate how one may derive the special case of this that we need from the classical case: Let Z be the A-algebra obtained by specializing v to 1. Choose also q E k (where k is an algebraically closed field of characteristic p > 0 as in previous sections) such that q is a primitive lth root of unity (so p ~ 1). Specializing v to q, we have made k into an A-algebra. 11.3 There exists a Hopf algebra homomorphism .7":Uk Uz ® k determined by
Proposition
E} rn/l) (~ 1, ~(E[m))-
if lira , otherwise,
0,
{ F{ ~/t) ® 1, -
if lira,
0,
5~(K?l) - K ? I ( ~ 1, i-
1,...,n.
Proof: [Lul, Lu2]. Let now U denote the enveloping algebra over Q for the semisimple Lie algebra corresponding to (aij). Denote the standard generators of 0 by /~i,/~i, i - 1 , . . . , n , and set /ti - [/~i,/~i]. The Kostant Z-form of U- is the Z-subalgebra of 0 generated by /~r),/~i!r), i - 1 , . . . , n, where E~) - E~/r! and similarly for/~/(~);see [Hul, Section 26]. Proposition
11.4
There is a Hopf algebra isomorphism (]z--%Uz/({Ki- 1 1 i -
1 , . . . , n})
given by /~r) H image of E~r), /~(~) ~ /ti ~
image of F (~), image of [K/l; 0 ] .
Chapter 1 Modular Representations
43
Proof: [Lul, Lu2]. [--1 Let now 9~: Ua. ---, Uz ® k be the composite obtained from Propositions 11.3 and 11.4. If we then start by a representation V of the algebraic group G over k corresponding to (aij), then this representation is also a module for the so-called hyperalgebra Uz ® k. Hence V gives rise via Y to a module for Uk. We denote this Uk-module by V It]. Imitating the arguments for (10.1) we obtain
Proposition 11.5
H q i(vl;3)
~ H~(V)I;3
,
i _> o.
We can now prove the quantized version of the special case H1 = B, He - G Theorem
Proof:
of (4.4)" 11.6
i Let A e X(T). Then Hq(A) - O f o r i > N.
Let uk be the subalgebra of Uk generated by E i , F i , K ~ 1, i U~u~,
1 , . . . , n. (This corresponds to GI C G.) Set Zq -IndcU/-g°A
" T h e n Zq is
an exact functor, and we have
i H~(a) - H i ( U ~ / U [ U ~ °u~= Zq(A)). ,
Let Lq(#) be a U~U°uk-composition factor of Zq(A).
Then Lq(#) Lq(# °) ® l# 1, where # - # o + 1#1 with #o E Xz (i.e., 0 < #o < l, i 1 , . . . , n). Hence, by Proposition 11.5, we have
H i ( U k l U k U ~o k , L q ( ~ ) )
~ L~ (/_to) @ H i ( #1 ) [l] .
Now (4.4) gives H i ( p 1 ) = 0 for i > N, and we are done. [Z] Remark
From Theorem 11.6, it is a standard base change argument to
derive
H~(~)-
0
for ~ni > N.
It follows that the vanishing H~(A) - 0 for i > N also holds for q any nonzero element in any field F. Let us return to the case where F - C
and q E C\{0}. I f q i s not a
root of unity, the Borel-Weil-Bott theorem holds for Uq and hence Lq(A) -
H. H. Andersen
44
H°(A) for all A e X ( T ) +. Moreover, in this situation ~q is semisimple (as it follows from that theorem combined with quantized Serre duality, cf. Theorem 7.6). So suppose q is a primitive /th root of unity.
Then we have as an
analog of Problem 3.11
Problem
11.7
Determine ch Lq()~), A C X +.
Now Lusztig has conjectured that if 1 - p _> h (h denoting the Coxeter number of R), then ch Lq()~)
-
ch L(A)
for all A e Xp(T).
(11.1)
This conjecture has recently been proved ([AJS]) for p >> 0. Moreover, work of Kazhdan and Lusztig [KL] relates Problem 11.7 to a similar problem for affine Lie algebras. Early in 1994 Kashiwara and Tanisaki announced a solution of this latter problem [KT], and hence Problem 11.7 is also solved (although there still seems to be some problem if (aij) is not symmetric) for all 1. The proof of (11.1) in [AJS] (for p :>> 0) involves the "dual" approach of injective modules, :see Problem 9.7. We conclude this section with a few remarks on injective modules for Uq. Set Stz = L q ( ( l - 1)p) (We are again in the situation where q E C is a n / t h root of 1. For simplicity, we shall assume that 1 is odd.) Then Stl is injective in ~q (this is in contrast with the situation for representations of G in characteristic p > 0, where no finite dimensional representation is ever injective). Considering Stl ® Lq()~), )~ E Xz, we obtain (compare Corollary 9.8)
For each )~ e Xt, the injective hull Qq(A) of Lq(£) in Eq is finite dimensional. In fact, Qq()~) is a distinguished summand of Stq ® Lq(wo)~ + ( 1 - 1)p).
Proposition
11.8
We also have a Brauer-Humphreys reciprocity law for the Qq()~):
Chapter 1 D e f i n i t i o n 11.9
Modular Representations
45
A Uq-module M is said to have a good filtration if there
exists a filtration by Uq-submodules
O=FoCFlC...cF~=M with Fi/Fj_ 1
~
H°(/~i) for certain hi e X +, i -
1 , . . . ,s. The number of
times a given H°(#) occurs in such a filtration is denoted [ M ' H ° ( # ) ] .
Let A C X1. Then Qq(A) has a good filtration and
P r o p o s i t i o n 11.10
[Qq()~) . H°q(#)] - [H°(p) . Lq()~)].
12
Tilting modules In this section we shall give a short account of the theory of tilting
modules for algebraic groups, as well as for quantum groups. We will use the language of algebraic groups leaving it in most cases to the reader to quantize the results by adding the index q. Whenever this quantization is not straightforward, we have added a note or a remark. We shall assume that G is a connected reductive group over an algebraically closed field k. The corresponding quantum group is Uq, where q E C is a n / t h root of unity, 1 odd. D e f i n i t i o n 12.1 A finite dimensional representation M of G is called a tilting module if both M and M* have good filtrations (see Definition 11.9). (The dual module M* in the quantum case has Uq-action given by u f ( m ) = f ( S ( u ) m ) , u e Uq, f e M*, m e M.) Examples
12.2
Since H°(0) - k, the trivial module is a tilting mod-
ule. The same is true more generally whenever H°(A) is irreducible (because then H°(A) * = L(A)* ~_ L(-w0A) = H°(-w0A)).
Elementary
weight considerations show that H°(£) is irreducible for all p whenever )~ is minuscule. By the linkage principle [An1], the same is true, e.g., for = (pC={ucX
1)p and for ~ in the upper closure of the fundamental alcove + I ( u + p , aV) < p f o r a l l a c R + } .
H. H. Andersen
46
Let us denote by £t the category of tilting modules. It is easy to see that £t is closed under finite direct sums and under extensions (but not under formation of submodules and quotients). It is not trivial to see that it is also closed under tensor products: T h e o r e m 12.3 If_M1, M2 E £t, then also M1 ® M2 E G. (The Uq-action on tensor products is obtained via the comultiplication A" Uq ---, Uq C~ Uq.) This theorem was proved by Wang [Wa], Donkin [Do], and Mathieu
[Ma]. A proof (taking care of also the quantum case) using Lusztig's canonical bases was written down by Paradowski [Pal. T h e o r e m 12.4 i) For each A C X ( T ) +, there exists an indecomposable tilting module D(A) with highest weight A. Moreover, dimD(A)~ - 1. ii) The modules {D(A) [ A e X ( T ) +} .form a complete set of inequivalent indecomposable tilting modules.
Proof: (This theorem is due to Ringel [Ri] in a somewhat more general context.) We give a proof which only works for p not too small, namely we assume that H°(a;) C G for all fundamental weights a~. For p > h + 6 this is OK by Example 12.2, because then all fundamental weights ~ belong to the fundamental alcove C). (i) Write A - - E i=1 n a i, ~ , where ~Zl,.. . : a;,. are the fundamental weights By our assmnption and Theorem 12.3, H°(a;1)®al®...QHO(~dn)®an belongs to G, and since this module has highest weight A and this occurs with multiplicity 1, we find D(A) as an indecomposable summand. (ii) Let M E £t, and pick a maximal weight A of M. Thus, we have a B-homomorphism M ~ A (see Corollary 3.4), and by Frobenius reciprocity (Proposition 1.10) this gives a G-homomorphism M - G H ° ( A ) . Since M has a good filtration, this must be a surjection. On the other hand, we 71- t
also have a surjection D(A)--* H°(A). Since E x t ~ ( H ° ( # ) *, H°(A)) - 0 (see Corollary 7.5), we obtain G-homomorphisms ¢: M --* D ( A ) a n d ¢': D(A) M making the diagrams
M
~
H°
¢
D(a)
H°
Chapter 1 Modular Representations
47
commutative. Since ¢ o ¢~ is nonzero on the 1-dimensional space D(A)A, we conclude that D(A) is a summand of M, and we are done. [-1 E x a m p l e s 12.5
i) D(0) = k.
ii) D((p- 1 ) p ) = St1. iii) D ( A ) = H°(A) for all ~ E C. iv) Let A~ E C ~, where C ~ is the alcove obtained from reflecting C in the {~ E X ] (~ + p,a~)} - p }
wall H short root
(here, a0 E R+ is the highest
we are assuming that R is indecomposable). Then we
have an exact sequence 0 ~ H°(A)~
D ( , ~ ' ) ~ H°(A ') ~ 0,
where/~ = S~o.A' + pet. Problem
12.6
Determine ch D(A) for all ~ E X ( T ) +.
To convince you that Problem 12.6 is an interesting (and hard) problem, we prove P r o p o s i t i o n 12.7 Let )~ E Xl. Then Qq(£) = Dq(2(1- 1)p + w0A). (Contrary to our usual convention we have stated this result in its quantized version. The reason is that we have no finite dimensional injective G-modules although it is possible to define G-modules that behave like the Qq()~) by putting an upper bound on the weights allowed.) I
Proof: Proposition 11.10 says that Qq()~) has a good filtration. Since Stl is self-dual, we see that Qq(/~)* is also injective (cf. the construction of Qq()~) in Proposition 11.8), and hence Qq()~)* has also a good filtration. Thus, Qq(A) E ~t. By construction, we furthermore see that Qq()~) has highest weight 2 ( / - 1)p + w0£ and that this weight occurs with multiplicity 1. [2] We conclude by some results whose quantized version play a role in the construction of invariants of 3-manifolds by Reshetikhin and Turaev [RT]. The key is Theorem Enda(M). Tr(f) =0.
Let M E et, and suppose f e If M has no summands of the form D(,~) where /~ E C, then
12.8 ((JAn4, Theorem 3.4]))
H. H. Andersen
48
(In the quantum case, we (of course) have to use the quantum trace Trq. It is defined by T r q ( f ) = Tr(K2pf).)
Proof." By Theorem 12.4 we immediately reduce to the case M = D(A) with A C X(T)+\C. Moreover, since D ( A ) i s indecomposable, any f C E n d a ( D ( A ) ) may be written f = a . Id + f ' for some a E k and some nilpotent f ' E Enda(D(A)). Since T r ( f ' ) = 0, we have reduced the theorem to the following statement: If A e X(T)+\C, then Pl dim D(A).
(12.1)
By Weyl's dimension formula, [Hul, Corollary 24.3], we have dim H ° (#) -- 1-I
+ p' c~v) ~eR+ (#
In particular, we see that if # is p-singular (i.e., PI(# + P, c~v} for some c~ e R+), then pldimH°(#) (we assume
For any two finite dimensional G-modules M and N with M indecomposable, we have
Lemma
12.9
p[ dim M =~ Pl dim Q
for all G-summands Q of M ® N.
Proof: Exercise (Hint: look at Tr(¢) for ¢ E E n d a ( M ® N)). [--] Theorem 12.8 allows us to define a new tensor product on {t. First, let for M C {t the integers (a~(M))~X(T)+ be determined by
M-
@ AEX(T)+
Then ha(M) = 0 for all but finitely many A E X(T) +.
Chapter 1
Definition 12.10 fined by
Modular Representations
49
Let M1, M2 C ~t. The reduced tensor product is de-
M1 ~ M2 -- ( ~ D(,~) a~(Ml®M2) AEC
Proposition 12.11
® is associative.
Proof: Use Theorem 12.8. E] For further results in this direction, see, e.g., [AP].
References
[Anl]
H. H. Andersen, The first cohomology group of a line bundle on G / B , Invent. Math. 51 (1979), 287-296.
[An2]
~ , The strong linkage principle, J. reine angew. Math. 315 (1980), 53-59.
JAn3]
~ , The Frobenius morphism on the cohomology of homogeneous vector bundles on G / B , Ann. of Math. (2) 112 (1980), 113121.
[An4]
, Tensor products of quantized tilting modules, Comm. Math. Phys. 149 (1992), 149-159.
[AJSI
H. H. Andersen, J. C. Jantzen, and W. Soergel, Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: independence of p, Asterisque 220 (1994).
[AP]
H. H. Andersen and J. Paradowski, Fusion categories arising from semisimple Lie algebras, Comm. Math. Phys. 169 (1995), 563-588.
[APW]
H. H. Andersen, P. Polo, and K. Wen, Representations of quantum algebras, Invent. Math. 104 (1991), 1-59.
[Bo]
R. Bott, Homogeneous vector bundles, Ann. of Math. (2), 66 (1957), 203-248.
[Cu]
C. W. Curtis, Representations of Lie algebras of classical type with applications to linear groups, J. Math. Mech. 9 (1960), 307-326.
[De]
M, Demazure, A very simple proof of Bott's theorem, Invent. Math. 33 (1976), 271-272.
50
H. H. Andersen
[Do]
S. Donkin, Rational representations of algebraic groups, Lecture Notes in Mathematics 1140, Springer-Verlag, 1985.
[Ha]
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, 1977.
[Hull
J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics 9, Springer-Verlag, 1972.
[H,,2]
, Linear Algebraic Groups, Graduate Texts in Mathematics 21, Springer-Verlag, 1975.
[Ja]
J. C. Jantzen, Representations of Algebraic Groups, Academic Press, 1987.
[KT]
M. Kashiwara and T. Tanisaki, Characters of the negative level highest weight modules for affine Lie algebras,Int. Math. Res. Not. 1994, 151-160.
[KL]
D. Kazhdan and G. Lusztig, Tensor structures arising .from. afifine Lie algebras, J. Amer. Math. Soc. 6 (1993), 905-947, 949-1011:7 (1994), 335-381,383-453.
[Kel]
G. Kempf, Linear systems on homogeneous spaces, Ann. of Math. (2), 103 (1976), 557-591.
[Ke2]
~ , Representations of algebraic groups in prime characteristics, Ann. Sci. Ecole Norm. Sup. 14 (1981), 61-74.
[Lull
G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990). 257-296.
[Lu2]
~ , Introduction to quantum groups, Progress in Mathematics, BirkhSuser, 1993.
[Ma]
O. Mathieu, Filtrations of G-modules, Ann. Sci. Ecole Norm. Sup. 23 (1990), 625-644.
[Pa]
J. Paradowski, Filtrations of modules over quantum algebras, Proc. Symp. Pure Math. 56 (1994), Part 2, 93-108.
[Hi]
C. M. Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), 209-223.
[RT]
N. Reshetikhin and V. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103, 547-597.
Chapter 1 Modular Representations
[Sw]
51
M. Sweedler, ttopf Algebras, Benjamin, 1969. J.-P. Wang, Sheaf cohomology of G/B and tensor products of Weyl modules, J. Alg. 66 (1982), 162-185.
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Chapter 2 Orbital Varieties, Goldie Rank Polynomials and Unitary Highest Weight Modules ANTHONY JOSEPH
Weizmann Institute of Science
CONTENTS
Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture
Lecture 1
1 2 3 4 5 6 7 8
Introductory lecture The geometry of orbital varieties Goldie rank and characteristic polynomials Unitarizable highest weight modules The Parathasarathy equality The last place of unitarity Quantization of orbital varieties Further results and perspectives Comments and further reading Acknowledgments References
I n t r o d u c t o r y lecture
These lectures combine a number of related topics inspired by the orbit method and a desire to complete our knowledge of the primitive spectrum of the enveloping algebra of a semisimple Lie algebra. Let ~ be a complex finite dimensional Lie algebra with enveloping algebra U(~). Its dual g* admits the action of an irreducible algebraic group G, and the orbit method suggests the existence of a bijection between ~*/G and Prim U(~). This works rather well for g solvable [D], and it is now even known that the method of induction provides a homeomorphism 53
54
A. Joseph
[Ma] of these topological spaces. For g semisimple, however, this point of view is not as good. Basically, the greater noncommutativity of g forces discrepancies between these objects, which are a priori unrelated. If we consider that our basic aim is to determine Prim U(g) along with some interesting (for example, unitary) modules, then it will be enough that the orbit method provides a (key) part of Prim U(g). For example, the zero orbit in g* should correspond to the primitive ideals of finite codimension. which for g semisimple is an infinite set containing nevertheless just one completely prime member. One could therefore just replace Prim U(g) by Prime U(g) := {J E Prim U(g) 1 U ( g ) / J is a domain}; however, it would be a mistake to ignore the structure imposed by the presence of the remaining ideals. This is indicated by the following discussion. Here and from now on, it will be assumed that g is semisimple. Identify g* with g through the Killing form. Call x E g* nilpotent if ad~ x is a nilpotent derivation. This property is preserved under the action of G, and so it is natural to define a nilpotent orbit in g* to be one consisting of just nilpotent elements. T h e o r e m 1.1 ([BBr2, J5, Vo3]) For each J e PrimU(g), the associated variety V(gr J) of J is the closure of a nilpotent orbit. Thus, for each nilpotent orbit 49 in g* it makes sense to consider the subset X(O) of X - Prim U(g) of primitive ideals whose associated variety is 49. Now for J E X one may form Fract(U(g)/J), which is a matrix algebra over a skew field. The rank of this matrix algebra is denoted rk(U(g)/J) and called the Goldie rank of U(g)/J. Rank 1 corresponds to U(g)/,] being a domain or, equivalently, to P being completely prime. Let D denote a Cartan subalgebra of g. For each A E [~* one may associate a simple highest weight module L(A) of highest weight A - p (where p is the sum of the fundamental weights), and hence a primitive ideal J ( A ) " - Annu(~)L(A). After Duflo [Du], the map A ~ J(A) of [1" into X is surjective. In other words, the elements of X (and of X(O)) are parametrized by [?*. One may ask, how does the Goldie rank depend on this parametrization? The answer is that this dependence is essentially a polynomial one; that is, there is a finite set of polynomials the values of which determine these Goldie ranks. Moreover, for each X(O) it is appropriate to select a subset of these polynomials. Remarkably, this subset forms a basis for an irreducible representation of the Weyl group W for the pair (g, D), and this representation can be identified with the Springer representation
Chapter 2
Orbital Varieties
55
attached to (_9 defined by quite different (geometrical) considerations. It would clearly be almost a ridiculous coincidence if just one of these polynomials took the value 1 and moreover just once in the appropriate parameter set. These considerations will be made more precise in Lecture 3. For the moment, let us just stress a few salient points. 1) Up to scale factors, the Goldie rank polynomials are known and indeed given by a simple formula involving the Kazhdan-Lusztig polynomials (or rather their values at q = 1). 2) The scale factors can be largely determined just by knowing that X((_9) A Prime U(I~) ~: 0 for each (9. Hence an interest in the foregoing question. So far, one can really only check this case by case. In other words, we have no satisfactory general theory for constructing Prime U(~). 3) The Goldie rank polynomials are intimately related to the geometry of the nilpotent orbits via what was at first just a strange coincidence with the Springer theory. However, one can now directly attach to a nilpotent orbit polynomials that are not quite the same, but that span the same space. These are the characteristic polynomials of the orbital varieties attached to a given orbit. It turns out that these are given by exactly the same formulae, but involving geometric analogs of the Kazhdan-Lusztig polynomials that can be and are different. This gives an entirely new twist to the orbit method and insight into geometry from representation theory rather than vice versa. For example, one can conjecture that the inclusion relation of orbital variety closure is determined by these geometric Kazhdan-Lusztig polynomials (the values of these are as yet unknown) in a manner analogous to the known ordering [J1, Vol] in Prim U(g). 4) The relationship between Goldie rank and characteristic polynomials allows one to conclude that the latter (and hence the former) can be expressed as a positive linear combination of products of (distinct) positive roots. Consequently, Goldie rank 1 occurs only at small parameters. This is also true for annihilators of unitary modules, though the reason is not so clear. Yet D. A. Vogan [Vo2] has noted that if the simple quotient U(~)/J is a unitary module for g × ~ (in the appropriate sense), then it is necessarily a domain. This establishes at least
56
A. Joseph one link between unitarity and small parameters.
In the case when
L(A) is unitary, it need not be that Ann L(~) E Primc V(g); yet there is a natural bound (Theorem 8.2 and remarks following) on the Goldie rank, which fails completely in the nonunitary case. So far we have gained at least one motivation for studying orbital varieties. Let us give their precise definition. Fix a triangular decomposition ft = n - ® D ® n +, where n + (resp., n - ) denotes the span of the positive (resp., negative) root vectors. Set b + = D ® n +. Then, for each nilpotent orbit (9, the intersection (9 n n + is a locally closed subset of n + and, therefore, may be expressed as a union of its irreducible components, which by definition are just the orbital varieties attached to (9. As we shall see in the next lecture, the Spaltenstein-Steinberg theory allows one to describe them more precisely as follows. Let B be a Borel subgroup corresponding to b +. Then, for each w C W, there is a unique orbital variety V(w) the closure of which coincides with the B saturation set of n + N w(n+); that is, V(w) = {b(n + n w(n+))b-1 : b C B}. Moreover, the map w H V(w) of W to the set 12 of orbital varieties is surjective. Finally, there is a unique nilpotent orbit O(w) the closure of which coincides with the G saturation set of n + n w(n +), and the map w ~ O(w) of W to the set N" of nilpotent orbits is surjective. Thus, one has surjections W---~12---~N', and the orbital varieties attached to a given orbit are just fibers of the second map. The fibers of the first map are called the geometric cells of W. They differ ever so slightly from the left cells of W defined by the theory of primitive ideals. One may note that an orbital variety is Lagrangian and therefore a natural subject for quantization, a process that may be interpreted as follows. Can one give the algebra of regular functions A[V] on the closure V of an orbital variety a U(g)-module structure that at least preserves (up to a shift) its D-module structure? A more demanding version of this question is whether one can find a highest weight module N(A) such that g r A n n u ( n - ) e~ coincides with the ideal of definition of V. This is quite possibly so, and also one would expect N(A) and in particular A to be completely determined (modulo some obvious reservations).
A different but
related approach is to study the algebra D[V] of differential operators on V. Obviously, A is a faithful D-module, and so one can ask if there is a natural map of U(I~) into D making A a U(g)-module of not worse than finite length? Since V can be singular, it is not immediate (and not known)
Chapter 2
Orbital Varieties
57
if A[V] is simple as a D[V]-module. These questions have particularly nice answers when V lies in the nilradical m + of a parabolic subalgebra p of ft with m + commutative. Then A[V] has even the structure of a unitary highest weight module. Moreover, except for easily described special cases, D[V] coincides with
U(g)/Anna[V]. Conversely, every unitary highest weight module L is associated to a parabolic subalgebra with a commutative nilradical m +, and remarkably Annu(m-) L is a prime ideal and hence the ideal of definition of the closure of an orbital variety contained in m+. The proof of these results will be indicated in Lectures 3-6. Some further perspectives including the use of q u a n t u m groups and of differential operators are given in Lectures 7 and 8.
Lecture 2
The geometry
of o r b i t a l v a r i e t i e s
A key observation of the orbit method of Kirillov-Kostant-Souriau is that a G-orbit in g* admits the structure of a G-equivariant symplectic variety. Let us brush aside the geometric sophistication for the moment and consider what this means in just algebraic terms. Recall that S(g) identifies with the algebra of polynomial (and hence regular) functions on ft*. It is a commutative algebra and admits a Poisson bracket { , }, which we recall for any algebra A is a bilinear antisymmetric map A x A >A satisfying the Jacobi identity such that for each a C A the map b ~-+ {a, b} is a derivation of A. Clearly, { , } is defined by its value on generators, and for S(g) one just takes the Lie product on IJ. A basic question is to determine the prime Poisson ideals of A, namely those prime ideals I satisfying { A , I } C I. Notice one can take { , } to be the c o m m u t a t o r bracket, and this is one reason that their study for S(t~) and for V(g) are supposed to be related. On the other hand, S(g) being commutative gives rise to ideals that are not obviously present in U(t~/, that is, the ideal I of definition of any closed subvariety V of g*. If the variety in question is irreducible and G-invariant, then I is prime and Poisson. An important special case is when V is the closure of a G-orbit (9. Then, the Poisson bracket gives a nondegenerate antisymmetric bilinear form on the cotangent space T~, V to any point x C O. Indeed, for all a E A / I , x E O, define 2 where mx is the ideal of definition of {x}. Then d x a - a - a(x) mod rex, d~a e T ' v , and d~a x dxb H {a, b}(x)is the required form.
58
A. Joseph The construction of prime (or primitive) ideals in U(0) follows a quite
different path. For the latter, one constructs a simple U(0)-module M. In terms of growth rate, M should be one-half the size of U(O)/Ann M. Thus, a geometric equivalent should be a subvariety of the corresponding orbit, in the sense of Theorem 1.1 having half the dimension. It turns out that there is a natural candidate, namely the variety of zeros of grAnnU(0 ) rn, which is independent of the choice of 0 =/= m E M. (See exercises at the end of this lecture.) It is called the associated variety V(M) of M and is trivially contained in V(Ann M). Moreover, by a deep result of O. Gabber [G], it is an involutive subvariety of V ( A n n M ) , and this remembers to a large extent the U(0) module structure of M.
Here, being involutive
can be expressed by saying that the ideal I(M) of definition of V(M), namely v/grAnnU(0)m, satisfies { I ( M ) , I ( M ) } C I(M). This is equivalent to (dxa, dxb) := {a, b}(x) = 0 for all a, b E I(M), x E V(M). Thus, the subspace {dxa • a C I(M)} of T*x,V(Ann M) is isotropic and so can have at most half the dimension. Its orthogonal in Tz,V(AnnM ) identifies with 1 dim V(Ann M). So far no appeal T~,V(M), and this forces dim V(M) >__-~ to the proposed simplicity of M has been made, and this should provide the reverse inequality. One calls a subvariety of an orbit (9 (or more generally, of a symplectic variety) Lagrangian if it is involutive and of exactly half the dimension. The preceding discussion suggests that one should be able to attach to a Lagrangian subvariety 12 a simple module M satisfying Ann M E X(O) and with associated variety /2. However, it is too much to expect a bijective map, as involutivity cannot be expected to capture the full meaning of M being a U(0)-module, nor need a simple module necessarily have the correct dimension. On a finer level, modules may differ appreciably in size and yet still have the same associated variety because the latter only sees the radical of grAn n U(0) m. Recall V(w) of Lecture 1. The involutivity of V(w) has an easy proof based on the observation that the orthogonal of n + N w(n +) in n - is a subalgebra. That V(w) is Lagrangian results from the Spaltenstein-Steinberg equality (see Theorem 2.1), which asserts that dim(O M n +) - 1 dim (_9 for any nilpotent orbit (9. A deeper analysis of orbital varieties results [St] from considering the Steinberg variety 8. This is defined as follows. Let b/ denote the set of all unipotent elements of G, and B the set of all Borel subgroups of G.
Chapter 2
59
Orbital Varieties
By definition, S "-- {(u, U l , B 2 ) E U x B x B I u E B 1 n U2}. For each conjugacy class d c U, let $(C) denote the inverse image of C in $ under projection onto the first component. Obviously, 8 is a disjoint union of the $(C), and since the number of conjugacy classes is finite, this is a finite union. Under conjugation, B identifies with G / B for some fixed Borel subalgebra B of G. Set g B g -1 - 9B for all g E G. Bruhat decomposition implies that under diagonal action, the G orbits Z~ in B x B have as representatives the (B,'~'B) • w E W .
Set 8w(C) -
{(u, B 1 , B 2 ) E 8 ( C ) ]
(B1, B2) E Z~}. These form a finite disjoint union of 8(C). Set U - B N b/ and U~, - U n ~U. The fibers of the projection 8 ~ ( d ) isomorphic to d n U~.
~ 13 x 13 are all
Since Zw identifies with G / ( B n ~B), it follows
that dim 8~(C) - d i m ( G / B N ~ B ) + dim(C N U~) _< dim G - rk G, with equality if and only if C N U~ is dense in U~, equivalently if (GU~) - C. We denote the subset of w E W satisfying this last condition by St -1 (C). In this case, C n U~ is irreducible. Now S~(C) can be identified with the set of G translates of the fiber C n U~ over (B, WB). Since G is irreducible, it follows that so is S~(C). Its closure is hence an irreducible component of ,5(C) when w E S t - I ( c ) . At least for characteristic zero, the B a l a - C a r t e r - D y n k i n theory gives a specific procedure for determining for each conjugacy class d an element w E W such that d N U~ is dense in Uw. The idea is as follows. Consider e Efl nilpotent. By the Jacobson-Morosov theorem, there exists an s[(2) triple (e, h, f) containing e. Let g-
(~gi iEX
be the decomposition of g into its t? eigenspaces with gi corresponding to h-eigenvalue i. Now [e, go] = g2 from s[(2) theory, and so dim g2 < dim go implies g~ -76 0. Moreover, g~ - go/, and so this subalgebra is reductive. Let ~0 C g~ be a Cartan subalgebra. Then gbo is a proper Levi factor of g containing e. This reduces the problem to a strictly lower rank case. If dim g2 = dim go, then gl
-
-
0 by the B a l a - C a r t e r - D y n k i n theorem.
This was at first checked case by case, but then J. C. Jantzen found the following "one-line" proof. Consider the Dynkin parabolic p = (~i>0gi, whose nilradical is rn = 1~i>1 gi. By Richardson's theorem (which is also m
proved using Bruhat decomposition), the corresponding parabolic group
A. Joseph
60 P has a dense orbit in m.
In Lie algebra terms, this means t h a t there
exists x C ra such t h a t [p,x] - m. Decomposing x into h eigenvectors xi, one obtains [g0,Xl + x2] + [~1,Xl]
--
~1 O ~2. The first t e r m on the left
h a n d side has dimension _< dim S0 - dim g2, whereas the second t e r m has dimension < dim ~1 unless ~1
--0.
Hence the assertion.
On the other hand, setting m ~ - ~ i > 2 gi one checks again from ~[(2) theory t h a t d i m G e -
dimg-dimg~
and dim Pe >_ dim G e - dim m present case ~1
--
-
dimm+dimm
t. Yet Pe C m ~
dim m t, and so Pe is dense in m ~. In the
0, SO t h a t Pe is dense in m. Finally, for an appropriate
choice of w C W, one has Lie U~ - m. T h e r e is a second n a t u r a l way to obtain irreducible components of 8(C).
Fix u E C, and set B~ -
{B E B I u C B}.
An a r g u m e n t of
N. Spaltenstein IS] based on B r u h a t decomposition (which we use later in a related context) shows how to move from component to component of B~. From this, one concludes t h a t Bu is equidimensional. Now let X be a S t a b c u orbit of an irreducible component of Bu × B~, hence in particular a finite union of pairs of irreducible components of B~. Let Y be the inverse image of X under the continuous map (guy- 1, B1, B2) ~ (g- 1B1, g- 1B2) of $(C) onto Bu × B~. T h e n Y is closed, irreducible, and has dimension dim C + dim X = dim C + 2 dim Bu. Since 8(C) is a finite irredundant union of such Y, the latter run over its irreducible components. On the other hand, we have seen t h a t at least one such component has dimension equal to dim G -
rk G. Consequently, every component has this
dimension and furthermore, (2.1)
2 dim Bu + dim C = dim G - rk G.
A further comparison of these two descriptions of irreducible components shows t h a t the S t a b a u orbits of an irreducible component of B~ are the closures of t h e C w ' St-I(c).
{gB I v g E G
such t h a t g ( B M W B )
~u}'wc
Finally, let V1, V 2 , . . . , Vn be the irreducible components of C M U C M B, and C1, C 2 , . . . , Cm the irreducible components of B~. Consider the surjective morphisms
71"l ( g )
--
g--lug o f
e
onto C and ~2(g) - gB of G onto
~. Identifying B with G / B identifies ~21(Ci) with CiB. Since C~ and B are irreducible, so is 71-21 (Ci), and so these sets form the irreducible components of ~ - l ( B u ) -
~11(C M V).
Thus, the Vi • i -
1,2,...,n
are among the
Chapter 2 7r17r~-l(cj) " j -
1,2,...,m.
Orbital Varieties
Moreover,
7r17121(Ci) is
61 independent of the
choice of Ci in its S t a b a u-orbit. Thus, the V/ are among the closures of the 7tl(Tr21(Cw)) - B(Uw n C ) . Notice t h a t from the definition of Zrl, zr2 one has d i m C j + d i m B - dimzrlTr~-l(cj) + dim S t a b o u.
(2.2)
Combined with (2.1) and the equidimensionality of Bu, this implies t h a t U n C is equidimensional and of dimension dim(U n C) -- dim Bu + dim B - dim S t a b a u - 1 dim C. The preceding result may be summarized by the following. Theorem
2.1
Let 0 be a nilpotent orbit of a complex semisimple Lie
algebra g with triangular decomposition g -
n + N 1? n n - .
Then the irre-
ducible components of n + n O are Lagrangian subvarieties of 0 and take the form V(w) "- B(n+ n w(n+)) N O" w e s t - l ( o ) . Note t h a t V(w) ¢2 B(n+ a w(n+)). This point must be carefully taken into account in what follows. Spaltenstein's operation on the components of Bu caries over to the components of n + n O and takes the following form. Let ~r be the set of simple roots defined with respect to the foregoing triangular decomposition. Let V be a component o f n + n O ,
and take a C zr. Let Ps denote
the parabolic subalgebra b + ® g - s , and ms its nilradical. Let Ps be the connected subgroup of G with Lie algebra Ps.
The T-invariant of V is
defined to be T(V) "-- {a e 7r I V c m~}. Since ms is Ps-stable, it follows that a C T(V(w)) if and only if n + N w(n +) C ms, which in turn is equivalent to w - l a C -NTr or that s s w < w for the Bruhat order. In particular, V is Ps stable if a E ~-(V). Suppose a ~ T(V). Let us show that
m,~ m "~(V(w)) c V(w) u V(s~w) u (
U
zEwls~z
V(z)).
(2.3)
Set Vl(W) =B (n + n w(n+)). By Bruhat decomposition and the hypothesis, one has P s w B -
B w B U B s s w B . Then, since ms is Ps stable,
ms N P°'(VI(w)) -- P~(ms n w(n+)) =
=
n
u
n Vl (w))U Vl
(SAW).
A. Joseph
62
Set V2(w) - Vl(w) N O. Then ms n P~(V2(w)) -- (ms n V2(w)) U V2(s~w). Now consider V(w) \ V2(w). This has codimension _ 1 in V(w) and is B-stable. Hence dimP~ (V(w) \ V2(w)) _< dim V(w). Since O N n + is equidimensional, it follows that ms N P~(V(w) \ V2(w)) is precisely those components of O n n + that lie in ms and whose union contains Together with the first observation, this proves (2.3).
V(w) \ V2(w).
Notice it is not guaranteed that V(saw) is actually a component of n + N O, as it may be of codimension _> 1. The contribution in the far right hand side of (2.3) may be quite sizable. Starting from a given orbital variety V, the operation defined by (2.3) iterated sumciently often eventually gives all the orbital varieties lying in the unique dense orbit O contained in
GV. This was originally proved by
Spaltenstein [S] using an argument of Steinberg involving Bruhat decomposition. It implies that O n n + is equidimensional. A similar argument can be used to show that O n n + is equidimensional. Using the characteristic polynomials introduced in Lecture 3, we can follow this operation more closely. By this means, it is possible to generate interesting examples of what can occur on the right hand side of (2.3). However, a complete description is still an important open problem. Exercises and results on associated varieties Let {~-~n(U(g))}mcN be the canonical filtration [D, 2.3] of U(g). By the Poincard-Birkhoff-Witt theorem, gr 7 U(g) identifies with S(g). Let M be a finitely generated U(g)-module with generators ml, m 2 , . . . , ms. Then M can be made a filtered module for (U(g),J c) through
.
-
i--1
1) Show that
V(Anns(g) gr 7
M) - V(gr(fi Annv(g) mi)),
(2.4)
i=1
and that both sides of this expression are independent of the choice of generators. (It is important in Lecture 3 to know that these assertions also hold in the scheme theoretic sense.)
Chapter 2
Orbital Varieties
63
2) One writes V(M) for the variety in (2.4), and, by a slight abuse of notation, V ( A n n M ) for V(U(g)/ Ann M). Show that V(Ann M) = V(gr Annv(~) M) D GV(M). One knows Suppose M ple, if M is that V(M)
3) 4)
(2.5)
that equality holds in (2.5) under certain circumstances. is homogeneous for Gelfand-Kirillov dimension, for examsimple. Then a result of O. Gabber [LS, Appendix] asserts is equidimensional. For further results on associated vari-
eties, see [BE]. Show that gr~= Ann M prime implies that Ann M is completely prime [J7, 8.1]. Assume g semisimple. Given J E PrimU(g), show that V(grj= J) is contained in the cone of nilpotent elements of g*.
Lecture 3
Goldie rank and characteristic polynomials
Let us describe more precisely how rk U(g)/J(A) depends on A E b*. First some notation is needed. Let R (resp., R +,Tr) denote the set of nonzero (resp., positive, simple) roots relative to our triangular decomposition of ~. For each c~ E R, set av = 2c~/(a, a). The lattice of integral weights is defined to be P ( R ) : = {A E [~* ] (c~V,A) ¢ Z, V c~ E R}. The Weyl group W leaves P(R) invariant, and this action admits the subset P(R) + := {A c P ( R ) ] (c~,A) _> 0, V c~ c R +} of dominant weights as a fundamental domain. Finally, a weight A is called regular if (c~, A) ¢- 0, V c~ C R. A regular dominant weight is just one which belongs to the subset P(R) ++ := P(R) + + p. The fibers of the map w ~ J(w)~) of W into Prim U(~) are known after W. Borho and J. C. Jantzen [BJ] to be independent of the choice of
A C P(R) ++ (with some degenerations for A C P(R)+). They are called the left cells of W. Their conjectured [J1] description in terms of the Jordan-HSlder multiplicities [M(yA): L(wA)] was settled by D. A. Vogan [Vol], a result that was one of the motivations for the Kazhdan-Lusztig polynomials [KL1] now known to determine these multiplicities. This result leads naturally to a conjectural description [J4, 9.9] of the geometric cells of W (Lecture 1), but one that is as yet unresolved. (There is unfortunately a gap in the proof in [T, 2.11].) One calls the set of ideals {J(A) : A C P ( R ) }
A. Joseph
64
the integral fiber (of the Duflo map). Except in ~[(n), not all nilpotent orbits occur as associated varieties (cf. Theorem 1.1) of ideals in the integral fiber. Those that do coincide with the special orbits of G. Lusztig, which were defined in a rather different manner [Lu]. Notice that this already means that left cells and geometric cells must differ outside ~[(n). If we let a(w,y) denote the coefficients of the matrix inverse to [M(yA) : L(w)~)]y,~w (known by [Janl] to be independent of A C P ( R ) + + ) , then the formal character of L(w)~) takes the form
ch L(w)~) - ~
a(w' y)eY:~-P
yCW
ii~eR+(1 _ e_~).
(3.1)
It turns out that the required Goldie ranks are obtained by studying the asymptotics of the right hand side of (3.1). Indeed, fix u C P(R) ++. Then it makes sense to consider e ( ~ ' ' ) as a genuine exponential and to expand the function
E
a(w'y)e(Y~'V)
yEW
in powers of (yA, u). The first nonvanishing term in the expansion has the remarkable property that it factors as a product of two polynomials; namely, one has
E
a(w, y)(y)~, u) ~ -- p~()~)q~(u).
(3.2)
,yEW
This fact (observed by a student, D. King, of D. A. Vogan) resulted from the interpretation of the first factor p~ (A) as the required Goldie rank, namely, one has [J3, Theorem 5.1] T h e o r e m 3.1
For all )~ C P ( R ) +, w C W, rk(U(9)/J(w)~)) - p~(£).
Notice one can express this result by saying that
p~ - ~ yEW
a(w, y)y-lpm
(3.3)
Chapter 2 Orbital Varieties
65
up to some scalar. These scalars are not yet completely known [J6]. However, since Pw takes integer values on P ( R ) +, it must take integer values on the whole of P(R). Then the scalar would be determined by knowing t h a t it takes at least once the value 1. I conjectured this to be always true [J7, 8.4(i)]. Parts (ii), (iii) of the cited conjecture have counterexamples
[B, T]. The factorization (3.2) has some further remarkable consequences, which are quite easy to prove [J3, Section 2].
QWpw is a simple W-module. Moreover, this simple module
A)
occurs with multiplicity one in the space of homogeneous polynomials on O* of degree m and not in lower degree. To prove this, one should observe that replacing p by zp: z C W in the expression (3.3) for p~ only changes the latter by a scalar. This result generalizes an observation of I. G. Macdonald [M]. We call such modules univalent. Not all W-modules have this property. B)
p~ is a W-harmonic function on 0", that is, it satisfies the relation
E
p(w# + ~) -[W]p(~),
for all #, ~ C O*.
wOW
For this, one should recall the vanishing of the sum
Y~y~w a(w, y)(y)~, ~,)n
for n < m. The second polynomial q ~ ( , ) occurring in the factorization (3.2) has also an i m p o r t a n t interpretation and leads to a determination of the associated variety of L(w)~). Formalizing this leads to the notion of the characteristic polynomial of an orbital variety, and this has some important geometric consequences. This will be examined. First note, however, after [J2, 5.4] one has a(w, y) = a(w -1, y - l ) and so C)
q~-i = pw up to a scalar.
Let V be a closed irreducible O-stable subvariety of n +. View its ideal of definition
I(V) as an O-invariant ideal of S ( n - ) .
From the previous
discussion, it is natural to study the asymptotics of the formal character of
S(n-)/I(V). Indeed, fix v C P(R) ++ and consider E
dim(S(n-)/I(V))_~
(~,~,)~_n
as a function of n. From Hilbert-Samuel theory, it is a polynomial in n the degree of which is the dimension of V. Moreover, the leading coefficient
A. Joseph
66
rv({) is a function of {, and the simple calculation following will show that this takes the form =
for some polynomial Pv on 11", called the characteristic polynomial of V. If V is the associated variety of L(wA) and the latter is irreducible as in, say, ~[(n) by [Mel], then comparison with the previous analysis shows that
py - Pw- ~ up to a nonzero scalar. More generally, pw-1 is a sum with strictly positive rational coefficients of the characteristic polynomials of the components of the associated variety of L(wA). The general form that Pv can take is easily deduced using hyperplane intersection [J8, Section 8]. Let n + -
nl+Dn2~ . . . .
0 be any l)-stable
flag of n +. It will be sufficient to suppose that V C ni and to relate rw to rVnni+l" Here, V n ni+l is interpretated scheme theoretically; that is, V n rti+l is a linear combination of the components Vj of V n hi+l, each possibly occurring several times; then, rvnn~+l
--- ~_~ r V 5 • J
It remains to prove the following lemma. Let c~i be the weight of a vector in ni/ni+l. Lemma ii) ry -
i) rvn,i+l - r v , if !OLi r y n , i + l , otherwise.
3.2
V C ni+l.
Proof: Only (ii) is nontrivial. Write a - c~i, n - hi, m - hi+l, I - I(V), and
Rv(u)-
E
dim(S(n*)/I)-~e(~")
~ ENrr
for all L, E P(R) ++. Set J - I + S(n*)x_~. For each { E NTr, choose a subspace V_ e of ( S ( n * ) / I ) _ e that under the canonical projection S ( n * ) / I S ( n * ) / J maps bijectively to ( S ( n * ) / J ) _ ~ .
,
Set V - ~ V_~, which as an
l~-module is isomorphic to ¢o(m*)/I N S(m*).
Chapter 2
67
Orbital Varieties
The hypothesis of (ii) implies that x_~ is a nonzero divisor in S ( n * ) / I , and so the sum }--~kV x ~ is direct and equals S(n*)/I. One concludes that
Rv(u)- Z
Z (dim V_~)e(~-k~'')
(1)
kEN ~ENTr
and hence the assertion of (ii) follows by a simple asymptotic analysis [J4, 2.3]. I-I This simple argument gives the following significant positivity result [J8, 8.6], not apparent from our previous explicit formula. C o r o l l a r y 3.3 The Goldie rank polynomial pw is a linear combination with positive rational coefficients of products of distinct positive roots. This helps [J8, 8.6] to locate completely prime primitive ideals. For example, one can show that U ( g ) / A n n L ( w A ) : A E P ( R ) ++ is a domain if and only if L(wA) is induced from a one-dimensional representation of a parabolic subalgebra. Unfortunately, the restriction on A is rather strong in this context. (See [J9, 5.3] for a slight improvement.) Examples
1) Take g - s[(4) and V the orbital variety closure with ideal
of definition (e21, e43, e31e42-e32e41). Then Pv - c~1c~3(al +2c~2+a3). 2) Take g - s[(6) and V the orbital variety closure with ideal of definition (C21, e43, e54, e53, e31e42--e32e41, e41e52--e42e51, e 3 1 e 5 2 - - e 3 2 e 5 1 ) . Then pv
-
+
+
+
+
OL3 -31-OZ4)] .
3) Take ft - ~0(7) and :r = { a l , a 2 , a 3 } with a3 short. Then 7r' = {a2,a3} is a simple system for a Lie algebra g' of type so(5). This admits a unique orbital variety of dimension 2, the closure V' of which has characteristic polynomial pv, - a3(a2 + a3). Set V - n + + V', which is the closure of an orbital variety of ft. Then Pv - a3(a2 + a 3 ) . Neither pv' nor Pv can be proportional to Goldie rank polynomials attached to ideals in the integral fiber. However, in the so(7) example one has the additional surprise that G V is the associated variety of a primitive ideal in the integral fiber. This in turn implies that the associated varieties of the corresponding highest weight modules cannot all
A. Joseph
68
be irreducible and illustrates the discussion in (3) following Theorem 1.1. This example was developed from a note of T. Tanisaki, and we refer the reader to [J9, 8.6] for further details. Determining all such occurrences is still very much an open problem. In general, pv need not be W-harmonic, nor need it generate a simple W-module. To prove that these properties do hold when V is an orbital variety closure motivates a similar but independent presentation of these characteristic polynomials analogous to that already established for the Goldie rank polynomials. This is provided by [JS] using the geometric considerations sketched briefly in what follows. Fix ~ C O*, h E O. One knows that expressions of the form
E
A(w,y)e (y~)(h) "A(w,y) C Z
can be obtained as integrals. Then the key factorization property should be obtained by reexpressing this integral in product form by some change of variables. Now, the orbital variety V(w) can be viewed as the image under the moment map of the conormal
Y(w) "- {bwB, b(n+ N w(n+))b -1" b e B} in T*(G/B). Consequently, one should integrate over Y(w). To determine the relevant integrals, one appeals to the remarkable theory of M. F. Atiyah and R. Bott as developed more recently by N. Berline and M. Vergne [BGV] and W. Rossmann [R]. Essentially, the integrand is chosen so that Stokes theorem can be applied. The integral over V(w) becomes Pv(w) by [J8, 5.7]. Explicitly, this means that, up to a nonzero constant,
Pv(w) - E A(w' y)YPm'
(3.4)
yEW
where m is the least integer _> 0 for which such an expression is nonvanishing. This result was first obtained by W. Rossmann using Hotta's result [Ho] and answered a conjecture of [J4]. A key point in the required factorization is that the integral over a fiber of the moment map is independent of which fiber is chosen. This is because the integral represents the class of the fiber in homology [J8, 3.6],
Chapter 2
Orbital Varieties
69
which through the G action is independent of the fiber chosen. Finally, it is possible to express the A(w, y) in terms of Euler numbers [J8, 4.8] and in particular to show that A(w, y) - A(w -1, y - l ) . Then by the analog of (C) following Theorem 3.1, the Pv(w) must form a basis for (Springer's) W-module Mo(w) attached to (9(w) lying in the homology space of a fixed point set Bu " u E (9(w). This recovers the aforementioned result of R. Hotta [Ho]. Set A(w) - E y ~ W A(w,y)y. As in (A) following Theorem 3.1, the factorization of the first nonvanishing term (A(w)A, u)~n implies that Mo(~) is invariant. In particular, an isomorphism Mo(~)-Z-,Mo(~,) of W-modules must be an equality. Moreover, according to Springer's well-known result, one has T h e o r e m 3.4
The M o • (9 C A/" are pairwise nonisornorphic.
As first noted by Kazhdan-Lusztig [KL2], following a suggestion of Springer, this may be obtained from a study of the Steinberg variety (see also IT] and references therein). It gives the following important and a priori surprising result. C o r o l l a r y 3.5
The p~. • V E Y are linearly independent.
E x e r c i s e Take u E V(w). Show that the fiber over u under the moment map Y(w) ~ V(w) is just a Stabc u-orbit of an irreducible component of B~. (See Lecture 2.) Except possibly in type s[(n), the A(w, y) and a(w, y) do not quite coincide [J8, 4.9]. Thus, we obtain the following rather subtle version of the orbit method correspondence. Orbital varieties and primitive ideals satisfy exactly analogous theories, but the data that must be used to calculate their properties (viz. left cells or geometric cells, Goldie rank, or characteristic polynomials) differ slightly. Again the A(w, y) contain data (for example, the characteristic polynomials) concerning all the nilpotent orbits, whereas a(w, y) concern only those occurring in the integral fiber (Lusztig's special orbits). For example, take g = s0(5). Then 1~ admits four nilpotent orbits of dimensions 8, 6, 4, 0. Exactly the four-dimensional one does not occur in the integral fiber. The unique orbital variety attached to it has the characteristic polynomial given in example (3). Again using t~(.) for reduced
70
A. Joseph
length and > for B r u h a t order, one has a(w, y) a n d is zero otherwise.
( - 1 ) ~(w)+~(y) if y > w
One concludes from T h e o r e m 3.1 t h a t the Goldie
r a n k polynomials are p r o p o r t i o n a l to 1, a2, a3, a 2 a 3 ( a 2 + c~3)(c~2 + 2c~3). In particular, the p o l y n o m i a l of example (3) does not appear. Yet it m u s t a p p e a r in the right h a n d side of (3.4). Consequently, A(w, y) must differ from a(w, y) in this case. F u r t h e r reasons t h a t A(w, y) and a(w, y) m u s t differ are given in [J8, 4.9]. Let us go into how the i n t e g r a n d is chosen in a little more detail. Let M be a C ~ manifold with symplectic form a, and x a vector field on M .
If f is a s m o o t h function on M , the compatibility condition we
need is t h a t df + c(x)a - 0, where c(x) denotes contraction by x. T h e n
dx "- d + c(x) is just the "covariant differential." One has dx(~ + f ) - 0, a n d so dx(a + f ) n _ 0 for all integer n > 0. T h e integrand is chosen to be e x p ( a + f ) - exp a exp f , with the convention t h a t only the t e r m O"n i s r e t a i n e d if the i n t e g r a n d is over a variety Y of real dimension 2n. U n d e r suitable conditions (in p a r t i c u l a r t h a t the set ~ of zeros of x is finite), the A t i y a h - B o t t - B e r l i n e - V e r g n e - R o s s m a n n t h e o r y asserts t h a t
y fO'n -- E ef(W)n(w) wE~ for certain real n u m b e r s n(w). In the present case we identify G / B with the c o m p a c t real variety K / T , where K is a m a x i m a l c o m p a c t s u b g r o u p (see Lecture 4) of G viewed as a real Lie group and T a m a x i m a l torus for K . Set t - Lie T. T h e n T* ( G / B ) identifies as a real manifold with K ×T t ±. W i t h respect to the elements x~ • a E R of a Chevalley basis for ~c, one has Cartisian coordinates
es "- i ( x s + x - s ) ,
fa "- Xs - x - s " a C R + on t ±. Fix h E b, i~ C t*
regular. T h e n one takes f to be the function on K X T t ± defined by 1
f (k, a) - (h, k~) - -~ E
2
2
(es + f s ) ,
sER+
where (es, f s ) " a C R + denotes the coordinates of ka" k C K u n d e r the p r o j e c t i o n onto t ± defined by the d e c o m p o s i t i o n ~ - t ± ® t. This function satisfies the preceding (integrability) condition for a " - a~ +Th, where a~ is the 2-form on K / T defined t h r o u g h a~(x, y ) ( k T ) and where
1 sCR+
(des A d f s ) .
([x, y], kA), V x, y E [,
Chapter 2
Orbital Varieties
71
Examples and exercises Let 7r~ be a subset of 7r, and set R '+ = NTr~ N R +. One easily checks that there exists an orbital variety the closure V of which has characteristic polynomial p~, := H~cR'+ c~. According to the preceding theory, the W-module generated by p v has a basis formed by the characteristic polynomials of the orbital varieties attached to the unique dense orbit in GV. Moreover, these polynomials are given by (3.4), though here the A(w, y) are unfortunately not yet known. However, in low rank cases one can obtain these polynomials just by following the rules imposed by Spaltenstein's operation described in Lecture 2 in the following manner. Let V be an orbital variety, and recall (Lecture 2) the definition of the T-invriant T(V). 1) Set 7r' = T(V). Show that p~, divides p~. 2) Suppose c~ ~ 7r~. Show that s ~ p ~ - p~ is a linear combination with nonnegative rational coefficients of characteristic polynomials of orbital varieties. 3) Assume there exists c~ C :r \ 7r', /~ C 7r' such that (c~,c~)= (/~,/3)= -2(a,/3). Show that there exists a unique irreducible component V' in rn~ N P~ V such that c~ C ~-(V'), /3 ~ T(V). For more details see [J4, Sect. 9]. 4) Calculate p~ for each of the 26 orbital varieties V occurring in ~[(5) using the information provided by (1)-(3). Lecture 4
Unitarizable
highest weight m o d u l e s
Let g be a complex semisimple Lie algebra. The notion of a complex unitarizable g-module depends on the choice of a real form go of g. Let j: g ~ g denote complex conjugation with respect to g0. Let {x~ :c~ C R; h~ := [x~,x_~] : /3 C 7r} be a Chevalley basis for t~. This basis generates a real Lie subalgebra of g, and we let jc denote complex conjugation with respect to this basis. The (compact) Chevalley antiautomorphism ~rc associated to this basis is defined by Crc(X~)= x_~. Define an involution 0 through O ( x ) = - j ~ ( x ) , V x E 1~. One may check that the Killing form is negative definite with respect to the fixed points of 0, and this defines the compact real form of ~. By, say [H, Chapter 2, Theorem 7.1], one may choose the Chevalley basis so that Oj = jO. Consequently, g0 is 0-stable. 0 is called the Cartan
72
A. Joseph
involution of go and gives g0 a 7-a grading. Indeed, setting
~o = {z e go l e ( x ) = one
~},
po - {x e ~o l e ( x ) =
-x},
has
~o
=
~o • po,
[~o, ~o]
c
~o,
[~o, po]
c
po,
[po, po]
c
~o.
The adjoint action of g0 on itself defines a Lie algebra homomorphism x H ad~o x of g0 into g[(n,R) : n = dimg0, which is injective since g is semisimple.
The compact real form can be viewed as {~0 ® ip0. W i t h
respect to the (anti)orthonormal basis it defines for the Killing form, one may identify 0 with x ~ - t x , where t denotes transpose of a matrix. Let G be the connected (real) subgroup of GL(n,R) of elements of the form exp x : x C go. Then Lie G = go. Let K be the subgroup of G of all orthogonal matrices (that is, tg = g - l ) . Then K is a compact subgroup of G, in fact, maximal compact. Moreover, Lie K = ~0. After Harish-Chandra (see D. A. Vogan's lectures in Chapter 5), classifying irreducible unitary representations of G is equivalent to classifying simple admissible (g,K)-modules admitting a positive definite invariant Hermitian form ( , ) .
If K is connected, then a (g, K)-module M is simply
a g-module, admissibility means that M is a direct sum of simple {~-modules each occurring with finite multiplicity, and invariance of the form means that
O= (xm, n) + (m, xn),
V m, n E M, x E go.
We shall call such modules unitarizable. Let g = m + ® ~ ® m - be a decomposition of g into Lie subalgebras such t h a t p := m + ® ~ is a parabolic subalgebra with nilradical m + and Levi factor {~ leaving m + stable.
One may assume that ~ D ~, and then
any such decomposition corresponds to writing 7r
=
71-1 ]_I 71"2
taking the
root vectors in ~ (resp., m +, m - ) to have roots in the subset R N ZTrl (resp., R + \ (R A NTrl), R - \ (R A - N 1 r l ) ) . ~(~+)
In particular, crc([~) = ~ and
= ~-.
If m + (and hence m - ) is commutative, one may define an involution 0 on g taking the value 1 on ~ and - 1 on m - ® m +. Suppose this holds, and set cr = 0ac. Then j ( x ) = - j c a ( x ) defines a new complex structure on g. Set g0 = {x C g I j ( x ) = x}. Explicitly, g0 has an R-basis formed
Chapter 2
Orbital Varieties
73
from { i h ~ ' ~ E 7r, i ( x a + x - a ) , (xa-x-a)'c~ER + ' - R +MNTrl} and from {xa + x - a , i ( x a - x - a ) " c~ E R + " - R + \ ( R + N NTrl)}. Moreover, the former (resp., latter) gave an IR-basis for t~0 - t~0 N t~ (resp., P0) taking 0 Igo to be the Cartan involution. The preceding condition on m + implies that R + N 7r is a singleton, called the noncompact simple root, which must furthermore occur with multiplicity one in any root/3 of m +. Let Q(~) be a highest weight module with highest weight ~, that is, a quotient of a Verma module M(~).
(Note our change of convention
compared with Lecture 3.) By the Poincar~-Birkhoff-Witt theorem, the triangular decomposition of 1~ gives a direct sum decomposition U(g) - U(O)® (n-U(f;) + U(f;)n+). Let p be the projection of g(g) onto U(O) defined by this decomposition. Define a Hermitian form on U(I~) through (a,b) ~ p((j~a)(a)b)(,~). One easily checks that its kernel contains Annu(~) e~, where e~ denotes a highest weight vector for M(A). Writing rn - ae~, n - be,x, one thus obtains a Hermitian form on M(A) satisfying (m, x n ) - (j~cr(x)rn, n) for all m , n E M(A), x E g. One easily checks that its kernel is precisely the unique maximal submodule of M(A), and so the form passes to Q(A) and in particular to L(A). Notice that j~a - - 1 on go, and hence ( , } is the unique (up to scalars) form on Q(A) satisfying the required invariance condition for unitarity. In particular, Q(A) is simple if this form is positive definite. Again, L(A) being admissible implies that its highest weight vector generates a finite dimensional p-module V. Then L(A) is an image of the induced module U(g) ®u(p) V. Thus, L(A) is admissible and hence is unitarizable if and only if the form is positive definite. All unitarizable highest weight modules for noncompact connected real Lie groups arise in this fashion. Let c~ E 7r~ "-- R + N 7r denote the noncompact simple root and aJ the fundamental weight corresponding to c~. Set 7r~ - 7r \ 7rn. In P ( R ) , P ( R ) + let a subscript c denote the condition that the corresponding weight is orthogonal to c~. Given 7- E P ( R ) +, let V(T) denote the simple finite dimensional t~-module with highest weight T. For all u E C, set A~, "= ~- + ua~. The simple highest weight {~-module it defines can be identified as V(~-) ® C ~ , which we view as a p-module by letting rn + act by zero. The induced module N(A~) "- U(f;)®u(p)(V(~-)® C u ~ ) i s isomorphic to S ( m - ) ® V(~-) ® C ~
as a p-module.
For its simple quotient to be
A. Joseph
74
unitarizable, one requires u E IR. It remains to describe for which pairs (~-, u) C P ( R ) + × IR this quotient L ( ~ ) is unitarizable. In this, it is enough to assume g simple. One defines a totally ordered sequence ~1,/32,...,/~t of (strongly orthogonal) noncompact positive roots as follows. Let
~1 be the highest root.
Delete the simple roots in the Dynkin diagram of ~t not orthogonal to 13, and take the unique connected component containing a. Let ~2 be the unique highest root of this component, and continue until a itself is deleted. One may show that the partial sums i
E/3j
"i-
1,2 . . . . ,t
j=l
belong to P ( R ) +. Moreover, up to scalars there exists for each i a unique nonzero vi E S(m)"
of weight - # i . Let Vi denote the simple ~-submodule
of S(m) generated by vi under adjoint action. Fix ~- C P ( R ) +. According to a result of K. R. Parthasarathy, R. Ranga Rao, and V. S. Varadarajan [PRV], there exists a unique component Pi of V(~-) ®Vi having extreme weight T - # i . Call this the PRV component. Finally, define ~- to be of level s if (~i - / 3 j , T) - 0, V i, j - 1, 2 , . . . , s, and s is maximal _< t with this property. The classification of unitarizable highest weight modules now goes as follows [EJ]. Take ~- C P ( R ) + and suppose it is of level s. Take i E {1, 2,.. • , s}. Then there is a unique value u iT of u such that S(m)P~ ® Cu~ is a V(g)-submodule of N ( / ~ ) .
The u~ • i -
1,2,...,s
are decreasing,
and for u < u~ the induced module N(~u) is unitarizable. The remaining unitarizable modules are its quotients by the previous submodules.
One
~ 1 " 2 <_i < s may also calculate the u i . Notably, the differences u i - ui_ are independent of ~- and of i. One calls u~ the first reduction point, and u[ the last place of unitarity. One may show, [J10, 2.3] for example, that the S(m)V~" i -
1, 2 , . . . , t
run through the ideals of definition of the closures of the orbital varieties lying in m +. Thus, when 7- - 0, the corresponding unitarizable highest weight modules identify with the spaces of regular functions on these varieties. Their highest weights are the u°w • i - 1, 2 , . . . , t. Example
Take tt - ~[(n), and choose m E { 1 , 2 , . . . , n -
1}. One may
take a - am to be the noncompact simple root. Then, t - min{m, n - m } .
Chapter 2
75
Orbital Varieties
Moreover, vi : i = 1, 2 , . . . , t is the lower left hand corner i x i minor, and V/is the subspace spanned by all i x i minors lying in the b o t t o m left hand rn x ( n - m) corner. In particular, if n = 4, m = 2, one recovers examples (1) and ( 2 ) o f Lecture 3. In the last example, take ~- = a~l. Then V(T) is a two-dimensional ~-module (here, t~ ~_ t3[(2) x ft[(2)). One may choose a basis {x, y} for V(T) and a basis { a , b , c , d }
for rn- such that v2 = a d -
bc and P1 has basis
a y - bx, cy - d x . Then,
( a d - b c ) x -- c ( a y - bx) - a ( c y - d x )
(4.1a)
( a d - bc)y = d ( a y - bx) - b(cy - d x ) ,
(4.1b)
and
which shows that v 2 V ( T )
is contained in rnP1.
Consequently, we have
Annv(m-) L ( ~ ' ) - U(m-)v2, which identifies with a p r i m e ideal of S ( m - ) . Of course, (4.1a, b) is just Cramer's rule of linear algebra, which here translates to a statement about Annu(rn-) L(A) being prime [J10, 5.16] whenever L(A) is unitarizable! An interesting example also occurs in ET, and the cotresponding polynomial v3 C V3 is of degree 3 in 27 variables [J10, 8.1]. It was studied in an old paper of A. Freudenthal [F] concerning geometry over an octionic field.
Lecture 5
T h e P a r t h a s a r a t h y equality
Recall the setup and notation of Lecture 4. Take A - ~- ÷ ua~ with ~- C P ( R ) +, u E C, and let N(A) be the induced module N(A) = U ( g ) ® v ( p ) V(A). Let e be a t~ highest weight vector of highest weight p. Let ( , ) denote the C a r t a n inner product on [?*, and set I]ull2 = (u, u). The key to studying when L(A) is unitarizable obtains from the following result due to K. R. P a r t h a s a r a t h y [P]. Lemma
5.1
(ll x ÷ pll 2 aER +
76
A. Joseph
Proof." Let I (resp., J) denote the Casimir invariant for t~ (resp., 2). Then, up to appropriate choices, Ie - I A + pll2e, Je - I]# + Pll 2e. On the other hand, for these choices I-
J-
2 ~_, X--aXa. hER +
Recalling that a(x_~) - - x ~ for all a E R +, the assertion results. Z] Remark
Of course, for the "compact" Chevalley antiautomorphism one
has a ( x _ ~ ) = x~, and the resulting conclusion would lead to quite different results. A similar result holds for any real form of 9, and fairly easy calculations similar to those following can be used to show that only the real forms we are considering lead to unitary highest weight modules. It is clear that the assertion of the lemma applies to any quotient of N(A), in particular to L(A). This gives the following important C o r o l l a r y 5.2 The Hermitian form on L(A) is positive definite if and only if I1# + Pl > IIA + Pll for every 2 highest weight # ~ A occurring in L(A).
P r o o f : Let 9~m(L(A)) denote the image of ~ m ( U ( m - ) ) ® Y(A) in L(£). Obviously, x ~ m ( L ( A ) ) C 9~m-1 (L(A)) for all a E R +. Since K is compact and 20 - Lie K, an invariant form on any simple finite dimensional 2 (20)c-module is either positive or negative definite. In particular <, } can be assumed to be positive definite on 9~°(L(A)) - Y(£). Take m e N, and assume <, } is positive definite on 9~m(L(A)). Assume e e ~'m+l(L()~)). The sum occurring in the right hand side of the assertion of the lemma is >__0, with equality if and only if x~e - O, V a E R +, which forces A - p. Hence the required assertion follows. [:] For each dominant weight ~, let V(~) denote the simple finite dimensional {~-module with highest weight y. One calls ~ the 2-type of V(~). Let ~1, ~2 be dominant weights, and let V(~) be a composition factor of V(~I) ® V(~2), which one may recall is completely reducible. One calls a minimal 2-type if I1~ ÷ Pl] takes its minimal value among those 2-types occurring in V ( ~ I ) ® V(~2). The PRV component of V ( ~ I ) ® V(~2) is by definition the composition factor with extreme weight ~1 + w0~2. L e m m a 5.3
~-type.
The P R V component of V (111) ~
V
(/]2) is its unique minimal
Chapter 2
Orbital Varieties
77
Proof: Let u be a tNtype of V(ul) ® V(u2). By complete irreducibility, one has a surjective map V ( u l ) ® V(-2) , V(u). Now e. 1 ® ewo.2 is a cyclic vector for the preceding tensor product and hence has a nonzero image in V(u). Let u0 be the highest weight of the PRV component. This takes the form y(ul + w0u2) for some y E W. Then
i) I1~11 > 11-1 -+-wo~211- II~oll. ii) ( u - uo, p) > 0, with equality if and only if u - uo. Hence II~ + pll > I1~o + pll, w i t h equality if and only if u - uo. This gives the lemma. Kl Exercises
Let A1, A2, A3 be dominant weights.
1) One knows (but this is quite difficult) that Annu(n+)
g--)~l - - E
U(l~nt-)x~O~V'/~l)nt-l"
c~Err
2) Recall that ea~ ® e_~ 1 is a cyclic vector for V(A3) @ V(')~I). (Here, V(-A1) denotes the simple module with lowest weight -A1. It is isomorphic to V(A1)*.) Let V(A2)~;_Xl denote the subspace of vectors in V(A2) of weight A a - A1 and annihilated by Annu(,+)e_a I . Construct an isomorphism Hom,(V(A3) ® V ( - A , ) ,
V(A2))--7-+ V(A2)~;_A1.
3) Deduce from Frobenius reciprocity that dimHom,(V(A1) ® V(A2) ' V(Aa)) - dim V(A2) ~1 A3 --A 1 " 4) Define -1, "2, u0 as in Lemma 5.3. Using (1), verify that
V(-o) ul - ~ nt-wo ° ~ v2 -- - V ( - o ) ~ + ~ o ~ 5) Deduce from (1)-(4) that dimHom~(V(ul) ® V(u2), V(u0)) - 1. This proves the existence and uniqueness of the PRV component. (See [W2, Section 9] for further details.) Recall the notation of Lecture 4. Suppose r c P(R) + has level s. Take j E { 1 , 2 , . . . , s } , and let uj denote the highest weight of the PRV component Pj of Vj ® V(7). Set
.
~J =
II-j + pll 2 -I1~ + pll 2 2j(~, ~)
"
78
A. Joseph
Take A - ~- + Theorem
T
U j 03.
5.4
N(/~)'-
U ( m ) ( P j ® Cu;~) • j -
1,2,...,s
is a U(g)-
submodule of N (~).
We give the proof for j - 1, which is a little easier. By definition, the simple I-module P1 has highest weight # - Ul + u ~ . I ~ + pll ~ -
I~ + pll ~ =
Then
I1-~ + p I~ - I 1 ~ + p I~ - 2(~ - -1, ~ ) ~ Y 2~
(~ -
"1 -
~, ~).
Yet c~ occurs with multiplicity 1 in any root - ~ of m - , and so ( r - u l - ~ , w ) = 0. L e m m a 5.1 shows that y ~ a +
( x ~ e , , x~eu} - O. Yet x~eu E V(A),
on which the form is positive definite.
Thus, x~eu, -
O, V c~ C R +, as
required. [5] Remarks For the general case, one notes that ( r - u k Moreover, the u~ • k - 1, 2 , . . . , s are strictly decreasing and
kcx, w) -
[m+, Vk] c [~Vk-1 ÷ Vk-1
0.
(5.1)
by [EJ, 2.5, 3.6]. One proves inductively that ( , ) is positive definite on Vk ® V~ ® Cu~ for k < j and then the required assertion of the theorem by the procedure used for the case j - 1. W h e n 7 - - 0, one has
0
[]uj + pl[2 _ [[pl[2
uj --
2j(c~, ~)
[[pjl[2 + 2(p, #j)
=
2j(c~, ~)
"
Now the/3/ are always long roots and pairwise orthogonal. Thus, Ilpjll 2 = j (/31,/31). Consequently, 0
0
Uj_ 1 -- Uj --
J
1
j ( j _ 1)(c~ ~) E ( / 3 i - / 3 j , p) > 0, '
i=1
and so the fact that the uj0 are strictly decreasing is quite easy.
Thus,
apart from verifying (5.1), we have also proved the theorem for 7 - 0. This proves that the space of regular functions on the closure of an orbital variety contained in m + admits the structure of a U(g)-module of highest weight, namely N()~)/N()~) where ~ - u°~ • i -
1 , 2 , . . . , t. Further estimates of
this type show that this quotient is unitarizable and hence simple [E J, 8.2].
Chapter 2 Lecture 6
Orbital Varieties
79
T h e l a s t p l a c e of u n i t a r i t y
The description of the unitary module for 7- - 0, namely the L(A °) • i - 1 , 2 , . . . , t , was given by N. Wallach [W1]. The set A° - 0 > A° > ..- > At° is often called the Wallach set.
Their relation with orbital varieties
(using a slightly different language) is due to Harris and Jakobsen [H J]. It was realized (particularly by M. Kashiwara and M. Vergne [KV]), that the tensor product of unitary modules is again unitary. In particular, given that the L(A[) are unitary, then so are the L()~[) ® L()~°) • i -
1, 2 . . . . , t,
and moreover, for i _< s (the level of T) one recovers the L(A~) • i 1, 2 , . . . , s in this fashion as the orthogonal direct s u m m a n d with highest weight /~ + A°.
(The L(A~)" i > s are also unitary but correspond to
members of the continuous family of unitary modules occurring before the first reduction point is reached.) Moreover, one can show that all unitary modules are obtained in this fashion (excluding the continuous families of induced ones). u~-u~
This can be regarded as explaining the independence of
on T and on i, j, since A~ - A ~ + A °. Note again that 1 ° - 0
and corresponds to the trivial module, whereas tensoring L(A °) with itself i times gives L(A°+I). One may call L()~ °) the generalized metaplectic representation. Notice we have now solved the purely algebraic problem of quantizing all orbital varieties lying in any nilradical (of a parabolic subalgebra) that is commutative. In this, the positive definiteness of the form leads to a significant simplification. However following [E J, Section 2] and as noted in Lecture 7, one also obtains this result by more naive algebraic calculations. It thus essentially remains to establish unitarity at the last place A[, and this was a point of view emphasized by H. P. Jakobsen [Ja]. However, his analysis was case by case and rather complicated. easy intrinsic proof.
Here, we give an
Simultaneously with to the Jakobsen classification,
one should also mention, the unitary highest weight modules were classified by T. J. Enright, R. Howe, and N. R. Wallach [EHW]. The present analysis comes from joint work with T. J. Enright [E J]. Set A - A[. Theorem 5.4 established the existence of the quotient N ( A ) / N ( A ) . Now we prove its unitarity. Set
R< - { . e
+
v,
+ p) < 1}
A. Joseph
80
and Ft = UieN f~i, where Fti = { r - 71 - 7 2 . . . . . 7i17i E R<}. (Here, the 7i need not be distinct.) Let ft + denote the set of {~-dominant elements in f~; that is, a + = { h C a [ ( c x v,8)_>0, g a C r r ~ } . Our aim is to prove the following [E J, 7.9]. P r o p o s i t i o n 6.1
Every ~-type of N(A)/N(/~) belongs to f~+ + u[aJ.
Let us first show how this establishes unitarity at the last place.
Take 7 e P(R) +, and set ~ - 7 + u[w. Then N()~)/N()~) is unitarizable (and simple).
T h e o r e m 6.2
Proof: If N(A)/N(A) were not simple, then some highest weight ~ of a submodule would have to satisfy I1~ c + P]I = IIA + PlI. Further, taking account of Corollary 5.2, it is enough to show that I1" + Pll > IIA + Pll for every ~-dominant w e i g h t , ¢ A of N()~)/N()~). By the proposition, we can w r i t e , = 7 - 5 + u [ ~ with 5 C NR< \ {0}. Then the required inequality will result from 2(5, A + p) < (5, 5). Now, 5-
E
7i
"~i E R<
and (7i, 7j) ~ 0, because in R + the sum of two roots is not a root. Hence,
> Z
"~iER<
+ p),
by definition of R<. Moreover, the second inequality is strict, as 5 ~: 0. [3 We now sketch a proof of the proposition. Set R> = Rn \ R<, and let m< (resp., m>) denote the subspace of m := m - spanned by the root vectors x_~ : 7 E R< (resp., R>). An easy verification gives Lemma6.3 i)[n + , m < ] C m < . ii) [n~-,m>] C m>. Let Me(v) denote the Verma module for ~ with highest weight ~-. Consider M := m ® Mc(7) as ~-module for the diagonal action. It has a Verma flag [D, 7.6.14] with quotients isomorphic to the M ~ ( T - 7 ) " 7 C R +. Moreover, writing My = U(n~-)(x_~ ® er) and
-~ER<
~,ER>
Chapter 2
Orbital Varieties
81
then by Lemma 6.3(i), M< identifies with a submodule of M and M> can be viewed as the quotient M/M<; that is, we have an exact sequence 0
> M<
>M
> M>
>0
(6.1)
of U(£)-modules. One checks that [ I T - ' 7 + P~ll # IIw- ~ ' + pcll for all 7 E R< and "7' C R>. Using the center of U(t~), it follows that (6.1) splits. Let M~ denote the image of M> in M. L e m m a 6.4
P1 is the image of M~ under the map m ® Mc(~)
, r. ® v ( ~ ) .
Proof: Observe that M r identifies in the Verma flag of M with Mc(~--'7). NOW,
I1~ + p - ~11~ = I1~ + pll ~ + (~, ~ ) - 2(~, ~ + p) ~ I1~ + pll 2 if and only if ~/C R>. Then the assertion follows from Theorem 5.4.71 it is not hard to deduce Proposition 6.1 from Lemma 6.4, though in working out the details it is convenient to work with the tensor algebra of m rather than the symmetric algebra, as the appropriate quotients are easier to identify. This completes the proof of unitarity in the last place and with it the classification of unitarizable highest weight modules. Of course, far more detailed results are available [E J, E]. For example, one has [E J, 4.2] an explicit general formula for u i • i - 1 , 2 ,.. . , s, w h i c h f o r i - l g i v e s u I - 1 + ~ 1 IS1,~1 + 2(pc,/3 v ) - 0 - + p , /3v ), where/3 is the unique highest root and
S1,7 - {~ e Rc+ [ (T, .)iv)_ 1, (/3, 7 v) -- 2}. (This is empty in the simply laced case.) That N(A) is in fact the maximal submodule of N(A) for an arbitrary unitary parameter I results from [E] or [DES].
A. Joseph
82 Lecture 7
Q u a n t i z a t i o n of o r b i t a l v a r i e t i e s
Let V c n + be the closure of an orbital variety and I(V) C S ( n - ) its ideal of definition. As noted in Lecture 1, one may interpret the quantization of V as the construction of a highest weight module N(A) : )~ E b* with highest weight vector e~ of weight A such that gr A n n u ( , - ) e), = I(V).
(7.1)
A particularly intriguing aspect of (7.1) is the determination and meaning of ~ + p, which in general can neither be integral nor regular. Physicists may recognize ~ as the analog of the zero point energy that arises in the quantization of a Hamiltonian system. One may remark that (7.1)implies [J7, Section 8.2] that A n n N ( ~ ) i s completely prime. Thus, a positive answer to quantization would construct a large part if not all of Primc U(fl). As noted in Lecture 6, the preceding problem was completely solved in the case when N(~) can be a unitarizable module. Then, the possible values of ~ define the Wallach set. For fl = s[(n) : n _< 6, this problem was solved by E. Benlolo in her thesis. To appreciate the amount of work involved, we remark that for ~[(6) there are 76 orbital varieties that have to be individually quantized. An important result of hers [B] is that there are two of these orbital varieties in ~[(6) that cannot be made to satisfy (7.1) if N(~) is required to be simple, but which paradoxically do satisfy (7.1) for some nonsimple N(~). This introduces some obvious difficulties, in particular the significance of ~, which can now also serve for some other orbital variety. However, one can "canonically" replace ~ by the highest weight of the necessarily unique simple submodule of N(~) whose associated variety is V. A sophisticated approach to (7.1) would no doubt cunningly exploit the fact that V is Lagrangian and is even (see Lecture 8) the image under the moment map of a conormal to a B-orbit in G/B. At present, we do not even understand the possible nonintegrality of ~ in geometric terms. Thus, one can at present only propose the naive approach indicated in the following. The solution to (7.1) naturally decomposes into three stages. 1) Find a system of generators for I(V).
Chapter 2
Orbital Varieties
83
2) Find a left ideal J(V) of U ( n - ) such that gr J(V) = I(V). 3) Give U(n-)/J(V) a left U(g)-module structure. The work of Benlolo indicates that (1) has the following solution when ft = s[(n). Let M be the n x n matrix with 0 above the diagonal, 1 on the diagonal, and for i > j having eij C n- c S ( n - ) as its ijth entry. Then
I(V) should take the form gr(al, a 2 , ' " , an}, where the ai are among the minors of M. Notice that then I(V) contains (gr al, gr a 2 , " - , gr an), and we remark t h a t there was one case found by Benlolo in s[(6) where this inclusion was strict. In the preceding, we can also interpret the eij as elements of U ( n - ) . This makes it possible to write down elements bl, b2,..., bn C U ( n - ) as corresponding "minors," though there is an obvious and significant ambiguity in doing so. These elements should generate J(V) as a left ideal. Whereas one may anticipate no difficulty in checking that gr J(V) D I(V), equality should only be a deeper consequence of the involutivity of V. Finally, for (3) it is sufficient to linearly order the bi (possibly using degree) and to solve the system of equations i-1
x~bie,x = 0 mod
a E 7r, i = 1 , 2 , . . . , n , j=l
for A. Heuristically, this should be possible because V is B-invariant. It leads to a condition on (A, c~v) that if not empty should be (miraculously) independent of i. This point of view does not integral or regular, nor that U(n-)/J(V) be the unitary case, one finds that (/~,c~v) = 0 noncompact root one has -(A,c~ v) = m - 1,
imply that A + p be either a simple U(g)-module. In if c~ E 7rc, whereas for the where m is the size of the
minor t h a t (Lecture 4) is the unique highest weight vector of the ~-module generating J(V). It turns out that the use of q u a n t u m groups facilitates steps (1) and (2). First, it gives a natural interpretation of the mysterious M matrix. Second, no ambiguities arise in step (2). Finally, the formalism naturally generalizes to arbitrary ft semisimple. All this is now detailed, the notation being essentially that of [Jll]. Let Uq(ft) denote the Drinfeld-Jimbo quantization of the enveloping algebra U(ft) of a complex semisimple Lie algebra g. It is a Hopf algebra over C(q) that has essentially the same finite dimensional modules as U(t~).
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Set A - C[q,q-1], and identify C with A / ( q - 1}. Let UA(~) be the Asubalgebra of Uq(ft) generated by the obvious "canonical generators" of Uq(~). Up to a minor triviality, UA($)®A C identifies [Jll, 4.1.15] with
u(g).
One may adopt the construction of the algebra R[G] of regular functions on G to construct the Hopf dual of Uq(g) as follows. For each w 6 P ( R ) +, let Y(~) be the simple finite dimensional Uq($)-module with highest weight ~. For all ~ 6 Y(w)*, v e V(~), let c~,v be the element a ~-. ~(av) of Uq(9)*. Then the c~v generate a Hopf subalgebra Rq[G] of Uq(9)*. We write c~,v - c~,~ when ~ (resp., v) has weight # (resp., ~). Define an order relation on P(R) through # _> ~ if # - ~ 6 NTr. Set RA[G] - {c 6 Rq[G] I c(UA(g)) C A}. One easily checks that Corresponding to the triangular decomposition 9 - n + ® ~ ® n - , one has subalgebras Uq(n +) of Uq(9). Let 7 be an isomorphism of the additive abelian group P(R) to a multiplicative abelian group T. In what follows, we must assume that Uq(g) is its so-called simply connected version with torus T, that is, Uq(9) - Uq(n-)TUq(n+). The subalgebras Uq(n +) "Uq(n+)T are Hopf subalgebras. Writing UA(n +) -- UA(I~)CI Uq(n+), one has UA(n +) @A C - U(n +) by [Jll, 3.4.5]. The advantage of quantum groups arises from the Drinfeld isomorphism, first given in the context of topological Hopf algebras but reinterpreted in a purely algebraic context in [Jll, 9.2.12] as follows. Consider the map Rq[G] ---+Uq (b +)* defined by restriction, the kernel of which we denote by J ± . To describe its image, we remark (see, for example, [Jll, 9.2.10]) that there exists a nondegenerate skew-Hopf pairing ~: Uq (b-) × Uq (b +) C(q). This defines in particular an injection O"b H (a H ~(a,b)) of Uq(b-) into Uq(b+) *, which is a homomorphism of algebras and an antihomomorphism of coalgebras. The result in [Jll, 9.2.11] asserts that Im(~ ' - Im(Rq[G] ~ Uq(b+)*). This and a corresponding result with + interchanged eventually gives T h e o r e m 7.1
There is an isomorphism ~+" Rq[G]/J + ~ ~Uq(b+) of Hopf
algebras. In what follows, it is convenient to consider the algebra isomorphism • '" Uq(b-) ~ ~Rq[G]/J +, since this more closely corresponds to the vector space isomorphism b- ~ >(b+)* obtained from the Killing form and which
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allows one to consider S ( n - ) as the algebra of regular functions on n + and as the graded dual of U(n +) viewed as a graded algebra through weight space decomposition. Set [3~ - {~ E 13" ] (~, c~v) C Q, V c~ E 7r}. By adjoining nth roots of q to C(q), one may view A E ~ as the character on T given by T(p) H q(',a). From this, one may define a highest weight module Nq()~) of Uq(g). Such a module is given by a left ideal L of Uq (b-). Through (I)', one may identify L as a left ideal of Rq[G]/J +. Moreover, the latter identifies with the algebra of matrix coefficients c~,. in which the entries strictly above the diagonal (i.e., when p ;~ ~,) are set equal to zero. Again, the torus element c cO, , , must act by a scalar on the highest weight vector ex of Nq(A) (specifically, the scalar q - ( " ~ ) via [Jll, 9.2.11(ii)]). Thus, L contains the left ideal L~
of I~q[G]//J + generated by the c ~ , , - q-("'a)l • co C P(R) +, # C P(R). Consequently, the diagonal matrix coefficients specialize to 1 in the q ~ 1 limit. Finally, when G = SL(n), the algebra Rq[G] is generated by the cO , where w corresponds to the n-dimensional defining matrix coefficients c~,, representation. These matrix coefficients may be written a s Cij : i,j = 1, 2 , . . . , n and then, for an appropriate ordering, J + is generated by the Cij : j > i. For the k-fundamental weight a;k (corresponding to the kth exterior power of w), the matrix coefficients C~cOkcO ,YcOk k " x,y C W are the q u a n t u m k x k minors in the cij, and of course these specialize to ordinary minors in the q ~ 1 limit. From all this, we can understand that our previous conjectured description of I(V) has a more intrinsic formulation. Our program for quantizing an orbital variety closure V can be expressed as determining some A e 13~ and a left ideal Jq(V,A) of Uq(b-) containing J~ " - }-~,eP(R)Uq(b-)(T(p)- q(a'")l) such that 1) The Uq(b-)-module structure on Nq(A) := Uq(b-)/Jq(V,A) extends to a Uq(g)-module structure. 2) I(V) = g r ( J ~ ( V ) ® A C), where J'A(V) = rb'(Jq(V, A))A I m R a [ G ] . For (2), we note that (I)'(Uq(n-))N ImRA[G] = {c e O'(Uq(n-)) ] c(UA(n+)) C A} and so coincides with the A-subalgebra S A defined analogously to S + of [Jll, 3.4.5]. Thus, S A ®A C coincides with S ( n - ) viewed as the graded dual of U(n +) (see the paragraph following Theorem 7.1). Since ~ ' ( J ~ ) - L~, it follows that the (c~,u - 1) belong to J~A(V)®A C, which may therefore be viewed as an ideal of S ( n - ) , which is [~-stable but not necessarily graded for gradation by degree. Defining formal character
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ch as in [Jll, 3.4.7], one obtains from (2) that =
(s(.-)/i(v)).
(7.2)
On the other hand, we can define a second specialization J(V) of Jq(V,A) by setting J(V) = JA(V)®A C, where JA(V) = Jq(V,A) N UA(n-)T. It identifies with a left ideal of U ( n - ) . Moreover, N(A) := V ( n - ) / J ( V ) has by (1) the structure of U(g)-module of highest weight A, and ch Nq ( A ) = ch N(£).
(7.3)
The obvious diagram involving the preceding two specializations fails to be commutative. Yet (7.2) and (7.3) imply that PV(N(~,)) = PV. Now, s u m o f the characteristic polynomials of the orbital varieties occurring in V(N(A)), and so by their linear independence (Corollary 3.5)
PV(N(A)) is a
we deduce that V ( N ( A ) ) = V. Thus, gr J(V) C I(V), and equality holds by (7.2) and (7.3) again. Although (1) and (2) can be expected to essentially uniquely determine Jq(V, A) if it exists, we would clearly like to have a formula for it analogous to that for I(V). Toward this aim we note that ~' gives Uq(b-) a Uq(b +) bimodule structure! Then to satisfy (1) we need only require that O'(Jq(V, A)) be invariant with respect to the adjoint action of Uq(b+) (defined by (a,b) H axbS(a2) for all a E Uq(b+), b C Rq[G]/J +, where A(a) = al ® a2 in the sum convention of [Jll, 1.1.8] and $ is the antipode). This is the quantum analog of the orbital variety closure V being B-stable. Finally, ~'(Jq(V, A)) is a left ideal of Rq[G]/J + ~-~ Uq(b+) *, and so it is natural to think of it as an ideal of zeros of some subset. We already require it to contain L~ and to be ad Uq(b+)-stable. If V = V(w), one should therefore further require it to vanish on some quantum analog of n + n w(n+). This should be a subspace of Uq(n +) with basis formed by choosing for each 7 C R + n w - l R - a weight vector a~ of weight 7. Such vectors can be obtained using the Lusztig-Soibelman automorphisms [Jll, 10.2], but this procedure is not canonical. An alternative would be to make the choice specified by the conclusion of [J13, 4.5]. Finally, A must be chosen to ensure that Jq(V, A) is big enough to satisfy (2). Although this point of view may seem a little far-fetched, it is nevertheless true that the generic prime (two-sided) ideals of Rq[G] can be viewed as ideals of zeros [J14, Theorem 1].
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Exercises 1) Assume N(A) satisfies (7.1). Show that N(A) has a simple socle, say L(A'), and that V ( N ( ) ~ ) / L ( ) ~ ' ) ) C V ( N ( A ) ) . 2) Take 9 of type G2 with 7r - {c~,/3}, where c~ is the short root. Let V be the closure of the unique 3-dimensional orbital variety. Show that I ( V ) is generated by x_~ and by three further quadratic generators that span the unique 3-dimensional representation of M x M, where M =
CX-/3 0 CX-(c~+~) (~ CX-(2c~+~) 0 CX-(3c~+~) viewed as an [~® Cx_~ ® Cx~module. Let a-2z-2~ denote its highest weight vector. Solve the equation xf~a_2z_2~e), = 0 for (~, ~v) and hence determine the solution to (7.1) in
this case. Show that N(A) = L(A). This constructs a completely prime ideal, namely Ann L(A), the associated variety of which is the unique 6dimensional orbit. In this example, one fails [Z] to have an embedding U ( ~ ) / A n n N ( ~ ) ~ D ( V ( N ( ) ~ ) ) ) unless one localizes at the Ore set generated by the image of the highest weight vector.
Lecture 8
F u r t h e r results and p e r s p e c t i v e s
Take A - A° • i - 1, 2 , . . . , t, that is, in the Wallach set. One has [LS, Chapter IV for ~ classical; J10, 4.2 in general]. T h e o r e m 8.1
Ann L(A) is a maximal ideal.
This is false for arbitrary unitary parameters. It can happen that Ann L(A) is strictly contained in the augmentation ideal. E. Gvirsk (unpublished) classified all such occurrences for ~ = ~[(n). One has a simple sumciency [J10, 6.8] criterion for AnnL(A) not to be maximal, but this does not quite seem to be a necessary one. Take A in the Wallach set. That Ann L(A) is completely prime is an easy consequence of the fact that Annv(m-)e~ is prime. Again, this fails for arbitrary unitary parameters.
However, one does have the following
curious fact [J10, 5.16]. Theorem
8.2
A s s u m e )~ a unitary parameter, say )~ - ~- + u iT ~.
Then
Q(A) := A n n u ( m - ) L ( ~ ) is a prime ideal. As a fairly easy consequence, one can show [J10, 6.5] that the rank r k ( U ( g ) / A n n L ( A ) ) divides dim V(r), but it need not be 1 nor dim V(T).
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It can be 1 even if dim V(~-) > 1 by [J10, 8.91. A table describing Q(A) as a function of A is given in [J10, 7.13]. We remark that the proof of Theorem 8.2 is a rather delicate matter. The result is quite false for nonunitary parameters and especially those which are unitary for the compact real form. This seems to exclude any easy shortcuts. One has a remarkable surjectivity theorem [J7]. Call a simple highest weight module L quasi-induced if it is a submodule of some E ® M with E of finite dimension and M a simple highest weight module induced from a proper parabolic subalgebra p of 0- One may easily show that L, M have the same associated varieties, and the latter is in some obvious sense nontrivially induced. Thus, it is easy to find L that are not quasi-induced. For example, L(A) for A in the Wallach set is quasi-induced if and only if its associated variety is the nilradical of a parabolic subalgebra, and in 5[(n), for instance, this only happens when the noncompact simple root is either
OL1 o r Ctn_ 1. L e t
F(L) (resp., A(L)) denote the set of g (resp., n - )
locally finite elements of End L. T h e o r e m 8.3
Assume L is not quasi-induced. Then A ( L ) = F(L).
If Ann L is maximal, then F(L) can only differ from U ( g ) / A n n L if their rings of fractions differ. Using this, one may check that F(L(,~)) = U(O)/AnnL(A) for A in the Wallach set. On the other hand, in this case A(L()~)) is just the algebra of differential operators on V(Anngrj: L(A)). This gives the following result [J10, 4.5], due to T. Levasseur and J. T. Stafford for 0 classical [LS]. C o r o l l a r y 8.4
Except if 0 = s[(n) and a
=
o~ 1 o r
an-l,
one has
U(o)/L()~) ~, D [ P ( A n n g r y L ( A ) ) ] for all )~ in the WaUach set. A key step in the proof of Theorem 8.3 comes from the study of (9algebras, that is, U(0)-algebras that as modules belong to the (9 category. There are surprisingly few such objects, especially if one demands that the algebra in question be semiprime [J12, 5.6]. In the completely prime case, this can be generalized to the case when 0 is Kac-Moody as in what follows. Let m be a Lie algebra and A an associative algebra. We say that an algebra homomorphism ~: U(m)
~ End A is a Leibniz action of U(m) on
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A if ¢(x) E Der A for all x c m. By definition, this makes A a U(m)-algebra. We say that the action is nondegenerate if A m reduces to scalars. Take g a symmetrizable Kac-Moody Lie algebra with triangular decomposition = n + ® 0 ® n - . Let ac be the Chevalley antiautomorphism of g (defined on generators through crc(x+~) = xT~, V a C ~). Given M C Ob O, consider M* as a left s-module through ac and set 5M C {m C M* ] dim U(O)m < c~}. Then 5M is a U(~)-submodule and belongs to (.9.
Let ~p,¢1 be ~ equivariant nondegenerate Leibniz actions of U(n +) on S ( n - ) . Then there is an algebra isomorphism intertwining ~p, ~p' and commuting with O.
L e m m a 8.5
Proof: Let c" S ( n - ) ~ (E be the augmentation, and define a pairing U(n +) x S ( n - ) ~ C by (x,a) - c(¢(x)a). Then (xy, a) - c(~p(xy)a) (x,~p(y)a) for all x , y e U(a+), a e S ( n - ) . Clearly, (1, 1) - 1 and ( , ) respects weight space decomposition. Let ~ E Nrr be minimal such that some nonzero a_ z C S ( n - ) of weight - Z lies in k e r ( , }. Then Z > 0, and the foregoing formula shows that a_ z is invariant hence zero. Since d i m U ( n + ) ~ - d i m S ( n - ) _ z < c~, V Z e N~, one concludes that {, ) i s nondegenerate. Similar assertions hold for the pairing defined by (x, a ) ' - c(¢'(x)a). We may therefore define an [}-linear isomorphism ~a of S ( n - ) by (x, ~a(a)) = (x, a}'. Then, ( x , ' @ ( y ) ( ~ ( a ) ) ) - ( x y , ~ a ( a ) ) - (xy, a)' -- (x, ¢' (y)a)' - (x, ~a(~p'(y)a)), which shows that ~a intertwines the actions. Clearly, p(1) = 1. To establish ~a(ab) - ~(a)~(b) = 0, it is enough to show that the left hand side is annihilated by ¢(x) : x E n + when x, a, b are all weight vectors. This results from ¢(x), ¢'(x) being derivations and from ~ intertwining ~, ~ / u s i n g an obvious induction on weight. E3 Lemma 8.5 is needed to justify the claim just preceding (.) of [J13, 4.4]. Such an action obtains from, say, the conclusion of [Jll, 3.4.6], which holds without any assumptions on the Cartan matrix (in the sense of [Jll, 3.1.2]). However, we shall also want S ( n - ) to have a U(g)-algebra structure. For this recall that, after Kac, ~ admits a nondegenerate invariant bilinear form K that identifies (b-)* with b+ (this only needs the Cartan matrix
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to be symmetrizable [Jll, 4.2.2]). The Verma module M(0) of highest weight 0 is just U(g)/U(g)b + ~ U(n-). Since U(g)b + is a coideal and a left ideal, (U(g)/U(g)b)* is a subalgebra of U(g)* admitting an action of g on the right through the coproduct [Jll, 1.1.6] and hence by derivations. This gives M(0)* a U(g)-algebra structure. The submodule 6M(O) is also a subalgebra and identifies with Crc(S(n+)) - S ( n - ) as a U(O)-algebra. Finally, S ( n - ) n+ - H°(n+,6M(O))- H0(n-, M ( 0 ) ) - C by [Jll, 4.2.9]. This result can be improved as follows. By transposing the adjoint action of b+ on itself under K, one obtains a linear action of b+ on b- and hence an [3-equivariant Leibniz action of U(n +) on S(b-). For each positive root ,3~ let {a~} be a basis for n~- and {bJz} a basis for n-z. One checks that a}(bJ_z) - /36ij. Let I be a maximal ideal of S([3) not containing any positive root. Then the Leibniz action of U(n +) on S(b-) factors to an [3-equivariant Leibniz action of U(n +) on S ( n - ) " ~~ S(b-)/I S ( n - ) , in which the pairing n - × n + --+ C, defined by restricting ( , } of the lemma, is nondegenerate. From this, one easily checks that S ( n - ) n+ reduces to scalars. By the lemma, any such action is equivalent to the foregoing (linear) one. In particular, it follows that 6M(O) admits a subspace V isomorphic to n - as an O-module satisfying U(b +)V C V • C. Moreover, given any compatible nontrivial U(b+)-algebra structure on 6M(O) (that is, compatible with its U(g)-module structure and for which (6M(0)) "+ - C as an algebra), it follows from [Jll, 3.4.4(i)] that 6M(O) is isomorphic to S(V) as a U(b+)-algebra. The foregoing may be generalized as follows. Given 7r~ C 7r, let p~, b + be the parabolic subalgebra of g whose Levi factor r~, contains (3 and the root subspaces corresponding to R ~ := ZTr~ A R. Let m~+ be the nilradical of Pr', and set m~, - ac(m +,). Set M r , ( 0 ) " - M(O)/~-]~cr' M(-a), which is just the generalized Verma module obtained by inducing from the trivial one-dimensional pr,-module. Then 6Mr,(O) is isomorphic to S(m~,) as an [3-module and hence is actually a U(g)-subalgebra of 6M(O), which we can view as S(m;,). Again, (6Mr,(O))m+~' identifies with (Mr,(O)/m;Mr,(O))* and hence reduces to scalars. On the other hand, taking I to contain just the positive roots in R '+ gives an 0-equivariant Leibniz action of V(m+,) on S(m~,) with a nondegenerate pairing mr-, x mr+ --+ C. Then, by the +
analog of Lemma 8.5 with n + [3 replaced by mr, , rr,, we conclude that 6Mr,(O) admits a subspace V isomorphic to m~-, as an rr,-module and satisfying V(pr,)V C V ® C. Then, for any compatible nontrivial V(pr,)-
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algebra structure on 5M~,(O), we conclude from [Jll, 3.4.4(i)] that 5M~,(O) is isomorphic to S ( V ) as a U(p~,)-algebra. Let us now show that one may obtain a nondegenerate pairing rn~, x rn~+ --~ C from the adjoint action of rn~, on ~ (rather than the coadjoint action we have considered). This requires that the Cartan matrix be integrable in the sense of [J12, 3.1.2] and not just symmetric. Set f i t ; , - ad U(n-)(EE~c~E~\ ~, C x _ ~ ) c rn;,. Lemma
8.6
One has r~-~, - m-~, .
If not, there exists/3 E R + \ R I+ minimal and a nonzero vector bz E + of weight/3 such that K ( b z , ~n-~,) - O. Certainly,/3 ¢ 7r. Assume c~ E 7r 1TtTr~ Proof:
is such t h a t / 3 - ( ~ E R + \ R I+. Then, the invariance and nondegeneracy of K combined with the choice of/3 imply that (ad x_~)b z - 0. Setting b_ z (7~(bz) gives (ad x ~ ) b _ z - O. Let e0 be the canonical generator of M~,(0) of weight 0. The preceding property of b_ z implies that x ~ ( b _ z e o ) - 0 for all c~ E 7r. Then ( p - / 3 , p - / 3 ) - (p, p) by [J 1 l, 4.2.9], which is impossible by, say, [Jll, A.1.3]. [5] For each c~ E 7r \ 7r~, one may choose p _ ~ E 5M~,(O) of weight -c~ such that x ~ p _ ~ - K ( x _ ~ , x ~ ) . Take /3 E R + \ R '+, c~ G 7r\Tr'. One checks that K((ad a_(z_~))x_~,bz) for all a_(z_a) E n of weight - ( / 3 - a ) (ad a _ ( z _ ~ ) ) x _ ~ - ~ ( a _ ( z _ ~ ) p _ ~ ). -
L e m m a 8.7
(8.1)
and bz E rn~, + of weight /3. Set
n extends to a linear surjection of ~ - ~ 6 ~ \ ~, U ( n - ) p _ ~ onto
m~, s a t i s f y i n g / ( ( n ( P - Z ) , Proof:
- bza_(z_~)p_ ~
bz) - DZP-Z"
Suppose that ~--~6~\~' a - ( Z - ~ ) P - ~ - O, and set a-z--
E
(ad a - ( z - ~ ) ) x - a "
~E~\~'
The formula (8.1) shows that K(a_;~, bz) - 0 for all bz E m~+ of weight /3. Then a_z - 0 by the nondegeneracy of K. Hence ~ is a well defined map, which by Lemma 8.6 is surjective. By construction, it satisfies the required identity. [5] Let V I be an O-stable complement of ker ~. The identity of Lemma 8.7 and the nondegeneracy of K imply that we have an injection S ( V I)
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5M~,(O) of O-modules. Yet as [3-modules both sides are isomorphic to S(m~-,), and so S(V')--%,SM~,(O). Remark
It does not seem obvious that we can choose V = V'.
8.8 Let A be an O-algebra such that A n+ = C as an algebra. Then there exists 7r' C zr such that A ~_ 8M~,(O) as a U(9)-algebra.
Proposition
Proof:
One defines 7r' := {a e ~r ] A_a = 0}.
Since H o ( n - , S A ) =
C by the hypothesis, it follows that 5A is a quotient of M~,(0).
Then
A ~-~ 5M~,(O). Yet A contains V' in the conclusion of L e m m a 8.7, and so A ~ >SM~,(O). It remains to check that 5M~,(O) can admit only one (nontrivial) V(9)-algebra structure. In this it is enough to show that the V(p~, )algebra isomorphism ~: S(V) ~ ~SM~,(O) satisfies x _ ~ ( b _ ~ ) = ~(x_~b_~) for all a C 7r and all b_z E 5M~, (0) of weight -/3. This follows by the obvious induction on weights applying x~ : 7 C 7r to both sides and using that (SM~, (0))"+ reduces to scalars. D Remark. Exercise.
This gives an alternative proof of [J12, 3.6]. Let ~ be a nondegenerate O-equivariant Leibniz action of
V ( n - ) on S ( n - ) .
Let ¢ ' be an 0-equivariant action of U(n +) on S ( n - )
commuting with ~. Show that ¢~ is Leibniz. One may place the question of quantizing orbital varieties in a wider context as follows. Let 7) denote the (possibly twisted) sheaf of differential operators on the flag variety G / B . The Beilinson-Bernstein theory [BB] relates U(g)-modules with an appropriate central character to coherent 7)-modules on the flag variety G / B . This allows one to cut down to the family of U(9)-modules corresponding to holonomic/)-modules, which seem to be the most interesting.
Now the associated variety 1;(j£4) in
T * ( G / B ) of a coherent 7)-module A4 can be defined in an analogous way to the associated variety I;(M) C 9" of a U(9)-module M, using the filtration on differential operators defined by degree. As before, 1;(A/I) is homogeneous and involutive. By definition, A/i is holonomic if dim ~;(J~/[) = ! dim T* ( G / B ) - dim G / B Then 1;(A/I) must be the closure of the conormal C ( Y ) of some smooth subvariety Y C G / B . By definition, this is the subvariety of T * ( G / B ) with base Y such that the fiber over each y E Y is just the cotangent space T*y,G/B . Obviously, C(Y) has dimension n and is involutive and homogeneous. Let 7r:T*(G/B) ' 9" be the moment 2
"
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93
map. Expressing T*(G/B) as G XB n*, this is just the map (g, x) ~ gx defined by the (coadjoint) action of G on ~*. Suppose M is the U(~)module corresponding to M under the Beilinson-Bernstein equivalence. By a result of W. Borho and J.-L. Brylinski [BBr, Theorem 1.9c], one has
~(V(M)) = V(M). Holonomic D-modules arise in the study of (ft, K)-modules. Here, K may be (the complexification of) a maximal compact subgroup of (a real form of) G or, more generally, a closed subgroup of G for which K \ G/B is finite (for example, B). The classification of simple (ft, K)-modules in terms of holonomic 79-modules was indicated by A. A. Beilinson and J. Bernstein [BB]. For such a module, the variety Y is a union of K orbits in
G/B.
The foregoing discussion (see also [Vo3]) motivates the following general question. Let K be as before. Let Y be a K-orbit in G/B, and set V = 7r(C(Y)). Can one give R[V] a natural structure of an admissible (fl, K)-module of finite length? This generalizes our previous question when V is orbital. One may anticipate that this will not be too easy to solve.
C o m m e n t s and further reading The main content of Lectures 1-6 and 8 was derived from the author's papers given in the bibliography. Lecture 7 was partly based on [B] and some new results on quantum groups [Jll]. A further study of characteristic polynomials and cells is made in [BBM], [Lu], and IT]. For an introduction to enveloping algebras, see [D] and [Jan2]. For the orbit method, one may consult the introduction to [BBG], and for a further point of view on the geometric approach to representation theory, the book of N. Chriss and V. Ginzburg [CG]. The integration method used in Lecture 3 is described in [BGV] and has been used notably in [R] and [V] to obtain results of a similar nature. For recent results on completely prime ideals, see [MG]. For the combinatorics of orbital varieties in s[(n), see [Me2]. Finally, the primitive ideal theory of quantum groups is developed in [Jll].
Acknowledgments I should like to express my appreciation to Professors Bent Orsted and Henrik Schlichtkrull for organizing a delightful meeting in Sandbjerg
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in which I had the opportunity to give these lectures. I should also like to thank Dr. Anna Melnikov for useful comments on the manuscript. References
[BB]
A. Beilinson and J. Bernstein, Localisation de ~ modules, C.R. Acad. Sci. Paris (A) 292 (1981), 15-18.
[B]
E. Benlolo, Sur la quantification de certaines varidtds orbitales, Bull. Sci. Math. 3 (1994), 225-241.
[BGV] N. Berline, E. Getzler, and M. Vergne, Heat Kernels and the Dirac Operator, Springer-Verlag, Berlin, 1991.
[BE]
J.-E. Bjbrk and E. K. Ekstrbm, Filtered A uslander-Gorenstein rings, pp. 425-448 in Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Progress in Mathematics 92, Birkh£user, Boston, 1990.
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Chapter 3 Discontinuous Groups and Clifford-Klein Forms of Pseudo-Riemannian Homogeneous Manifolds TOSHIYUKI KOBAYASHI
University of Tokyo
CONTENTS
Introduction Clifford-Klein forms Discontinuous actions (discrete and continuous version) Calabi-Markus phenomenon Generalized Hirzebruch's proportionality principle and its application 5. Compact Clifford-Klein forms of non-Riemannian homogeneous spaces 6. Open problems Acknowledgment Postscript References 1. 2. 3. 4.
Introduction
Let G be a Lie group and H a subgroup. A Clifford-Klein form of the homogeneous manifold G / H is a double coset space F \ G / H , where F is a subgroup of G acting properly discontinuously and freely on G / H . For example, any closed Riemann surface M with genus _> 2 is biholomorphic to a compact Clifford-Klein form of the Poincar~ plane G / H PSL(2, IR)/SO(2). On the other hand, there is no compact Clifford-Klein form of the hyperboloid of one sheet G / H - PSL(2, R)/SO(1, 1). Even 99
T. Kobayashi
100
more, there is no infinite discrete subgroup of G that acts properly discontinuously on G / H (the Calabi-Markus phenomenon). We discuss recent developments in the theory of discontinuous groups acting on G / H where G is a real reductive Lie group and H a noncompact reductive subgroup. Geometric ideas of various methods together with a number of examples are presented regarding the fundamental problems:
Which homogeneous manifolds G / H admit properly discontinuous actions of infinite discrete subgroups of G ? Which homogeneous manifolds admit compact Clifford-Klein forms ? 1 1.1
Clifford-Klein forms Homogeneous manifolds First, we consider the setting:
GDH. Here, G is a Lie group and H is a closed subgroup of G. Then the coset space G / H equipped with the quotient topology carries a C ~ - m a n i f o l d structure, so that the natural quotient map
7~: G ---, G / H is a C ~ - m a p . We say G / H is a homogeneous manifold. 1.2
Clifford-Klein forms Second, we consider a more general setting:
FcGDH. Here, G is a Lie group, H is a closed subgroup, and F is a discrete subgroup of G. Then one might ask
Does the double coset space F \ G / H equipped with the quotient topology carry a COO-manifold structure so that the natural quotient map, w: G --, F \ G / H ,
Q u e s t i o n 1.1
Chapter 3 Discontinuous Groups
101
is a C~-map ? If yes, we say F\G/H is a Clifford-Klein form of the homogeneous
manifold G/H. Unfortunately, the action of a discrete subgroup F on G/H is not always properly discontinuous (see Definition 2.1) when H is not compact, and the quotient topology is not necessarily Hausdorff. Thus, we cannot always expect an affirmative answer to Question 1.1.
This is the main
difficulty of our subject. However, leaving this question aside for a moment, we first discuss the geometric aspect of Clifford-Klein forms in this section.
1.3
C l i f f o r d - K l e i n f o r m s f r o m the viewpoint of g e o m e t r y From the viewpoint of differential geometry, the important point in
considering a Clifford-Klein form F\G/H is that we have a local diffeomorphism p: G/H --. F\G/H, with the following commutative diagram: G
7r/ G/H
P
,r\G/H
Therefore, any G-invariant local structure (e.g., affine, complex, symplectic, Riemannian, or pseudo-Riemannian) on a homogeneous manifold G/H induces the same kind of structure on a Clifford-Klein form F\G/H. In other words, the double coset space F\G/H is a manifold enjoying the same local properties as G/H, as long as the discrete subgroup F allows an affirmative answer to Question 1.1. Conversely, let us start from a differentiable manifold M endowed with some local structure T and then explain how a Clifford-Klein form arises. Let M be the universal covering manifold of M. Then, the local structure T is also defined on M through the covering map M ~ M. We set G = A u t ( M , T ) " - {~ e Diffeo(M) • ~ preserves the structure T}. We note that G is the group of isometries if T is a Riemannian structure; it is the group of biholomorphic automorphisms if T is a complex structure. We fix a point o C M and write 5 E M for its image under the covering map M ~ M. Then, the fundamental group ~I(M, 5) acts effectively on
102
T. Kobayashi
M as the covering automorphism. We write F for the image of the injection 7]"I ( M , 6) ~
G - Aut(M, T). That is, F is a discrete subgroup of G in the
compact open topology, which is isomorphic to 71"I ( M , O). Then we have a natural diffeomorphism M _ F \ M . Assume G - A u t ( M , T )
is small enough to be a Lie group (see a
textbook by Shoshichi Kobayashi [Ko-S] for some sufficient conditions) and is large enough t o a c t transitively on M. Then, M is represented as a homogeneous manifold M ~ G / H with the isotropy subgroup H at o E M. Consequently, M is naturally represented as a Clifford-Klein form of G / H : M ~
7r1(M)~
GIH
Ir
M ~_ r \ G / H
We remark that Question 1.1 has automatically an affirmative answer in this case. We shall give a number of examples of simply connected manifolds M "~ G / H equipped with some local structure in Section 1.4; and Clifford-
Klein forms of these manifolds in Section 1.5. 1.4
E x a m p l e s of h o m o g e n e o u s m a n i f o l d s
First, we present some examples where G - A u t ( M , 7-) acts transitively on a simply connected manifold M, so that M is represented as a homogeneous manifold of G. E x a m p l e 1.2
Let
M " - R ~, 7"
the canonical affine connection V.
Then Aut(R n, V) _~ GL(n,R) ~< IRn is the affine transformation group, acting transitively on IRn by (a,b). x := ax + b
f o r ( a , b ) c G L ( n , R ) b
x C I R ~.
The isotropy subgroup at the origin 0 E R" is isomorphic to GL(n,R), and therefore the manifold R n is represented as the homogeneous manifold GL(n, R) ~< R n / G L ( n , IR).
Chapter 3
E x a m p l e 1.3
{
~r
:=
Discontinuous
103
Groups
Let
R ~ ,
T := standard Riemannian structure g.
Then A u t ( R n, g) _~ O(n) ~
for (a, b) E O(n) ~< IR~,
x C ]~n.
Thus, IRn is represented as the homogeneous manifold O(n) ~< R n / O ( n ) . E x a m p l e 1.4
Let
{ ~'i "-- {X C i~n+l : X2 ÷ . .
" ÷ X n2- - X n +2 1 = 1}
(n _> 3),
T " - the Lorentz metric g induced from dx~ + . . .
+ dx 2 -dX2+l
.
Let O(n, 1) be the indefinite orthogonal group preserving the quadratic form x~ + • • • + x n2_ Xn+2 1" Then O(n, 1) acts transitivelv~ and isometrically on the hyperboloid of one sheet M, and it coincides with the group of isometrics of M. Thus, A u t ( M , g) - O(n, 1) and M ~ O(n, 1 ) / O ( n - 1, 1). E x a m p l e 1.5
Let
M " - C, T"
the standard complex structure J.
Then the group of biholomorphic automorphisms is given by Aut(C, J) ~_ C x ~< C, a semidirect product of C x and C with C normal. The action of C x ~< C on C is given by the complex affine transformation: (a,b).z=az+b
for (a,b) E C x ~
This action is obviously transitive, so that the complex plane C is represented as the homogeneous manifold C x ~< C / C x .
T. Kobayashi
104
Example 1.6
{
Let
/ ~ := C P 1
(the complex projective space),
7" "- the standard complex structure J.
The natural action of SL(2, C) on C 2 \ {0} induces a transitive action of PSL(2, C) on the projective space C P 1 = (C 2 \ {0})/C × _~ CtJ{c~}, which is given by the linear fractional transformation:
(ab) c d
•z -
az+b cz+d
for
(a c
b)
d
e PSL(2, C) = SL(2, C ) / { ± I 2 } .
Then PSL(2, C) ~_ A u t ( C P 1 , J ) , the group of biholomorphic automorphisms. The projective space CP 1 is represented as a homogeneous manifold PSL(2, C)/B, where B is a Borel subgroup of PSL(2, C).
Example 1.7
{
Let
M "- 7 / 7" :
{z e C : Imz > 0}
(the Poincar4 plane),
the standard complex structure J.
Then the group of biholomorphic automorphisms is given by A u t ( ~ , J) _~ PSL(2, I~), where the action of PSL(2, I~) is also defined by the linear fractional transformation:
(ab) c d
•z -
az+b cz+d
for
(a c
d
e PSL(2, R) - SL(2, R ) / { ± I 2 } .
This action is transitive, so that the Poincar4 plane 7-/is represented as the homogeneous manifold PSL(2, I~)/SO(2).
1.5
Examples of Clifford-Klein forms Next, we present a number of examples of Clifford-Klein forms of the
homogeneous manifolds given in Examples 1.2-1.7. We also explain that these examples are closely related to the following interesting topics: Examples 1.8 and 1.9 Example 1.10 Example 1.11
The Auslander Conjecture on the fundamental group of compact complete affine manifolds The Calabi-Markus phenomenon for relativistic spherical forms The uniformization theorem of Riemann surfaces due to Klein, Poincar4, and Koebe
Chapter 3
Discontinuous Groups
105
E x a m p l e 1.8 (see Example 1.2) An affine manifold M is a manifold that admits a torsion free affine connection the curvature tensor of which vanishes. It is said to be complete if every geodesic can be defined on all time intervals. Then it is known that the universal covering of any complete affine manifold M = M n is isomorphic to (R n, V) as affine manifolds. Therefore, it follows from Example 1.2 that M can be represented as a Clifford-Klein form M _~ F \ G L ( n , ]~) ~< R n / G L ( n , ]1{), where F is a discrete subgroup of GL(n, ]1{) ~<]~n that is isomorphic to the fundamental group ~1 (M). E x a m p l e 1.9 (see Example 1.3) Retain the notation in Example 1.8. If M n is a Riemannian manifold and if M n is a complete affine manifold (see Example 1.8) for the Levi-Civita connection, then the universal covering of M is isometric to R n endowed with the standard Riemannian metric g. Therefore it follows from Example 1.3 that M is represented as another Clifford-Klein form
M _~ F \ O ( n ) ~, ]~n/O(n)
with F C O(n) ~< ]~n. Similarly, if M n is a Lorentz manifold (namely, M carries a pseudoRiemannian metric of type ( n - l , 1)), and if M is a complete affine manifold for the Levi-Civita connection, then M is reduced to be a Clifford-Klein form M _~ F \ O ( n with F C O ( n -
1, 1) ~< ] ~ n / O ( n - 1, 1)
1, 1) ~< R n.
R e m a r k (the Auslander conjecture) Regarding to Examples 1.8 and 1.9, we mention the Auslander conjecture, which asserts that the fundamental
group Th of any compact complete affine manifold is virtually solvable (see [Au], [Mi], [Ma], and references therein). In view of Example 1.8, this is equivalent to the conjecture that a discrete group F is virtually solvable if F \ G L ( n , 1~) ~( ]~n/GL(n, R) is a compact Clifford-Klein form of a homogeneous manifold GL(n,I~) ~< R n/GL(n,]~) _~ R n. Auslander's conjecture remains open except for some special cases such as F C O(n) ~
106
T. Kobayashi
also [To] for a generalization to rank one groups). The geometric meaning of F C O(n) ~ 3 with constant sectional curvature K = +1. Any relativistic spherical space form M is represented as a Clifford-Klein form: M _~ F \ O ( n , 1)/O(n - 1, 1), where F is a discrete subgroup of O(n, 1) that is isomorphic to the fundamental group 7rl(M). We shall see that F must be a finite group in Section 3 (the Calabi-Markus phenomenon [CM]). E x a m p l e 1.11 (the uniformization theorem of Riemann surfaces: see Examples 1.5-1.7) The uniformization theorem of Riemann surfaces due to Klein, Poinca%, and Koebe asserts that a simply connected Riemann surface M is biholomorphic to one of the following complex manifolds: C,
C P 1,
or 7-L
We recall that the Riemann mapping theorem is the special case obtained by assuming M to be a domain of C. We refer the reader to [Sp], [AhSa] for details. The point here is that any of simply connected Riemann surfaces C, C P 1, and 7-/ has a transitive transformation group of biholomorphic automorphisms, as we have seen in Examples 1.5, 1.6, and 1.7, respectively. Consequently, any connected Riemann surface M is represented as a Clifford-Klein form M ~_ r \ a / H ,
where ( G , H ) i s (C x ~< C, CX), (PSL(2, C),B), or (PSL(2, R),SO(2)), and F is a discrete subgroup of G that is isomorphic to 7rl(M). In particular, suppose M is compact. The rank of the first homology group H i ( M ) is always even, and it is denoted by 2g. Then, g is said to be the genus of the Riemann surface M, and such a Riemann surface is usually
Chapter 3 Discontinuous Groups
107
denoted by My. Then the universal covering of Mg is uniquely determined up to biholomorphic automorphism by the genus g. In fact, we have
Mg _~ F\PSL(2, C)/B,
r = ~ (M~) = {~}
(g = 0),
M~ ~_ r \ c
F = 7rl(Mg) -~ Z 2
(g = 1),
r = ~I(M~)
(g > 2).
x ~ C/C x ,
Mg _~ F\PSL(2, R)/SO(2),
2
Discontinuous actions (discrete and continuous version) Let G be a Lie group, H a closed subgroup, and F a discrete subgroup
of G. In Section 1, we have defined a Clifford-Klein form of a homogeneous manifold G/H as the double coset space F\G/H if it is Hausdorff and carries naturally a C ~ structure (see Question 1.1). This condition is satisfied if F acts properly discontinuously and freely on G/H (see Definition 2.1), as we shall see in Lemma 2.2. The purpose of this section is to understand properly discontinuous actions by exhibiting a number of typical bad features of the action of noncompact groups. 2.1
C l i f f o r d - K l e i n f o r m of PSL(2, R)/SO(2)
First, we recall Example 1.11. Let Mg be a closed Riemann surface with genus g > 2. The universal covering manifold of My is biholomorphic to the the Poinca% plane 7-/. So My is represented as a compact CliffordKlein form F\PSL(2, R)/SO(2) of the Poincar6 plane PSL(2, R)/SO(2), where F is an infinite discrete subgroup of PSL(2, R) isomorphic to 71"1 (Mg). There are two directions of generalization of this classical example, namely, i) generalization to higher dimensions (e.g., Theorem 5.2) ii) generalization to noncompact isotropy subgroups Here, we are interested in the latter direction. 2.2
C l i f f o r d - K l e i n f o r m of PSL(2,R)/SO(1, 1) There is no compact Clifford-Klein form of the hyperboloid of one
sheet PSL(2, R)/SO(1, 1). This fact can be proved in various ways, which
108
T. Kobayashi
are of importance as typical examples of later arguments. We indicate here four different proofs, which we shall elaborate in the following sections. i) A direct calculation (this section). Assume that F is a discrete subgroup of PSL(2, I~) such that ~F = co. Then we can show that the action of F on PSL(2, I~)/SO (1,1) is never properly discontinuous, and the corresponding quotient space F\PSL(2, I~)/SO(1, 1) is not Hausdorff. We consider first the most nontrivial case, where F consists of unipotent elements. For example, let us consider the case where F is a discrete subgroup of PSL(2, I~) generated by a single unipotent element. Then, the quotient topology of F\PSL(2,]~)/SO(1, 1) is locally Hausdorff but nonHausdorff, as we shall explain in this section (see Example 2.8 and Exercise 2.9(iii),(v)). On the other hand, if F does not contain a nilpotent element (still we assume ~F = c~), then the action of F has an accumulating point in PSL(2, R)/SO(1, 1) (see Definition 2.4), so that F\PSL(2, ]~)/SO(1, 1) is not Hausdorff, too (see Example 2.7(ii) and Exercise 2.9(iv),(v)). ii) Calabi-Markus phenomenon (Section 3). We will give a necessary and sufficient condition in terms of the ranks of G and H for a homogeneous space G/H of reductive type to admit an infinite discontinuous group (see Theorem 3.10 for a criterion of the so-called Calabi-Markus phenomenon; see also Example 1.10). The idea of proof is to study an analogous problem in a continuous setting and to look at the infinite points by means of the Cartan decomposition. iii) Hirzebruch's proportionality principle (Section 4). A CliffordKlein form M of PSL(2, R)/SO(1, 1), the hyperboloid of one sheet, carries an indefinite-Riemannian metric of type (1, 1) induced from the Killing form. This metric gives rise to a nonvanishing vector field on M. If M were compact and orientable, this would imply the vanishing of the EulerPoincar~ class of M thanks to a theorem of Poincar~-Hopf. On the other hand, the Euler-Poincar~ class of a "compact real form" S 2 ~_ SU(2)/SO(2) does not vanish. This leads to a contradiction by a generalized Hirzebruch's proportionality principle (see Corollary 4.6). The idea here leads to a necessary condition for the existence of compact Clifford-Klein forms (see Corollary 4.12 and Example 4.13). iv) Semisimple orbits, invariant comple¢ structure (Section 5). The hyperboloid of one sheet is realized as a semisimple orbit of the adjoint representation of PSL(2,1~). We shall see in Section 5 that a semisimple
Chapter 3
Discontinuous Groups
109
orbit having a compact Clifford-Klein form must be isomorphic to an elliptic orbit, so that it carries an invariant complex structure (see Corollary 5.13). But this is not the case for the hyperboloid of one sheet.
2.3
P r o p e r l y discontinuous action
A distinguished feature in the setting Section 2.2 is that the isotropy subgroup H _~ SO(l, 1) is noncompact. As a consequence, the action of a discrete subgroup F on G / H is not automatically properly discontinuous. Here, we recall the definition of properly discontinuous actions: Suppose that a discrete group F acts continuously on a locally compact Hausdorff topological space X. For a subset S of X, we put
r s := {~ E F : ~ S n S # O } . Note that if S is a singleton {p} (p E X), then Fs is nothing but the isotropy subgroup at p. In general, F s is not a subgroup.
Definition 2.1
The action of F on X is said to be
i) properly discontinuous, if Fs is a finite subset for any compact subset S of X; ii) free, if F{p} is trivial for any p E X. Then we have the following standard fact: L e m m a 2.2 Suppose that a discrete group F acts on a [C ~ , Riemannian, complex, ... ] manifold X properly discontinuously and freely [and
smoothly, isometrically, holomorphicaUy, ... ]. Equipped with the quotient topology, F \ X is then a Hausdorff topological space, on which a manifold structure is uniquely defined so that 7r : X ~ F \ X is locally homeomorphic [diffeomorphic, isometric, biholomorphic, ... ]. Although this lemma is well known, we give a sketch of its proof so that readers get used to the definition of properly discontinuous actions.
S k e t c h o f proof'. 1) (Hausdorff) (We use only the assumption that the action is properly discontinuous.) Take x, y E X so that 7r(x) ~: 7r(y).
110
T. Kobayashi
We want to find neighborhoods V, W of X such that x C V, y C W, and 7r(V) N 7r(W) = q). First, we take relatively compact neighborhoods V1 and W1 of X such that x C V1, y C W1. As the action of F on X is properly
discontinuous, Fvluw 1 -- {')/C F : ")/(V1 U W1)N (V1 U Wl) ~ 0} is a finite set, say, {~/1,..., ~/k}. Second, we take neighborhoods V and W such that x E V C V1, y E W C W1, a n d ~ j V n W =
0
(j = 1 , . . . , k ) .
Then, we
have FV N F W = 0, that is, 7r(V)n 7 r ( W ) = 0, which we wanted to prove. 2) (manifold structure)
The proof is quite similar to that of (1). In
fact, by using the assumption that the action is properly discontinuous and free, we can find a neighborhood V C X at each point x E X such that e r:-yv
n v
O} =
Such an open set V is homeomorphic to 7r(V) C F \ X , and these sets form a basis for the open sets giving local charts of F \ X .
[-1
Here is a necessary condition that the action of F on X is properly discontinuous. L e m m a 2.3
Suppose a discrete group F acts properly discontinuously on
a locally compact, Hausdorff topological space X . Then we have i) There is no accumulating point of the action of F on X . ii) Each F-orbit is a closed subset of X . Here we recall D e f i n i t i o n 2.4
For each element x of X, the F-orbit through x is a
subset of X given by r.x
:= { ' y x : 7
r}.
We say y (E X) is an accumulating point of the F-orbit F. x if e r.
=
for any neighborhood U of y in X. We say y (E X) is an accumulating point
of the action of F on X if there exists x E X such that y is an accumulating point of the orbit F . x.
P r o o f o f L e m m a 2.3:
i) We want to show that the action of F on X
is not properly discontinuous if there exists an accumulating point of the
Chapter 3 Discontinuous Groups action of F on X .
111
Suppose t h a t y E X is an a c c u m u l a t i n g point of the
action of F on X . T h a t is, there exists x E X such t h a t ~(U n F - x) = oc for any n e i g h b o r h o o d U of y. We choose U to be relatively c o m p a c t and define S := U U {x} and F' := {~/E F : ~x E U}. T h e n we have ~F' = oc, because ~(U n F - x ) = oc. T h e n we have ~Fs = co, because
Fs = {'y E F : "yS N S # O} D F ' . Hence, the action of F on X is not properly discontinuous. ii) We want to show t h a t the action of F on X is not p r o p e r l y discontinuous if t h e r e exists a nonclosed orbit F . x. Take y E F . x \ F . x and relatively c o m p a c t , open n e i g h b o r h o o d s Uj (j = 1, 2 , . . . ) of y such t h a t U1 D U2 D • .. and t h a t n j
uj
-
{y}. Then, we can take ~j E F such t h a t %j-x E Uj for
each j E N, because F . x n uj 7~ O. T h e n we have ~ {~j : j = 1, 2 , . . . } = oo, because n j u j - {y}. Finally, we p u t S " - Vl U{x}, and we have ~Fs - oc, because
U
c
aj} D
5 - 1, 2 , . . . } .
J H e n c e , the action of F on X is not properly discontinuous.
2.4
Discontinuous groups acting on homogeneous
W1
manifolds
A discrete s u b g r o u p F of G is said to be a discontinuous group acting on G/H if the action of F on G/H is p r o p e r l y discontinuous.
D e f i n i t i o n 2.5
If F is a discontinuous group acting on G/H and if the action of F on G / H is free, t h e n the double coset space F \ G / H carries a n a t u r a l C ~ manifold s t r u c t u r e from L e m m a 2.2, so t h a t F \ G / H is a Clifford-Klein form of the h o m o g e n e o u s manifold G/H.
Thus, we have an affirmative
answer to Q u e s t i o n 1.1 in this case. Here, the point is t h a t F is a s s u m e d to be a s u b g r o u p of G, so t h a t
F\G/H inherits
any G-invariant (local) s t r u c t u r e on the h o m o g e n e o u s m a n -
ifold G/H. One should keep in m i n d the essential difference b e t w e e n the action on G/H of a discrete group F C G (our case) and a discrete group F C Diffeo(G/H). For example, if G/H = SL(2, R ) / S O ( 1 , 1 ) (a hyperboloid of one sheet), t h e n G/H is diffeomorphic to S 1 x R (a cylinder),
T. Kobayashi
112
which admits a properly discontinuous action of Z ( C Diffeo(G/H)) along the direction of JR. But this action does not come from SL(2, R) and is not isometric with respect to a natural indefinite Riemannian metric on the hyperboloid. 2.5
A remark
In this section, proper discontinuity is essentially important, but freeness is less important. In fact, if a discrete group F acts on a manifold X properly discontinuously, then the isotropy group F{p} at p E X is indeed not necessarily trivial, but it is always finite. Correspondingly, F \ X is not necessarily a smooth manifold but still has a nice structure called
V-manifold in the sense of Satake [Sa] or called an orbifold in the sense of Thurston (see also [Car]). Moreover, if F acts properly discontinuously on X and if F' C F is a torsion free subgroup (i.e., x E F', x n = e (n > 1) =~ x = e), then the action of F ~ on X is properly discontinuous and free. In view of this, the following result due to Selberg is quite useful. Theorem
2.6 ([Sel, Lemma 8])
A finitely generated matrix group has a
torsion free subgroup of finite index. 2.6
Examples and exercises
E x a m p l e 2.7 Suppose a discrete group F : - Z acts on a manifold X :--IR in two different manners: i)
FxX---~X,
(n,x)~-.x+n.
ii)
F x X ~ X,
(n,x) ~ 2nx.
The action in (i) is properly discontinuous and free.
The resulting
quotient manifold F \ X is diffeomorphic to S 1. On the other hand, the action in (ii) is not properly discontinuous: in fact, F s = Z is not a finite set if we take S := [0, 1], the unit interval in R. The resulting quotient space F \ X has a non-Hausdorff topology. T h a t is, F \ X is homeomorphic to S 1 U {point} U S:, which is topologized to be connected! In Example 2.7(ii), we easily see that the action of F is not properly discontinuous because the origin 0 is an accumulation point of the action of F. The next example is more subtle, without accumulation points.
Chapter 3 Discontinuous Groups
113
E x a m p l e 2.8 Suppose a discrete group F := Z acts on a manifold X := I~2 \ {0} in the following manner:
(n, (x, y)) H (2nx, 2-ny).
FxX~X,
Then this action is not properly discontinuous. In fact, we let B~(x, y) be a ball of radius c with the center (x, y) and put S := B1/2(1, 0) t2 B1/2(0, 1). Then, it is an easy exercise to see that F s - {7 C Z : ~S A S # 0} is equal to Z. We note that there is no accumulation point of F. In fact every F-orbit is closed in X. The resulting quotient topology of F \ X is not Hausdorff, though it is locally Hausdorff in the sense that one can find a Hausdorff neighborhood of each point of F \ X . A picture of a similar topology as that of F \ X is illustrated by the following one-dimensional example:
E x e r c i s e 2.9
In the setting as in Example 2.8, prove the following:
i) F s = Z if S = B1/2(1, 0) U B1/2(0, 1). ii) Let G = SL(2, I~) act naturally on R 2 \ {0} from the left. This action is transitive, and the isotropy group at 1
(1)0 is
X
- {(01) so that we have a diffeomorphism:
G/H _~ ]t{2 \ {0}. Let 0
0)
y-1
"y>0
}
,
Lz'-
0
0)
2 -x
.xEZ
}
_~Z.
Via the isomorphisms G/H ~- ]~2 \ {0} and Lz "~ F (= Z), the action of Lz on G/H coincides with that of F on R 2 \ {0} in Example 2.8. Thus,
114
T. Kobayashi
the action of Lz on G/H is not properly discontinuous, and the quotient topology on L z \ G / H is not Hausdorff. iii) Let Hz'-{(0
1
x 1)"xEZ}.
Then the action of Hz on G/L is not properly discontinuous, and the quotient topology on Hz\G/L is not Hausdorff. iv) The action of Lz on G/L has an accumulating point and is not properly discontinuous, either.
v)
We define a quadratic form on R 2 b y Q ( X ) "-xy. The polarization \ Y/ of Q gives an indefinite metric of signature (1, 1). Then L is the identity component of
Thus, G/L ~_ SL(2, R)/SO0(1, 1). Hint: The topology whereas that of Lz\G/L and in (iii) are in a kind that of H on G/L. That
of Hz\G/L has similar features as Example 2.8, is similar to Example 2.7(ii). The actions in (ii) of duality, between the action of L on G/H and is, in the setting Lz c L c G D H D Hz
with L/Lz and H/Hz compact, the action of Lz on G/H is properly discontinuous if and only if that of Hz on G/L is properly discontinuous (see Lemma 2.16(i); see also Observation 2.12 and [Bou, Chapitre 3]). 2.7
B a s i c p r o b l e m s in a d i s c r e t e s e t t i n g Here are the basic problems in the theory of discontinuous groups
acting on a homogeneous manifold: P r o b l e m 2.10
Suppose G is a Lie group and H is a closed subgroup.
i) Which homogeneous manifold G/H admits an infinite discontinuous group ? ii) Which homogeneous manifold G/H admits a compact Clifford-Klein form?
Chapter 3
Discontinuous Groups
115
(i) is a first step in the study of discontinuous groups acting on homogeneous manifolds. The existence of a compact Clifford-Klein form in (ii) would be interesting not only from the viewpoint of geometry but also from that of harmonic analysis and representation theory (see Open problem (x) in Section 6). We will study (i) in Section 3, and (ii) in Section 5 (partly also in Sections 3 and 4). 2.8
as a c o n t i n u o u s a n a l o g of p r o p e r l y
Proper actions
discontinuous actions
In general, the study of a discrete group is quite difficult. Our approach is to approximate the action of discrete groups by that of connected Lie groups.
For this purpose, it is crucial to find a continuous analog of a
properly discontinuous action: D e f i n i t i o n 2.11 (see [Pal)
Suppose that a locally compact topological
group L acts continuously on a locally compact topological space X. For a subset S of X, we define a subset of L by
Ls = {'7 C L : "yS N S ~ O}. The action of L on X is said to be proper if and only if Ls is compact for every compact subset S of X. Compared with the definition of proper discontinuity (Definition 2.1), compactness in Definition 2.11 has now replaced by finiteness. We note that the action of L on X is properly discontinuous if and only if the action of L on X is proper and L is discrete, because a discrete and compact set is finite. 2.9
An observation
The following elementary observation is a bridge between the action of a discrete group and that of a connected group.
Suppose a Lie group L acts on Let F be a cocompact discrete subgroup of L.
O b s e r v a t i o n 2.12 ([Ko2, Lemma 2.3])
a locally compact space X . Then
1) The L-action on X is proper if and only if the F-action is properly discontinuous. 2) L \ X is compact if and only if F \ X is compact.
116
T. Kobayashi
P r o o f : 1) Suppose F acts properly discontinuously. Take a compact subset C in L so that L = C . F and C = C -1. Then, for any compact subset S in X, Ls = {g E L : g. S N S ~ O} C C . F c s . Because F c s is a finite set from the assumption, we conclude that the action of L is proper. In view of F s = F A Ls, the "only if" part follows immediately from the definition. 2) Suppose L \ X
is compact. We take an open covering X = ~ s Us by
relatively compact sets Us. Then X - ~Js LUs gives an open covering of
L \ X . By the compactness of L \ X , we can choose finitely many Uj among {Us} so that X - Uj L . Uj - U j
(F.
C ) . Uj - U j F . ( C . Uj),
F \ X is compact. The converse statement is clear. 2.10
P r o b l e m s in a c o n t i n u o u s
showing
[3
analog
In view of Observation 2.12, we pose the following analogous problems in a continuous setting. P r o b l e m 2.13
Let G be a Lie group and H and L closed subgroups.
1) Find the criterion on the triplet (L, G, H) such that the action of L on G / H is proper. 2) Find the criterion on the triplet (L, G, H) such that the double coset L \ G / H is compact in the quotient topology. We will give a complete answer to Problem 2.13 in terms of Lie algebras in the following cases: i) Problem 2.13(1) when G is reductive (see Section 2) ii) Problem 2.13(2) when the groups G, H, L are real reductive (see Section 5). In preparation for the next section we introduce here some notations that are useful for a further study of Problem 2.13(1). 2.11
Relations ~ and
Suppose that H and L are subsets of a locally compact topological group G. D e f i n i t i o n 2,14 ([Kol0, Definition 2.1.1])
We denote by H ~ L in G if
there exists a compact set S of G such that L C S H S -1 and H C S L S -1. Here, S H S -1 "- {ahb -1 E G" a, b 6 S, h C H } . Then the relation H ~ L in G defines an equivalence relation.
Chapter 3
Discontinuous Groups
117
We say the pair ( H , L ) is proper in G, denoted by H rfi L in G, iff S H S -1 n L is relatively compact for any compact set S in G. Definition 2.14 is motivated by the following: 2.15 Let H and L be closed subgroups of G, and F a discrete subgroup of G.
Observation
1) The action of L on the homogeneous manifold G / H is proper if and only if H rh L in G. 2) The action of F on the homogeneous manifold G / H is properly discontinuous if and only if H rh F in G. Here are some elementary properties of the relations ~, rfi : L e m m a 2.16 ([Kol0]) Suppose G is a locally compact topological group and that H, H t and L are subsets of G.
i) H ~ L if and only if L ~ H. ii) If H ~ H' and if H rfi L in G, then H' rh L in G. Now we are ready to give a reformulation of Problem 2.13(1) as:
Problem 2.17 (a reformulation of Problem 2.13(1))
Let G be a Lie group, and H and L subsets of G. Find the criterion on the pair (L, H) (or on the pair of their equivalence classes with respect to ~ ) such that L dl H in
G.
2.12
Property (CI)
If a discrete group F acts on X properly discontinuously, then every isotropy subgroup is finite and every F orbit is closed (Lemma 2.3). The latter condition corresponds to the fact that each point is closed in the quotient topology of F \ X . In general, the converse implication does not hold (cf. Example 2.8). We have a similar picture in a continuous setting. In fact, let H, L be closed subgroups of a locally compact topological group G. If L acts properly on G / H , then any L-orbit L g H "~ L / L N g H g -1 C G / H is a closed subset, and each isotropy subgroup L n gHg -1 is compact. In general, the converse implication is not true. However, we focus our attention on the latter property, that is, each isotropy subgroup is compact.
118
T. Kobayashi
D e f i n i t i o n 2.18 ([Ko4, Ko6]) Suppose that H and L are subsets of a locally compact topological group G. We say that the pair (L, H) has the property (CI) in G if and only if L n gHg -1 is compact for any g E G. Here (CI) stands for that the action of L has a compact isotropy subgroup L n g H g -1 at each point gH E G / H , or stands for that L and gHg -1 has a compact intersection (g E G) (see also [Lip]). If H M L in G, then the pair (L, H) has the property (CI) in G. The point is to understand how and to what extent the property (CI) implies the proper action. P r o b l e m 2.19 ([Ko6, Problem 2]; see Open problem 6(ii))
For which
Lie groups, does the following equivalence (2.1) hold? H M L in G e, the pair ( L , H ) has the property (CI) in G.
(2.1)
In the next section, we shall see that the equivalence (2.1) holds if G, H, L are real reductive algebraic groups (see Theorem 3.18). R. Lipsman [Lip] has pointed out that it is likely that the equivalence (2.12.3) holds as well if G is a simply connected nilpotent group. There are some further cases, especially in the context of a continuous analog of the Auslander conjecture (see the remark following Example 1.9), where the equivalence (2.1) is known to hold. See Example 5 and Proposition A.2.1 in [Ko4, Ko6]; Theorem 3.1 and Theorem 5.4 in [Lip]. However, we should note that the equivalence (2.1) does not always hold. For instance, if G = K A N is an Iwasawa decomposition of a real reductive group G and if we put L := A and H := N, then (L, H) has the property (CI) in G, while L ~ H in G as we saw in Exercise 2.9 when G = SL(2, R).
3 3.1
Calabi-Markus phenomenon E x i s t e n c e of infinite d i s c o n t i n u o u s g r o u p s
Let G be a Lie group and H a closed subgroup. In the section, we focus our attention on the first basic problem:
Which homogeneous manifold G / H admits an infinite discontinuous group?
P r o b l e m 3.1 (see Problem 2.10(i))
We first note that if M is a compact manifold, then there is no infinite discrete group (C Diffeo(M)) acting on M properly discontinuously. So, we
Chapter 3
Discontinuous Groups
119
might expect that the answer to Problem 3.1 should be related to certain aspect of compactness of a homogeneous manifold G / H with respect to the transformation group G C Diffeo(M). One observes that a criterion given in the reductive case (Theorem 3.6) has such an aspect. Our main interest here is the (non-)existence of an infinite discontinuous group in the reductive case. However, we include a quick review for some other typical cases as well.
i)
relativistic spherical space form (Section 3.2),
ii)
G / H with H compact (Section 3.3),
iii)
G / H with G solvable (Section 3.4),
iv)
G / H with G reductive (Section 3.5- Section 3.10).
3.2
R e l a t i v i s t i c s p h e r i c a l s p a c e form In 1962, Calabi and Markus discovered a surprising phenomenon on
the fundamental group ~1 of a Lorentz manifold with constant curvature"
Every relativistic spherical space form is noncompact and has a finite .fundamental group 7q (see Example 1.10 for definition). T h e o r e m 3.2
([CM])
As we saw in Example 1.10, the Calabi-Markus theorem is reformulated in a group language as"
If n > 3, then there does not exist a discrete subgroup F of O(n, 1) acting on O(n, 1 ) / O ( n - 1, 1) properly discontinuously and freely such that the fundamental group ~ I ( F \ G / H ) is infinite.
T h e o r e m 3.3
We note that if n - 2, then G / H is diffeomorphic to a cylinder and has the fundamental group ~ Z. Theorem 3.3 is also reformulated as
Any discontinuous group acting (see Definition 2.5) on a homogeneous manifold SO(n, 1 ) / S O ( n - 1, 1) (n > 2) is finite.
T h e o r e m 3.4
This result will be generalized in Section 3.5 for homogeneous spaces of reductive type (Definition 3.12). Setting the main subject aside for a moment, we mention a related problem in non-Riemannian geometry based on the following observation:
T. Kobayashi
120
i) Any complete Riemannian manifold M n (n ~_ 2) with constant sectional curvature +1 is compact with finite fundamental group. ii) Any complete Lorentz manifold M n (n >_ 3) with constant sectional curvature +1 is noncompact with finite fundamental group.
O b s e r v a t i o n 3.5
We recall again in a group language that M _~ F \ O ( n + 1)/O(n) in the first case, whereas M _~ F \ O ( n , 1 ) / O ( n - 1, 1) in the second case. A classical theorem due to Myers may be interpreted as a "perturbation" of Riemannian metric in the first statement:
If the Ricei curvature of a complete Riemannian manifold M satisfies K(U, U) >_ c > 0 for all unit vectors, then M is compact with finite fundamental group.
Theorem
3.6 ([My])
The author does not know a result concerning a "perturbation" of the Lorentz metric in Observation 3.5(ii). So we pay an attention on the following problem in pseudo-Riemannian geometry:
Suppose M is a complete Lorentz manifold (or more generally a complete manifold equipped with an indefinite Riemannian metric). Find a sufficient condition in terms of a local property of M that assures that M is noncompact with a finite fundamental group. In particular, is there a sufficient condition given by some positiveness of the curvature of M?
P r o b l e m 3.7
3.3
H is c o m p a c t Suppose that H is compact.
Then it follows immediately from the
definition of properly discontinuous actions that The action of F on G / H is properly discontinuous ¢=~ F is a discrete subgroup of G. Moreover, if F is a torsion free discrete subgroup of G, then the action of F on G / H is properly discontinuous and free whenever H is compact. Therefore, we have the following:
If G is a noncompact connected linear Lie group and H is a compact subgroup, then there always exists an infinite discrete subgroup F of G acting on G / H properly discontinuously and freely on G / H . P r o p o s i t i o n 3.8
Chapter 3 Proof:
Discontinuous Groups
121
It suffices to show t h a t any noncompact linear Lie group G con-
tains an infinite torsion free discrete subgroup. Assume first t h a t G is a noncompact semisimple Lie group; we write
G - K A N for an Iwasawa decomposition. We note t h a t G is noncompact if and only if I~-rank G - dim A > 0. So we can take a lattice of A, which is isomorphic to %dim A and is an infinite and torsion free discrete subgroup of G. Assume second t h a t G is a noncompact solvable Lie group. Let G be the universal covering group of G, and Z the kernel of the covering map G --. G.
It follows from Theorem 1 and R e m a r k of [Ch] t h a t we find
(0 _<) r _ n - dim G and a basis X 1 , . . . , Xn of the Lie algebra of G which has the following properties: i) R n -~ G,
(tl,...,tn)
H
exp(tlX1)...exp(tnXn)
is a surjective
homeomorphism. ii) Z is isomorphic to the free abelian group generated by the elements exp X 1 , . . . , exp Xr. We note t h a t G is noncompact if and only if r < n. Therefore, if G is noncompact, we put F " - { e x p ( n X r + l ) E G ' n
E Z}, which is an infinite
and torsion free discrete subgroup of G. Now, we suppose G is a noncompact connected linear Lie group. Let g -
[ + ~ is a Levi decomposition, where ~ is the radical of 9 and [ is a
semisimple Lie algebra. We write L, S for the analytic subgroups of G with Lie algebra [, ~, respectively. Then, L and S are closed subgroups of G. If G is noncompact, then L or S must be noncompact. In either case, we can find an infinite and torsion free discrete subgroup of G. 3.4
[3
Solvable groups
In the case of simply connected solvable homogeneous spaces, it turns out t h a t the C a l a b i - M a r k u s phenomenon does not occur. The following result is proved based on a structural result of a simply connected solvable groups due to Chevalley [Ch].
Suppose G is a solvable Lie group and H is a proper closed subgroup of G. Then there exists a discrete subgroup F of G acting on G / H properly discontinuously and freely such that the fundamental group Trl ( F \ G / H ) is infinite. Theorem
3.9, ([Ko8, T h e o r e m 2.2]; see also [Lip])
T. Kobayashi
122 3.5
Reductive
cases
The Calabi-Markus phenomenon in the reductive case has been studied by Calabi, Markus, Wolf, Kulkarni and Kobayashi, and it has been settled completely in terms of a rank condition:
Let G / H be a homogeneous space of reductive type. The following statements are equivalent: T h e o r e m 3.10, ([CM; Wol; Wo2; Wo3; Zu; Zo2])
1) Any discontinuous group acting on G / H is finite. 2) •-rank G = R-rank H. We will explain some terminology used here in Section 3.6 Section 3.6, and then give some examples and a sketch of the proof in Sections 3.7 and 3.8. 3.6
Homogeneous
spaces of reductive type
We set up some notation of real reductive groups.
Good references
for what we need are [He, Chapter VII and [War, Chapter I]. Let G be a connected real linear reductive Lie group. Fix a maximal compact subgroup K in G. We write ~ = ~ + p for the corresponding Cartan decomposition of the Lie algebra of G. Then, the homogeneous space G / K equipped with a G-invariant Riemannian metric is said to be a Riemannian symmetric space. We fix a maximal abelian subspace a C p. Then, a is said to be a maximally split abelian subspace
for G. If we want to emphasize the group G, we write aG for a. We write A for the corresponding connected Lie subgroup. We define E(~, a) : the restricted root system, Wc : the Weyl group associated to E(~, a), rank G := the dimension of any maximal semisimple abelian subspace of 0, c-rank G := rank K, ]R-rank G := dim a = rank G / K ,
d(G) := dimp = d i m G / K . We note that c-rank G _< rank G >__ JR-rank G <_ d(G).
Chapter 3 Example
3.11
Let G -
Discontinuous Groups
123
SO0(p, q) (p > q) be the identity component of
the indefinite orthogonal group of signature (p,q). Then K _~ SO(p) x SO(q) ' A -~- R q ' c - r a n k G [~] + [~] ' rank G [P+ql R-rankG - q' 2 J' d(G) - pq, and Wa ~ ®q ~ (Z//2Z) q. Here, Gq denotes the qth symmetric group. D e f i n i t i o n 3.12
Suppose that H is a closed subgroup in G with at most
finitely many connected components. If there exists a C a r t a n involution of G that stabilizes H, then H is said to be reductive in G and G / H is said to be a homogeneous space of reductive type. If G / H is a homogeneous space of reductive type, then G and H have a realization in GL(n, N) such that H C G C GL(n, R) are closed subgroups and that H - tH and G - tG. Here tG "- { t g . g E G}, the transposed set of G in GL(n, R). Example
3.13
Suppose G is a real reductive linear group.
i) If H is compact, then G / H is a homogeneous space of reductive type. ii) ([Yo]) If H is semisimple, then G / H is a homogeneous space of reductive type. iii) Let a be an involutive automorphism of G. If H is an open subgroup of G ~ " - {g C G" c r g - g}, then G / H is a homogeneous space of reductive type. The homogeneous space G / H is said to be a reductive symmetric space. If G is semisimple, G / H is said to be a semisimple
symmetric space. iv) If X E 1~ is a semisimple element, namely, ad(X) E End(g) is semisimple, then the semisimple orbit G/ZG(X) "~ A d ( G ) . X (C g), where Z c ( X ) "- {g C G ' A d ( g ) X - X}, is a homogeneous space of reductive type. v) ([Mos]) If Go D G1 D . . . D Gn and if Gi-1/Gi (1 _< i <_ n) are all homogeneous spaces of reductive type, then so is Go/Gn. Suppose G / H is a homogeneous space of reductive type.
There is
a nondegenerate Ad(H)-invariant bilinear form B on g that is positive definite on p and negative definite on 2. (Actually, there exists B t h a t is Ad(G)-invariant.) nondegenerate.
Because b is 0-stable, the restriction of B to ~ is also Therefore, B induces a nondegenerate Ad(H)-invariant
bilinear form g/[~ ~ To(G/H), the tangent space of G / H at o - e H
C G/H.
T. Kobayashi
124
It induces a G-invariant (indefinite-)Riemannian metric on G / H by the left translation. The signature of this metric is (dim(p/p N O), dim(~/~ N O)). In analogy with the polar coordinate in the Euclidean space R n , there is a polar coordinate in a Riemannian symmetric space G / K .
In group
language, this means that a real reductive linear Lie group G has a Cartan decomposition
G = KAK. In this decomposition, there is an element a(g) E A, unique up to conjugation by WG, such that g C K a ( g ) K for each g 6 G. D e f i n i t i o n 3.14 ([Ko2], [Kol0])
For each subset L of G, we define:
A(L) := A N K L K = {w. a(g) : w e W a , g e L} C A, a(L) := logA(L) C a. Here, log: A ~ a is the inverse of the diffeomorphism exp: a ~ A. If L is a subgroup of G that is reductive in G, then we can take a maximal compact subgroup K of G such that L N K is also a maximal compact subgroup of a reductive Lie group L. Let aL be a maximally split abelian subspace for L. Then there exists an element g of G such that
Ad(g)aL c aG. Then a(L) is a finite union of subspaces in aG: a(L) = W a . Ad(g)aL C aa. In this case, the notation a(L) coincides with that in [Ko2] as a subset of
at~We. Remark
Some remarks are in order.
i) If G / H is a homogeneous space of reductive type, then both G and H are real reductive Lie groups. Note that the converse statement is not always true.
ii) We avoid the terminology reductive homogeneous space,
which is usually used in the following sense: the Lie algebra g may be decomposed into a vector space direct sum of the Lie algebra ~ and a H-stable subspace m (see, for example, [KoN, Chapter X, Section 2]). This notion is wider than that of homogeneous spaces of reductive type in Definition 3.12. In particular, neither G nor H itself is required to be reductive in this usual definition of a reductive homogeneous space.
Chapter 3 Discontinuous Groups 3.7
E x a m p l e s of t h e C a l a b i - M a r k u s
125
phenomenon
We present here some examples of Theorem 3.10. E x a m p l e 3.15 Let G/H = SO(p + 1, q)/SO(p,q). Then R - r a n k G = min(p + 1, q) and R-rank H = min(p, q). Therefore, we have there is no infinite discontinuous group acting on ¢~
R-rank G = R-rank H
ca
q<_p.
G/H
The case q = 1 is the original result of Calabi-Markus (see Theorem 3.4). E x a m p l e 3.16
i)Any para-Hermitian symmetric space does not admit
an infinite discontinuous group (see [Lib] for the definition of para-complex structure; see also [KaKo] for the definition and a classification of irreducible para-Hermitian symmetric spaces), ii) Any hyperbolic orbit does not admit an infinite discontinuous group. Here, we say a semisimple orbit G/Za(X) ~- A d ( G ) X (see Example 3.13(iv)) is a hyperbolic orbit if X C g is a hyperbolic element, that is, if all eigenvalues of ad(X) E End(fl) are real. We note that a para-Hermitian symmetric space is a reductive symmetric space that can be realized as a hyperbolic orbit. E x a m p l e 3.17
There does not exist an infinite discontinuous group act-
ing on the following homogeneous manifolds: i) reductive symmetric spaces GL(n,
C)/GL(n,R),
SO(2n + 1,C)/SO(2n,
C),
U(m,n)/O(m,
n).
ii) para-Hermitian symmetric spaces
Sp(n,R)/GL(n,R),
SO*(nn)/SU*(2n) x R.
iii) hyperbolic orbits GL(nI + . . . +
nk,R)/GL(nl,R) x . . . x GL(nk,R).
On the other hand, there exists a discontinuous group that is isomorphic to Z n acting on the following homogeneous manifolds:
Sp(2n, R)/U(n,n),
n)
GL(2n, IR)/GL(n, C),
(n _<m).
T. Kobayashi
126 3.8
S k e t c h of p r o o f of t h e c r i t e r i o n for C a l a b i - M a r k u s First we explain a continuous analog, that is, Problem 2.13(1) for
the criterion for a proper action in a general reductive setting (Theorem 3.18). The criterion for the Calabi-Markus phenomenon (Theorem 3.10) is obtained as a very special case of the continuous analog, combined with Observation 2.12. 3.9
C r i t e r i o n of p r o p e r a c t i o n s in a c o n t i n u o u s s e t t i n g
Let G / H , G / L be homogeneous spaces of reductive type. Then the following five conditions are equivalent:
Theorem
3.18 ([Ko2])
1) L acts on G / H properly. 1)~ H acts on G / L properly.
1)" H ~ L
inG.
2) The triplet (L, G, H) has the property (CI). That is, for any g C G,
L N g H g - 1 is compact. 2)' a(L) N a(H) = {0}. See Section 2.11 for the definition of ~ , Section 2.12 for the property (CI), and Definition 3.14 for a(L). In the theorem, (1) w (1)' ¢:~ (1)", (2) ~:v (2)' and (1) =~ (2) are easy. The nontrivial part is the implication (2) =v (1). The proof of (2) =~ (1) is divided into two steps: i) Reduction to the case where H and L are abelian. This is an easy step that can be proved by using the Caftan decomposition. ii) Proof of the abelian case. This is done by looking at the infinite points in a Riemannian symmetric space based on some structural results on parabolic subgroups and nilpotent elements. 3.10
C r i t e r i o n of p r o p e r a c t i o n s in a d i s c r e t e s e t t i n g Similar techniques lead us to a generalization to the case where H and
L are not necessarily reductive: Theorem
3.19 ([Kol0])
Let H , L be subsets of a real reductive linear
Lie group G. i) H ~ L in G ¢v a(H) cfi a(L) in a. ii) H-.~ L in G ¢=~ a ( H ) ~ a ( L ) in a.
Chapter 3
Discontinuous Groups
127
If H and L are reductive in G, then it is easy to see t h a t Theorem 3.19(i) implies Theorem 3.18. If G = GL(n,N) and H = G L ( m , N ) , then Theorem 3.19(i) implies a result of Friedland ([Fr, Theorem 3.1]). For more details, we refer to [Ko2], [Kol0]. Example
3.20
Let G -
SO(2m, 2n), L -
SO(p,q) ~ 12m-p x SO(p,q) x 12n-q C G.
U(m,n)
C G, and H -
Here, we assume 0 < q <
p, 0 < n < m, p < 2m, and q < 2n for simplicity. Then, with a suitable coordinate, we can identify aa with N 2n and WG ~ G2, ~< (Z/2Z) 2n in GL(aG). Up to the conjugacy by an element of WG, we have
a(L) - { ( a l , a l , a 2 , a 2 , . . . , a n , a n ) a(H) - { ( b l , b 2 , . . . , b q , O , . . . , O ) ' b i
" ai E R (l _< i <_ n)}, C ]R (1 __ i _< q)}.
So the condition (2)' in Theorem 3.18 amounts to q -
1. Therefore, we
conclude that U(m, n) acts on SO(2m, 2n)/SO(p, q) properly if and only if q-1. Exercise
3.21
Let G / H = SO(2p, 2q)/U(i,j)
(i < p, j < q, i < j).
Prove that if i < min(p, q) then there exists a discontinuous group F (C G) acting on G / H such that F is isomorphic to 7rl(Mg). Hint: Show first that there is a subgroup of G isomorphic to PSL(2, N) that acts properly on G / H (use Theorem 3.18). Then use the fact that there exists a discrete subgroup F of PSL(2, R) such that F _~ 7rl(Mg) (see Example 1.11). 3.11
Historical notes
neductive case:
The implication (1) =~ (2) in Theorem 3.10 was first
proved by E. Calabi and L. Markus in the case G / H = S O ( n + l , 1)/SO(n, 1) in [CM] (see Theorem 3.2). Then J. Wolf extended their result to the case
G / H = SO(p + 1, q)/SO(p, q) (q <_ p) in [Wol]. After finding some other similar results for symmetric spaces of rank one (e.g., [Wo2]), he finally obtained the sufficiency of the real rank condition in the case of semisimple symmetric spaces in the 1960s (see [Wo3]). His idea is also applicable to our more general setting. On the other hand, the proof of the necessity of the real rank condition given there was incomplete because of some confusion with the definition of properly discontinuous actions (see Example 2.8 for the difference between local Hausdorff topology and Hausdorff topology).
128
T. Kobayashi
The converse implication (2) =v (1) in Theorem 3.10 is more difficult because we have to show the existence of an infinite discontinuous group acting on G / H if ]R-rankH < R-rankG. It took about 20 years before the converse implication was first proved in the rank one case G / H SO(p + 1, q)/SO(p,q) (q > p) by R. Kulkarni ([Ku, Theorem 5.7]).The method there is based on a study of quadratic forms of type (p + 1, q). with the aid of a lemma due to N. Wallach. The general case is due to T. Kobayashi [Ko2] as an application of Theorem 3.18 with dim L - 1. Solvable case:
It was proved by Kobayashi that there always exists a
Clifford-Klein form of a homogeneous manifold G / H with infinite fundamental group (Theorem 3.9) if G is a solvable group and H ¢ G ([KoS]). Lipsman made a further study of properly discontinuous actions and the property ( C I ) i n [Lip]. General case: It is still an open problem to find a criterion on (G, H) for a general Lie group G such that a homogeneous space G / H admits an infinite discontinuous group.
G e n e r a l i z e d H i r z e b r u c h ' s proportionality principle
and its application 4.1
H i r z e b r u c h ' s proportionality principle In 1956, Hirzebruch proved
T h e o r e m 4.1([Hi]) Let D be a bounded Hermitian symmetric domain, F a torsion free discrete cocompact subgroup of the automorphism group Aut(D) of D, and M the compact Hermitian symmetric space dual to D. Then there is a real number A = A(F) such that c"(F\D)[F\D] = Ac~(M)[M] for any c a, where a = ( a l , . . . , a k ) is a multi-index and c a - c ~ 1 U . . . U c ~k k is a monomial of Chern classes. In this section, we shall clarify this principle by eliminating unnecessary conditions. The setting is generalized as follows: Hermitian symmetric spaces tangent bundles characteristic numbers
homogeneous spaces of reductive type homogeneous vector bundles characteristic classes
Chapter 3 Discontinuous Groups
129
As an application, we find an obstruction to the existence of compact Clifford-Klein forms of homogeneous spaces of reductive type by means of the Euler-Poincar~ class (see Corollary 4.12). 4.2
S k e t c h of i d e a
It has been a classical and standard technique, in particular, in representation theory, to compare two objects through a holomorphic continuation: For instance, i) Weyl's unitary trick - - finite dimensional representations of reductive groups ii) Flensted-Jensen duality
eigenspaces on semisimple symmetric spa-
ces (e.g., IF1]) The argument in this section lies in the same line: iii) (generalized) Hirzebruch's proportionality principle classes of homogeneous spaces of reductive type
characteristic
Let us explain the idea briefly in the case of Theorem 4.1, taking G = SL(2, R) as an example. In this case D = SL(2, N)/SO(2)
(the Poincar~ upper half plane),
M = SU(2)/SO(2)(~_ C P 1)
(the projective space).
There are two ways to compare the two manifolds D (or F \ D ) and M: i) D C M ii) D C SL(2, C)/SO(2, C) D M
(the Borel embedding), (complexification).
The proportionality principle for Hermitian symmetric spaces (Theorem 4.1) can be proved based on the Borel embedding (i) as well as based on the complexification (ii). However, we will see that the argument using (ii) has a wider application, even when there is no natural map between D and M. Returning to our special example, we note that Mc := SL(2, C)/SO(2, C) is a common complexification of M = SU(2)/SO(2) and D = SL(2, R)/SO(2) in (ii), if we forget the original complex structures and look upon M and D simply as real manifolds. Now, we can compare differential forms that represent characteristic classes of two manifolds D and M through the holomorphic continuation on Mc.
This is the main
idea of the proof of a generalized Hirzebruch's proportionality principle, which we discuss in Sections 4.3 and 4.4.
130
T. Kobayashi
4.3
C o m p l e x i f i c a t i o n a n d a s s o c i a t e d R i e m a n n i a n s p a c e s of compact type
Let us introduce the setting to realize the idea in Section 3.2. Let G be a connected real reductive linear group and H a connected subgroup reductive in G. We assume that there is a connected complex reductive Lie group Gc and a connected closed subgroup Hc with Lie algebras gc and I?c respectively, such that G C Gc. Take a Cartan involution 0 of G such that OH = H. We write the corresponding Cartan decomposition as g = [~+ p. Let Gu be the connected subgroup of Gc with Lie algebra gv := t~ + x/%--lp, and Hu the connected subgroup of Gc with Lie algebra I?v := t) n t~+ x/r2-1I~ np. Then, Gu is a maximal compact subgroup of Gc, and so is Hu of Hc. The homogeneous manifold G u / H u is said to be an associated Riernannian space of compact type. In summary, we have the following setting:
G
C
u
H
Gc
D
u C
Hc
Gu u
D
Hu
Remark The argument in this section is still valid for H with finitely many connected components, if we replace Hc by Hc(H n K) and Hu by
Hu(HnK). E x a m p l e 4.2
Here are some typical examples of Go~He and G u / H u .
i) Let G / H = SL(n,]R)/SO(p, n - p). Then Go~He - SL(n, C)/SO(n, C) and G u / H u - SU(n)/SO(n). ii) Let G / H = GL(n, IR)/(IR x)n. Then Go~He - GL(n, C)/(C x )n and G u / H u = V ( n ) / V ( 1 ) n. iii) Let G / H = Sp(2n,]R)/Sp(n, C). Then Go~He = Sp(2n, C)/Sp(n, C) x Sp(n, C) and G u / g u = Sp(2n)/Sp(n) x Sp(n). E x a m p l e 4.3
The foregoing assumption (i.e., closedness of He) is satis-
fied for (G, H) in Example 3.13(i)-(iv). We recall that G / H is a reductive symmetric space in Example 3.13(iii); G / H is a semisimple orbit in Example 3.13(iv). The corresponding associated Riemannian space of compact
Chapter 3 Discontinuous Groups type
4.4
131
Gu/Hu is Gu/Hu = a compact symmetric space
for Example 3.13(iii),
Gu/Hu = a generalized flag variety
for Example 3.13(iv).
A homomorphism
between cohomology rings
In the setting of Section 4.3, suppose a discrete subgroup F of G acts
G/H properly discontinuously and freely, so t h a t F\G/H is a smooth manifold. (We do not require that F\G/H is compact. F is allowed to be on
the trivial subgroup {e}.) We define
Mu := Gu/Hu C Mc := Go~He D MR := G/H. Then, to compare characteristic classes of the two manifolds
Mu and MR,
we use the following restriction maps:
(.9 /
rest.
"~ rest.
Gu and G acting on Mu and MR, respectively, have the common complexification Go, which acts holomorphically on Me. Because Gu
The groups
is compact, we can find a Gv-invariant differential form as a representative of an arbitrary element of the de R h a m cohomology group
H*(Gu/Hu; C). Go~He, and
This element extends to a holomorphic differential form on then we can restrict to
G/H and to F\G/H. Thus, the diagram induces a
h o m o m o r p h i s m on the cohomology level: Theorem
4.4
We have a natural C-algebra homomorphism ~7: H*(Gu/Hu;C) ~ H*(F\G/H;C).
If F\G/H is compact and H is connected, then ~ is injective. The last s t a t e m e n t follows from the Poincar@ duality.
132 4.5
T. Kobayashi
Real homogeneous vector bundles We review the definition of an associated vector bundle. Let p" H ~ GLR(V)
be a representation of H on a real vector space V.
Suppose a discrete
subgroup F acts on a homogeneous space G / H properly discontinuously. Associated to the principal H-bundle H --. F \ G ~ r \ a / H
is a real homogeneous vector bundle over F \ G / H , r E " - F \ G x o V, defined by the set of equivalence classes with respect to the action of H , that is, (~, v), (g--7,v') E F \ G x V are equivalent ¢ , - g - g-Th and v - p ( h - 1 ) v ~ for some h C H.
The projection on the first component F \ G x V ~ F \ G gives rise to the projection r E - F \ G x p V ~ F \ G / H with typical fiber V. Similarly, an associated real vector bundle E u := G u x pv Vu over G u / H u
is defined if
a representation p v : H u --* GLR(Vu) is given. 4.6
Weyl's unitary trick The idea of Weyl's unitary trick is a holomorphic continuation of finite
dimensional representations. Here, we set up some notation of finite dimensional representations that fits into Hirzebruch's proportionality principle. Let H C He D H u be as in Section 4.3, namely, Hc is a complex reductive Lie group, H is a real form, and H u is a compact real form. Let p" H ~ GL~(V), Pu" H u --~ GL•(Vu)
Chapter 3
Discontinuous Groups
133
be finite dimensional representations over R. We say that the complexifications of p and pv are isomorphic if there are i) a complex vector space Vc ii) a holomorphic representation Pc: He ~ GL(Vc, C) iii) isomorphisms ~b: V ® C -% Vc and Cu: Vu ® C -% Vc such that the following diagram commutes:
4.7
L
H
¢-~
Hc
~
Hu
GLut(V)
~
GLc(Vc)
~-"
GLR(Vu)
principle
Hirzebruch's proportionality
T h e o r e m 4.5 ([KoO]) Retain the setting of Section ~.3. Let F be any discrete subgroup of G acting on G / H freely and properly discontinuously from the left. Suppose that the complexifications of p: H ~ GL~(V) and pu : Hu ~ GL~t(Vu) are isomorphic. Then the ith Pontryagin class satisfies V(p~(Eu)) = p~(rE) e H4~(F\G/H;I~). In particular, if there is a relation ~ a~pa(Ev) = 0 in H * ( G u / H u ; ] ~ ) , then the equation ~-]a~pa(rE) = 0 in H * ( F \ G / H ; I ~ ) holds. Here c~ = (C~l,..., c~k) is a multi-index and pa p~l U ... U Pk~k is a monomial of _
Pontryagin classes. S k e t c h o f Proof'. We take an invariant connection (the canonical connection of the second kind on G / H in the sense of [No]) on the principal bundles G ~ G / H , Gu ---* G u / H u , respectively. Then the curvature forms are represented in terms of the Lie algebras. By the Chern-Weil theory (see [D], [NON], [We]), characteristic classes are represented by using curvature forms. Now, we have the theorem by the usual Weyl's unitary trick for finite dimensional representations and by using the holomorphic
continuation through r/.
D
We note that in the case of Riemannian symmetric spaces (in particular, Hermitian symmetric spaces), we can prove the theorem without using a holomorphic continuation. That is, we pull back directly the curvature forms via the R-linear bijection ~ v / O v ~~g/[}. This map is different from r] in Theorem 4.4 only by a constant multiple depending on degrees.
T. Kobayashi
134
4.8
Tangent bundle The characteristic classes of a manifold are, by definition, those of the
tangent bundle. The tangent bundle T ( F \ G / H ) is associated to the adjoint representation AdlH" H ~ GL~(g/O), that is,
T ( F \ G / H ) ~_ F\G
x g/l],
AdlH
and similarly
T ( G u / H u ) ~ Gu
x
AdIHu
gu/i)u.
Since AdiH" H ~ GL~(9/[3) and AdiH u • Hu ~ GL~(gu/[~u) have isomorphic complexifications, we have now relations of Pontryagin classes (of the tangent bundle) between G u / H u and F \ G / H as follows.
In the same setting as Theorem 4.5, we have
C o r o l l a r y 4.6
rl (pi(Gu/Hu)) - p i ( F \ G / H ) E H4i(F\G/H;N). Furthermore, if H is connected, then we have a relation of Euler-Poincard classes:
( x ( G u / H u ) ) - X(F\G/H) e H~(F\G/H;R). We note that both F \ G / H and G u / H u are orientable if H is connected. 4.9
Complex homogeneous vector bundles Let p: Hc ~
GLc(Vc) be a holomorphic representation on a finite
dimensional vector space Vc over C. Associated to the principal H-bundle
H ~ F\G ~ F \ G / H , we define a homogeneous complex vector bundle over F \ G / H by
rEc:=r\G
x Vc.
Pig
Similarly, we define a homogeneous complex vector bundle over G u / H u by
Euc := Gu x Vc. P[Hu
Chapter 3 4.10
Discontinuous Groups
135
C h e r n classes
Retain the same setting as before. Let F be any discrete subgroup of G acting on G / H freely and properly discontinuously from the left. Then the i th Chern class satisfies Theorem
4.7 ([KoO])
~(cg(Euc)) = c g ( r E c ) ~ H2~(F\G/H;R). In particular, if there is a relation ~-~a~c~(Ev) = 0 in H*(Gu/Hu;I~), then the equation ~-~a~c~(rE)= 0 in H*(F\G/H;I~) holds. I f F \ G / H is compact and H is connected, the converse statement also holds. E x e r c i s e 4.8 Formulate an analogous result to Corollary 4.6 in the case of Chern classes of homogeneous manifolds. (Hint: The assumption will be that G / H is an elliptic orbit. See Section 5.12. See also [KoO, Corollary
4]) 4.11
Some examples
E x a m p l e 4.9
We consider a semisimple symmetric space
G / H = SO(p, q)/SO(p - 1, q). All Pontryagin classes of any Clifford-Klein form F \ G / H of a homogeneous manifold G / H vanish in H* (F\G/H; ~) because we know the corresponding result holds for G u / H u ~- S p+q-1. In particular, all Pontryagin classes of a Riemannian manifold of constant sectional curvature vanish (see also [Su]). We mention that there exist compact Clifford-Klein forms of SO(p, q)/SO(p - 1, q) if (p, q) = (1, n), (2, 2n), (4, 4n), (8, 8) (n e N) (see Example 5.14). E x a m p l e 4.10
We endow C p+q+l with an (indefinite) Hermitian metric
of type (p + 1, q), that is, ( Z , Z ) :-- Z l - ~ - J r ' ' " + Z p + l Z p + l -- Zp+2Zp+2 . . . . .
Zp+q+lZp+q+l.
Let X(p, q) be the open subset of the projective space CP p+q, which consists of the complex lines on which the restriction of the indefinite Hermitian
136
T. Kobayashi
metric is positive definite. Then U(p + 1, q) acts transitively on X (p, q), so that we have a diffeomorphism X(p, q) "" U(p + 1, q)/U(1) x U(p, q) -" G / H _~ U(n + 1)/U(1) x U(n)
C C.P n
-" G u / H u .
Here, we put n - p + q. We note that X ( n , O ) - CP n ~ G u / H u and that X(0, n) is the dual Hermitian symmetric domain of the noncompact type (see [He] for the terminology). Let F be a discrete subgroup of V(p ÷ 1, q) acting on X(p, q) freely and properly discontinuously, so that F \ X ( p , q) is a Clifford-Klein form of X(p, q). Then, we have a relation among Chern classes:
cj(r\X(p,
q))
-
n
n+ +1 -11
el (1-'\X (p, q) )J
(1 ~ j ~ n) .
This follows from the corresponding fact for X ( n , O) - CP n ~ U(n + 1)/U(1) x U(n), that is, the total Chern class c(CP n) - 1 + c~ (CP ~ ) + . . . + Cn (CP n) of the complex projective space C,P n is given by c(CP ~) - (1 + x) n+l
mod x ~+1,
where x is the first Chern class of the hyperplane section bundle (see [BoHi, 15.1], [MiSt, Theorem 14.10]). If F \ X ( p , q ) i s a compact Clifford-Klein form of X(p, q), then cj(r\x(p, q)) ¢ 0 for any j with 1 _< j <_ n. There exists a compact Clifford-Klein form of X (0, n), X (n, 0) (Pdemannian case), and X(1,2r) (see Corollary 4.7), whereas any discrete subgroup acting properly discontinuously on X(p, q) with p >_ q is finite (see Theorem a.10). E x a m p l e 4.12
Let M be a compact Clifford-Klein form of a complex
manifold SO(p, q + 2)/SO(p, q) × SO(2). Then the Chern class cj(M), for any j with 1 <_ j _< p + q, of a complex manifold M does not vanish, because we know that the corresponding result holds for the Hermitian symmetric space SO(n + 2)/SO(n) × SO(2). There exists a compact Clifford-Klein form for (p, q) - (n, 0) (n E N) and (4, 1) (see Corollary 5.6).
Chapter 3 Discontinuous Groups
4.12
137
Compact Clifford-Klein form We mentioned in Section 2.2 a third proof that PSL(2, R ) / S O ( 1 , 1)
does not admit a compact Clifford-Klein form by using a nonvanishing vector field. Now we are ready to state a generalization of this result to a higher dimensional setting:
Corollary 4.13 Let (G, H) be as in Section ~.3. Denote by K a maximal compact subgroup of G such that H N K is also a maximal compact subgroup of H. If r a n k G = rank H and dim K / H n K is odd, then G / H admits no uniform lattice, that is, there exists no compact Clifford-Klein form of G/H. Sketch of proof:
We may and do assume H is connected. Then, G / H
admits a G-invariant orientation, so that a Clifford-Klein form F \ G / H is an orientable manifold. The tangent bundle T(F\G/H) splits according to the H n K - m o d u l e decomposition q = q n ~ + q n p . Then,
X(F\G/H) = 0
because dim• q n t = dim K / H N K is odd. On the other hand, as H is of maximal rank in G, so is Hu in Gu. As H is connected, so is Hu. Therefore,
Gu/Hu is a compact orientable manifold with nonvanishing Euler number X(Gu/Hu) ([HoS]). Now it follows from Theorem 4.4 t h a t x ( G u / H u ) ~ 0 contradicts the fact that x(r\a/H) = 0 if F \ G / H is compact. Hence, F \ G / H cannot be compact. [2]
Example 4.13 (see Section 2.2) We know that the hyperboloid of one sheet G / H = SL(2, R)/SO(1, 1) does not admit a compact Clifford-Klein form. This fact was explained by using nonvanishing vector fields in the item (iii) of Section 2.2, which is a geometric idea of Corollary 4.12. In this case, we have rank G - rank H = 1 and dim K / H N K = 1, so t h a t the assumptions in Corollary 4.12 are satisfied.
Exercise 4.14 (see also Example 5.14)
Prove that a semisimple sym-
metric space S O ( / + k,j + l)/SO(i,j) × SO(k,/) does not have a compact Clifford-Klein form if exactly one element among i, j, k, 1 is even. At the end of this section, we pose the following conjecture:
Conjecture 4.15 (see [Ko3, Conjecture 6.4]) Let G / H be a homogeneous space of reductive type. It is likely that the inequality rank G + r a n k ( H n K ) _> rank H + rank K
T. Kobayashi
138
holds if G/H admits a compact Clifford-Klein form. The case with rank G = rank H can be proved based on Corollary 4.6 and on an argument of the cohomological dimension of a discrete group, as a slight improvement of Corollary 4.12 (see [Ko2, Corollary 5]). If H is compact satisfied. holds for compact
or if G/H is a group manifold, then the inequality is obviously As far as the author knows, the inequality in Conjecture 4.15 all homogeneous spaces of reductive type that are proved to admit Clifford-Klein forms (cf. Corollary 5.6).
N o t e s 4.16
i) Most material in this section is taken from [KoO].
ii) To establish the holomorphic continuation, we calculate characteristic forms in terms of Lie algebras. Similar calculations are also carried out in [Bo2], [CGW] for Riemannian symmetric spaces. iii) There exist different generalizations of Hirzebruch's proportionality principle, due to D. Mumford [Mu] for the noncompact Clifford-Klein forms of Hermitian symmetric spaces, and due to J. L. Dupont and W. Kamber [DK] for the secondary characteristic numbers. iv) F. Labourie informed me that Corollary 4.12 is also valid for a manifold modeled on a homogeneous space G/H (by e-mail, 1994). v) A similar idea (at least implicitly) to Corollary 4.12 is used in [Zu] based on the Gauss-Bonnet theorem in the case of rank 1 symmetric spaces G/H = SO(p + 1, q)/SO(p, q) (cf. Exercise 4.13).
C o m p a c t C l i f f o r d - K l e i n f o r m s of n o n - R i e m a n n i a n homogeneous spaces In Section 5, we focus our attention on compact Clifford-Klein forms of homogeneous spaces of reductive type. This section is organized as follows: Subsections 5.1 - 5.9
Homogeneous spaces with compact Clifford-Klein forms.
Subsections 5 . 1 0 - 5.12
Homogeneous spaces without compact Clifford-Klein forms.
5.1
U n i f o r m lattice, compact Clifford-Klein form
D e f i n i t i o n 5.1 Suppose G is a Lie group and H is a closed subgroup. Suppose a discrete subgroup F C G satisfies the following two conditions:
Chapter 3
Discontinuous Groups
F acts properly discontinuously and freely on G/H. F \ G / H is compact.
139 (5.1a) (5.1b)
Then F is said to be a uniform lattice .for the homogeneous space G/H. The double coset space F \ G / H is a compact Clifford-Klein form of the
homogeneous space G/H. Suppose there exists a G-invariant measure on G/H. (This is the case for a homogeneous space of reductive type.) Then it induces a measure on a Clifford-Klein form F\G/H. The discrete subgroup F is said to be a
lattice for the homogeneous space G / H provided both (5.1a) and F \ G / H is of finite volume
(5.2)
are satisfied. If H is compact (in particular H = {e} or H = K), Definition 5.1 coincides with a usual one. 5.2
Riemannian
cases
If the isotropy subgroup H is compact, then any discrete subgroup of G acts properly discontinuously on G/H, as we saw in Section 3.3. This case, reffered to as the Riemannian case, allows a compact Clifford-Klein form. We recall the following important theorem due to Borel, Harish-Chandra, and Mostow-Tamagawa.
Suppose G is a real linear reductive Lie group and H is a compact subgroup of G. Then G / H admits compact Clifford-Klein forms. Furthermore, G / H has noncompact Clifford-Klein forms with finite volume if G contains a noncompact semisimple factor. T h e o r e m 5.2 ([Bol],[BoHa],[MoTa])
We shall not prove this theorem, and instead we refer the reader to [Ra], [Zil] and the original papers. We shall mention, however, some typical examples. i) A compact Riemann surface Mg (g _> 2 ) i s a compact Clifford-Klein form of the Poincar(~ plane PSL(2, R)/SO(2) (see Example 1.11). ii) If F = 7/~n C G = l~n D H = {0}, then F \ G / H __ ~ 1 x . . . x ~ 1 is a compact Clifford-Klein form of G / H ~_ I~n . iii) If F = G L ( n , Z ) C G = GL(n,R) D H = {e} (n >__ 2), t h e n F \ G / H is a noncompact Clifford-Klein form of G / H ~_ G L ( n , ~ ) with finite volume.
T. Kobayashi
140 5.3
Moore's ergodicity theorem A simple remark here is that if F is a uniform lattice for a group
manifold G = G/{e}, then the quotient topology of the double coset space F \ G / H is always compact. However, if H is noncompact, the double coset space F \ G / H does not have a good topology in general. In fact, the action of F on G / H is not properly discontinuous. We leave it to the reader as an easy exercise: E x e r c i s e 5.3
Prove that the action of F on G / H is not properly discon-
tinuous if F is a uniform lattice of G and if H is noncompact. Furthermore, the action of F on G / H can be ergodic: 5.4 ([Moo], see also [Zil, Chapter 2]) Let G be a noncompact simple linear Lie group. IS F C G is a lattice and H C G is a closed subgroup, then
Theorem
The action of F is ergodic on G / H
¢:~
H is noncompact.
Here, we recall that the action of F is said to be ergodic if every F-invariant measurable set is either null or conull. Thus, a uniform lattice for G / H must be smaller than a uniform lattice for G in some sense. In this respect, the cohomological dimension of an abstract group is a nice measure for the "size" of a discrete group (see [Ko2, Corollary 5.5]). We use it in the proof of Theorems 5.7 and 5.8. Basic references for the material we use on the cohomological dimension of groups are [Ser] and [Bi].
5.4
Indefinite-Riemannian
case
Suppose that G is a real reductive linear group and that H and L are both reductive in G. If the triplet (G, L, H) satisfies both conditions
T h e o r e m 5.5 ([Ko2, Section 4])
a(L) n a(H)
= {0},
d(L) + d(H) = d(G),
(5.3b)
Chapter 3
Discontinuous Groups
141
then G / H admits compact Clifford-Klein forms. Furthermore, G / H admits noncompact Clifford-Klein forms that have finite volume if L contains a noncompact semisimple factor. The same results hold for G/L because of the symmetric role of H and L. 5.5
W e l l k n o w n e x a m p l e s (I)
Riemannian
cases revisited
Suppose G / H is of reductive type. We know t h a t there exist compact Clifford-Klein forms of G / H if H is compact (Theorem 5.4). This fact is explained in the context of Theorem 5.5 as follows. Suppose H is compact. Then we have a(H) = {0} and d(H) - 0. If we take L = G, then the conditions (5.3a,b) are satisfied. Thus, Theorem 5.4 (Riemannian case) is a special case of Theorem 5.5. But this does not give a new proof of Theorem 5.4 because we shall use T h e o r e m 5.4 as a starting point for the proof of Theorem 5.5. 5.6
W e l l k n o w n e x a m p l e s (II)
group manifold cases
Let G' be a noncompact real reductive Lie group. Suppose (G, H ) -
(G' × G',diagG'), so t h a t G / H ~_ G' is a group manifold. This case is trivial because it is obvious t h a t G / H admits a compact Clifford-Klein form by T h e o r e m 5.4.
But it is instructive to see how T h e o r e m 5.5 is
applied in the group manifold case because H ~_ G' is noncompact. Let g' = Lie G' = ~' + p' be a C a r t a n decomposition, and ca, C p' a maximally abelian subspace. We write WG, for the Weyl group of the restricted root system E($, a). Then, we have aG -- aG, •
aG,,
Wa = Wa, × Wa,. Let us take L := G' x 1 (or 1 x G'). Then, we have
a(H) = {(X, wX) : X e ca,, w e W e , } , a(L) = {(X, 0 ) : X e c a , } , d(H) = d(L) = d(G'),
d(G) = 2 dim G'.
Hence, the conditions (5.3a,b) are satisfied.
T. Kobayashi
142 5.7
E x a m p l e s of i n d e f i n i t e - R i e m a n n i a n h o m o g e n e o u s s p a c e s that have compact Clifford-Klein forms As a corollary of Theorem 5.5 we have
C o r o l l a r y 5.6 The following homogeneous spaces (six series and six isolated ones) admit compact Clifford-Klein forms. Also, they admit noncompact Clifford-Klein forms of finite volume. W su(2, 2n)/U(1, 2n), a) SU(2, 2n)/Sp(1, n), i) b) SO(2, 2n)/SO(1, 2n), a) SO(2,2n)/U(1, n), i~) b) SO(4, 4n)/SO(3, 4n), a) SO(4, 4n)/Sp(1, n), iii) b) SO(8, 8)/Spin(8, 1), a) SO(8, 8)/SO(8, 7), iv) b) SO(4, 4)/Spin(4, 3), a) SO(4, 4 ) / s o ( n , 1) × SO(3), v) b) SO(4, 3)/G2(2). a) SO(4, 3)/SO(4, 1) × SO(2), vi) Here, (a) and ( b ) i n each line forms a pair (G/H, G/L) that satisfies the assumptions (5.3a,b) of Theorem 5.5. For example, the first line means ( G , H , L ) = (SU(2,2n),Sp(1, n), U(1,2n)) satisfies (5.3a,b). The signature of the indefinite metric on G / H (and also on CliffordKlein forms) induced from the Killing form is given by (4n, 3n 2 - 2n), (4n, 2), (2n, n 2 - n), (2n, 1), (12n, 7n 2 - 4n + 3), (4n, 3), (8,7), (56,28), (12, 3), (4, 3), (8, 2), and (4, 3), respectively. 5.8
S k e t c h of p r o o f
The idea of Theorem 5.5 is illustrated by the abelian case. Let G R n D H - R k (n > k). Take a complementary subspace L ~ N n-k of H in G, and choose a lattice F _~ Z n-k of L. Then, r\a/H
~_ Z~-k \R ~ /R k ~- s* x . . . x s ~
is a compact Clifford-Klein form of G/H. In this case (5.3a) is satisfied because L __ R n-k is complementary to H _~ R k, and (5.3b) is satisfied because d(L) + d(H) = (n - k) + k = n = d(G). For the general case, the proof of Theorem 5.5 is divided into the following three steps: i) The criterion for proper actions (continuous analog of discontinuous groups) (Theorem 3.18). ii) The criterion for compact quotient (continuous analog of uniform lattice) (Theorem 5.7).
Chapter 3
Discontinuous Groups
143
iii) Existence of uniform lattice in the Riemannian case (Theorem 5.4). If both (5.3a,b) are satisfied, then L acts properly on G / H by Theorem 3.18 and the quotient topology of L \ G / H is compact by Theorem 5.7. On the other hand, Theorem 5.4 assures the existence of a torsion free cocompact discrete subgroup F of L. Then, F turns out to be a uniform lattice for G / H thanks to Observation 2.12. This shows the first half of Theorem 5.5. Similarly, a torsion free covolume finite discrete subgroup F of L is also a lattice for G/H.
5.9
C o n t i n u o u s analog A continuous analog of a compact Clifford-Klein form F \ G / H of a
homogeneous manifold is a compact double coset space L \ G / H in the quotient topology where L, H are closed subgroups of G such that L acts properly on G / H (see Definition 2.11).
Let G be a real reductive linear Lie group, H and L closed subgroups that are reductive in G. Under the equivalent conditions in Theorem 3.18 (i.e. L ~ H in G in the sense of Definition 2.1~), the following two conditions are equivalent:
Theorem
5.7
L \ G / H is compact in the quotient topology. d(L) + d(H) = d(G).
(5.4a) (5.4b)
A f l a v o r of the proof:
We recall that d(G) = dim~p if we write a C a r t a n decomposition as ~ = t + p. In view of the Cartan decomposition of Lie group G ~_ K × exp(p), we may regard
d(G) = the dimension of "noncompact part" of G.
(5.5)
A homogeneous space of reductive type G / H has a vector bundle structure over a compact manifold K / H N K with typical fiber p/p A [} (e.g., [Zo2, Appendix]). In view of dim~ p/p A ~ = d(G) - d(H), we may regard
d ( G ) - d(H) = the dimension of "noncompact part" of G/H.
(5.6)
Now, one might expect that d(G) - d(H) - d(L) = the dimension of "noncompact part"of L \ G / H ,
(5.7)
T. Kobayashi
144
and t h a t (5.7) would lead to Theorem 5.7. However, (5.7)is not always "true" if we do not assume that L rh H in G. (We remark that the dimension of "noncompact part" of L \ G / H is not defined yet,) The simplest and illustrative observation is when G = R n and H, L are vector subspaces of G with dimension k, l, respectively. Then d(G) - d(H) - d(L) = n - k - l,
L \ G / H ~ R n-k-1 + dim(HAL) Therefore, if we assume d i m ( H n L) = 0 (or, equivalently, L ~ H in a ) , then we have
d(G) - d(H) - d(L) = 0 ~
L \ G / H is compact.
This explains why the assumption L rh H is necessary in Theorem 5.7, and how (5.4a) and (5.4b) are related under this assumption. For the general case where G is a real reductive linear group, the underlying idea is similar but we need some more work. This is carried out in a framework of a discrete analog of Theorem 5.7, where L is replaced by an arithmetic subgroup F of L and d(L) is replaced by the cohomological dimension of F. See [Ko2] for details. 5.10
[3
N e c e s s a r y c o n d i t i o n s for t h e e x i s t e n c e of c o m p a c t Clifford-Klein forms
For the existence of compact Clifford-Klein forms of homogeneous spaces of reductive type, we have presented two necessary conditions so far: i) Calabi-Markus phenomenon (Theorem 3.10) ii) Hirzebruch's proportionality principle (Corollary 4.12) Here, we give a third necessary condition for the existence of compact Clifford-Klein forms of homogeneous spaces:
Let G / H be a homogeneous space of reductive type. G / H does not admit a compact Clifford-Klein form if there exists a closed subgroup L reductive in G satisfying the following two conditions:
Theorem
5.8
a(L) C a(H),
(5.8a)
d(L) > d(H).
(5.8b)
Chapter 3 Discontinuous Groups
145
The proof of Theorem 5.8 is similar to that of Theorem 5.5 and Theorem 5.7, by using the cohomological dimension of an abstract group together with Theorem 3.18. See [KoT] for details. For an application of Theorem 5.8, we need to find a suitable L satislying (5.8a,b), provided G/H is given. This is done systematically for some typical homogeneous space of reductive type in the next two subsections. 5.11
Semisimple symmetric
spaces
To apply Theorem 5.8 to a semisimple symmetric space, we need some results on the root system for semisimple symmetric spaces. First of all, we give a brief review of the notion of e-families introduced by T. Oshima and J. Sekiguchi [OS]. Let g be a semisimple Lie algebra over R, cr an involution of g, and 0 a C a r t a n involution of g commuting with a. Then, cr0 is also an involution of g, because (or0) 2 = a202 = 1. Let g = ~ + p = O + q = ~a _~_ qa be direct sum decompositions of eigenspaces with eigenvalues +1 corresponding to 0, cr and aO respectively.
The symmetric pair (g, Oa) is said to be the
associated symmetric pair of (g, 0). Note that Oa = t~ A O + P A q and that
(oa) a --
O.
Fix a maximally abelian subspace ap,q of p A q. Then
E(ap,q) - E(g, ap,q) satisfies the axiom of root system (see [Ro, Theorem 5], [OS, Theorem 2.11]) and is called the restricted root system of (g, O). As a0 - id on ap,q, we have a direct sum decomposition of the root space g(ap,q, A) = (g(ap,q, A) A Oa) ~- (~](flp,q, A) ~ qa). We define a map
(m +, m - ) : ~(%,q) ~ N x N, by m + ( A ) : =
dim(g(ap,q,A) N0a), r e - ( A ) : =
dim(g(ap,q,)~) Nqa).
(5.9) Note
t h a t if (g, 1)) is a Riemannian symmetric pair, then ~a = g and m - - 0. A map ~: E(ap,q) U {0} ~ { 1 , - 1 } is said to be a signature of E(ap,q) if e is a semigroup homomorphism with e(0) - 1. It is easy to see t h a t a signature is determined by its restriction to ~, a fundamental system for E(ap,q), and t h a t any map • ~ { 1 , - 1 } is uniquely extended to a signature. To a signature ~ of E(ap,q), we associate an involution er~ of g defined by a~(Z):= ~(A)a(X)if X e g(ap,q; A), ~ e E(ap,q)U{0}. Then, a~ defines a symmetric pair (g, O~). The set F ((g, b)) := {(g, I~): ¢ is a signature of E(ap,q)}
146
T. Kobayashi
is said to be an e-family of symmetric pairs ([OS, Section 6]). Among an s-family, there is a distinguished symmetric pair, called basic, characterized by
.~+(~) > m-(~)
for any A E E(ap,q) such that A
It is known that there exists a basic symmetric pair of F - F ((g, l?)) unique up to isomorphisms ([OS, Proposition 6.5]). E x a m p l e 5.9
i) { ( s [ ( n , N ) , s o ( p , n - p)): 1 < p < n} is an e-family for which the pair (~[(n,R),~o(n)) is basic. This family is an example of the so-called K~-family, which is a special case of an s-family. ii) For F = R, C or H (a quaternionic number field), we write V(p, q;F) for O(p, q), V(p, q), and Sp(p, q), respectively. We fix r < q < p. Then { ( u ( ~ , p + q - ~; ~ ) , u(k, p - k; ~) + u(~ - k, q - ~ + k; R) • 0 _< k _< ~}
is an s-family with (u(r,p + q - r;]F),u(r,p - r;F) + u(q; N)) basic. Now we are ready to state an application of Theorem 5.8 to symmetric spaces: C o r o l l a r y 5.10 ([Zo7, Theorem 1.4]) If a semisimple symmetric space G / H admits a compact Clifford-Klein form, then the associated symmetric pair (g, Oa) is basic in the s-family F ((g, Oh)). S k e t c h o f proof: We apply Theorem 5.8 with "H" := H~ and "L" := H a. Then the assumptions (5.Sa,b) follow from the following lemma (see [No7, Lemma 4.5.2]). E] L e m m a 5.11 With notations as before, let (g, [~) be basic in the s-family F = F ((g, ~)) and (g, ~ ) be not basic in F. Then we have
i) a(H a) -- a(Hea), ii) d(H a) > d(H~a). Here are some examples of symmetric pair (g, ~) such that the associated pair (g, [~a) is basic in its s-family.
Chapter 3
Discontinuous Groups
147
E x a m p l e 5.12 As usual, we write (g, t~) for a Riemannian symmetric space if g is a Lie algebra of noncompact reductive Lie group G. In the following table, F denotes R, C, or ]HI.
(~,~)
(~t, 0 a) is basic
(g,g)
(g,~)
(g,~)
(~,~)
(g + g, diag g)
(~c,~c) (u(p, q; F), u(m, q; F) + u(p - m; F))
(g + g, diag g)
(gc, g) (u(p. q; y). u(.~; y) + u(p - .~. q; y))
An obvious remark is that the associated symmetric pair (g, 0 a) of a Riemannian symmetric pair (g, t~) is (g, g), which is obviously basic, and that of a group manifold (gP + gP, diag g~) is again a group manifold, which is also basic. This is consistent with the fact that both Riemannian symmetric spaces and group manifolds admit compact Clifford-Klein forms (cf. Section 5.5 and Section 5.6). See [Ko7, Table 4.4] (and also [Ko5, Table 5.4.3]) for a list of semisimple symmetric spaces that are proved by this method not to admit compact Clifford-Klein forms. 5.12
S e m i s i m p l e orbits
Another typical example of a homogeneous space of reductive type is a semisimple orbit Ad(G). X ~_ G / Z c ( X ) , where X C g is a semisimple element (Example 3.13(iv)). We apply Theorem 5.8 to semisimple orbits. C o r o l l a r y 5.13 ([Ko7], [BL]; see also [Ko4, Ko6])
Let G be a real reduc-
tire linear Lie group, and X a semisimple element of g. If G . X ~_ G / Z G ( X ) admits a compact Clifford-Klein form, then the orbit G . X carries a Ginvariant complex structure.
T. Kobayashi
148
Before giving a sketch of proof, we remark that a simple group G cannot be a complex group if there is a nonzero semisimple element X E g such that Ad(G). X ~_ G/ZG(X) admits a compact Clifford-Klein form. In fact, since rank G = rank Z a ( X ) , we have N-rankG = N-rank ZG(X) if G is a complex reductive Lie group. Then there is no infinite discontinuous group acting on G/ZG(X) by Theorem 3.10 (the Calabi-Markus phenomenon). Therefore G/ZG(X) does not admit a compact Clifford-Klein form. This means that G itself is not a complex group. Nevertheless Corollary 5.13 asserts that the homogeneous space G/ZG(X) carries a G-invariant complex structure. This is because G/ZG(X) is realized as an elliptic orbit, which is a crucial point of the proof.
Sketch of proof." Let X = Xe + Xh be a decomposition of a semisimple element X C 0, where Xe is elliptic and Xh is hyperbolic such that [Xe, X h ] - 0. Applying Theorem 5.8 with L "- ZG(Xe) and H "- ZG(X), we obtain Corollary 5.13. An alternative proof is given in [BL] based on symplectic structure.
7-1
Let X be an elliptic element of 0. We give here a quick review of the rich geometric structure of an elliptic orbit A d ( G ) . X ~_ G/ZG(X). It is well known that an elliptic orbit G/ZG(X) carries a G-invariant complex structure via the generalized Borel embedding into the generalized flag variety Gu/ZGu (X):
G/ZG(X) c av/zG
(x).
See, for example, [KoO, Appendix] for a proof. (This embedding is a usual Borel embedding if G/ZG(X) is a Hermitian symmetric space, equivalently, if G/Za(X) is ~ symmetric space and if ZG(X) is compact; see [He].)
G/ZG(X) is said to be dual manifolds of K5hler C-space in the sense of Griffiths-Schmid if ZG(X) is compact; G/Za(X) is said to be a ½-Kahler symmetric space in the sense of Berger if G/ZG(X) is a symmetric space and if Za(x) is noncompact.) A G-invariant symplectic structure on the elliptic orbit is induced from the one on the coadjoint orbit through the identification g ~- 0", which is given by a nondegenerate G-invariant bilinear form B on g (e.g., the Killing form if g is semisimple). The orbit
G/Za(X) also
carries a G-invariant
(indefinite) Kiihler structure induced by B. The indefinite K/ihler structure is then compatible with the symplectic structure.
Chapter 3
Discontinuous Groups
149
We note that Clifford-Klein forms of an elliptic orbit inherit these structures. Corollary 5.6 asserts that the homogeneous manifolds
U(2, 2n)/U(1) × U(1, 2n),
SO(2,2n)/U(1, n),
so(4, 3)/s0(2) × so(4,1) admit compact Clifford-Klein forms. Furthermore, these are realized as elliptic orbits of the adjoint action. So we obtain new examples of compact indefinite-K/ihler manifolds. Finally, in sharp contrast, recall that a hyperbolic orbit does not admit an infinite discontinuous group (Example 3.11).
5.13
Some examples
We give here a number of examples of homogeneous spaces that are proved not to have (or to have) Clifford-Klein forms by the method of this section. These examples may help to reveal the applications and limitations of various methods known so far (e.g., [Kol], [No2], [Ko5], [KoO], [No4, Ko6], [KoT], [BL], [Zi2], [LaMZ], [Koll], [Co])in studying the existence problem of compact Clifford-Klein forms, which has not yet found a final answer (see also Notes in Section 5.14).
Example 5.14
First, we consider a semisimple symmetric space
G / H - SO(i + j, k + l)/SO(i, k) x SO(j, l)
(i < j,k, 1).
It follows from Theorem 5.8 (or Corollary 5.10) that if G / H admits a compact Clifford-Klein form, then G / H is compact, or H is compact, or 0 - i < 1 _< j - k. Moreover, we have jkl - 0 mod 2 as an application of Hirzebruch's proportionality principle in Section 4 (see Corollary 4.12 and a remark after Conjecture 4.15). Conversely, if (i,j,k, 1) - (0, 2n, 1, 1), (0, 4n, 1, 3), (0, 4, 2, 1), (0, 8, 1, 7), or if i - 1 - 0, then there exists a compact Clifford-Klein form of G / H (see Corollary 5.6).
T. Kobayashi
150 E x a m p l e 5.15
A semisimple symmetric space
G/H - SO* (2n)/U(1, n - l) does not admit a compact Clifford-Klein form if 31 < 2n < 61 and if n > 3.
We explain this in Example 5.18 together with similar
examples. It admits compact Clifford-Klein forms if (n,1) - (4,1) , (4,3), (3,1), (3,2), (2,1), 1 - 0 ,
or 1 - n .
In fact, we have a local isomorphism SO* (8)/U(1, 3) ~ SO(2,6)/U(1,3), which is proved to admit compact Clifford-Klein forms in Corollary 5.6. It is trivial in the cases (3, 1), (3, 2), and (2, 1) because G/H is then compact. G/H is a Riemannian symmetric space in the cases 1 - 0 or 1 - n (see Theorem 5.2). E x a m p l e 5.16
A semisimple symmetric space
G/H-
SO*(4n)/SO*(4p+ 2) x SO*(4n - 4 p - 2)
does not admit compact Clifford-Klein forms if 1 _< p _< n - 2 , by Corollary 5.13. We note that SO*(8)/SO*(6) x SO*(2) (namely, in the case ( n , p ) (2, 1)) admits compact Clifford-Klein forms. Similarly, there exist compact Clifford-Klein forms in the case (n,p) - (2, 0). In fact, we have a local isomorphism SO*(8)/SO*(2) x SO*(6)..~ SO(2,6)/U(1,3), which is in the list of Corollary 5.6. E x a m p l e 5.17
Suppose that Gc/Hc is a complex irreducible semisimple
symmetric space. Then, we have a conjecture that Gc/Hc admits compact Clifford-Klein forms if and only if Gc/Hc is locally isomorphic to a group manifold. One can prove that Gc/Hc does not have compact CliffordKlein forms unless Gc/Hc is locally isomorphic to either a group manifold, SO(2n + 2, C)/SO(2n + 1, C), SL(2n, C)/Sp(n, C) (n >_ 2), or E6,c/F4,c (see [Ko7, Example 1.9]).
Chapter 3 Discontinuous Groups
151
E x a m p l e 5.18
G/H
does not have a compact Clifford-Klein form H
G
Range of parameters
SL(n, IR)
Sp(l,]R)
0 < 21 < n
2
SL(n, C)
Sp(I,C)
0 < 2l < n
1
SL(n, C)
SO(l, C)
0 < 1< n
SO*(2n)
U(/,n
l)
31<2n<61,
n>3
Sp(n, IR)
Sp(I,C)
0 < 21 < n
su*(2 )
SO*(2/)
1 < 1< n
SL(2n, C)
su(;,q)
p+q<norp=q
(pq > O)
SL(2n, R)
so(;,q)
p+q<norp=q
(pq > O)
These examples, except for Sp(n,R)/Sp(1, C) with n = 2l, are proved by Theorem 5.8. In applying Theorem 5.8, the choice of "L" is not unique. Here are examples of the choice of "L" for G/H:
SO(1, n-1) U(1, n-1) U([½],n - [½])
SO* (41 + 2)
U(1, n -1)
1 l Sp([5], n - [~]) U(p, 2n - p) or Sp(p, C) SO(p, 2n - p) or Sp(p, R)
for for for
SL(n,R)/Sp(n,R), SL(n, C)/Sp (l, C),,
SL(n,C)/SO(1, C), for SO*(2n)/U(I,n- l), for Sp(n,R)/Sp(1, C), for SU*(2n)/SO*(21), for SL(2n, C)/U(p,q), for SL(2n, R)/SO(p,q).
The proof for the non-existence of compact Clifford-Klein form of the symmetric space G/H := Sp(21,R)/Sp(1, C) is different. In this case we have rank G = 21 = rank H and can apply an argument of Hirzebruch's proportionality principle for the Euler-Poincar@ class (see Section 3). Then,
152
T. Kobayashi
in view of K / H N K = U(21)/Sp(l), the rank condition rank K = 21 > r a n k ( H N K) - l implies the non-existence of compact Clifford-Klein form of G / H (see [No2, Example (4.11)] for details). For more examples of the foregoing type, we refer to Tables 4.4 and 5.3 in [KoT].
E x a m p l e 5.19 G / H does not have a compact Clifford-Klein form
Range of parameters
H
G
m is even
(n>m>l)
m is odd 3 n > -~m + -~
SL(n, R)
SL(m,N)
n > 3m
SU* (2n)
SU*(2m)
3 n > ~m
1
n > -~m+ 2
SO* (2n)
SO*(2m)
3 n > 7m
1
n > -~m
Sp( ,m
Sp(m,N)
n > 3m + 1
SL(n, C)
SL(m, C)
n > 3m
1
n > 3 m + - ~1
SO(n, C)
SO(re, C)
n > 23-m
1
n>3m
Sp(n, C)
Sp(m, C)
3 n > 5m
3
1
3
2
1
n>3m+3+6m,3
21
n > 23-m+ g1 +
(~m,3
Table A
We indicate the proof of Table A for G = SL(n, N) D H = SL(m, N) with m even. We want to show that G / H does not admit a compact 2 Clifford-Klein form if gn > m. We take a subgroup
L'=SO
-~,n-
cG.
We identify a(G) with N n by choosing a suitable coordinate of aa, so that the action of the Weyl group W a ~- ®n is given by permutation of
Chapter 3
153
Discontinuous Groups
coordinates. Then, we have a(L) a(H) -
(Yl,..., Y ½ r n , - - Y l , . . . , - - Y ½ m , 0 , . . . , O ) ' y j E IR (1 _< j _< l m ) } ,
{
( X l , . . . , x m , 0 , . . . , 0 ) ' x j EN, ( l < j < m ) ,
xj --O
,
.__
regarded as subsets in a a / W a . Therefore, we have a(L) c a ( H ) . On the other hand, rn (n _ rn 1 d(L) - d ( H ) - --ff ~ ) - (-~m(m + 1) - 1)
= m(n-~rn-1)+l>0 2 Now Theorem 5.8 shows that G / H does not admit a compact CliffordKlein form. Other cases in Table A are proved similarly from Theorem 5.8. A choice of L for each (G, H) is listed in Table B. A choice of L for Table A H
G (n>m>
1)
miseven
rn is odd
SL(n, N)
SL(m, N)
SO( ~m, n
~_) m
SU* (2n)
SU*(2m)
Sp(mT,n
_ff)m Sp(m~---!,n
SO*(2n)
so*(2m)
u( -ff,n
~)
Sp(n, N)
Sp(m, N)
U( ~-,n m
,~ ~-)
m-1 V( ,n -T)
SL(n, C)
SL(m, C)
U( ym, n
m ~-)
U( C,n
SO(n, C)
SO(m, C)
SO(-ff,nm _ff)m
Sp(n, C)
Sp(m, C)
Sp( ~-,nm Table B
SO(~l,n
-~) -~)
SO( C,n-
_ff)m Sp(-~A, n
-~)
T. Kobayashi
154
Example 5.20
G/H does not have a compact Clifford-Klein form H
G (j > _ i > 0 ,
range of parameters
p>_i, q>_j) j C - q or p > q
or ( p + l ) i q - 1
O(p,q)
O(i,j)
U(p, q)
U(i, j)
jT~q or p > q
Sp(p, q)
Sp(i, j)
jT~q or p > q
mod2
For IF - N, C or H (a quaternionic number field), we write U(p, q;IF) for O(p, q), U(p, q) and Sp(p, q), respectively.
Sketch of Proof:
To prove the results in the table, we first choose L - V(i,q;IF) (IF - R, C, H). Then a(H) - a(L) because i _< j _< q, and d(H) <_d(L). Here, the equality holds if and only if j - q. Therefore, it follows from Theorem 5.8 that U(p,q;IF)/U(i,j;IF) admits a compact Clifford-Klein form only if j - q. Now, U(p, q; iF)/U(i, q; iF) admits a compact Clifford-Klein form if and only if so does a reductive symmetric space U(p,q;iF)/U(p -/;IF) x U(i,q;IF). Then, by Corollary 5.10, we have p > q. See Example 5.14 for the condition (p + 1)iq- 1 mod 2 in the case where IF-R.
K]
Example 5.21 (k times)
i) Suppose G - SL(n, R) D H - SL(2, R) × - . . × SL(2, R) (n _> 2k > 0). If n > 3, then there does not exist a compact
Clifford-Klein form of G / H . ii) Suppose G SL(n,C) D H SL(2, C ) x . . . × SL(2, C) (k times) (n >_ 2k > 0). If n >_ 3, then there does not exist a compact Clifford-Klein form of G/H. In particular, SL(3, C)/SL(2, C) does not admit a compact Clifford-Klein form ([No6, ExampleT]).
Proof: i) Take L - SO(k, n - k). Then a(H) - a(L) and d ( L ) - d(H) k(n-k)-2k - k(n-k-2) > 0 i f n > 4 and i f n >_ 2k > 0. Take L - Sp(2, N) for n - 4.
Chapter 3 Discontinuous Groups ii) Take L -
SU(k,n-k).
2k(n-k)-3k-k(2n-2k-3)
Then a(H) - a(L) and d ( L ) - d ( H ) >0ifn_>2k. [Z]
155
-
For more examples of homogeneous spaces without compact CliffordKlein forms, we refer to [KoT, Table 5.3].
5.14
Notes
The existence problem of a compact Clifford-Klein form of a homogeneous space G / H has its origin in the uniformization theorem of Riemann surfaces due to Klein, Poincar~, and Koebe (Example 1.11). In this case, the homogeneous space is the Poincar~ plane G / H = PSL(2, IR)/SO(2), which is the simplest example a Riemannian symmetric space of the noncompact type. At the beginning of the 1960s, the existence problem of a compact Clifford-Klein form of any Riemannian symmetric space was settled affirmatively by Borel, Harish-Chandra, Mostow, and Tamagawa ([Bol], [BoHa], [MoTa]; see Theorem 5.2). Contrary to this, around the same time, it was found by Calabi, Markus, and Wolf that certain pseudo-Riemannian symmetric spaces (of rank 1) do not admit compact Clifford-Klein forms (Theorem 3.2; [CM], [Wol], [Wo2]; see also Section 3.10). Clifford-Klein forms of the real hyperbolic space SO(p, q ) / S O ( p - 1, q) were studied by R. Kulkarni in the begining of 1980s, and in particular, the Calabi-Markus phenomenon of this special case was settled [Ku]. It was in the late 1980s that a systematic study of the existence problem of compact Clifford-Klein forms was basically begun by T. Kobayashi [Kol-8] for a general homogeneous space of reductive type, which is a wide class of homogeneous spaces containing Riemannian symmetric spaces, reductive group manifolds, pseudo-Riemannian symmetric spaces and semisimple orbits of the adjoint action. An overview is given in [Ko4, Ko6]. A sufficient condition for the existence of compact Clifford-Klein forms (Theorem 5.5) was proved in [Ko2]. The proof rests on an argument in the continuous setting, namely the criterion for proper actions (Theorem 3.18) and the criterion for the compactness of a double coset space (Theorem 5.7). Among six series and six isolated homogeneous spaces that admit compact Clifford-Klein forms in Corollary 5.6, (ii-b) and (iii-b) (i.e., in the case S O ( p + 1, q)/SO(p, q)) were first proved by Kulkarni ([Ku]) and
156
T. Kobayashi
other cases were by Kobayashi (see [Ko2] for (i-a), (i-b), (ii-a), (iii-a); [Ko4, Ko6] for (vi-a)and (vi-b); and [Koll] for (iv-a), (iv-b), (v-a), (v-b)). Conversely, necessary conditions for the existence of compact CliffordKlein forms have been also studied since the late 1980s by various approaches. That is, i) The Calabi-Markus phenomenon and the criterion for proper actions ([Ko2], [Fr], [Kol0]) ii) Hirzebruch's proportionality principle ([KoO]) iii) Comparison theorem ([Ko7]) iv) Construction of symplectic forms ([BL]) v) Ergodic theory ([Zi2], [LaMZ]) We have explained (i), (ii), and (iii) in Section 3 (e.g., Theorem 3.10, 3.18, and 3.19), Section 4 (e.g., Corollary 4.12), and Section 5 (e.g., Theorem 5.8, Corollary 5.10, Corollary 5.13), respectively. We have not dealt with (iv) and (v) here, which are quite different from other methods. However, examples obtained so far by other methods (e.g. [BL], [Zi2], [LaMZ], [Co]) are not necessarily new in the reductive case; most of them are also proved (sometimes in a stronger form) by (i), (ii), and (iii). To clarify the applications and limitations of various methods, we will examine some typical classes of homogeneous spaces. We note that the setting of the foregoing results due to Benoist, Labourier, Zimmer, Mozes, Corlete ([BL], [Zi2], [LaMZ], [Co]) requires either
Zc(H) contains I~
(5.10)
or
Zc(H) contains a semisimple Lie group with I~-rank _> 2.
(5.11)
First, suppose that G/H is a semisimple symmetric space; we assume G/H is irreducible for simplicity. We have seen that the methods (i), (ii), and (iii) give rise to necessary conditions for the existence of compact Clifford-Klein forms, namely, Theorem 3.10, Corollary 4.12 and Corollary 5.10. On the other hand, no new results are obtained for semisimple symmetric spaces if we assume (5.10) or (5.11). In fact, the assumption (5.10) is satisfied if and only if G/H is a para-Hermitian symmetric space. And a para-Hermitian symmetric space does not admit compact Clifford-Klein
Chapter 3 Discontinuous Groups
157
forms (Example 3.15(i)) by the Calabi-Markus phenomenon. The assumption (5.11) is never satisfied in the symmetric case. Second, suppose that G/H is tion (5.10) is satisfied. In this case, (iv), gave an alternative proof of strengthened form ([BL, Theorem
a semisimple orbit. Then, the assumpBenoist and Labourie, using the method Corollary 5.13 (the method (iii)) in a 1]).
Typical examples of homogeneous spaces satisfying the assumption (5.11) are those listed in some parts of Examples 5.18-5.20 (these are not symmetric spaces as we already remarked.) Zimmer proved in particular that SL(n,R)/SL(m,R) (which is referred as the "basic test case" in [Zi2]) does not admit a compact Clifford-Klein form if ~n 1 > m. His approach is based on Ratner's theorem and ergodic theory ([Zi2, Corollary 1.3]). Labourier, Mozes, and Zimmer simplified and extended Zimmer's approach to manifolds locally modeled on homogeneous spaces ([LaMZ]). Their result 1 - m in this example. On the other hand, our method allows the case ~n (iii) (see Section 5; [Ko4, Ko6], [Ko7]) applied to this special case shows that SL(n,R)/SL(m,R) does not admit a compact Clifford-Klein form if 2 2 5n > m for even m (or if gn > m + 1 when m is odd), which gives a sharper result in most cases (when n and m are large enough; see Example 5.19). Similarly, the indefinite Stiefel manifolds U(p,q;F)/U(i,j;F) (1F = R, C,H) (j _> i > 0, p _> i, q _> j) do not admit compact Clifford-Klein forms if j ¢ q or if p > q, as we explained in Example 5.20 as another example of our method (iii). Again, the very special case with (p,q,i,j) = (n, 2, m, 1), n > 2rn, and F = SI implies a result that Sp(n, 2)/Sp(rn, 1) (n > 2m) does not admit a compact Clifford-Klein form, which was announced by Corlette in ICM-94 ([Co, Theorem 12]). Thus, Corlette's results were not new, but his approach apparently differs from methods (i)-(v). Since the beginning of the 1990s, the existence problem of compact Clifford-Klein forms of homogeneous spaces with noncompact isotropy subgroups has interacted with other branches of mathematics. For example, in the classification of certain Anosov flows, Benoist, Foulon, and Labourier [BFL] encountered the nonexistence problem of compact Clifford-Klein forms of
G/H where G is real reductive and H is a semisimple part of
a Levi subgroup of a maximal parabolic subgroup.
Also, symplectic ge-
ometry [BL], ergodic theory and Ratner's theory [Zi2], [LaMZ], harmonic maps [Co], restriction of unitary representations [Ko9]
have come to be
T. Kobayashi
158
interacted with the existence problem of compact Clifford-Klein forms. Many basic questions about Clifford-Klein forms of non-Riemannian homogeneous spaces have not yet found a final answer. As observed in the recent developments just mentioned, it is fascinating that different areas of mathematics seem to be closely related to the existence problem of compact Clifford-Klein forms.
6
Open problems We collect some open problems. i) (Problem 2.10(ii)) Find a criterion for the Calabi-Markus phenomenon for a general Lie group. ii) (Problem 2.17)
Find a criterion for L ch H in G if
L,H are closed
subgroups of G. iii) (Problem 2.19) In which class of Lie groups, does the following equivalence hold? H ch L in G ¢, the pair (L, H) has the property (CI) in G.
iv)
v) vi)
(Problem 3.7) Is there a sufficient condition on some positiveness of the curvature of a pseudo-Riemannian manifold M (in particular, Lorentz manifold) assuring that M is noncompact with a finite fundamental group? (Problem 2.10 (ii)) Find a criterion on G/H that admits a compact Clifford-Klein form. Also, find a criterion on G/H that admits a noncompact Clifford-Klein form of finite volume. Suppose that G/H is a homogeneous space of reductive type that admits a compact Clifford-Klein form. Is there a subgroup L reductive in G which satisfies both of the conditions (5.3a) and (5.3b)?
vii) (Conjecture 4.15)
Does the inequality
rank G + r a n k ( H N K) >__rank H + rank K hold if
G/H admits a compact Clifford-Klein form?
viii) Is there a Teichmiiller theory for a compact Clifford-Klein form?
Chapter 3 Discontinuous Groups
159
ix) If there exists a compact Clifford-Klein form of G/H, then does there also exist a noncompact Clifford-Klein form of G/H of finite volume, and vice versa? x) ([Wal]) Is there an analog of Eisenstein series that describes the decomposition of or if
F\G/H
L2(F\G/H)if F\G/H
is a compact Clifford-Klein form
is a noncompact Clifford-Klein form of finite volume?
We have already explained most of these problems.
Here are some
short comments for the convenience of the reader. Problem (i) is solved for the reductive case and the simply connected solvable case as we explained in Section 3. Problem (ii) is solved if G is a reductive group. Problem (iii) is a subproblem of (ii), and in particular, the property (CI) might be the final solution for a simply connected nilpotent group (the conjecture of Lipsman). Problem (iv) is a question in pseudo-Riemannian geometry. It is regarded as a "perturbation" of the Calabi-Markus phenomenon, and should be in a good contrast to a classical theorem of Myers [My]. Problem (v) has been one of the main subjects of this chapter. For example, SO(i +
j,k + 1)/SO(i, j)
× SO(k,/) remains open (see Example
4.13.1 for partial results obtained so far). Problems (vi) and (vii) are subproblems of (v). Problem (vi) asks if a converse of Theorem 5.5 holds. This is a delicate problem because the Zariski closure of a uniform lattice for G/H of reductive type does not always satisfy the conditions (5.3a,b). We have found that an analog of the Weil rigidity does not always hold for semisimple symmetric spaces of higher ranks, and that an analog of the Mostow rigidity does not always hold for semisimple symmetric spaces of higher dimensions [KoS]. There seems to be much to study in the deformation of uniform lattices, in which area we pose Problem (viii). Regarding Problem (x), we recall that an existence result for compact Clifford-Klein forms (or noncompact ones of finite volume) of Riemannian symmetric spaces (Theorem 4.2) has opened a theory of Eisenstein series in harmonic analysis on square integrable functions over the double coset space
F\G/H.
It is natural to expect that an existence result for pseudo-
Riemannian symmetric spaces could open a theory of harmonic analysis on such nice double coset spaces.
T. Kobayashi
160 Acknowledgment
The author would like to express his sincere gratitude to Professor Bent Orsted and Professor Henrik Schlichtkrull for inviting me to give lectures and for their heartful hospitality in European School of Group Theory, August 1994.
Postscript
December 21 1995
There has been some recent progress on the open problems in Section 6. Y. Benoist gave a different proof of Theorem 3.19(i) and proved a nonexistence theorem of compact Clifford-Klein forms of some homogeneous spaces such as SO(4n, C ) / S O ( 4 n - 1, C) (see Example 5.16) in [1] (see Open Problem (v)). t~. Ghys studied a fine structure of the deformation of a lattice for a group manifold G' × G'/diag G' with G' = SL(2, C) in [2], and T. Zobayashi studied for which homogeneous manifolds of reductive type local rigidity of a uniform lattice fails in [3] (see Open Problem (viii)). Y. Benoist, Actions propres sur les espaces homogenes reductifs, Preprint. [2] t~. Ghys, Ddformations des structures complexes sur les espaces homog~nes de SL(2, C), J. reine angew. Math. 468 (1995), 113-138.
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~ , @aces of Constant Curvature, 5th ed., Publish or Perish, Inc., Wilmington, Delaware, 1984.
[Yo]
K. Yosida, A theorem concerning the semisimple Lie groups, Tohoku Math. J. 44 (1938), 81-84.
[Zil]
R. J. Zimmer, Ergodic Theory and Semisimple Groups, Birkh~user, Boston, 1984.
[Zi2]
~ , Discrete groups and non-Riemannian homogeneous spaces, J. Amer. Math. Soc. 7 (1994), 159-168.
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Chapter 4 The Method of Stationary Phase and Applications to Geometry and Analysis on Lie Groups V. S. VARADARAJAN
University of California, Los Angeles
CONTENTS
1. Introduction and examples 2. Oscillatory integrals with nondegenerate phase functions 3. Additonal remarks on singularities and oscillatory integrals 4. Phase functions on homogeneous spaces 5. Spherical oscillatory integrals References
Introduction
It is well known that many problems arising in optics and electromagnetic theory, number theory, probability theory, and quantum physics lead to questions involving integrals of rapidly oscillating functions, especially the asymptotic behavior of these integrals as the frequency parameter becomes large. The method used in these studies is invariably what is known as the method of stationary phase that goes back to Stokes and Kelvin. It is equally the case, although less widely known, that oscillatory integrals and their asymptotics play an important role in many more modern parts of mathematics such as topology of manifolds, geometry and analysis on homogeneous and locally homogeneous spaces, differential equations on complex manifolds, and so on. In these notes, we shall attempt to give 167
V. S. Varadarajan
168
a brief introduction to oscillatory integrals and the method of stationary phase, and explain how it relates to the various areas of mathematics we have mentioned.
1
Oscillatory integrals. Introduction and examples
1.1 T h e p r i n c i p l e of
stationary phase
Let us begin with a classical situation in optics, namely the problem of determining the radiation at a point x in space, coming from a surface S that is emitting monochromatic radiation of very short wavelength isotropically from each of its points (see [GS]). If the frequency of radiation is T, it is easy to conclude by looking at spherically symmetric outgoing wave solutions of the wave equation that the total radiation at x (at a fixed time t, which we shall promptly forget) is a superposition of the radiations associated to the various points of the surface and so can be written as an integral
I(x " T) --
~S eirlx-y[ IX Z y[ a(y)dy.
The problem is that of examining the behavior of this integral as 7> c~. In optics, this is the problem of determining the intensity [I(x : T)[ 2 of the radiation at x in the regime when wave optics is replaced by geometrical optics. The integral I(x : T) is an example of an oscillatory integral because the function
eir[x--Y[
oscillates very rapidly when T is very large. The real-valued function =
ix - yl
(y E S, x ~ S)
is called the phase, and the possibly complex-valued function
a(y) ax(y) - I x _ y[
(y e S, x ~ S)
is called the amplitude. Note that both the phase and amplitude, especially the phase, depends on x. Analytically, x plays the role of a parameter that governs, in a way that we shall explain now, the behavior of this integral.
Chapter ~
Method of Stationary Phase
169
The basic method of dealing with this integral is to appeal to the principle of stationary phase of Stokes and Kelvin, which asserts the following: The principal contributions to I ( x : z) come from those points of S that are critical points of the phase ~ , namely, those points y at which the first order derivatives of p~ (along S) all vanish.
If, for instance, there
are no critical points, then for amplitudes that are smooth and compactly supported, the integral is a rapidly decreasing function of z; this depends of course on the location of x, and such x are said to be in the shadow or the dark zone:
I ( x " z) - O ( z -r)
V r >_ 1
(x in shadow).
Suppose now that there are critical points in S for ~;~ but that these are all nondegenerate in the sense that the Hessian matrix, namely the matrix of second derivatives (along S again) at each critical point, is nonsingular. It is then possible to show that the radiation is of the order of magnitude of T -1 at these points x. Since the intensity of light at these points x is much larger than at the points in the shadow, these points are said to form the light zone. More precisely (this is the technical content of the principle of stationary phase),
I(X " T) , ~ / T -1
(x in light zone).
where -y is a constant depending on the amplitude ax and is not identically 0. However, as x varies in space, it often happens that there will be points for which the associated phase ~x has degenerate critical points. Typically, this has the consequence that the radiation at x is of the order 7 -d where d < 1. Such points form a surface C in space and have neighboring points that are in the light zone; since d < 1, the intensity at their location is very much higher than at the neighboring points of the light zone. Light burns at these points, and so they have been known traditionally as caustics. The surface C is called the caustic surface of the phase ~x. The foregoing classification of points of space into light, dark, and caustic zones (see [AGV]) can be refined a little more if we assume that the surface is opaque to the radiation. The total radiation at x is then an integral, not on the entire surface S but only on the part visible from x. This is a region R~, which now has possibly a boundary, and there will be
V. S. Varadarajan
170
locations x at which I ( x : T) is of the order 7"-3/2. These are called points in the twilight zone: I ( x : 7-)~ ~T -3/2
(x in twilight zone).
In this example, it is easy to check that the time dependent radiation is
eirtI(x : 7-), and that it satisfies the wave equation exactly; in fact, it was derived from this point of view.
One can turn this around and ask, for very general
hyperbolic equations, whether asymptotic solutions can be expressed as such integrals, where we now allow amplitudes that depend on 7- and have asymptotic expansions in T.
Here, asymptotic solution means that the
integral satisfies the equation in question to greater and greater accuracy as 7 becomes large. Thus, the study of oscillatory integrals is i m p o r t a n t from the point of view of classical hyperbolic partial differential equations. Summarizing the preceding discussion, we may say the following. In problems of high frequency optics and classical partial differential equations, one comes across integrals of the form
I ( x " T) - I ( x " a " 7 ) -
/ e i~¢x'y)a(x,y, T)dy.
Here, T is called the frequency; a is the amplitude, which is a smooth complex valued function, defined on X × Y × ]t~ where X C ]Rk, Y C ]~n are open sets and ~
is the set of positive real numbers; p is the phase,
which is a smooth real function defined on X × Y; and there is a compact set K C Y such that a(x, y, T) = 0 if y ~ K. The problem is to understand the limiting behavior of the integrals I ( x : T) when 7 is very large. This is mathematically formulated as the problem of studying the behavior of
I ( x • T) as T
~ C~. Since for large T the function e i~(x'y) oscillates
rapidly, t h e integrals such as I are called oscillatory integrals. Typically, the amplitudes do not depend on v. But if they do, it is natural to assume t h a t their dependence on 7 is expressed by asymptotic expansions
ar(X, y)T t~-r,
a ~
r)0 ~
r~>O
Chapter ~
Method of Stationary Phase
Here, ~ refers to asymptotics as ~-
171
~ oc and means that for any integer
N _> 0, the difference of the left side and the sum up to r - N of the right side is O(~-~N), where #N ) oc with N, and the O is locally uniform in x and c~; the subscripts p, q denote differentiations with respect to x (p times) and y (q times). The simplest special case of such integrals are those of the form P
I(~-) - /eir~(U)a(y)dy
(a e C ~ ( K ) )
where there are no parameters. Using partitions of unity in the amplitudes, it is clear that we need to study this problem only locally, i.e., at each point of Y for amplitudes supported within arbitrarily small neighborhoods of this point. The Principle
of stationary phase in this context asserts that the essential contribution to the integrals comes from the stationary (critical) points of the phase function ~p; the other points contribute only terms that are rapidly decreasing in ~. We can state this in more precise language (see [D2])" P r i n c i p l e of s t a t i o n a r y
phase:
Let ~p be a phase function, and let
- { z e K Id
(z)-
0}.
Then, modulo rapidly decreasing functions of T, the behavior, as ~of the integral P
- /
~ eo,
a(y)dy
depends only on the behavior of the amplitude a in arbitrarily small neighborhoods of S~. Moreover, if a is localized at a critical point of ~p that is nondegenerate in the sense that its Hessian matrix (the matrix of second derivatives) is nonsingular, and if n is the number of integration variables, the integral is asymptotic to (~)
n/21det Hl-1/2ei~/nsgn(H)eir~(°)a(O),
where sgn(H) is the signature of H. Note that there is a change of phase in the asymptotic form of the oscillatory integral as the signature of the Hessian changes. In particular, from the m i n i m u m point (sgn = n) to the m a x i m u m point (sgn = - n ) the phase changes by n~/2; for n = 2, this is a phase shift of 180 °, the classic case in geometric optics.
V. S. Varadarajan
172 1.2 M o r s e t h e o r y
In spite of the rather specialized nature of the origins of the principle of stationary phase, it has become one of the main tools in modern analysis and its applications have become extraordinarily diverse. In this subsection we shall try to elucidate this a little more. First of all, the study of the behavior of oscillatory integrals leads directly to questions involving the local structure of smooth functions and maps.
Indeed, since integrals transform very simply under diffeomor-
phisms, we see that there is a direct relationship to the problem of describing the simplest possible forms of functions and maps when we are allowed to make arbitrary C a change of coordinates. The first such theorems were proved by Marston Morse in his epochmaking paper [Mo] when he showed that in the neighborhood of a nondegenerate critical point a smooth function can be transformed to a quadratic form, namely to its Hessian form, by a C a change of variables. As is well known, Morse used this result as a stepping stone to his famous theorems relating the structure of critical points of a function to the topology of the space on which the function is defined, and he used these in turn to obtain his fundamental theorems on the calculus of variations in the large. Since then, Morse theory has become a fundamental tool in analysis and geometry, and it has been shown to have many interesting applications to questions of geometry of homogeneous spaces. While Morse theory deals with the relationship between the topology of a manifold (both finite and infinite dimensional) and the structure of the critical points of a generic function defined on the manifold, the optical example discussed at the beginning focuses on an entirely different theme! Here we have a manifold M on which is defined somehow a family of real smooth functions ~(x, m) (x E X, m E M), X being the space of
parameters. For generic values of x, the function ~x "- ~(x, .) has only nondegenerate critical points on M (is a Morse function, as it is nowadays called); but for special values of x, it may happen that the critical points become degenerate. The behavior of the oscillatory integral I(x
" T) --
/M eiT~(x'm)a(m)dm
changes dramatically as x passes through these points, which are called, in analogy with the optical example, caustics.
Chapter ~ Method of Stationary Phase
173
As long as we are dealing with a single function, it is not an essential loss of generality to assume that it is a Morse function. This is due to the result that the Morse functions on a compact smooth manifold M form a dense open set; thus, a small perturbation (which can be attributed to imperfect measurement) will change any given function into a Morse function.
This is true locally also.
Thus, the function x 3 on R is not
Morse, as its critical point at 0 is degenerate; but the neighboring function x 3 - 3~2x for small real but nonzero c is Morse, because its two critical points are at +~ and are both nondegenerate. We thus see that a small perturbation has made the degenerate critical point 0 of x 3 split into a pair of nondegenerate critical points. We speak of this as a
Morsification
of the
function x 3. This might lead one to suppose that there is no physical need to study the phenomena of degenerate critical points. But to do so would be an error. The point is that, as in the optical example previously discussed, phase functions of physical problems often arise in a
family
depending on a
parameter of physical significance (such as the location of a point in space), and one has to perturb
this family
to see if the degenerate critical points
will disappear. It can be shown that this cannot always be done. Thus, in the example given just now, the family x 3 - 3c2x has a degenerate critical point at 0 when c -- 0; but, in every neighboring family, there will be a parameter value for which the phase has a degenerate critical point. This is not surprising, since, in the optical example, the degenerate critical points correspond to the caustics and so have real physical significance. In this way, one is led to the problem of describing and classifying, up to local and global diffeomorphisms, smooth functions and families of smooth functions, and to study how the degenerate members of the classification change into generic members under perturbations. This is nothing but the modern theory of singularities of smooth functions and maps, to which many mathematicians, most notably,
Mather, Arnold, Gel'land, Bernstein,
Whitney, Thorn, Malgrange,
and many others, have made remark-
able and profound contributions, both conceptually and technically.
1.3 Some e x a m p l e s We shall illustrate the preceding remarks with some examples. i)
Shadow
The oscillatory integral is
I(T) -- JRfnei~Xla(x)dx
(a e C~ (Rn)).
V. S. Varadarajan
174
This is rapidly decreasing in r by the fundamental result of Fourier transform theory. The phase has no critical points. ii) Light zone
The classic example is that of the Fresnel integral, namely
I(r) -- ~ ei~X2/2a(x)dx
(a E C ~ (N)).
This is evaluated using Fourier transforms. For any Schwartz function f on R, we define its Fourier transform 9rf by
cR). The Fourier inversion formula says that for two Schwartz functions f, g we have
f (x).q(x)dx - ~
:Ff (-~)~g(~)d~.
We now take f(x) = exp(-crx2/2) where a > 0 and get, for our amplitude a,
-
e-~2/2aT'a(~.)d~.
Now, both sides are analytic functions of a in the domain ~ ( a ) > 0, and so this equality persists in this domain; the function ~-1/2 is defined as r-1/2e ie/2 in polar coordinates with a = re i°, 101 < ~/2. If we now let cr converge to - i T (v > 0), we get (using the dominated convergence theorem on both sides)
~ eirx2/2a(x)dx - 7-1/2ei~/4
e-i~2 /2r.~a(~)d~.
(1.1)
The right side has 7 in the denominator in the exponential, and so, on expanding the exponential we get an asymptotic expansion in powers of r -1. From the Taylor expansion of e iu (u E R), we get TM
e iu --
1 + iu +
2!
+"" +
rn!
)
]U] rn+l (m + 1)!'
whereas the formula ~ ' ( a ( 2 m ) ) ( ~ ) - (--1)m~2m~a(~) gives a(em)(o) -- (-1)m2~- ~~2m'7"a(~)d~"
Chapter 4
Method of Stationary Phase
175
Putting these together and integrating the expansion
e-i~ 2/2~- = Z (--i)kT"-k~2k 2 k k! k>0 we get
I(7")- - (271") __ 1/2 e i ~ / 4 7"
E
ik
~ a ( 2 k ) ( 0 ) 7 " -k + O(7"-('~+(1/2))). O
In particular, taking m = 1 we get
(~) I(7.)
1/2
--
eiTr/4a(O)
-+-
0(7" -3/2)
(7"
> c~).
If Em (7. : a) is the error term in the first expansion, it is easy to show from our remarks that we have the estimate
IEml _< ~ll?(a(2~))ll~ T--(m+1)~2
v~2m
iii) Twilight zone We now consider a Fresnel type integral taken over a domain with boundary, namely, the half-space {x _> 0} in the plane IR2 with coordinates x, y: I(7" "a) -- f f J~~>o eiT(x+(1/2>y2)a(x, y)dxdy
(a e Cc~(R2)).
We now integrate by parts with respect to x to obtain
I(7. " a) -- i
ei'y2/2a(O, y)dy +
7"
e ir(x+(1/2)y2)ax(x, y)dxdy, 7"
>0
which leads to
I(7. " a)
-
-
ix/~7.-3/2eiTr/4a(O, O) + 0(7- -5/2)
and
I(7." a)
-
i i~//~7"-3/2e i7r/4 a(0, 0) + -(ax(0, 0) + 1 7"
-~ a y y
(0, 0)) ] + 0(7. -7/2)
,
V. S. Varadarajan
176
and so on. We leave it to the reader to verify that the general expansion is
I(T " a) -- ix/r~T-3/2e i~/4
E
i~A~(a)T -~ + O(T -(m+3/2))
O
with
At(a) -
E
2-ka(r-k'2k)(O' 0).
O'~ k "~ r
iv) Oscillatory integral associated to x k in I~: caustics
This is the most
basic case in which we encounter degenerate critical points. The family of phase functions is " x)
-
x 3 +
ax
(a, x E R).
For a > 0, the phase is strictly increasing and has no critical points; for < 0, there are two critical points, at + V / l a l / 3 , both of which are nondegenerate, where the Hessians have opposite signs. For very small a, these are very close to each other. When a = 0, the critical point is 0 and is degenerate.
From our classification, we see that the region c~ > 0 is the
shadow or the dark zone, the region c~ < 0 is the light zone, and the origin = 0 is the caustic. The oscillatory integral is
~ eir(~ 3+~x) a(x)dx.
(1.2)
It was first considered by Airy in 1836 in a famous paper JAil, perhaps the first to investigate the behavior of light in the neighborhood of a caustic. Airy mentions that the study of this integral is at the foundation of the theory of the rainbow, and that he has been unable to integrate it in terms of the functions known at that time. We now know not too much more, except that we call it the Airy function and denote it by Ai(x): Ai(t) = j f e i(x3 +tx) dx. The singularity of x 3 at 0 is called a fold, and it splits into two nondegenerate singularities =i=V/lal/3 when we go over to the nearby function
x 3 + ax. The family x 3 + ax is thus called an unfolding (or a Morsification in this case) of the function x 3. There are asymptotic expansions of the
Chapter ~ Method of Stationary Phase
177
integral (1.2) uniform in the parameter c~, but they will not be in powers of 7-1; rather they are in terms of the Airy function (1.3) (see [D3]). In terms of powers of T, one can get only uniform estimates
~ eir(x3+ax)a(x)dx
< #(a)7 "-1/3
(a e C ~ ( R ) ) ,
where # is a continuous seminorm on the Schwartz space of N. Of course, for (1.3) there will be an asymptotic expansion in powers of T -1. We shall derive this now; it goes back, as far as I know, to Erdelyi [Er]. We consider
I(~-" a) - ~ ei'~ka(x)dx
(a e Cc~(N)).
To treat the neighborhood of 0 separately, we introduce a function b E C ~ (N) that is 1 in a neighborhood of the origin, and we write
I--I1 +I2, where 11 - -
f ei~ka(x)b(x)dx,
h - / ei'Xka(x)(1 - b(x))dx.
The amplitude a(1 - b ) i n / 2 vanishes in a neighborhood of the origin, and so we can integrate by parts using the differential operator D" 1 D
Let us write
tD
__
d
kX k- 1
dx '
D (e irxk )
- (iT)-le/rxk
for the transpose of the operator D. Then,
is still in C ~ ( R ) , so that
I2(T" a) -- (iT)-N fR ei~xk(tDg (a(1
--
which shows t h a t / 2 is rapidly decreasing in T:
112(
-a)l _<
--N
VN>I,
b)))dx,
tDN(a(1 -- b))
V. S. Varadarajan
178
where #N are continuous seminorms on the Schwartz space of IR. The treatment of I1 requires more care. Again we integrate by parts, integrating the function e i~xk repeatedly. We split the integral into two, I ~ , defined by the integrations over the positive and the negative semiaxes, and rewrite the integral I~- as an integral over the positive semiaxis. Thus,
I1 ('7-" a) --
~o~
eir~ka(x)b(x)dx +
fo ~
e (-1)ki'zk a ( - x ) b ( - x ) d x .
Note t h a t for k even the integrals are of the same nature, whereas for k odd they have r entering with opposite signs.
The basic idea in [Er] is
to define the repeated integrals of the exponential as an integral in the complex domain; the line of integration depends on the sign in front of r. For any integer m >_ 0, we define 4-
fm+l (U) -- (--1) m+
1 ~u u+e
"+~Tr/ 2k (:XD
(z
~m' ~+~Z~dz~ m!
(~ > O, ~" > 0),
where the notation means that the integration is over the ray
{v ,
>u + ve +i'~/2k [ v > 0}.
It is clear that
f(m+l) rn+l ( U ) - ei~-~
(U > 0).
Moreover, f m + l (0)
-
-
( - - 1 ) m + l e 4.i(m+1)Tr/2k
1
kin!
k
To estimate the function f m + l itself, we note that +
k -
+i~
(±iv ~ +...),
where the ellipses indicate terms the imaginary part of which are of the form +iwsin(sTr/2k) with w > 0 and 0 <_ s <_ k. Hence, on the line of integration, we have
le±i~zk l <_ e-~lz-~lk.
Chapter ~ Method of Stationary Phase
179
Hence, 1
_(m+l)/kF ( m + 1 )
Ifm+l(U)l <_ -ff~mw.~-
k
(u > 0).
After these preliminaries, we can now integrate by parts the integral I1 and get the following: If
Ilt:(¢ " a) -
e+i~Xka(x)b(x)dx,
then
Ili(~ - " a) -
E
~s.vleii(s+l)~/2kF (s+k 1 ) a(~)(0)~__(8+l)/fi + Era(z-"a),
O<_s
where IEm(~ : a)l <_ pm(a)T -(m+l)/k pm being a continuous seminorm on the Schwartz space of R. Write now
Ik(~ - " a ) -
jfei"~ka(x)dx
(a e C~(N)).
We consider the case of even and odd k separately. The #m denote continuous seminorms on the Schwartz space of N. k even
We have
ik(~_ : a) •
_- 2 k o_
-~Le ~1( 2 ~ + l ) / 2 k r ( 2 s ;
1 ) a(2S)(O)T -(28+1)/k
m (2s)!
+ Em(~- " a),
where IEm(T : a)l <_ #m(a)T -(2m+l)/k It is not difficult to verify that this agrees with the asymptotic expansion derived earlier by direct use of the Fourier transform when k = 2. We leave it to the reader to do this.
V. S. Varadarajan
180 k odd
Define Cks
_
Then,
1 ~
Ik(T " a) - -~ O<~<m
ei(s+l)/2k + (--1 )s e -i(s+
1( s+l)
~r
k
where I E m ( T " a)l
1)/2k.
~ksa(S)(O)7"-(s+l)/k + Era(7" "a),
~_~m(a)T-(m+l)/k.
In particular,
I3(T " a) -- F
4) a(O)T_l/3 + E,
-~
IEI <_#(a) T-2/3
The leading term of the asymptotic expansion is now ~_-1/3 instead of the
T -1/2
that is characteristic of the points for which the phase is a
Morse function.
The reader should also note that the derivation of the
asymptotic expansion is considerably harder than in the generic case of Morse functions. This is fairly typical. 1.4 Phase
functions on homogeneous
spaces
When working in the geometric context of homogeneous spaces, there is a great deal of symmetry, and the phase functions that are significant display this symmetry in their definition. The typical situation is the following. We have a connected Lie group G and a linear representation of G in a finite dimensional vector space V over the real or the complex field. We suppose that there is a bilinear form (., .) defined on V × V that is G-invariant. If G is compact, such a form always exists, and we may even suppose that it is positive definite. Let v E V be a point such that the orbit My of v under G is closed in V; if Gv is the stabilizer of v in G, we have an analytic diffeomorphism
G / G ~ ~ M~, and
so
~(~: gG~):= (g.~,~>
(w e V)
Chapter 4
Method of Stationary Phase
181
is a well-defined phase function on G/G~ ~ My. Let us write
L~(u) = (w - u, w - u)
(u E M~).
Then, it is immediate that
L~(g.v) = (w, w) + (v, v) - 2 ~ ( w : gGv), showing that, except for the possibility that (u, u / may be an indefinite
metric, these are just the generalization of the phase functions coming from the optics example; and, in the case when G is compact and the metric is positive definite, they are exactly the phase functions defined by taking the distance from an arbitrary point. When G is compact or semisimple, this method leads to some very interesting phase functions that have a lot of significant information that is useful in the topology and analysis on the homogeneous spaces G/Gv. Later, we shall study these cases in greater detail. We shall obtain many well-known results from this point of view: the conjugacy of maximal abelian subspaces of s, where ~ - [~ ® s is the Cartan decomposition of a real semisimple Lie algebra ~; the beautiful theorem of Kostant t h a t the Iwasawa projection of (exp H ) . K in a is the convex hull of the Weyl group transforms of H; computations of the cohomology of complex flag manifolds; estimates for the matrix elements of the principal series representations of a connected real semisimple Lie group with finite center; applications of these estimates to the problems of error estimation of spectra in locally symmetric spaces of negative curvature; and so on. It may not be out of place to discuss briefly an interesting nonclassical aspect of the phase functions defined on the flag manifolds via the Iwasawa decomposition. Let G be a connected real semisimple Lie group with finite center, and let G = K A N be an Iwasawa decomposition of G with a the Lie algebra of A. We then have an analytic map H:G
~a,
x,
~H(x),
where H(x) is defined by x = u exp H ( x ) n
(u E K, H(x) E a, n E N).
V. S. Varadarajan
182
The matrix elements of the principal series representations of G are of the form
K eir(H(xk))g(k)dk, where ~- C a*. Although the integral is written over K, it is really over
K/M, where M is the centralizer of A in K, so that we have oscillatory integrals on the flag manifold K/M. But there is a new element; the frequency variable is now vector-valued since it varies in a*, and so the asymptotics are much more subtle than in the classical case where there is only one frequency variable. This is not very surprising; for example, there is a deep analogy between problems of scattering theory and problems of harmonic analysis on G/K; but when the rank of the symmetric space
G/K is > 1, the past and future of conventional scattering are replaced by the chambers defined by the Weyl group, and the single scattering operator of the conventional theory is replaced by a family of scattering operators, going from an initial chamber to a final chamber. The situation in the case of oscillatory integrals with several frequency parameters is approximately similar. This is a situation (together with the scattering analogy just mentioned) that seems to merit some further examination from a general point of view.
2
Oscillatory integrals with nondegenerate phase functions
2.1 L o c a l s t r u c t u r e of s m o o t h m a p s We begin with a brief discussion of the local structure of smooth functions and maps between smooth manifolds. These are always assumed to be second countable and without boundary unless otherwise stated. Let M, N be smooth manifolds, and let f be a smooth map of M into N: f: M
~ N;
and let
xeM,
f(x)=yeN.
A basic problem is to understand the structure of f in a neighborhood of x, especially the problem of describing the simplest possible forms of f that
Chapter ~ Method of Stationary Phase
183
arise when we subject f to an arbitrary smooth change of coordinates in M and N. We write f" (M, x)
, (N, y)
and keep in mind that we are only interested in the germ of the map f. There is an obvious notion of isomorphism for such maps. Let G~ be the group of germs of diffeomorphisms g" (M, x)
ho f og -1 for the (germ of the) map t ,
(g~G~,
~ (M, x). If we write h~Gy)
, h(f(g-l(t))), we see that
((h,g),f),
,hofog
-1
gives an action of Gy x Gx on the set of germs of the maps f. in the same orbit are the ones that are isomorphic to each other.
Maps The
problem is thus one of classifying the orbits for this action, or at least to describe generic orbits, where generic is to be interpreted appropriately and reasonably. Note that this problem makes sense in the complex analytic
category also. Let Tx(M) be the tangent space to M at x. We denote as usual by dfx the tangent map of f at x:
dfx" Tx(M)
, Ty(N).
Write m - d i m ( M ) , n - dim(N). i) If dfx is surjective (then m > n, and f is called a submersion at x), then f is isomorphic to the projection (Xl,...,Xm)
I
, (Xl,...,Xn).
ii) If dfm is injective (then rn _< n, and f is called an immersion at x), then f is isomorphic to the injection
(Xl,...,Xm)
'
)
(Xl,...,Xm,O,...,O).
Both (i) and (ii) are valid in the complex analytic category. If m - n, in either of these cases, f is a local diffeomorphism (or a local analytic isomorphism in the complex analytic case).
V. S. Varadarajan
184
iii) By a critical point of a map f ( M
~ N) we mean a point x where f is neither immersive nor submersive, i.e., dfx is neither injective nor surjective; f ( x ) is then called a critical value. If N = R so t h a t f is a real function, these definitions agree with the usual definitions of critical points and critical values of a function. The f u n d a m e n t a l result
on critical values is Sard's theorem: 2.1 (Sard) The set of values of a smooth map at points at which it is not submersive has measure zero.
Theorem
We shall not give a proof here but indicate it in the exercises.
In
local coordinates, measure zero is u n d e r s t o o d to mean measure zero in the Euclidean space defined by the coordinates; since this p r o p e r t y is preserved under diffeomorphisms, it makes sense to speak of a set of measure zero in a coordinate neighborhood without specifying the chart; and as the manifold can be covered by a countable number of coordinate neighborhoods, it is clear t h a t measure zero makes sense globally: A set is of measure zero if and only if its intersection with each coordinate open set has measure zero. For arbitrary dimensions of M and N, even the local classification of maps under local diffeomorphisms is a very difficult problem. Nevertheless, a great deal of u n d e r s t a n d i n g has been gained in recent years on w h a t h a p p e n s generically. For functions (real s m o o t h or complex analytic) the situation is much better. For a function f or a family of functions, it is a question of finding canonical forms in the neighborhood of a critical point. T h e most f u n d a m e n t a l result of this type goes back, as we mentioned in the introduction, to Morse. It is called Morse's lemma. To formulate it we first define the notion of the Hessian form of a function at a critical point. Definition-Proposition
2.2
Let f be a s m o o t h function on M , and let
x be a critical point of f . T h e n the Hessian form of f at x is the symmetric bilinear form Hz defined on Tx(M) x T z ( M ) as follows: If u, v E T z ( M ) , and U, V are vector fields defined around x such t h a t Uz - u, Vz - v, t h e n
= (UV y)(x). We must show t h a t this makes sense. First, note t h a t there is symmetry in u, v because the difference ( U Y f ) ( x ) - ( V U I ) ( x ) is ([U, Y ] f ) ( x ) = O, since x is a critical point. To see t h a t Hx is well defined, note t h a t if u = 0,
Chapter ~ Method of Stationary Phase
185
the expression for Hx (u, v) is 0 because Ux = 0; if v = 0, the same conclusion is valid, because we can replace (UVf)(x) by (YUf)(x). If t l , . . . , tm are local coordinates around x taking x to 0, then the matrix of Hx in the basis cO/Otj (1 _< j < m) is
02f cOti Otj / o '
where the subscript 0 means that the derivatives are calculated at 0.
D e f i n i t i o n 2.3
Let f ( M
~ R) be a smooth function and x C M a
critical point of f. Then x is said to be nondegenerate if the Hessian form Hx at x is nondegenerate. If all the critical points of f are nondegenerate, f is said to be a Morse function.
A nondegenerate critical point is always isolated. In particular, for a Morse function, the critical set is discrete, and so is finite if M is compact. Proposition
2.4
Proof: The first assertion is local, and so we may assume t h a t f is a map of a neighborhood of 0 in It(TM and has 0 as a critical point. We then have the gradient map
Vf'(tl,...,tm),
of
of )
> ( Otl , . . . , Ot---~ '
and the matrix of its tangent map at the origin is just the Hessian matrix
02f O t i O t j ) O"
So if this matrix is nonsingular, which is the content of the assumption t h a t 0 is a nondegenerate critical point, V f is a local diffeomorphism in a neighborhood of 0, and so, in that neighborhood, 0 is the only point that maps to 0. Since the points where V f = 0 are precisely the critical points of f, it follows that there are no critical points in a sufficiently small neighborhood of 0. The remaining assertions are trivial. [:]
V. S. Varadarajan
186 2.2 M o r s e ' s l e m m a
We shall now state and prove Morse's fundamental lemma, which describes completely the local structure of a map in a neighborhood of a nondegenerate critical point: It says that in a neighborhood of a nondegenerate critical point, the map can be written as a quadratic form in a suitable chart. We shall actually treat the case with parameters for greater flexibility in applications. T h e o r e m 2.1 (Morse's lemma with parameters) Let f E C ~ ( X x Y), where X x Y is an open neighborhood of (0,0) in IRk x R n, and write
fx(Y) = f ( x , y ) . Suppose that
d 2 fo(O)
d fo(O) - o,
(that is, 0 is a nondegenerate critical of fo). Then: i) 3 an open neighborhood X1 x ]/1 of (0, 0) and a smooth map x . ~ y(x) of X1 into Y1 such that y(O) = 0 and y(x) is the unique critical point of fx in Y1; moreover, the Hessian matrix Q(x) (in the coordinates yj) of fx at y(x) is invertible, so that this critical point is nondegenerate. ii) 3 a diffeomorphism
(x, y),
, (x,
y))
of X1 x ]/1 with u(O, O)= 0 such that, for all (x, y) C X1 x ]/1, 1
f ( x , y) - f ( x , y(x)) + -~(Q(x)u, u). P r o o f : i) Consider the map (x, y) ,
~ (x, V y f x ( y ) ) ,
which takes (0, O) into (0, 0). Its tangent map at (0, O) has the matrix
(, 0) 0
Q(0)
'
where Q(0) is the Hessian matrix of f0 at 0. Since Q(0) is invertible, this map is a local diffeomorphism around (0, 0), mapping an open neighborhood Xl x ]/1 of (0, 0) onto an open neighborhood W(X1) of (0, 0) that projects onto X1. By shrinking X1 and Y1, we may assume that the matrix
OyiOyj
Chapter ~ Method of Stationary Phase
187
is invertible t h r o u g h o u t X1 × Y1. Obviously, there is a unique point of the form (x, Clearly,
y(x)) t h a t maps to (x, 0), the dependence of y(x) on x being C ~ . y(x) is the unique critical point of fx in Y1, and it is nondegenerate. ~
( x , y - y(x)) takes (0,0) to (0,0) and is a
diffeomorphism around (0,0).
If f goes to g under this map, it is clear
ii) T h e m a p (x,y) ,
t h a t we need to prove the t h e o r e m with g in place of f (with smaller X1 and Y1). For g, 0 is the unique critical point of gx in ]I1, and (invertible) Hessian m a t r i x of
Q(x) is the
gx at 0.
For any function F of a real variable u defined on an interval containing 0, its Taylor expansion at 0 is
F(u) - F(O) + uF'(O) + u 2 ~o 1(1 - v )F" (vu)dv g(x, uy), we get, as 0 is a critical
Applying this at u - 1 to the function point for gx for all x E X1, 1
hij(x, y) - 2 with with
l~_i,j~_n Ji 1 (1 -
v)gij(x, vy)dv,
gij being w r i t t e n for the partial 02g/OyiOyj. Let H(x, y) be the m a t r i x hij(x, y) as its entries. We now search for a m a p of the form (x, y) ,
n(x, y)y),
R(o, o) - z,
where R is a s m o o t h n × n m a t r i x and y is viewed as a column vector. This is a diffeomorphism around (0, 0) for any R and sends (x, 0) to (x, 0). T h e idea is to see if we can choose R so t h a t the requirements of Morse's l e m m a are met, i.e.,
1
g(x, y) -- g(x, O) + -2(Q(x)Ry, ny) for all (x, y) near (0, 0). For this it is enough to have
tR(x, y)Q(x)R(x, y) - H(x, y), where
t R is the t r a n s p o s e d of the m a t r i x R.
(2.1)
V. S. Varadarajan
188
To this end, we consider the linear space M of real n × n matrices and its linear subspace S of symmetric matrices. Let so be an invertible element of S, and let x ~ > s(x) be a smooth map of X into S with s(0) -- so. Let us now consider the map ~ o f X × M into X × S given by 7r(x, m) = (x, t m s ( x ) m ) .
We assert that this map is a submersion at
(0, I). For this, we must check that the tangent map at (0, I) is surjective. If L is the tangent map in question, L takes (0, Z) to (0, t Zso + soZ), and by choosing Z to be l s o l W , where W is any n × n real symmetric matrix, we see that the range of L contains all elements of the form (0, W). On the other hand, if ej are the standard vectors in I~k, L maps (ej, O) into (ej,Os/Oxj(O)) for any j.
From this, it follows at once that L is
surjective. Since submersions are isomorphic to projections, we can find a smooth map p of an open neighborhood X0 × So of (0, so) in X × S into an open neighborhood of (0, I) in X z M such that ~ o p is the identity map. Obviously, p(x, W) = (x, r(x, W)), and so we have the equation w) = w
for all (x, W) C X0 × So. This result is now immediately applicable to (2.1). We take s(x) = Q(x), and then define
R(x,y) = r(x,H(x,y)). Then (2.1) is satisfied. This completes the proof of the theorem. E] Remark
This proof is the same as in [D1].
Morse's proof of it when
there are no parameters [Mo] was different and was a variation of the usual proof of the reduction of a quadratic form to a linear combination with i l coefficients of squares. In the exercises, we indicate a sketch of this proof. We shall now treat the case of nondegenerate critical points in the setting of manifolds with a boundary. Locally, this comes down to working at a boundary point of a half-space. Let x, y l , . . . , Yn be the coordinates in
]~n+l,
and let H be the half-space {x >_ 0}. Let Y be the b o u n d a r y
of H, i.e., the subspace x = O. We are given a phase function ~ defined in an open neighborhood of 0 in H, and we shall investigate its structure near 0 under assumptions about its critical behavior at O. To say t h a t
Chapter ~
Method of Stationary Phase
189
is smooth means that there is a smooth function in a full neighborhood of 0 in It(n+l t h a t coincides with ~ when restricted to H; a similar remark applies to maps into a manifold; in particular, by a diffeomorphism of H at 0 we mean a diffeomorphism on a neighborhood N of 0 in R n+l such t h a t it fixes 0, takes N N H into H, and is a diffeomorphism on N A Y into Y.
Proposition 2.6
i) Suppose ~ is a smooth real function defined on a neighborhood of 0 in H. If 0 is not a critical point of ~lY, then there is a local diffeomorphism of the form
(x, y),
~ (x, ~(x, y)),
U - - (?.tl,... , u n), u(O, O) -- O,
such that
eft(X, y) : eft(O, O)-~- Ul. ii) If 0 is a nondegenerate critical point of ~lY but is not a critical point of ~, there is a local diffeomorphism of the form
(x, y) ,
~ (t(x), ~(x, y) ),
with u = ( U l , . . . , Un), u(O, O ) = O, t ( O ) = O, t'(O) > O, such that
1
~(x, y) - ~(0, 0) + ~t + ~(Q~, ~), where c = +1 is a constant, and Q is a constant n x n real invertible symmetric matrix. Proof:
We note that in case (ii) the partial derivative 0 ~ ( 0 , 0) in any
direction ~ directed interior to H is nonzero, and its sign does not depend on the choice of ~. It is clear that ~ is this sign. i) We assume that ~ is defined and smooth in a full neighborhood of 0. By permuting the coordinates yj, we may assume t h a t O~/Oyl(O, O) ~ O. Then,
(x, y),
, (x, ~(x, y) - ~(0, 0), y~,..., y~)
is a local diffeomorphism of H at 0 with the required property. ii) We regard ~ as a function of y with x as a parameter and apply Morse's lemma with parameter. Write ~ ( y ) = a local diffeomorphism at 0 of the form
(x, y) ,
~ (x, v(x, y) )
~(x, y). Then, we can find
V. S. Varadarajan
190 such t h a t
y) -
z(x)) +
1
v),
where z(x) is a s m o o t h map of a neighborhood of 0 in R into R n taking 0 to 0 with the p r o p e r t y t h a t z(x) is the unique critical point of ~x near O. T h e m a t r i x Q(x) is real, symmetric, and invertible, and so it has the same signature as Q(O) for all small x. By elementary Lie theoretic arguments, it is easy to see t h a t there is a s m o o t h map o of a neighborhood of 0 in R into the orthogonal group of n variables such t h a t o(0) = I and to(x)Q(O)o(x) =
Q(x) for all x near O. The diffeomorphism ( x , v ) , )
(x,w), w = o(x)v,
has the p r o p e r t y y)
-
z(x))+
1
We now observe t h a t as 0 is not a critical point of ~ but of ~0, the derivative of ~ ( x , z ( x ) ) with respect to x at x = 0 is nonzero, so t h a t we can write ~(x, z(x)) - ~(0, O) = st(x) where s = ±1 and t'(O) > O. The diffeomorphism
v) is now seen to have the required property. [i] 2.3 C o n s t r u c t i o n
and genericity of Morse functions
How can we produce Morse functions on arbitrary, or at least compact, manifolds? We shall presently indicate a very simple m e t h o d of constructing Morse functions which shows t h a t they are dense in the space of s m o o t h functions. For another m e t h o d of constructing Morse functions and showing t h a t they are generic, see [GP, pp. 43-48]. Later on we shall see t h a t for homogeneous spaces, the structure of their groups of motions allow us to define certain phase functions t h a t are in the nicest possible position and so yield a great deal of information on the geometry and topology of these homogeneous spaces. This m e t h o d of constructing Morse functions goes back to the example of optics considered in the introduction. There we encountered the case of a surface M imbedded in R 3 from which isotropic m o n o c h r o m a t i c radiation is issuing; the radiation at a point not on M is given by an oscillatory integral the phase function of which is the distance of the point from a variable point on the surface. We adapt this m e t h o d to a general manifold
Chapter 4 Method of Stationary Phase
191
by first imbedding it in a Euclidean space of sufficiently high dimension and working with the square of the distance rather t h a n the distance itself for obvious reasons (to get a smooth function everywhere). If M is a smooth manifold of dimension rn, compact or not, we know by the famous theorem of H. W h i t n e y t h a t M can be imbedded as a regular closed submanifold of IR2m+l. Hence, there is no loss of generality in supposing t h a t M is contained as a closed submanifold of some R n. We write II" II for the usual norm on R n and consider, for any p E R n, the function Lp t h a t is the square of the distance of p from a variable point of M:
Lp(q) = l i p - ql] 2
(q E M).
In the example from optics discussed in Section 1, it was a question of rather t h a n
L 1/2
Lp. However, for p ~ M, L 1/2 and Lp have the same critical
set and proportional Hessians at the critical points. So we may work with the
Lp, which are smoother. This definition gives us a family of smooth functions on M parame-
trized by points of R n. It can be shown (see the exercises) t h a t given p, a
Lp if and only if the line q-p is "normal to M at q," i.e., if and only if q - p is orthogonal to the tangent space Tq(M) regarded as a subspace of R n. Let N be the total space of the normal bundle of M , i.e., the set of all (q, v) such t h a t q E M and v is orthogonal to Tq(M). It point q is critical for
can be shown t h a t N is a submanifold of dimension n in R 2~. We have the m a p E t h a t takes (q, v) to the "endpoint" of the vector based at q in the direction v, i.e.,
E(q,
(q c M,
±
This is obviously smooth, and it can be shown t h a t for (q, v) E N and
Lp if and only if (q, v) is a critical point of E. In other words, if p is not a critical value of E, then Lp is a Morse function on M. By SaM's theorem, we now see t h a t for almost all p E R n the function Lp is Morse on M. It is even possible to use the Lp to show t h a t the Morse functions are dense in the space C~(M). A critical value of E is, heuristically, a point where infinitesimally near normals intersect. So they are called focal points. It can be shown t h a t
p = q + v, q is a degenerate critical point of
for a given (q, v) E N, the focal points on the normal to M at q along v are the points
q + K~-lv where Ki are the principal radii of curvature
192
V. S. V a r a d a r a j a n
(with multiplicities) of M at q in the direction of v.
The set of focal
points is called the c a u s t i c v a r i e t y of M, and the foregoing is the method of constructing it. It is clear from the preceding description of the caustic variety that for a curve M imbedded in I~2 the caustic variety is the locus of the centers of curvatures at its points or the envelope of its normals, and so it is the so-called evolute of the curve M, the study of which goes back to H u y g e n s (in connection with his studies on the theory of pendulum clocks).
For
example, for the ellipse with equation x2 y2 a2 + ~-~ -- 1, the evolute is a curve called t h e astroid, which has 4 cusps. In parametric form, it is given by x = A cos 3 0,
y = B sin 3 0,
7r and it is easy to check that the cusps arise for 0 - 0, 3,7r, ~.
At each
of these points, the cusp is diffeomorphic locally to the cusp at the origin of the s e m i c u b i c a l parabola given by the classical equation y2 = x 3. It is remarked by Arnold et al. (see [AGV]) that this is a stable situation; if the ellipse is perturbed a little, its evolute is still a curve with four cusps of the type in a semicubical parabola that resembles the astroid. For the ellipsoid x2
y2
z2
a--~ + -~ + --~ -- 1,
the caustic surface was investigated by C a y l e y as far back as 1873 (see [AGV]). For the symmetric case of the sphere S when a = b = c = 1, the distance function becomes Lp(q) -
1 + u 2 + v 2 + w 2 - 2p.q
(p = (u,
w), q e s).
In view of the rotational symmetry it is enough to consider the case pw = (0, 0, w),
w ~: 0, +1.
The corresponding phase function is c;~(q) = 1 + w 2 - 2 w z
(q = (x, y, z)).
Chapter ~
Method of Stationary Phase
193
Thus, it is essentially the height function (see [Mi, p. 1]), which has two critical points q+ = (0, O, +1), q+ being the maximum and q- the minimum. So the oscillatory integral
Iw(r) - f s eirlp~-qla(q)dq ~" LT-1
(w # 0 , +1).
The normals at all points of S pass through the origin, so that {0} is the caustic set. For w = 0 the phase is constant on S, and so
IO(T) -- 9/(a) is independent of r. This is a good illustration of the increase in intensity at a caustic in a simple everyday example. 2.4 O s c i l l a t o r y i n t e g r a l s w i t h n o n d e g e n e r a t e p h a s e f u n c t i o n s We begin with some preliminary remarks on asymptotics. Let F ( r ) be a complex-valued function on (0, c~). A statement such as
F(,) ~ Z
,
(2.2)
r>0
means the following: i) The exponents er are real numbers and el
<
<
e2
"'"
<
er
<
"'"
,
er
~ oo
as
r
) (:x~.
ii) c r c C f o r a l l r > _ 0 . iii) There are positive numbers fN such that fN -
-
, e c as N
~ co and
,
O
We refer to (2.2) as an asymptotic expansion of F. Given the exponents, it is clear that the coefficients cr are determined uniquely and inductively by
cr = lim r e~ T - - - - + CK)
O<j
V. S. Varadarajan
194
It is also obvious t h a t the exponents corresponding to those c~ t h a t are not zero are also uniquely determined. Note t h a t if
G(7) is rapidly decreasing
in 7, F and F + G have the same asymptotic expansion.
F(T) is a distribution on a s m o o t h manifold Y for F(T)(a) = F(7: a)(a C C~(Y)). T h e n by the a s y m p t o t i c
Suppose now t h a t each v, and write expansion
F ~ E CrT-e~
(T
, OC)
r>O
is m e a n t the following:
a E C~(Y), we have
i) For each
F(T'a)~Ecr(a)T-e~
(7
,oc),
r>O
where the exponents e~ are independent of a; in this case, the c~ are distributions on Y. ii) Given any compact set K C Y, there are numbers oc, and for all
fN (N >_ 1) and
C~(K) such t h a t fN
continuous seminorms PN on
~ oc as N
a C Cc~(K) and T >_ TO we have F(T'a)-
c~(a)T-~ < #g(a)r -fN.
E
(2.3)
O
Suppose, finally, t h a t F depends in addition on x C X , X C IRk an open set such that, for each x E X ,
F(x.v.a)~Ecr(x'a)T-e~
(T
~cc),
r>0
where the exponents are independent of a and x. The asymptotic expansion is said to be
locally uniform at xo if there is a compact neighborhood N of fN, Pn, TO independent of x E N. If
x0 such t h a t (2.3) is valid for suitable
this is so for each x0 E X , we say t h a t the asymptotic expansion is locally uniform in X. In the special case when all the cr are zero, F is rapidly decreasing and we write it as F ~0,
Chapter ~
Method of Stationary Phase
195
with the obvious interpretations when F is a distribution for each ~-, depending possibly on additional parameters. We now study the asymptotics of integrals of the form
I ( x ' T " a ) - ~ . e i~(x'y)a(y)dy
(xcX, r>0),
where X is open in R k 2.7 Suppose (xo, Yo) C K × I~n is such that dy~(Xo,yo) # O. Then we have the asymptotic expansion
Theorem
S(x:.:.)~0
(.
locally uniformly on X. Proof: We may clearly assume, by permuting the coordinates if necessary, that u : = / ) ~ / / 0 y l is bounded away from 0 on the closure of X0 x K0 for a suitable neighborhood X0 x K0 of (x0, y0). Hence if L := u-l(9/cgyl, then L is a smooth first order differential operator on X0 x K0 with bounded coefficients, and L~ = 1 on X0 x K0. So
ei~ = (iT)-lL(ei~) .....
(i~-)-~L~(ei~)
for all integers r :> 1. Integration by parts now gives us, with L t denoting the formal transpose of L,
I(x " ~" " a) - (iT) -~ / e i ~ (Ltra)dy. Since the coefficients of Lt ~ are bounded on X0 x K0, we have, for all a C Cc~ ( K o ) , x C Xo, I I ( x : T : a)l ~_ C#(a)T -r, where p(a) - ~--~0_<j_
[3
We shall now treat the neighborhoods of points where the phase has a nondegenerate critical point. In view of the application of Morse's lemma, the key step is to calculate the Fourier transform of
eir(Qx,x)/2,
V. S. Varadarajan
196
where Q is a nonsingular symmetric n × n matrix. Now this is a bounded continuous function and so defines a tempered distribution. It is the limit (in the sense of bounded convergence, hence in the weak topology of tempered distributions) lim
z-----~- i r The function of one variable
e -z(Qy'y)/2.
e-Zy2/2
is in the Schwartz space for ~(z) > 0; its Fourier transform for z > 0 is
(271")1/2 Z- 1/2 e-~ 2/2z. By analytic continuation, we therefore have
(e -zy2/2) (~) - (27~)1/2Z-1/2e-~2/2z, where the square root is the continuous one defined for ~(z) > 0, z ~: 0, in polar coordinates by
Z--1/2 __ r-1/2e-ie/2
(z-re
~° r > o
--iTq (q e
Taking the distribution limit as z
lel<~)7r
1~), we obtain (as T > 0)
(e '~qy'/2) (~) - (2~)l/21Tq[-1/2e('~/4)sgn(q>e-i~2/2~q" From this, we get at once the following n-variable formula:
-
--
T
Iql... qn1-1/2e(i'~/4) y'~
sgn(qj)e-i y'~ ~/2,-
In ~n we have an orthogonal matrix R such that
Q - R D R -1,
D-
diag
(ql,...,qn),
and so finally we have
I det(Q)l-1/2e(i~/4)sgn(Q)e -i(Q-lS'~)/2r
Chapter ~ Methodof Stationary Phase
197
where sgn(Q) = n+ - n_ with n+, n_ as the number of positive, respectively negative, eigenvalues of Q. Thus, for any function h in the Schwartz space of IRn, we see that
J eiT/2h(y)dy equals
27r)n/2 Idet(Q)1-1/2e(i~/4)sgn(Q)/ e-i/2rjZ(h)(~)d~" T
The exponential can now be expanded as a power series in
7" - 1
and leads
us to an asymptotic expansion. By the Fourier inversion formula, we have, for any polynomial P,
f P(~)~(h)(~)d~ - (P(iV)h)(O). Hence, we finally get the asymptotic expansion
f eir/2h(y)dy (27r) 1 (Rkh)(O)7.-k m n/2 ]det(Q)l-1/2e(i~/4)sgn(Q)E -£[. T
k>0 m
as 7
~ c~, R being the operator
R
i -
Z
v>.
The estimation of the error terms is not particularly difficult in this asymptotic expansion (see the analysis given in Section 1) and so is left as an exercise. We thus finally obtain the following key result.
V. S. Varadarajan
198
L e m m a 2.8 Suppose Q(x) is a family of nonsingular n × n symmetric real matrices depending continuously on x. Then, for all smooth a C Cc~(K), we have, with
I(x " T " a) - J e i~(Q(x)u'u)/2a(y)dy, the asymptotic expansion I detQ(x)l-1/2e(i'/4)sgn(Q(x)) E
1 -~. (nka)(O)T - k
k>O
as ~- ~ c<), locally uniformly in x E X ; here R is the second order constant coefficient (but x dependent) partial differential operator i Ry -- -~(Q(x)-lVy, Vy}.
We are now in a position to formulate the asymptotics of the integrals
I-
Theorem
2.9
I ( x " T " a) - / e i~:(x'y) a(x, y)dy.
Suppose that dy~(xo, Yo) = O,
2 dy~(xo, Yo) is invertible.
Then there is a compact neighborhood Xo × Ko of (xo, yo) such that for all amplitudes a with support in y contained in Ko, I has the asymptotic expansion ]det H(x)l-1/2e (i~/4) sgn(H(x°)) 1
× ei~-~(x'Y(x))E ~ ( R ~ a ) ( x , y(x))T -k k>O
as ~) c~, uniformly in x E Xo; here, R u is a second order partial differential operator, y(x) is the unique critical point in Ko of ~(x, .), and H ( x ) is the Hessian matrix of ~(x, .) at the critical point y(x).
Chapter ~ Proof."
Method of Stationary Phase
199
This follows at once from Morse's lemma with parameters and
L e m m a 2.8, by making the x dependent change of the integration variables that changes the phase function to the Hessian quadratic form at its critical point y(x). [-1 There is a variation of the nondegenerate situation that is encountered not infrequently and is called clean by Bott [Boll. If ~ is a smooth function on a smooth manifold M and C is its set of critical points, (p, M) is clean if i) C is a smooth, not necessarily connected, imbedded submanifold of M. ii) At each point c of C, the Hessian of ~ is transversally nondegenerate, i.e., nondegenerate on Tc(M)/Tc(C). One should note that ~ is constant on C, and so the Hessian form vanishes if one of the vectors is from Tc(C).
Hence, the Hessian form
descends to the quotient space Tc(M)/Tc(C), and the condition (ii) is t h a t this is nondegenerate. 2.10 If (~, M ) is clean, and c E C, there is a chart (y, v) at c, (y,v) = ( y l , . . . , y n , v l , . . . , V p ) at c, such that ~ takes the form
Proposition
-
+
1
y)
in that chart, Q(v) being a n z n real invertible symmetric matrix depending smoothly on v. Proof." There is a chart x = (u, v) = ( U l , . . . , Un, V l , . . . , Vp) at c such t h a t C is locally described by the equations u - 0. Then ~(0, v) is constant, thus equal to p(0, 0). We now treat ~(u, v) as a family of phase functions in u depending on the parameter v and apply Morse's lemma with parameters. For any v we have 0 as a unique critical point for ~(., v), and so there is a diffeomorphism of the form ,
such that
1 ~(u, v) - ~(0, 0) + ~ (Q(v)y, y).
Theorem 2.9 now leads to the following corollary.
[:]
V. S. Varadarajan
200
We have, for a sufficiently small open neighborhood U of c, an asymptotic expansion for
C o r o l l a r y 2.11
I(7 " a) - / e i r ~ ( X ) a ( x ) d x
of the form I ~
(a e C ~ ( U ) )
(2__~T)n/2~ Cr r>O
as T > c¢, uniformly in x E Xo, where the cr are distributions of order < 2r supported by C, co being therefore a measure on C. The local results lead to global results on compact manifolds in the usual way by localizing and using partitions of unity. We state the following result, which is an immediate consequence of the theorems that we have proved. We treat the case of Morse functions on a compact manifold M of dimension m. This is the generic case because it can be shown that they form an open set in C ~ (M) (see the exercises). Before formulating the global result, we note that the oscillatory integrals on M will be obtained by integrating with respect to an m-form on M. In local coordinates, this will give rise to an integration in Euclidean space with respect to some density function, and it is important to make sure that the factors in front of the coefficients of the asymptotic expansion make sense globally. Suppose V is an m-dimensional real vector space and B is a nondegenerate symmetric bilinear form on V × V. By det B we mean the determinant of the matrix (B(vi,
Vj)), where
(vi) is a basis of Y. This,
of course, depends on the basis. If, however, there is an exterior m-form L on V t h a t is nonzero, and we write ILl for I L ( v l , . . . , vm)l, it is obvious t h a t I det BI-1/~ILI is independent of the choice of the basis used in computing it, provided of course the same basis is used for both. We write it as
I det BI-1/21L I.
Chapter ~
Method of Stationary Phase
201
T h e o r e m 2.12 Let M be a smooth oriented compact manifold of dimension m, w a smooth m-form > 0 on M , X an open set in ]~k, and ~p a real
smooth function on X × M such that ~x(') := ~(x,-) is a Morse function on M for each x E X . Let (aeC~(M)).
I(~-" a) = / M eir~(Y)a(y)w
Then we have the following asymptotic expansion:
I(T " a ) ~
--
c~(a)T -~ r>O D
where the cr (.) are distributions supported by the critical set C of ~ of order <_ 2r, and co is the measure given by Co--
~--~ ] det H(t)l-~/2lwtle(~/4)sgn(H(t))ei~(t)St. tEC
Here, H(t) is the Hessian form of ~ at t, and 5t is the Dirac delta measure at t. If ~ depends smoothly on a parameter, this expansion is locally uniform in the parameter. Finally, if ~ is clean, the asymptotic expansion is valid again, but the cr are now distributions of order <_ 2r supported by the critical manifold, co being therefore a measure on the critical manifold.
2.5 Manifolds with (smooth) boundary We shall now consider the case when M has a smooth boundary. We write M ° for the interior of M and OM for the boundary of M; we recall t h a t OM is a smooth manifold without boundary.
As before, m is the
dimension of M. We assume that M is oriented and use the usual definition of a smooth m-form on M that is > 0. We refer the reader to [GP] for the basic definitions and results concerning manifolds with boundary. A real-valued smooth function ~ on M is said to be a Morse function if it has the following properties: i) The critical points of ~ in M ° are all nondegenerate. ii) ~ has no critical points on OM. iii) The restriction of ~ to OM is a Morse function on OM.
V. S. Varadarajan
202
It follows from our local study of phase functions on half-spaces in Section 2.2 that ~ has only finitely many critical points in M °. The same method of partitions of unity used in the boundaryless case now reduces the study of the integrals to the local case, which has one of the two following forms:
>0
e ~yl a(x, y)dy
or
/x
>0
ei~[x+(QY'Y)]a(x, y)dy,
where Q is a real invertible n × n symmetric matrix. We thus easily obtain the following theorem. Theorem
Let ~ be a Morse function on M, and let ~ be a smooth
2.13
m-form > 0 on M. Write I(T " a) -
/M
ei~:(U)a(u)w
(aeC~(M)).
Then we have the asymptotics I(~-'a)~
(~)
(m+l)/2 E
T-~ar(a)+
r>0 m
(27r)m/2E T-~Zr(a), T r>0 -
-
where ar and ~ are distributions, with a~ supported by the critical set of the restriction of ~ to OM of order <_ 2r, while/~ are distributions supported by the critical set of ~a in M °, of order <_ 2r there. The distributions c~o and/~o are of the form described in Theorem 2.12. 2.6 E x e r c i s e s
SARD'S
THEOREM
1. Let f ( M
([GP, pp. 202-207])
~ N) be a smooth map, and C C M the set of x in M such
t h a t dfx is not surjective. The following steps are designed to lead to a proof that f ( C ) has measure 0 in N. It is clear that we may suppose t h a t M (resp., N) is an open neighborhood of the origin in R m (resp., I~n) and f -- ( f l , . . - , fn) with fi(0) - 0. Let Co - C, and C1 the set of all x C Co such that all first order partial derivatives of the fi vanish; more generally, let Cp for p _> 1 be the set of all points of C where all partial derivatives of the fi of order _< p vanish. There are two steps to the proof: (i) to prove
Chapter ~ Method of Stationary Phase
203
f(Cp \ Cp+l) has measure zero for all p >__ 0, and (ii) to prove t h a t f(Cp) has measure zero if p is large enough, for example, if p + 1 > n/m.
that
a) Suppose x E Co \ C1 and Ofl/OXl(X) ~ O. Show t h a t the map (Xl,...,Xm)' ~ (fl(X),X2,...,Xm) i s a d i f f e o m o r p h i s m a r o u n d x . b) I f f has the form (Xl,...,Xm)' ~ (Xl,f2(x),...,fn(x)), prove that / ( x ~ , x ~ , . . . x ~ ) C C if and only if ( x ~ , . . . ,xm) is a critical point for he map ( X 2 , . . . , X m ) ' ~ (f2(x~I,X2,...,Xm),...,fm(x~,x2,...,Xm)). C) Deduce from (a), (b), induction on m, and the Fubini theorem t h a t f(C0 \ C 1 ) has measure zero. d) Suppose
x E Cp \ Cp+l and O/Oxl(OZf)(x) ~ 0, where/~ is a multi~ (OZf, x2,...,Xm)is a
index with [/3[- p. Show that ( x l , . . . , x m ) ,
diffeomorphism around x, and hence show that in the corresponding
Cp \ Cp+l are m a p p e d into and hence deduce that f(Cp \ Cp+l) has measure
new set of coordinates at x the points of a subset
o f Yl -
0,
zero. e) Prove by Taylor series expansion that if x e Cp and ~ :> 0, ]]f(y) f(x)][ _< L~p+I for ] ] y - x][ < ~, where L is bounded as x varies in compact subsets of
Cp.
f) Fix 5 > 0, and let R be a cube in M of side 5. For any integer k > 1, let
Rj (1 _~ j <_ Mn) be the cubes into which R is subdivided by dividing the sides into k equal segments. Prove that if Dp is a compact subset of Cp, the image under f of R A Dp is contained in a union of k m sets of diameter < Lmp+l(5/k) p+l and so has measure < L'km(5/k) p+I, where L' is another constant depending only on L and n. g) Use (f) to show that f(Dp) has measure O(k -(n(p+l)-m)) for k > 1, and hence that if n(p + 1) > m the set f(Dp), hence also f(Cp), has measure zero. M O R S E ' S L E M M A ([Mi, pp. 5-9]) 2.
Use the Taylor expansion with remainders in integral form to prove
t h a t if ~ is a smooth function defined around the origin in R n, then the following are true. a) If ~(0) = 0, there exists an open neighborhood U of 0 and ~i C C ~ (U) such that p = El
~ - ~-~i,jxixjhij.
hij = hyi e C~(U) as in (a)
V. S. Varadarajan
204 3. Suppose t h a t ~ ( 0 ) a) I f n
-
0, d~(0) - 0 ,
and t h a t d2~(0) is invertible.
x2h where h is smooth and h(0) ~= 0. I r e is
1, write ~ -
the sign of h(0), show t h a t h -
ck 2 where k is smooth and k(0) =fi 0.
Check t h a t y - xk(x) defines a diffeomorphism around 0, and ~ - ey2 around 0. b) Assume n _ 2 and t h a t in terms of a chart (ui) at 0 we can write 2
~-~ CiUi -~- ~ uiujkij~ i_r where kij -
kji are smooth and ei - +1. Show t h a t the m a t r i x (kij(O))i,j>_~ is invertible, and t h a t after a linear transformation of the ui it may be assumed t h a t k ~ ( 0 ) ¢ 0. Write kr~ - e~g 2, where g is smooth and g(0) ¢-0.
c) Define the vi by
ui, g [ur + ~ i > r u i k i ~ k ~ ] ,
if i 7~ r, if/-
r.
Show t h a t the (vi) define a chart at the origin and t h a t
)9 -- ~ CiVi2 -t- ~ ViVjtij ~ i<_r i,j>_r+l where tij -- tji are smooth. d) Show by induction on r t h a t (c) gives Morse's l e m m a without parameters. P H A S E F U N C T I O N S ON I M B E D D E D S U B M A N I F O L D S ([Mi, pp. 32-381) 4.
Let M be a smooth manifold of dimension m imbedded as a closed
regular submanifold of R n.
Let N C R n be the normal bundle of M,
namely, the set of points (q,v) C R 2n - R n x R n such t h a t q C M and
v ± Tq(M). (Xl(U),...Xm(U)) be a diffeomorphism of some ball B in R TM onto an open set M ( B ) in M. Show t h a t if B is sufficiently small, there are smooth maps w ~ ( B ~ R n) (1 _ c~ _ n - m) such t h a t wa.wz - ~ and wa ± Tx(u)(M) for all u C B.
a) Let u -
(Ul,...,Um) ,
> X(U)-
Chapter ~ Method of Stationary Phase
205
b) Show t h a t if N(B) - {(q, v) l q e M ( B ) } the map
((ui), (G)) - (u, t) ,
(x(~),
Z
t.~.(u))
l
defines a chart on N(B) C N, and that these charts give the normal bundle N the structure of a closed submanifold of dimension n of ]I~2 n . c) Show t h a t in the coordinates (u, t) the endpoint map E((q, v), becomes
~ q+v)
E(~, t) - x(~)+ ~ t ~ . C~
Taking the scalar products of OE/Oui and OE/OG with Ox/Ouj and the wz, show that the tangent map dE is singular at (u, t) if and only if
det
0,
(gij - v'hij)l<_i,j<_ m -
where
Ox Ox gij -- Oui
~2 x h i j ----
cOuj '
Hint" Use the identity hij.wa
OuiOuj
,
v--
-- -Ox/Ouj.Owc~/Oui
~
ta wa .
obtained by dif-
ferentiating Ox/Ouj.wa - O. 5. Let Lp for p E
I~ n
be the phase functions defined on M C I[~n by the
formula
Lp(q) - [ I q - Pll 2, where ][.ll is the usual norm on I~n. a) Show that
OLP
- 2
Ox )
(x(u) - P ) - 5 ~
'
and hence deduce that x(u) is a critical point of Lp if and only if
qp := p - x(u) E T x(u), ± that is, if and only if the line from q = x(u) to p is normal to M at q. b) Suppose that x(u) is a critical point of Lp. Show that 02 L p
OuiOuj
= 2
( g i j - - -qP
"hij
)•
V. S. Varadarajan
206
c) Hence deduce that q = x(u) is a degenerate critical point for Lp if and only if p is a focal point associated to q. d) Use Exercise 2 and the preceding in conjunction with Sard's theorem
1-1/2 are Morse to show t h a t for almost all p the functions Lp and ~.p functions on M.
F O C A L P O I N T S AND RADII OF CURVATURE
(v'hij) is the so-called second fundamental form in the direction v. Suppose q = x(0) and gij(O) -- 5ij. Show t h a t the focal points on the normal to M at q along v are the points q + K ~ l v (with 6.
For any v C
A n ,
multiplicity) where Ki, the principal radii of curvatures in the direction v, are the eigenvalues (with multiplicity) of (v.hij). Deduce also that the index of the Hessian form of Lp at a critical point q is the number of focal points (with multiplicities) t h a t lie in the segment from q to p. M O R S E F U N C T I O N S ON C O M P A C T M A N I F O L D S 7. Let M be compact, and regard C ~ ( M )
as a Frechet space with the
topology of uniform convergence of functions and their derivatives (make this precise using local charts). Let h E C ~ ( M ) . a) Show t h a t there is a smooth imbedding M ~ R n with h = Xl. b) Let c and ~i be parameters such t h a t c
~ cx~ and the ci
~ 0 such
t h a t for p = ( - c + ~ 1 , ~ 2 , . . . , ~ n ) , the phase function Lp on M is a Morse function. Define
g(x) - Lp(x) - c 2
2c
Show t h a t g
(x c M).
~ h in C ~ ( M ) .
8. Show t h a t Morse functions on M as well as the Morse functions on M the critical values of which are all different, form dense open sets in C ~ (M) (see [GP, p. 47, Exercise 19]). 9. Complete the details in the proofs of Theorem 2.12 and 2.13.
Chapter ~ 3
Additonal
Method of Stationary Phase
207
r e m a r k s o n s i n g u l a r i t i e s a n d oscillatory integrals
3.1 Morse theory In this section we shall make a few additional remarks on the general
theory of singularities of smooth functions and maps and its applications to topology and oscillatory integrals. We begin with Morse theory. We recall t h a t given any symmetric bilinear form B on a real finite dimensional vector space V we can find a basis c~i of the dual space V* such that, if r is the rank of B, (vCV, ei-+l). l
The number of ci t h a t are - 1 is called the indez of B. It is independent of the choice of the basis with the given properties. The original investigation of Morse involved how the topology of a space was related to the structure of the critical set of a Morse function defined on it. Let M be a smooth compact manifold of dimension rn, and f a smooth real function on M t h a t is a Morse function. Recall t h a t this means t h a t the only critical points of f are nondegenerate. We have seen t h a t in this case there are only finitely many critical points. At any critical point x of f , we write I(x) or I ( x " f) for the index of the Hessian form of f at x. If rn -- d i m ( M ) , the index satisfies 0 _< I(x) <_ rn, with I(x) - 0 (resp., rn) if and only if x is a local m i n i m u m (resp., local m a x i m u m ) . For any integer r (0 _< r _< rn), let us write Mr - M r ( f ) for the number of critical points of f with index r; it is natural to call this the r t h Morse
number of M. Let Br be the Betti numbers of M (0 <_ r _< rn). Then, Morse formulated the relationship between the critical locus structure of f and the topology of M in the following theorem.
Theorem 3.1 (Morse inequalities) Between the numbers Mr and Br, the following inequalities are always satisfied: M ~ - M ~ - I + M ~ - 2 . . . . +Mo >_ B ~ - B ~ - I + B r - 2 . . . . -t-Bo In particular, we have Mr >_ Br
(0 <_ r <_ m).
(0 <_ r < m).
V. S. Varadarajan
208
Moreover, Mm - Mm_ 1 + M ~ _ 2 . . . .
+ Mo - Bm - Bm-1 q- Bm-2 . . . .
+ Bo.
We may introduce polynomials associated to M , the Morse polynomial of f, written M ( f • M • t), and the Poincar@ polynomial of M , written P ( M • t), defined by
M(f'M't)-
E M~(f)tr'
P(M't)-
O
E
Brtr"
O
The Morse inequalities are proved by establishing the following result:
There exists a polynomial Q(t) with nonnegative integer coefficients such that M ( f " M ' t ) - P ( M • t) - (1 + t)Q(t). (3.1) Most modern proofs of (3.1) follow essentially the original ideas of Morse introduced in his fundamental paper [Mo]. Let Ma be the set {x C
M I f ( x ) <_ a}. If a is a regular value of f, this is a smooth manifold with boundary, and it is a question of how the topology of Mc changes as c increases from a to a value (regular) b, and an essential role in this analysis is played by Morse's lemma (see [Bo3]). It turns out that the torsion in the integral homology of M also imposes restrictions on the Morse numbers. Let g(r) be the number of cyclic groups in the canonical decomposition of the torsion subgroup of H r ( M : Z); then
([Sm]), M r ( f ) >_ Mr + g(r) + g(r - 1). It is natural to call a Morse function optimal if
M r ( f ) - Br + g(r) + g ( r - 1)
(0 <_ r <_ rn).
Optimal Morse functions always exist if d i m ( M ) > 5. A Morse function is called perfect if
It is clear that the existence of a perfect Morse function implies that there is no torsion and M(f" M't)
- P ( M • t).
Chapter ~
Method of Stationary Phase
209
As a very simple application, let us c o m p u t e the Betti numbers of complex projective space C P n . We regard CP n as the space of one-dimensional subspaces of C n+l , which is viewed as a Hilbert space with the usual scalar product ( - , . ) .
ho < hi < ' "
Let H be a Hermitian m a t r i x with distinct eigenvalues
< hn. The function Z'
>
(Hz, z)
(z,z)
(Z e C n+l \ {0})
is scale invariant and so defines a s m o o t h function fH on C P n. We will show first t h a t fH is a Morse function the critical points of which are the points t h a t correspond to the eigenvectors of H , and then we c o m p u t e the Morse numbers. We may assume t h a t H is diagonal in the s t a n d a r d basis
(ej)O<<_j
-- hj)ltkl 2 + . . . ,
kCj
showing t h a t t = 0, which describes the point pj defined by ej, is critical with index 2j.
We now show t h a t there are no other critical points.
If
z C C n+l with (z,z) = 1 defines a critical point, differentiation in the direction of v gives ~(v, Hz) - (Hz, z ) ~ ( v , z) = 0. As this is true for iv also and for all v, we get
H z = (Hz, z)z, which implies t h a t the point defined by z is some pj. Since the index is always even, Mr = 0 for odd r, hence Br = 0 f o r odd r. B u t then, the relations M2k _> B2k,
M0 + M2 + ' - - +
M2~ = B0 + B2 + ' "
+ B2m
imply t h a t Mr = Br for all r. Thus, fH is a perfect Morse function, C P n has no torsion, and
P ( C P n : t) = 1 + t 2 + . . . + t 2n.
210
V. S. Varadarajan
Using phase functions on Lie groups we shall extend this result to all flag manifolds. The argument we have used is perfectly general and is known as the lacunary principle; it already goes back to Morse.
Proposition
3.2
Suppose f is a Morse function such that the Morse
numbers satisfy the relations MrMr+I - 0 has no torsion, and
for all r. Then f is perfect, M
P ( M • t) - M ( f " M ' t ) . P r o o f : We shall show by induction on r _ 0 that Mrt - Br for all r. We have Mo > B0 _> 1. Hence, M1 - 0, and so - M o > B1 - B0 _> - B o , which gives Mo < B0. Thus, M o - Bo. Suppose that r > 0 and Mk -- Bk for k < r. We have Mr > Br, and so, if Mr - 0, we must have Mr - Br = 0. If Mr > 0, then, Mr+l - 0, so that - M r + M r - 1 . . . . >_ B r + l - B r + B r - 1 . . . . , which leads to - M r _> - B r or Mr < Br. Hence, Mr - Br. V1 The proof of the Morse inequalities is usually obtained by first giving M a Riemannian metric, and deforming M along the flow generated by the gradient of f. Witten [Wi] has generalized this idea and considered quantum dynamical systems defined by f, obtaining the Morse inequalities and much more. 3.2 S t r u c t u r e of d i f f e r e n t i a b l e m a p s
The study of smooth maps and their generic structure was begun by H. Whitney in 1955, when he gave a complete description of the generic maps (R 2, 0)
> (N2 , 0). Since then, the subject has grown enormously
and has an immense number of intersections with many areas of modern mathematics.
To get a feeling for this development, the reader should
consult the two beautiful volumes [AGV] or the very nice introductory book [M]. The central concept is that of stability: An object is stable if it is isomorphic to all nearby objects.
The study of stability of smooth
maps has led to a classification of generic singularities at least in small dimensions. It is impossible to give any reasonable idea of the subject in these lectures. We just content ourselves with mentioning two striking results.
Chapter ~
Tougeron~s t h e o r e m multiplicity #, i.e., if
if
211
Method of Stationary Phase ,
(R, O) is a smooth map with finite
# = dim~R[[xl,...,Xn]]/(Of/Oxl,...,cgf/OXn)<
c<~
(here, (...) refers to the ideal generated), then f is equivalent to its Taylor jet of order # + 1.
Mather's theorem
An infinitesimally stable map is stable.
3.3 B e r n s t e i n ' s t h e o r e m Apart from all the asymptotic expansions on oscillatory integrals that the classification of singularities leads one to, there is a very general theorem proved by Bernstein (and by Atiyah using the theorem of resolution of singularities of Hironaka): T h e o r e m 3.3 If f is analytic in a neighborhood of 0 in R n and 0 is a critical point, the integral f e i r f ( X ) g ( x l , . . . , x n ) d X l "'" dxn, where g is localized at O, has an asymptotic expansion of the form
ei~f(°) E
ak,a(g)ra(l°g 7)k
(r
> c~),
k,c~
where c~ runs through a finite set of decreasing arithmetic progressions of rational numbers.
The reader should view all the special and more explicit results in the background provided by this general theorem. _ In addition to the references [AGV] and [M] mentioned already, the reader should refer to the paper [D3], which was very influential in making the ideas of singularities and oscillatory integrals available to the general mathematician.
V. S. Varadarajan
212
4
P h a s e f u n c t i o n s on h o m o g e n e o u s s p a c e s
4.1 D i s t a n c e f u n c t i o n s in r e p r e s e n t a t i o n s p a c e s Let G be a Lie group and V a finite dimensional real representation of G admitting an invariant nondegenerate symmetric bilinear form (., "/. We have seen in Section 1.4 that the phase function
is, up to an additive constant, the distance function from the point u to the manifold G.v, the orbit of v; here, the distance may be an indefinite metric if the form (., .} is not positive definite. If G is compact, we can always choose the form to be positive definite and invariant, so that in this case the interpretation as distance function is exact. Note that the orbit
G.v is also closed in this case. In this section, we shall take a closer look at these types of phase functions for compact G and discuss briefly some very nice applications to the geometry and topology of compact homogeneous spaces. Before we discuss these examples, we shall note that this method in principle applies to any homogeneous space G / H where G is a compact Lie group. More precisely, we have the following variant of a well-known theorem of Chevalley proved by him for affine algebraic groups (see also [Mos] and [Pal).
P r o p o s i t i o n 4.1 Let G be a compact Lie group, and H C G a closed subgroup. Then there is a linear representation of G such that H is exactly the stabilizer of a vector in the representation space. Proof:
By Frobenius reciprocity, we have L 2 ( G / H ) =
( ~ i > 1 F i where
each Fi is a finite dimensional sub-G-module with F H ~ 0, the superscript H denoting the subspace of vectors fixed by H.
Let HN be the closed
subgroup defined by
H g - {g E G [ g u = u V u E F H, l < i <_ N } . Then H1 D /-/2 D " - D HN D " ", and so, for reasons of dimension and compactness, we have an No such that HN -- HNo for all N >_ No. The proposition will follow if we show that HNo -- H. Indeed, suppose we have
Chapter 4
Method of Stationary Phase
213
proved this. Let F -- (~i<_No Fi, let (Vj)I<_j<_ m be a basis of F H, and let Y = F ® F ® . . . • F (m copies of F). If v = ( V l , . . . , V m ) , the stabilizer of v in G is HNo -- H. Suppose t h a t HNo # H. Since Hgo contains H, we can find an element h C HNo \ H. Let X = G/H, and f the natural map G , X; write x0 = f(1), x = f(h). Since H is compact, H.xo = x0, and x0 # x, we can find an H-invariant continuous function g on X t h a t is 0 on a neighborhood of x0 and 1 at x. Then g C L2(X), and it is fixed by H but not by h. But the projection of g in Fi is in F H and so, as h C Hi, this projection is fixed by h. As this is true for all i, g itself is fixed by h, a contradiction. [] Remark
In Chevalley's theorem where G and H are affine algebraic
groups, it is only possible to guarantee t h a t H is the stabilizer of a point in the projective space of the representation space. This is because it may happen t h a t G / H is a projective variety (for instance, when H is a Borel subgroup) and so cannot have a realization as an affine orbit. But if we assume t h a t G / H is affine, then the algebraic version of Proposition 4.1 is true. 4.2 P h a s e f u n c t i o n s d e f i n e d b y t h e a d j o i n t r e p r e s e n t a t i o n For any Lie group U, we generally write u for its Lie algebra and Ad for the adjoint representation of U; for X C u, u C U, we write X u for A d ( u ) ( X ) . For any X E u, we write Ux for the subgroup of U fixing X. Let K be a compact connected Lie group with Lie algebra {~. For X, Y E {~, with (., .) as the Cartan-Killing form of {~, we write
hx,.(k) = (X, yk)
(k e K).
The function hx,u is a phase function on the orbit K . Y ~ K / K y . If t is a m a x i m a l abelian subalgebra of {~ and Y E t is a regular element of t, then
K y = T is the maximal torus defined by t, namely T = exp t; and K / T is the flag manifold of K. The use of these phase functions goes back to Bott
[Bo2], We shall consider a slightly more general situation, where K is a maximal compact subgroup of a connected semisimple Lie group G with finite center and Lie algebra ~. Let g = ~ • s be the C a r t a n decomposition of ~, where s is the orthogonal complement of ~ in g with respect to the C a r t a n Killing form (., .). Let a be a maximal abelian subspace of s, and ga the
214
V. S. Varadarajan
root spaces for the decomposition of g with respect to a. We write A for the set of roots, A+ for a positive system of roots, and n(a) = dim(g~). We denote by w the Weyl group of (~, a), and for any X C a we write w x for the centralizer of X in w. If M is the centralizer of a in K, then w is the normalizer of a mod M, and so each element w C w has a representative xw in K, even in K °. For X, Y E ~, we write
f x , v ( k ) - < X , Y k)
(ke K),
which defines a phase function on K . Y ~ K / K y . The study of these phase functions goes back to H e r m a n n [He] and Takeuchi-Kobayashi [TK]. If K is semisimple, then the classical theory of Weyl allows us to view K as a maximal compact subgroup of a complex group G; the Lie algebra of G is then viewed as a real Lie algebra, and the C a r t a n decomposition is just g - ~ ® (-1)1/2t~. Thus, the actions of K on t~ and s are isomorphic via the map of multiplication by ( - 1 ) 1/2. The natural choice of a is now ( - 1 ) 1 / 2 t where t is a maximal abelian subalgebra of t~. The complex root spaces are now of dimension 2, when considered as root spaces for a. Proposition
4.2
If X, H E a, the critical set of f X,H is *
KX,H- U KxWKHWEW
KxWKH
H Wx\W/W
H
In particular, it is smooth. Proof: First consider U, V C s. For u C K to be critical for fu, v, it is necessary and sufficient t h a t for all Z E ~ we have d) -t
(U, V kexptZ> - O. t--o
The derivative on the left reduces to
- ,
* For any subgroup L of K containing M, xwL does not depend on the representative xw of w in K and so is written wL.
Chapter ~
Method of Stationary Phase
215
and this is 0 for all Z if and only if [V k, U] = 0. Thus, k is critical for fu, v ~
[V k, U] = 0.
(4.1)
We now write U = X, V = H and suppose t h a t X, H C a. If k is critical, H k centralizes X and so, as the centralizer of an element of a has the same structure as g, we can find u E K x
such t h a t H uk C a. By a well-known
result ([Va, L e m m a 16, p. 288]) t h a t asserts t h a t if an element v of K takes an element H1 of a into an element/-/2 of a, then one can find w C w taking H1 to/-/2, we can find w C w such t h a t H uk = H TM. Hence, for some v C K H , u k v -- Xw for some representative x~, of W in K . In other words, k E KxwKH.
The converse t h a t any such element is critical is immediate,
since for such an element we have [H k, X] = 0. To express the union as a disjoint union, first observe t h a t as the representatives of a subgroup w v can be chosen in K v , it follows t h a t K x w K H
depends on w only t h r o u g h
the double coset WXWWH. Moreover, as we can move X and H into the closure of the positive chamber a + of a, there is no loss of generality in assuming t h a t X, H C Cl(a+). T h e n K x , K H are respectively contained in the parabolic subgroups defined by the simple roots vanishing at X, H , respectively, and the disjointness follows from the B r u h a t decomposition. [2] Remark
Let U E s be arbitrary and V E a be regular. Then, as K ° is
compact, there must be at least one critical point (say at a m i n i m u m ) of fv, v. Hence, by (4.1) there is some k C K ° such t h a t [V, V k] = 0. By the regularity of V, this means t h a t U k is in a. From this, it follows in the s t a n d a r d way t h a t all maximal abelian subspaces of a are conjugate under K °. This type of a r g u m e n t goes back to Hunt [Hu] for the conjugacy of m a x i m a l tori, and to Helgason [HI in the present context. We shall now calculate the Hessians of the fX,H. Let 0 be the C a r t a n involution of ~ defined by 2, so t h a t 0 is the identity on ~ and minus the identity on s. If we put
then it is easy to see t h a t the orthogonal projections from g+~ onto £a and ~
are bijective. We write F~ for the orthogonal projection
P~:g~ ® ~1-~
> t~.
216
V. S. Varadarajan
It is clear from w h a t we have said t h a t dim(t~s) - n(c~).
Let k - uxwv with u C K x , v C KH, and xw a representative of w in K. Let Tlk be the Hessian form of f X,H at k, viewed as a symmetric bilinear form on ~ × 2. Then
Proposition
4.3
7-lk(Z, Z ) - -
E
a(H)(wa)(X)[]Fs(ZV)]12
(Z e 2).
sEA+
In particular, f X,H is clean, and its index at k is
y~
n(~).
sEA+, s(H)(ws)(X)>O
Proof: Clearly, nk(z, z) -
(d2) d-~
,=0 f x , g ( k exp tZ)
But, as f x , H ( k e x p t Z ) -- ( X * , H exptZv) where X * - X
(Z e 2). w-1 , we have
,7-lk(Z,Z) - (X*, [Z v, [ZV, H]]) = - ( [ X * , ZV], [H, ZV]). We now write, for any U E {~,
U-
E
(Us + OUs) mod m
(Us e gs),
sEA+
and observe t h a t for any R E a, since 0gs - g - s ,
[R, U~ + OU~] - ~(R)(U~ - OU~). If we now r e m e m b e r t h a t (fls, fD) - 0 unless a - -/3, and t h a t [[SI[ 2 = - ( S , S) for S E {~, we get the desired formula. T h e formula for the index is i m m e d i a t e . VI
Chapter ~
Method of Stationary Phase
217
Proposition 4.4
Suppose that X is regular. Then f X,H is a Morse function on K / K H , the critical points of which are at the Weyl group points wKH. If X E a + and H E Cl(a+), the Hessian of fX,H at wKH has index
aEA+, waEA+, a(H)>0
In particular, the phase function has a unique local minimum and local maximum (which are global minima and maxima). If both X and H are in a +, then the index of fX,H at the point w M of the flag manifold K / M is
c~EA+, wc~CA+
Proof: Immediate from Proposition 4.3. [-1 C o r o l l a r y 4.5 Suppose that K is semisimple and that G is its complexification viewed as a real Lie group. Then, for X E a+, the Morse indices of fx,H are even; and if both X and H are in a+, and if P(w) is the number
of positive roots that remain positive after application of w, the index of fX,H at wKH is P(w), which is equal to 2g(w0w) where wo is the element of the Weyl group that takes A + to - A + and g(.) is the length function on the Weyl group (viewed as a Coxeter group). Proof." In this case, the numbers n(a) are all 2, and the result is immediate. [3
4.3 Applications Our first application is to the cohomology of the flag manifolds.
Let K be a compact, connected, semisimple Lie group and its Lie algebra. Then the flag manifolds K / K H (H E 2) are all without torsion, and the distance from a generic point is a perfect Morse function. Moreover, for the full flag manifold K / T , where T is a maximal torus of K , the Poincard polynomial is T h e o r e m 4.6
P(K/T : t)=
HI
(1-t )n
218
V. S. Varadarajan
where n is the rank of K, and di are the degrees of the homogeneous generators of the Weyl group-invariant polynomials on the Lie algebra t of T. Proof." We have already seen in Section 3.1 that if all Morse numbers are even, the Morse and Poincar~ polynomials are the same and there is no torsion. Now, when K is semisimple, we take G to be the complexification of K. In this case, s = (-1)1/2[~, and multiplication by ( - 1 ) 1/2 establishes an isomorphism of the K-actions on ~ and 5. Corollary 4.5 now gives the statements of the proposition except for the computation of the Poincar~ polynomial of the full flag manifold K I T = K / K H where H C a +. But, by Corollary 4.5, the index of fx,H at w T is 2g(wow), and so the Poincar~ polynomial of K I T is E
t2~(w).
wEw The desired formula now follows from a well-known result in the theory of finite reflection groups ([St, pp. 130-144]) that asserts
E t2g(w)-- I-Ii
Remark
[:]
For K = SU(n + 1), the subgroup T is usually taken to be the
diagonal subgroup, and so K I T is the manifold of flags (Li)l
Further, w is the permutation group
Cn+l,
dim(Li) = i.
~n+l of
{ 1 , 2 , . . . , n + 1}, and t~(w)
is the number of inversions in {w(1), w ( 2 ) , . . . , w ( n + 1)}. By Newton's theorem on the symmetric functions, one knows that the numbers di are 2 , 3 , . . . , n + 1, and so the result used is
E t~(w) wESn+l
I-I1
(1 -- t)n
Chapter ~
Method of Stationary Phase
This is trivial to verify when n -
219
1, and the general case follows by in-
duction on n; one has to establish the formula (denoting the left side by
Pn+l(t) - P n ( t ) ( l + t + t 2 + - . . + tn+l), which can be done by considering the possible values for w(n + 2). Indeed, if sk is the interchange k ~ n + 2, Sn+2 is the disjoint union of the cosets
Sn+lSk (1 <_ k _~ n + 2), and it is easy to check that f(wsk) - f(w) + (n + 2-
k) for w E Sn+l. Similar but somewhat more involved considerations
apply to the other compact classical groups. For the flag manifold K / K H , the foregoing procedure gives a very explicit recipe for calculating the Poincar~ polynomial. We assume that H C Cl(a +) and denote by Z(H) the set of simple roots vanishing at H. The subgroup WH is then generated by the reflections corresponding to the elements of Z(H). If A M,+ is the set of positive roots taking a value > 0 at H, then for any w C w we may define pH (w) to be the number of positive roots in A M'+ that remain positive after application by w; it is clear that
pH(w) -- pH(wwH), where WH is any element of W H; this is due to the obvious fact t h a t WH permutes the elements of A H'+. From the formula for the index of fX,H at the Weyl group points on K / K H , it is now clear that
P(K/KH't)-
E t2P'(~)" wEwlwH
We leave it as an exercise to deduce from this the formula for the Poincar~ polynomial of C P n computed earlier, as well as the Poincar(~ polynomials of the manifolds of other types of flags (for example, the Grassmannians). The Morse theoretic computations of the cohomology of the complex flag manifolds go back to Bott [Bo2]. As another application we mention the convexity theorem of Kostant [Ko].
The theorem has an infinitesimal and a global version.
infinitesimal version t h a t we consider here.
It is the
We go back to the general
setting of it, a. Then, Kostant's theorem states that ifpa is the orthogonal
projection s
~ a, the projection p~(K.H)
is the convex hull of the points of the Weyl group orbit of the point H C a. Heckman [Hec] gave a very simple proof of this and the corresponding
V. S. Varadarajan
220
global version by geometric methods based on the critical point and Hessian calculations of the phase functions fX,H and their global analogs Fa,H (which we shall introduce and study in the next subsection). We refer the reader to the work of Heckman for the details. 4.4 T h e p h a s e f u n c t i o n s c o m i n g f r o m t h e I w a s a w a d e c o m p o s i t i o n We work in the context of (G, K) and introduce now a new class of phase functions on K, or rather K / M , the real flag manifold associated to G.
They appear to have been studied first in [DKV1].
They arise
naturally from the infinite dimensional representation theory of G, and their study was suggested by some questions of spectral estimates on locally homogeneous spaces associated to G. Recall that we have the Iwasawa decomposition of G: G -
KAN
(A -
expa, N - exp( E
g~))"
(4.2)
sEA+
This means that any element x C G can be uniquely written as
x - k(x)exp H(x)n(x)
(k(x) e K, H(x) e a, n(x) e N).
(4.3)
The functions k(.), H(.), n(.) are all analytic, and the Iwasawa decomposition gives an analytic diffeomorphism of G with K × A × N. Let as write, as usual, M for the centralizer of A in K. Then M normalizes N, and
P = MAN is a parabolic subgroup of G, in fact, a minimal one. We have
G / A N ~_ K,
G/P ~_ K / M ,
(4.4)
canonically. When G = SL(n,]~), K / M is the manifold of full flags in I~n, and so one refers to K / M in the general case also as the flag manifold of G. The identification (4.4) allows us to have an action of G on K and a compatible action on K / M . Thus, for x E G, u E K,
x[u] = k(xu),
x[uM] = k(xu)M.
(4.5)
Chapter 4 Method of Stationary Phase
221
The phase functions we want to investigate are
F~,A(k) = )~(H(ak))
(aEA,
~ E a*).
Since M normalizes N, it follows that these phase functions are defined on
K/M. One could have used an arbitrary element x of G instead of a E A; but in view of the polar decomposition (Ul, U2 E K , a + C CI(A+), A + = exp(a+)),
x -- u l a + u 2
it follows that
H(xk) = H(a+u2k); thus, it is enough to restrict oneself to the study of the Fa,A. These phase functions arise naturally in the representation theory of G. Indeed, the matrix coefficients of the representations of the principal series of G are of the form K e(iA-p)(U(ak)) g ( k ) d k ,
where g is a smooth function on K, and so are oscillatory integrals(!) of the form
K eiFa'~'(k)g(a' k)dk' /M function on A x (K/M). The
where g is a smooth
study of these phase
functions and their associated oscillatory integrals is therefore very natural from our point of view. It turns out that the infinitesimal versions of these phase functions are the phase functions fx,g discussed earlier. Indeed, we have
( d) -t
t-0
FexptX,A(k)_ (X, Hk) '
(4.6)
where HA is the image of A in a under the canonical isomorphism of a* with a induced by the (nondegenerate) restriction to a x a of the Cartan-Killing form, so that (HA, H') = A(H')
V H'
e a.
The relation (4.6) is not completely trivial to derive; indeed, its derivation will reveal that some effort is needed in obtaining information about the Fa,A. Here it is question of showing that
( d~ \
/
t--O
H((exptX)k)- pa(Xk-')
(X E a),
(4.7)
222
V. S. V a r a d a r a j a n
pa being the orthogonal projection z ~
a. Put X k-1 - Z, p a ( Z ) - R .
The orthogonal complement of a in z is contained in t~ ® n (where n is the Lie algebra of N), and so, (V e [, W e n).
Z-R+V+W
Thus, (exp t X ) k - k
exp
t X k-1
=
exp t V exp t R e x p t W exp O(t2),
from which (4.7) follows easily. We shall give presently the main calculation on the derivatives of the Iwasawa projection H: G
> a, from which all
such differential calculations will follow rather systematically. For x , y
C G, we write x y -
yxy -1.
Let U(g) be the universal en-
veloping algebra of the complexification of 0- The adjoint representation of G on 0 extends to a representation of G on U(0) by automorphisms. We write u x for Ad(x)(u); we have _
__
( x , y C G, u , v E U(O)).
We shall view elements of U(O) as left invariant differential operators on G.
To define this interpretation, we should specify how an element
u - X1X2...
X v , where the X i are in 9, acts as a differential operator. We
shall define (O(u)f)(x)
" - f ( x ; u) -
( O r / O t l O t 2 '' " 0tr)0 f ( x e x p t l X 1 . . . e x p t r X ~ )
the subscript 0 denoting the fact that the derivatives are taken at t l = ....
t~ - 0. Now the Iwasawa decomposition 0 - t~ ® a ® n gives rise to
the decomposition u(0)
-
u(0).)•
u(a).
It therefore makes sense to speak of the projection u(0)
, u(a),
where the projection is taken mod (I~U(0)+ U(0)n). It is clear t h a t this projection preserves the degree filtrations on both sides, and that if u C
Chapter 4
Method of Stationary Phase
223
U(g) has zero constant term, the same is true of Ea(u); here, constant term is a homomorphism , c
c: u ( g )
t h a t sends all elements of g to 0. Note that since a is abelian, U(a) is isomorphic canonically to the symmetric algebra over a. Thus, on U(a) the degree filtration arises from a gradation. Thus, we may speak in U(a) of the homogeneous components of any element.
Let y E G, b c U(g). Then
Lemma4.7
H(y; b)
--
~(b)
+
(Ea(bt(y))) 1 .
(4.8)
Here the subscript 1 means the homogeneous component of degree 1, and t(y) = a(y)n(y) is the "triangular part" of y. In particular, for X E 9, H(y; X ) = X t(y).
(4.9)
Proof." The fact that H is left invariant under K and right invariant under N implies that H(1; u ) = 0
v
e
+ u(g),.
Now, for z C G,
H(yz) = H(t(y)z) = H(zt(y)t(y)) = H(z t(y)) + H(y). So
H(y; b) - H(1; bt(y) ) - a(b) + H(1; Ea(bt(y))). But as H ( e x p X1 • • • exp Xr) = X1 + ' " + Xr for Xi C a, it is clear that for any c e U(a), H(1; c ) = Cl. I-3
Remark
It follows from the last assertion of the lemma that for X C
a, k c K , d
H((exptX)k) -
d ) t=o H ( k e x p t X k - 1 ) -dr
= H ( 1 ; X k-1 ) = E,,(X --" pfl ( X k - 1 ) ,
)
224
V. S. Varadarajan
since, for any Y E ~, E a ( Y ) = p a ( Y ) because the orthogonal complement of a in s is contained in ~ + n. As we noted earlier, this gives (4.7). We now have the machinery to determine the critical sets of F~,A. From (4.9) we see that, for any Z C 2, k C K, t(ak)-i } - (Z, H~(a k ) -1 ), Fa,A(k; Z) - ( Z , H A
so t h a t k is critical for F~,A if and only if H'2 (ak)-I C '~. But it can be shown t h a t for any R E a, R N = R + n; so, for any n C N , R n E ~ if and only if R n - R C s M n = 0, i.e., if and only if n C N R , where N R is the centralizer of R in N. Thus, k is critical for F~,A ~
ak E K A N A .
Here we have abused the notation and written NA instead of NH~. For any R E a, we have GR = K R A N R that KANR
= KKRANR
= KGR
= K R e X p S R , which implies
= K e x p a R , and so the criterion for
k to be critical for FexpX,A is t h a t exp X k E K exp aHx, which reduces to [X k-1 , HA] -- O. But this is exactly the condition t h a t k is a critical point
for f x , H ~ . We have thus proved
Proposition 4.8
The critical set of Fexp X,A is the same as that of f X,H~
and so is
KX,H~ -
]_[ KxwKA. w~wx\w/w~
The formula (4.6) can be generalized by calculating the derivative for all t, not just at t = 0. By (4.9) we have, for any a E A, H ( ( a exp t X ) k ) - H ( a k ; X k - l ) - X t(ak)k-1
(d) t=0
But if ak = ubn is the Iwasawa decomposition of ak, t ( a k ) = bn, and so t ( a k ) k -1 = u - l a .
Thus, X t(ak)k-1 = X u-~. But u -
of k by a under the A-action, and so we get
(},)
t=O
a[k], the transform
Chapter ~ Method of Stationary Phase
225
Taking a = exp s X, we have d -~YexptX,~(k) - (X, Y-[exptX[ k]
(4.10)
Integrating this from 0 to 1, we finally obtain
Fexp X,A(]g) = f01 f X,H~, (exp sX[k])ds
(kEK, XEa,
A E a*). (4.11a)
Furthermore, from (4.9) we get
Fa,~ (k; Y) - (yt(ak), HA} -- ( y a k H~[k]} for all Y E ~. From this it follows easily that, if a = exp X, F~,~ (k; Y) = fX,H~ (a[k]; Z),
(4.11b)
if Y is related to Z by Y = Ad
]g-1 o (ad X~ sinh
ad X ) o Ad (a[k])(Z).
(4.11c)
We now discuss a remarkable right invariance property of the Fa,~. Recall that the functions fx,H are right KH-invariant for trivial reasons. However, in general it is not to be expected that this property persists for the Fa,~. So it is rather surprising that Fa,~ is right invariant under
K~ provided that ~ is in the closure of the positive chamber, i.e., )~ is dominant. This can be proved by direct analysis; but to illustrate the intimate relationship of the functions Fa,~ to representation theory, we shall briefly sketch an argument going back to ideas of Harish-Chandra that establishes this right invariance. Since we want to use finite dimensional representation theory, we extend a to a Cartan subalgebra 0 of fl that is stable under 0 (the Cartan involution defined by t0 and choose a positive system of roots of (go, 0c) compatible with the positive system A + chosen already. We may, without changing much, work with a linear group, so that we may assume that G is contained in its simply connected complexification. Then, for any dominant integral linear function A on 0, we have a finite dimensional irreducible representation 7rh that is a representation of G and has the property that
V. S. Varadarajan
226
elements of K map into unitary operators and elements of A map into positive self-adjoint operators; this is achieved most simply by noting t h a t tt
--
t~ ¢ ) ( - - 1 ) 1 / 2 ~
is a compact form of gC and so we may suppose that
7rA(Z) is skew Hermitian for Z ¢ u. Let 1A be the highest vector for the representation 7rh. Write N - 0(N), the unipotent subgroup defined by the system of negative roots.
The compatibility of the positive systems
on ~)c and a means that 1i is fixed by N, while for any element g E N we have 7rA(n)lA- 1A is orthogonal to 1A. This gives Harish-Chandra's famous formula
II A(x)IAII
(x E a).
(4.12)
Now, iffor a positive root c~ of (f~c, he) we have (c~, A) - 0, then 7rA(X-~)IA = 0, and so 7rA(n)lA -- 1A for ~ C NA. This leads easily to
For any A such that HA E Cl(a+), the function Fa,~ is right invariant under K~ = KH~.
P r o p o s i t i o n 4.9
To complete this brief treatment of the Fa,~, it is now necessary to calculate their Hessians at the critical points.
This again is done using
L e m m a 4.7, but the calculations are more difficult than before because we are calculating second order derivatives. We do not give t h e proof of the following result here, and refer the reader to [DKV1, pp. 343-346]. It can also be obtained from (4.11) and the corresponding formulae for the Hessians of fX,H. P r o p o s i t i o n 4.10 For any X E a, A E a*, w E w, the Hessian form 7-l~ of Fexp x,~ at k - xw is given by n
(z,z) -
1 2 E
o~EA+
(c~, A)(1 -
where F~ is as in Proposition 4.3. Moreover, throughout the critical manifold K x w K H ~ the value, the signature, and the rank of the Hessian remain constant and are the same as for fx,u~. In particular, the Hessian is transversally nondegenerate so that FexpX,~ is clean. Moreover, the value of Fexp x,~, and the rank and signature of the Hessian of Fexp x,~ at the points of K x w K u ~ are given respectively by
o~EA +,),
Chapter ~ Method of Stationary Phase 0",w =
-E
227
n(a) sgn((a, A)(wa)(X)),
o~EA+,~,
where -
c A+
# o}.
R e m a r k s i) Although there is no right invariance under K~ if A is not dominant, the second order data are invariant and are the same as for the f X,H~ •
ii) The global convexity theorem of Kostant [Ko] asserts that the Iwasawa projection maps the K-conjugacy class of any element exp X (X C a) onto the convex hull of the points of the Weyl group orbit of exp X. As we mentioned earlier, Heckman [Hec] obtained relatively easy proofs of this and the infinitesimal convexity theorem of Kostant by using properties of the phase functions Fa,)~ and fx,~ and the relationships between them. iii) The phase functions fX,H and Fa,~ form parametrized families. But it must be noted that they are highly nongeneric because of the tremendous rigidity in their behavior as the parameters approach the caustic sets. For instance, as long as X and H (or A) are regular, these are Morse functions, but their critical data do not change when the parameters vary within the regular domains. When the parameters enter some root hyperplanes, which are now the caustic sets, the critical points become enlarged in the simplest possible way; some of the critical points are joined by smooth submanifolds that now consist entirely of critical points, and the Hessians are transversally nondegenerate. Again, variation without enlarging the caustic set does not change the critical data until the parameters become more singular, and the phenomenon of enlargement of the critical manifold by linking repeats itself, and so on. The simplest special case is G = SL(2, I~). If x-
(a
~)
7
5
a-
'
(e t 0
e
0 ) -t ,
k-
( cos0 -sin0
sin0) cos0
then llog(a2+72)(1
H(x) - -~
0 )
0 -1
'
1 log(e2t cos2 0 + e -2t sin 2 0) ( H(ak) - -~
10)
0
-1
'
V. S. Varadarajan
228
For t ¢ 0, the critical points are 0 = 0,7r/2.
When t = 0, the phase
function becomes constant, and the whole circle is critical. This method of calculation of H(x) can be extended to the classical groups, using explicit representations of G.
For instance, for SL(n) one can use the defining
representations and its exterior powers. 4.5 P h a s e f u n c t i o n s o n c o n j u g a c y c l a s s e s
An entirely different class of phase functions are obtained if we restrict the same functions to conjugacy classes. For any X E fl (resp., -~ C G), we write
C ( X ) -~ { X g [ g e G},
C(~) ~- {"/g -- g~g-1 I g e G}.
It is known that these are closed if and only if X (resp., -y) is a semisimple element. In this case, they are closed imbedded submanifolds of fl (resp., G). The infinitesimal versions are the phase functions sz,y, which are defined on C(Y) by
sx,y(Z) = (X, Z)
(Z e C(Y)),
(4.13)
where X, Y E ft. The global versions are defined in complete analogy with what we did before. For A E a*, we define S~,~ by
S~,~(t) = A(H(t))
(t e C(~/)).
(4.14)
It is absolutely remarkable that for these phase functions one can prove the same type of results as before. We state these results without proofs. For the details, one may refer to [D2]. We just introduce one small bit of terminology: closed submanifolds A, B of a smooth manifold C are said to have a clean intersection if A M B is a submanifold, and at each point c of A A B we have
Tc(A M B) = Tc(A) N T¢(B). It is obvious that this is a generalization of transversal intersection, which arises when Tc(A) and Tc(B) are complementary subspaces in Tc(C).
Chapter ~
Method of Stationary Phase
229
i) Let X, Y E g be semisimple. Then the critical set of s z , y is gx N C(Y). This is a clean intersection, and it is a disjoint union of finitely many G°-conjugacy classes, all of the Same dimension, and all belonging to the same conjugacy class in (gx)c under the complex adjoint group of (gx)c. ii) Let )~ C a*, and let "7 C G be a semisimple element. Then the critical set of S~,~ is GH~ M C(~/). This intersection is clean and is the union of finitely many G°H~-conjugacy classes, all of the same dimension. iii) In both of these cases, the Hessians are transversally nondegenerate. P r o p o s i t i o n 4.11
Remark
The cleanness and finiteness parts of the proposition follow
from well-known results of Richardson [Hi]. In (i), if X is regular and D is the Cartan subalgebra containing it, we have 9x = ~ and b A C ( Y ) is a finite set that is a subset of the complex Weyl group orbit of the point Yc E Oc, which is conjugate to Y under the complex adjoint group of f~c. In this case, s x , y is therefore a Morse function. It is possible that it has no critical points; this suggests that one has to compactify the orbit and consider sx,y on the compactification.
5
Spherical oscillatory integrals
5.1 O s c i l l a t o r y i n t e g r a l s w i t h m a n y f r e q u e n c y v a r i a b l e s The study of the phase functions on homogeneous spaces carried out in Section 4 is the foundation on which a theory of oscillatory integrals with these phase functions can be built. These oscillatory integrals arise in many problems of analysis involving semisimple Lie groups, and so it is worthwhile discussing what one can and should say about such integrals. Whenever one has a family of phase functions (~x) depending on a parameter x varying in some manifold X, the phase functions themselves being defined on a manifold M on which there is a volume form w, we have the oscillatory integrals
/M eir~(y) a(x " y)w
(a e C ~ ( X x K), K c M, K compact),
and the problems one encounters are of the following type:
(5.1)
V. S. Varadarajan
230
i) To investigate the asymptotic behavior of these integrals as ~-
~ cx~
for each x E X. ii) To get upper bounds for the integrals t h a t are locally uniform in x. As long as the phase functions are Morse (or at least clean) for generic x, there is no difficulty in treating (i) at the generic points x. But when x enters the caustic set of the family, the critical set becomes in general more complicated, and the problem of uniform estimates in the neighborhoods of p a r a m e t e r points that are in the caustic set is usually difficult.
We
have examined this situation in some detail in Sections 1-3. However, the families of phase functions that we have introduced in Section 4 behave very differently from generic families in two ways, one for the better and one for the worse. On the good side, we have seen in Section 4 that their behavior as the parameters approach the caustic set is very rigid and very simple. One is always in the clean situation, where the critical sets are smooth and the Hessians transversally nondegenerate. But on the bad side, it is
no longer sufficient to estimate the integrals of the form (5.1). The point is t h a t these phase functions are really vector valued, so t h a t if V is the vector space in which they take their values, and A is an element of V*, we are really interested in integrals
/M ei(~x(y)'~) a(x " y)~
(a E C ~ ( X x K ) ,
K C M, K compact), (5.2)
and we want estimates for the behavior of (5.2) when A
~ c~ in V*.
Clearly, this is a very much more general problem than (5.1). Indeed, if ()~i) is a basis of V* and we write
i
then we may think of (Ti) as a collection of frequency variables, and it is a question of studying what happens to the integral
M e i ~-~ ~i(x,y) a(x y)w as I~-I
~ c~. As we mentioned in Section 1.4, this is a new type of situation
and appears to be worth exploring for its own sake. Another way to look
Chapter ~ Method of Stationary Phase
231
at this is to say that in this problem it is a question of the structure of smooth maps of M into V, and so the problem is much more difficult than in the case of scalar phase functions, where Morse's theory describes the generic situation very well. In the next subsection, we shall investigate the integrals
fKeiA(H(ak))g(]g)d]g
(a e A, )~ e a*, g e C~(K))
(5.3)
from this "vectorial" point of view. Since these integrals occur naturally in the theory of spherical functions on semisimple Lie groups
indeed,
the elementary spherical functions on these groups are represented as such integrals
we shall call the integrals (5.3)
spherical oscillatory integrals.
Here, one can completely solve the problem of vectorial asymptotic estimates when a is in compact subsets of A [DKV1]. We shall give a brief discussion of the ideas behind the solution. However, in some spectral problems it is also important to understand what happens to the estimates that we obtain for (5.3) when a
) c~.
This again is a new situation because, at infinity in the parameter space (here it is A), the phase functions do not in general have a limit, and so it is not immediately obvious what should be done.
In Section 5.3 we
shall state a conjecture about uniform estimates for (5.3) when a
) co.
The conjecture, if proved, will imply certain estimates for the elementary spherical functions p(A : x) on a semisimple Lie group when both A and x go to infinity. The conjecture can be verified in some cases: when the symmetric space
G/K
has rank 1, and when G is complex.
In the last
section, we shall discuss an application of the conjectured uniform estimates of the elementary spherical functions to the problem of error estimation in the spectra of spaces of the form
F\G/K sx,y and
discrete subgroup such that The phase functions
L2(F\G/K)
where F is a cocompact
is smooth. S~,~ on conjugacy classes in g and G
are clearly important in the theory of characters and orbital integrals. We shall not treat the questions concerning their oscillatory integrals here.
5.2 Uniform estimates for spherical oscillatory integrals: method of trigonalization
The
We begin with the asymptotic expansion of (5.3) of Section 5.1 when goes to infinity along a ray. This is of course the classical situation and
V. S. Varadarajan
232
can be handled without any problem in view of the results of Section 4 on the phase functions Fa,~ and the theory developed in Section 2. We get the following result. Theorem
Fix ao E A, )~o C a*. Then we have the following asymp-
5.1
totic expansion of the function
eirF~o,Xo as a distribution:
eiTFao,~o r,J E eirw)~°(l°ga°)E T--nw/2Cw'r' w
(5.4)
r>0
where cw,r is a distribution of order <_ 2r supported by KaoWKH~o, the integer n~ is given by Proposition ~.10, and the sum is over a complete set of representatives of Wao\W/W~o. Moreover, the asymptotic expansion is uniform if a and )~ vary around ao and )~o in an equisingular manner. If a and A are regular, this gives the asymptotic expansion of the elementary spherical functions: ~(i)~ " a)
eirW~(l°ga)ei~/4a~Sw(a)'
~ T - dim(N)/2 E
(5.5)
W
where 5~ is the function on A defined by (a,A} sinh wa(log
a)] -~(~)/2
aEA+
with ~ an absolute constant that depends on the choice of the Haar measures on K and M. It is not possible to overlook the formal similarity of Theorem 5.1 with the famous Harish-Chandra asymptotics of the elementary spherical functions. There it is a question of a it is the other way around, with A
~ c~ for each fixed )~, whereas here ~ c~ as a remains fixed. Actually, if
a and A are regular, the expansion (5.5) can be obtained from the HarishChandra expansion of the elementary spherical functions. But Theorem 5.1 is more general; it applies to all amplitudes, hence to all matrix elements of the spherical principal series representations, and also to singular a and A.
Chapter ~
233
Method of Stationary Phase
If we desire asymptotic results t h a t are uniform when a and A vary in full neighborhoods of a0 and A0, then we have to face the problem of caustics, as we have explained repeatedly.
Because of the simplicity
of the behavior of the phase functions around the caustic sets, one can use Morse l e m m a with parameters to handle this situation. k0 E K is any point of Ko -
In fact, if
Kao,H~o, we can select local coordinates
X l , . . . , Xq, Y l , . . . , ys on K around k0 in such a way t h a t K0 is given locally around k0 by the equations yl . . . . .
ys - O. The phase function Fa,~
is then a function of the xi and yj and has a nondegenerate critical point when Xl = . . . .
Xq - 0, a - a0, A - A0. Hence, treating x l , . . . , X q , a ,
as p a r a m e t e r s and applying the m e t h o d of stationary phase as discussed in Section 2, we get uniform estimates for the spherical oscillatory integral. If we now observe t h a t as A varies around A0, the rays ~-A will fill up a conical neighborhood of A0 except for a compact set, we obtain the following corollary: C o r o l l a r y 5.2
Fix ao C A, Ao C a*. Then we can find a neighborhood
w of ao in A, a conical neighborhood F of Ao on a*, and a continuous s e m i n o r m v on C ~ (w x K ) such that for all a C w, A C F, g C C ~ (w x g ) , we have
I/K ei~(H(ak))g(a,k)dkl P(ga)
E H (1 ÷ W~o\w/w), aEA+ a0, HA 0
(5.6)
Unfortunately, as we vary A0 over a*, this estimate becomes too weak and so cannot be effective in many applications. So there arises the problem of uniform estimates, where we try to keep the same form as for the majorants on the right side of (5.6), but have a larger set of roots over which the product is taken, so t h a t the decay in A is much greater at infinity in a*. To do this, we have to take a closer look at the structure of the phase functions Fa,~ as a function of A. In what follows, we shall forget about a; everything we do will remain valid in a small neighborhood of a.
In
order to explain the fundamental idea of the uniform estimates theorem, we assume first t h a t A varies in the closure A + of the positive chamber, where
A+ =
I
> 0
V
e
(5.7)
V. S. Varadarajan
234
Then we know that Fa,~ is right invariant under
KHx.
Let us select an
ordering O~1,0/2,
• . . , OZr
of the simple roots, and define the dual (ordered) basis (Aj) in a* by {Aj, ai} = 5ij
(1 <_ i, j <_ r).
Then, we can write
l <_j<_r
l<_j<_r
If we put Ky = centralizer of {A1,/~2,
• • • ,/~j
(5.9)
},
then we have
K = Ko D K 1 D K2 D ... D K~ = M
(5.10)
and Fj is right Kj-invariant. It mac appear that we have lost some information since Fj is right invariant under the much bigger group KH),j; but this is not serious, because we have selected an arbitrary ordering of the simple roots here, and in an ordering where cU is the first, the information on Fj will be recaptured. To illustrate how we put the information contained in (5.10) to use, let us consider the identity in K, which is a critical point of all the Fj. The group Kt is the maximal compact subgroup of the reductive group for which the simple (restricted) roots are O Z t _ t _ l , C t t + 2 . . . . , C i r . The theory developed in Section 4 now implies that the restriction to Kj-1 of Fj has 1 as a critical point, with a Hessian nondegenerate transversally to Kj, so t h a t the rank of the Hessian is the integer nj, where
nj-
E
aEA;a
A+'3,a -- { ct C
n(a),
A+
I (log a)
7£ O, a -- E rntat, mj > 0}. t>_j
We now select a local section X j through 1 for K j _ I / K j .
Then, the right
invariance under the Kj satisfying (5.10) allows us to set up an equivalence of functions (F1,F2,.
. . ,Fr)
"-'
(fl, f2,. . . , fr),
Chapter ~
Method of Stationary Phase
235
where i) T h e fj are defined on X = X1 × X2 × -.. z X~. ii) fj depends only on the variables x[J]- ( X l , X 2 , . . . , x j ) , xj C X j . iii) T h e function fj(1, 1 , . . . , 1, xj) has xj = 1 as a n o n d e g e n e r a t e critical point with Hessian rank nj. T h e properties (i)--(iii) constitute w h a t we m e a n when we say t h a t the system
(Fj)l<_j<_r has the p r o p e r t y of trigonalizability around the identity 1 of K . We now consider the oscillatory integral
where the a m p l i t u d e is localized at 1 as an integral
x ei~-~"JTJfJg ( x l , . . . , x j ) w ,
(5.11)
where w is a volume form on X and g is s m o o t h with c o m p a c t support. T h e idea is to apply Morse's l e m m a with p a r a m e t e r s and carry out the integrations successively with respect to the variables xj, beginning with
the last, namely, xr. Lemma
5.3
The integral (5.11) has the bounds
I J X eiEjTJfJg(xl,...,Xj))021 ~ l](g) I I (1 l~j~_r
-~- Tj) -nil2
where y is a seminorm on C ~ ( X ) , and the 7 are restricted to a domain of the form T1 >_ ~/T2 >_ ~/2~-3 > " " >_ ~ / ~ - ~ for some "7 ~ O. Proof: I n t e g r a t i o n with respect to xr, with fr t r e a t e d as a Morse function with p a r a m e t e r s x 1 , . . . , X~-l, will produce an e s t i m a t e t h a t decays in ~-~ like ~_~-,~/2, but the new phase function will now be
E
l_~j
Tj f j(x [j]) + Tr_lfr_l (X[r-1]) ~- Trgr(x[r--1]),
V. S. Varadarajan
236
where g~(x [~-1]) is the critical value of fr(x[~-l],x~) as a function of
Xr.
We rewrite this as
E
Tjfj(x [j]) -Jr-Tr--1 (fr--l(X [r-l]) nt- (Tr/Tr--1)gr(x[r--1])) ,
l<_j
and for the next integration
Tr/7"r_l
will be a parameter that has to be
small; i.e., for 7 r - l , we have to have a domain 7r-1 _> 3'7~. If we continue this process we arrive at the estimate stated in the lemma with the 7j restricted to a domain of the form stated. [-1 The domains that we obtain are only a small part of the positive chamber of the 7j. ways.
But we can modify this analysis in two i m p o r t a n t
First, we can change the ordering of the Fj so that the Tj get
permuted.
Second, we can group the variables vj into disjoint subsets
and then consider the regime where the Tj in each group go to infinity in a mutually comparable manner, but the Tj in different groups go to infinity in a domain of the form described in the lemma. It turns out that the positive
chamber of the Tj can be written as a union of these special domains where we have groups of frequency variables going to infinity each faster than the next, but with the variables in each group going to infinity in a mutually comparable manner. At this stage we will have the definitive uniform bound for the spherical oscillatory integral (5.6), but with A restricted to be in the positive chamber A +. To remove this restriction appears at first to be a formidable technical problem, because we lose the right invariance that played a crucial role in the "trigonalizability" of the system of phase functions with which we are working.
But this can be overcome in the following way.
Recall that associated to the Fa, )~ are the infinitesimal versions fx,H for which, because of the symmetric way in which X and H enter their definition, there is right invariance with respect to the subgroups KH for all H, not just for the H in the positive chamber. So for them, the property of trigonalizability will persist, and this can be established exactly as before, since the critical sets have the same structure in both cases. But we now use the information contained in the equations (4.11a-c) to transfer the trigonalizability from the fX,H to the Fa,~, using the flow generated by the A-action on K / M that enters in the relationships expressed by (4.11a-c). It was in this manner that the following theorem was proved in [DKV1].
Chapter ~
237
Method of Stationary Phase
It is the definitive uniform estimate for the spherical oscillatory integrals on the flag manifolds, when the variable a is in a compact subset of A.
T h e o r e m 5.4
Let w be a compact set in A, and for w E w let
A w , w - - { c ~ E A + I wc~(loga)~=0
VaEw}.
Then there is a continuous s e m i n o r m u on C ° ° ( K ) such that for all )~ E a*, a E w, g E C o o ( K ) , we have
~ P~(H(ak))g(k)dkl~ ~(g)~ wEw
II
(1
+
-n(~)/2
c~EA+,~
5.3 T h e p r o b l e m of u n i f o r m b o u n d s for the spherical oscillatory
integrals at infinity on A It is a natural question to ask what happens to the estimate in Theorem 5.4 when a is allowed to go to infinity. This question has proved difficult to resolve so far except in some special cases. From the evidence provided by these special cases, it is possible that one should be able to prove t h a t the bounds in Theorem 5.4 remain valid in the cone generated in A by w, provided we add a factor of the form
Ce,.yI I logall
('y a constant > 0)
to the right side. More precisely, one may make the following conjecture
(c): Let ao E A \ {1}, and for any compact set w C A, let
C(w) - { e x p t H I t >__ 1, e x p H E w}. For w E w , let
/x~0,~ - {~ E /X+ I w~(log a 0 ) ¢ 0}. Then there is a compact neighborhood ~ of ao in A, a constant -y > O, and a continuous s e m i n o r m u on C o o ( K ) such that for all/~ E a*, a E C(w), g E C o o ( K ) , we have
I jfK
(1+ II wEw o~EA+o, ~
.
V. S. Varadarajan
238
It is possible to prove this when G / K has rank 1 and also in the case when w is a compact regular set in the positive chamber A + and A is restricted to a closed conical region inside the open positive chamber in a*. But the general case appears difficult to settle. If we take the amplitude g defined by
g(a, k) -- ep(H(ak))
(k e K),
then the integral is just the elementary spherical function ~(A • a), and the foregoing estimates become
I~(A" a)l _< Ce "/Ill°gall Z
H (1 + I(a, A>)I-n(~)/2. wEw ~E/~+
(5.12)
Actually, I believe that a much stronger estimate is true: There are constants C > 0, q >_ 0 such that I~(~X" a)l
~ Ce-p(l°ga) (1 -+-IllogaII) q ~
H
wEw aEA~,~ +
(1 + I(a,A))l -n(~)/2. (5.13)
Indeed this can be verified to be true in the following cases. i) G is complex (because in this case there are explicit formulae for the elementary spherical functions that go back to Harish-Chandra and Gel'fand-Naimark in the early 1950s). ii) G / K has rank 1. iii) G arbitrary, but w is restricted to be a compact subset in the interior of the positive chamber A +. 5.4 S p e c t r a l a s y m p t o t i c s of F \ G / K One reason for the interest in the asymptotics of elementary spherical functions when the frequency variable A goes to infinity is that the knowledge of such behavior plays a crucial role in the determination of the orders of magnitude of the asymptotics of the spectra of the spaces F \ G / K where F is a discrete cocompact subgroup of G. The space F \ G / K need not be smooth, and to make it smooth we have to ensure that F contains no elements of finite order. It is a well-known result of Borel that this can always be done by going over to a normal subgroup of F of finite index
Chapter ~ in F.
Method of Stationary Phase
So we shall assume that the space F \ G / K
is smooth.
239 Then, the
representation of G in L 2 ( F \ G / K ) decomposes discretely, and there is a discrete set S(F) of A in a~: with the property that the representations t h a t occur are precisely the spherical ones with parameters A in S(F), with fi-
nite multiplicities re(A). The set S(F) is called the spectrum of F \ G / K . It was proved in [DKV2] that the A that are not in (-1)1/2a* which form the so-called exceptional spectrum Se(F) are asymptotically negligible (in a sense to be explained) compared with the A in (-1)1/2a* which form the principal spectrum Sp(F)
, and that at infinity on (-1)1/2a*
the principal spectrum is asymptotic to the Plancherel measure. One way to formulate these results is as follows. 5.5 Let ~(t)t_>l be a family of bounded open sets with smooth boundary in ( - 1) 1/2 a* such that
Theorem
i) 3 constants C1 > O~ C2 > 0 such that for all t >__1 the diameter of the set ~t(t) is at most C1T, and its volume is at least C2t r (r - dim a*). ii) The measure of the boundary of ~(t) 'is o(tr-1). Then, as t
~ cxD, .~(~) - o ( t ~ - ~ ) . A~s~(F), IlAll_
Furthermore, /3(u)dv + O(t n-l).
Here, /3(u) be the density of the Plancherel measure for the decomposition of the natural representation of G on L 2 ( G / K ) . The first statement is the negligibility of the exceptional spectrum, since the main term in the second statement is of the order of t n-1. The error estimate appears not to be very good, and it is an interesting question as to how much sharper one can make it. This is a question that has attracted some attention. One way to treat it is to observe that the starting point of all the investigations is the Selberg
trace formula, which consists of a main term and a supplementary part; the retention of the main term only leads to the preceding result. If we now use
V. S. Varadarajan
240
the full trace formula and use Theorem 5.4 in analyzing the supplementary part of the trace formula, we get the following result, where the O has been replaced by o:
E ~s~(r), ~(t)
m()~) - £
(t)
/3(tJ)dv+ o(t n-l).
To go beyond this, it becomes necessary to analyze the supplementary part of the trace formula more closely, and this depends on accurate estimates on the behavior in ,~ of the elementary spherical functions p()~ : a) when a and ,~ both go to infinity. There is a consequence of (5.12) which we shall now formulate t h a t is one of the keys in this type of analysis. In fact, one can prove the following:
Under the assumption that the estimates (5.12) are true, we have the following. There exist constants C, c, rl > 0 such that for all
Theorem
5.6
u C a*, a C A,
/3(~)IsD(~ "a)l ~
C e c l l l ° g a [ l ( 1 nc
II~ll)dim(G/K)-r-n.
(5.14)
In view of our comments earlier, this result is true when G is complex and when r = 1. If we now assume t h a t (5.14) is true, we can state a sharper result on the error estimates for the spectrum Sp(F):
tn_1 ) ~s~(r), ~(t)
13(~,)dv+O
(logt) ~
"
It is possible t h a t these are too conservative, and t h a t the true error estimates are much sharper. In any case, it appears almost inevitable t h a t in any t r e a t m e n t of this question, the simultaneous asymptotics of the elem e n t a r y spherical functions in both the space and the frequency variables will play a crucial role.
Chapter ~
Method of Stationary Phase
241
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[Bol]
R. Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math. 9 (1956), 171-206.
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~ , On torsion in Lie groups, Proc. Nat. Acad. Sci. USA 40 (1954), 586-588.
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[D1]
J . J . Duistermaat, Fourier Integral Operators, Lecture notes, Courant Institute, 1973.
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[GP]
V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, 1974.
[GS]
V. Guillemin and S. Sternberg, Geometric Asymptotics, Mathematical Surveys and Monographs 14, Amer. Math. Soc., 1977.
[H]
S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962.
[He]
R. Hermann, Geometric aspects of potential theory in symmetric spaces, Math. Ann. 153 (1964), 384-394.
[Hec]
G. Heckman, Projections of orbits and asymptotic behaviour of multiplicities of compact Lie groups, Thesis, University of Utrecht, 1980.
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G. A. Hunt, A theorem of Elie Caftan, Proc. Amer. Math. Soc. 7 (1956), 307-308.
[Ko]
B. Kostant, On convexity, the Weyl group, and the Iwasawa decomposition, Ann. I~cole Norm. Sup. (4) 6 (1973), 413-455.
[M]
J. Martinet, Singularities of Smooth Functions and Maps, London Math. Soc. Lecture Notes 58, 1982.
[Mi]
J. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, Princeton, New Jersey, 1963.
[Mo]
M. Morse, Relations between the critical points of a real function of n independent variables, Trans. Amer. Math. Soc. 27 (1925), 345-396.
[ os]
G. D. Mostow, Equivariant imbeddings in Euclidean space, Ann. of Math. 65 (1957), 432-446.
[Pa]
R. S. PalMs, Imbeddings of compact diferentiable transformation groups in orthogonal representations, J. Math. Mech. 6 (1957), 673-678.
[Hi] ¸
R. W. Richardson, Jr., Conjugacy classes in Lie algebras and algebraic groups, Ann. of Math. 86 (1967), 1-15.
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S. Smale, A survey of some recent developments in differential topology, Bull. Amer. Math. Soc. 69 (1963), 131-145.
[St]
R. Steinberg, Lectures on Chevalley Groups, Lecture notes, Yale University, 1967.
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M. Takeuchi and S. Kobayashi, Minimal imbeddings of R-spaces, J. Differential Geometry 2 (1968), 203-215.
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V. S. Varadarajan, Harmonic Analysis on Real Reductive Groups, Lecture Notes in Mathematics 576, Springer-Verlag, 1977.
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E. Witten, Supersymmetry and Morse theory, J. Differential Geometry 17 (1982), 661-692.
Chapter 5 The Orbit Method and Unitary Representations for Reductive Lie Groups DAVID A. VOGAN, JR. *
Massachusetts Institute of Technology
CONTENTS Introduction 1. Reductive groups 2. Representations and operator algebras 3. Primitive ideals and Dixmier algebras 4. Structure of coadjoint orbits 5. (Co)adjoint orbits for reductive groups 6. Interlude on the classification of (co)adjoint orbits 7. Hyperbolic elements, real polarizations, and parabolic subgroups 8. Taylor series and differential operators on R n 9. Twisted differential operator algebras 10. Elliptic orbits, complex polarizations, and admissibility Acknowledgments References
Introduction
Suppose G is a real reductive algebraic group. A unitary representation of G is a Hilbert space endowed with a continuous action of G by unitary operators. It is irreducible if the Hilbert space is nonzero, but it cannot be written as a direct sum in a nontrivial G-invariant way. The orbit method * Supported in part by NSF grant DMS-9402994. 243
D. Vogan, Jr.
244
pioneered by Kirillov and Kostant seeks to construct irreducible unitary representations by analogy with quantization procedures in mechanics. A classical physical system may sometimes be modeled by a configuration space M, a manifold the points of which represent the possible positions of the bodies in the system. The corresponding phase space is the cotangent bundle T ' M , the points of which represent the possible states (positions and momenta) of the bodies in the system. In this setting, a classical observable is a function on T*M. The quantum mechanical analog of this system is based on the Hilbert space L2(M) of square-integrable half-densities on M. A state of the quantum system is a unit vector v in L2(M); Ivl 2 is a probability density on M, usually thought of as describing the probability of observing the quantum mechanical state in a given classical configuration. A quantum mechanical observable is an operator T on L2(M); the scalar product (Tv, v) is the expected value of the observable on the state v. Some connection between classical and quantum observables is provided by a symbol calculus; an interesting quantum observable is often a differential operator on M, and the corresponding classical observable is related to the symbol of the differential operator (a function on T* M). The states of some more complicated classical mechanical systems may be represented as points of a symplectic manifold X. Here, X corresponds to the cotangent bundle T*M in the previous example, but there is no longer an underlying configuration manifold M. Classical observables are still functions on X. A quantization of this classical system is a Hilbert space T/(X) endowed with an algebra A(X) of operators. Here, A ( X ) is to be some noncommutative analog of the algebra of functions on X. It is not easy to say in general what ~-/(X) ought to be, except in some special classes of examples like the one in the previous paragraph. This relationship between classical and quantum mechanics suggests a classical analog of an irreducible unitary representation of G: a homogeneous space for G, endowed with an invariant symplectic structure. Such homogeneous spaces turn out to be very close to coadjoint orbits: orbits of G on the dual of its Lie algebra. The orbit method in representation theory seeks to attach to a coadjoint orbit X (regarded as the phase space of a classical mechanical system admitting G as a group of symmetries) a Hilbert space ~-/(X) (the state space for the quantized system). If this can be done in sufficiently natural way, then the action of G on X by symplec-
Chapter 5 Orbit Method and Unitary Representations
245
tomorphisms will give rise to an action of G on T/(X) by unitary operators, that is, to a unitary representation 7r(X) of G. In the case of reductive groups, the orbit method is fairly well understood for semisimple orbits X: That is, one can construct in a natural way a unitary representation ~(X). The purpose of this chapter is to outline this construction. Following an idea of Dixmier, we will emphasize not so much the Hilbert space 7-/(X) as the algebra of operators A(X). are several advantages to this approach.
There
First, the relationship between
X and .A(X) is somewhat more elementary and direct than that between X and 7-/(X). Second, A(X) depends only on the complexification of G. Because complex reductive groups are more or less combinatorial in nature, the operator algebras are necessarily uncomplicated. Here is an outline of the chapter. In Section 1, we recall the definition of real reductive groups and some of their structure theory.
Section 2
recalls some general facts about representation theory of Lie groups and enveloping algebras.
Section 3 outlines ideas of Dixmier about ideals in
enveloping algebras. At the end of Section 3, there is a moderately precise formulation of the problem that the orbit method seeks to solve. Some of the basic notions from the method of coadjoint orbits are summarized in Section 4. Sections 5 and 6 describe the classification of coadjoint orbits for reductive groups. The orbits are all built from three special classes: hyperbolic, elliptic, and nilpotent. The semisimple orbits are those built only from hyperbolic and elliptic pieces. It turns out (although we will not explain why) that the construction of representations attached to coadjoint orbits can more or less be reduced to these three classes. The first two have been treated completely (as we will explain), but the third is understood only in special cases. In Section 7, we construct the representations attached to hyperbolic coadjoint orbits. Roughly speaking, these are principal series and "degenerate series" representations induced from one-dimensional characters of parabolic subgroups. The operator algebras that we construct here are all differential operator algebras. Section 8 is an elementary discussion of such operators, arranged in a form that fits well with the orbit method. The actual construction of some algebras A ( X ) appears in Section 9. Finally, Section 10 contains a (very incomplete) outline of Zuckerman's construction of representations associated to elliptic coadjoint orbits.
A
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D. Vogan, Jr.
complete account will appear in [KV]. Some other expositions of related material include IV4], [G], and IV6].
1
Reductive
groups
A real reductive group is an abstract mathematical object of which there are relatively few examples. For this reason, the general theory is informed and guided by the examples to a remarkable extent.
This is
evident even in one of the simplest definitions of a real reductive group, which is based on some familiar properties of matrices. Write G = GL(n, IR) for the group of invertible n x n matrices with real entries, the general linear group over IR. If g C G, define Og - tg -1 ,
(1. la)
the inverse of the transpose of g. Since transpose and inversion are both antiautomorphisms of G (meaning that t(gh) = thtg, for example), the map 0 is an automorphism. Since transpose and inversion have order 2 and commute with each other, 0 has order 2. We call 0 the C a f t a n involution of G. Write K = G o for the subgroup of fixed points of 0. This is the group O(n) of n x n real orthogonal matrices, the orthogonal group. It is compact. Write fl0 = $[(n, IR) for the Lie algebra of G, consisting of all n x n matrices with real entries.
The differential of 0 at the identity is an
involutive linear automorphism of $0, also written 0. We have OX = - t x
(1.1b)
for X E go. (Using the same letter for the differential of 0 is an abuse of notation, since G C g0, but no confusion should result.) The +l-eigenspace of 0 is [~o = n × n skew symmetric real matrices;
(1.1c)
it is the Lie algebra of K. The -1-eigenspace of 0 is P0 = n x n symmetric real matrices.
(1.1d)
fto = [~o ® Po,
(1.1e)
We have
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247
the Caftan decomposition of ~[(n, IR). There is a corresponding decomposition for G. For that, we need a lemma. Recall that a real matrix g is called positive definite symmetric if the bilinear form on IRn defined by Bg(v, w) = (gv, w) is positive definite and symmetric. (Here we have written ( , ) for the usual inner product on
Rn.)
i) An n × n real matrix X is symmetric if and only if there is an orthogonal basis of tRn consisting of eigenvectors of X . ii) An n × n real matrix g is positive definite symmetric if and only if there is an orthogonal basis of IRn consisting of eigenvectors of g with strictly positive eigenvalues. iii) The exponential map is an analytic diffeomorphism
Lemma
1.1
exp: P0 -+ P
from symmetric matrices to positive definite symmetric matrices. The inverse log: P --+ P0 is also analytic. S k e t c h o f proof: Part (i) is standard linear algebra. For (ii), suppose first that g C P. The s y m m e t r y of the form Bg makes g symmetric. The existence of an orthonormal basis of eigenvectors follows from (i). If v is a nonzero eigenvector with eigenvalue I, then the assumed positivity of Bg gives 0 < By(v, v)/(v, v) = (gv, v)/(v, v} = (Av, v)/(v, v) = I. So the eigenvalues are positive. The converse assertion is elementary. For (iii), it follows from (i) and (ii) that exp is a one-to-one map of P0 onto P. The analyticity of log is slightly more subtle; it may be proved for example by calculating the Jacobian of exp.
[3
Suppose G = GL(n, R), K = O(n), and P0 is the space of n × n symmetric matrices.
Proposition
1.2 (Polar or C a r t a n decomposition for GL(n,]~))
Then the map Kxpo~G,
(k, x)
k exv(X),
D. Vogan, Jr.
248
is an analytic diffeomorphism of K x P0 onto G. The inverse is given by g ~-~ ( g e x p ( - ~1 log((Og)-lg)),-~1 log((Og)_lg )).
Proof." Suppose g e GL(n,]~). Define p form attached to p is =
=
=
(0g)--lg
=
(tg)g.
The bilinear
=
which is symmetric and positive definite. So p is positive definite symmetric. By Lemma 1.1, there is a unique symmetric matrix X with p = exp(2X). Define k = g e x p ( - X ) . Then,
tkk : e x p ( - X ) t g g e x p ( - X ) : e x p ( - X ) e x p ( 2 X ) e x p ( - X )
= 1,
and so k belongs to O(n). By construction g = k exp(X), which proves the surjectivity of the polar decomposition map. The construction also proves the formula for the inverse. V1 The polar decomposition allows one to study many structural problems about GL(n, I~) in two steps: one involving the compact group K = O(n), and one involving the vector space P0 (often regarded as a representation of K). Elie Cartan discovered that all real reductive groups share a similar property. It is so fundamental that it may be taken as the definition of the class. D e f i n i t i o n 1.3 group if
A subgroup G c GL(n, ]1{) is called a linear real reductive
i) G is closed. ii) If Z is a symmetric matrix, then Z E Lie(G) if and only if exp(X) E G. iii) G is preserved by the Cartan involution 0 of GL(n, IR). That is, a matrix g belongs to G if and only if t g belongs to G. Suppose G is such a group. The restriction of 0 to G (still denoted 0) is called the Cartan involution of G. Write g0
=
Lie(G) C g[(n,]~)
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249
for the Lie algebra of G; necessarily, it is preserved by the Cartan involution 0 of g[(n, I~). Accordingly, we can write go -
~0 ® P0
for the decomposition into +1 and - 1 eigenspaces (the skew-symmetric and symmetric matrices in g0). Put K - G e - G n O(n), a compact subgroup of G (by condition (i)); its Lie algebra is ~0. Proposition
1.4 (Cartan decomposition for linear real reductive groups)
In the setting of Definition 1.3, the map Kxp0~G,
(k, x)
k exp(X),
is an analytic diffeomorphism of K x P0 onto G. Proof: Suppose g C G. Write g = k exp(X) for the polar decomposition of g in GL(n,R). Then exp(2X) - tgg, which belongs to G by condition (iii) of Definition 1.3. By condition (ii) of the definition, X belongs to P0. It follows that k = g e x p ( - X ) belongs to G n O(n) = K. This proves the surjectivity of the Cartan decomposition map. The remaining assertions follow from Proposition 1.2. E] Many of the most interesting real reductive groups are nonlinear; that is, they do not appear as subgroups of GL(n, I~). The following definition is broad enough for us. D e f i n i t i o n 1.5 A real reductive group is a Lie group G endowed with a continuous homomorphism 7r:G --. GL(n,I~), subject to the following conditions: i) 7r(G) - G is a linear real reductive group (Definition 1.3). ii) The kernel of 7r is finite. Suppose G is such a group. We use the differential of Tr to identify the Lie algebra 90 of G with g0 - Lie(G) C g[(n,I~). Define ~0, P0, and K as in Definition 1.3, and put g - 7r-: (K). By hypothesis (ii), K is a compact subgroup of G with Lie algebra ~0.
D. Vogan, Jr.
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P r o p o s i t i o n 1.6 (Cartan decomposition for real reductive groups)
In the
setting of Definition 1.5, the map Kxp0~G,
(k, X) ~ k exp(X),
is an analytic diffeomorphism of K x P0 onto G. The map 0: G ~ G defined by O(k exp(X)) - k e x p ( - X ) is an involutory automorphism (the Cartan involution) with G ° - K . P r o o f : Given g E G, write ~(g) -- k exp(X), D
with X E P0. The identification of P0 with P0 provides an element X E P0The element k - g e x p ( - X ) satisfies ~(k) - k, and so belongs to K. This I
proves the surjectivity of the decomposition, and the analyticity of the inverse maps. Analyticity of the map to G follows from the analyticity of the exponential map and of multiplication. For the last assertion, 0 is obviously an analytic map of order two with fixed point set K. What must be shown is that O(gh) - O(g)O(h) for all g, h E G. To prove this, consider the function
F(g, h) -- O(gh)O(h)-lO(g) -1
from G x G to G. We want to show that it carries G x G to the point 1. Since 0 is an automorphism of G, F takes values in the kernel of ~. Since this is a finite set, F is constant on the connected components of G x G. Obviously, F is trivial on K x K; but by the Cartan decomposition, this subgroup meets every connected component of G x G.
[2
The definition of reductive group has the following "hereditary" property.
Chapter 5 Proposition
1.7
Orbit Method and Unitary Representations
251
Let G - K e x p ( p o ) be the Caftan decomposition of a
real reductive group with Cartan involution 0. Let H be a subgroup of G, and assume that i) H is closed. ii) If X E P0, then X C Lie(H) if and only if e x p ( X ) E H . iii) H is preserved by 0. Then H is a real reductive group with Cartan involution OIH. This is almost immediate from Definition 1.5. To construct interesting examples of reductive groups, we need a way to verify condition (ii) of Definition 1.3. Here is one. Lemma
1.8
Suppose A is an n × n matrix and X is a symmetric n × n
matrix. Then X commutes with A if and only if e x p ( X ) commutes with A. Proof." The first (resp., second) condition is equivalent to the assertion t h a t the linear t r a n s f o r m a t i o n A respects the decomposition of ]~n as a direct sum of eigenspaces of X (resp., e x p ( X ) ) . decompositions coincide. Lemma
1.9
By L e m m a 1.1, these
[5
Suppose G is a real reductive group and X E P0.
i) I f g E G, then A d ( g ) X - X if and only if g commutes with exp X . ii) If Y E ~o, then a d ( X ) Y - 0 if and only if Ad(exp X ) Y - Y . Proof:
The two s t a t e m e n t s are very similar; we consider only the first.
Write 7r:G --~ G C GL(n,I~) as in Definition 1.5, and ~ -
7r(g), X dTr(X). "Only if" is trivial, so suppose g commutes with exp X. Applying
7r, we find t h a t ~ commutes with exp X. By L e m m a 1.8, ~ c o m m u t e s with X; so A d ( ~ ) X - X. Consequently, A d ( g ) X - A d ( ~ ) X - X. Since dTr is one-to-one, it follows t h a t A d ( g ) X = X , as we wished to show. Proposition
1.10
Suppose G is a real reductive group with Caftan invo-
lution O. i) If S is a O-stable subset of G, then H-
[:]
Z c ( S ) - {g C G lgsg -1 - s, all s e S }
is a real reductive group with Cartan involution
OIH.
252
D. Vogan, Jr.
ii) If ~ is a O-stable subset of go, then H-
Z c ( s ) = {g e G I A d ( g ) Y -
Y, all Y e ~}
is a real reductive group with Cartan involution OIH. P r o @ We apply Proposition 1.7. Conditions (a) and (c) are immediate, and (b) is satisfied because of Lemma 1.9. [] A second way to verify condition (ii) of Definition 1.3 is using bilinear forms. L e m m a 1.11 Suppose B is a bilinear form on ]~n, and X is a symmetric n x n matrix. Then the following two conditions are equivalent: 1) For all v, w e ~n, B(X~), w) -97B('v, X w ) -- O. 2) For all v, w e ]~n, B ( e x p ( X ) v , e x p ( X ) w ) = B(v, w). P r o o f : The conditions may be checked for v and w belonging to eigenspaces of X or exp(X). Write Vs for the eigenspace of X with eigenvalue s C l~. Then (1) is equivalent to 1') Whenever s + t 7~ 0, B(Vs, Vt) = O. Similarly, write W~ for the eigenspace of exp(X) with eigenvalue A > 0. Then (2) is equivalent to 2') Whenever A# ~: 1, B ( W a , W~,) = O. Since Vt = Wexpt, conditions (1') and (2') are equivalent.
[-7
Any bilinear form B on ]~n may be represented by a unique n x n matrix A, by the formula B(v, w) : (Av, w);
(1.2a)
the form on the right is the standard inner product on R n. The symmetry group of the form is
G(B)
-
{g e
GL(n,
l B(gv, gw) : B(v, w), all v, w e mn}
: {g e GL(n,]R) ItgAg- A}.
(1.2b)
We are interested in constructing real reductive groups as symmetry groups of forms. In order to apply Proposition 1.7, we need to know conditions for G ( B ) to be 0-stable. Here is a simple one.
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Orbit Method and Unitary Representations
253
P r o p o s i t i o n 1.12 Suppose B is a bilinear form o n ~ n , represented by an n x n matrix A. If A 2 = cI is a nonzero scalar matrix, then the symmetry group G(B) is preserved by the Cartan involution 0 of GL(n,R). Consequently, G(B) is a real reductive group; the maximal compact subgroup K ( B ) is the centralizer of A in O(n).
Proof: Recall that Og - tg-1 for g E GL(n,R). The condition in (1.25) for g to belong to G(B) is tgAg = A. By inverting both sides, this condition is equivalent to g - l A - 1 t g-1 _ A_I. The hypothesis on A says that A -1 - c-lA; multiplying our condition on g by c gives g 1Atg - A, or _
-1
t(Og)A(Og) = A; and this is the condition for Og to belong to G(B). To see that G(B) is reductive, apply Definition 1.3: condition (i) is clear, (ii) is Lemma 1.11, and we have just established (iii). K] E x a m p l e 1.13 Suppose p and q are nonnegative integers, and n = p + q. The standard quadratic form of signature (p, q) on I~n is the form gp,q(V,
w)
- - V l W 1 -~- . . .
~- V p W p -
Vp+lWp+
1 .....
VnWn.
The corresponding matrix Ap,q is diagonal with p entries equal to 1 and q equal to - 1 . Obviously A2q - I, and so the symmetry group of Bp,q is a linear real reductive group. It is called O(p, q), the real orthogonal group of signature (p, q). The maximal compact subgroup is O(p) × O(q). For a second example, the standard symplectic form on R 2n is n ViWnWi
-
m VnWiWi"
i--1
The corresponding matrix is often called J; it is
('n 0) 0
--/n
This is precisely the matrix of multiplication by i in a certain identification of R 2~ with C n. A first consequence is that j2 = - I , and so the symmetry group of w is a linear real reductive group. It is called Sp(2n, 1~), the real
symplectic group. The maximal compact subgroup K(w) is the centralizer of J in O(2n). The centralizer of J consists precisely of the linear transformations that are complex linear when ]~2~ is identified with C n. We may therefore identify K ( w ) w i t h the unitary group U(n).
D. Vogan, Jr.
254
A third construction of real reductive groups is by changing the base field. Let F denote one of the three fields IR, C, or ]HI, and put d = dim~ F. The standard basis of IF (namely {1}, {1, i}, or {1,i,j,k}) provides an identification of the right vector space F n with •nd. In particular, each element z C F defines by right multiplication a linear transformation End(IRnd). The IR-bilinear form on F n defined by
p(z) C
is just the standard inner product on IRnd; here bar denotes the standard antiautomorphism of F (acting by +1 on the first basis element of F and by - 1 on the others). Obviously,
(p(z)v, w) - (vz, w} - Re (~--~ VpZWp) p--1
Vp(Wp-2) - (v, w-2)- (v, p(-2)w).
(1.3a)
p(-2) = t p(z),
(1.3b)
p(-2-1) - O(p(z))
(1.3c)
= Re p=l
So
and
for z C IF nonzero. Now the algebra g[(n, F) of n x n matrices over F may be identified with the algebra of F-linear transformations of F ~ , and so with the R-linear transformations of R nd commuting with right multiplications by F; that is, with the centralizer of p(F) (or p(F × ) ) i n ~[(nd, IR). In particular, GL(n, F) = centralizer of p(F × ) in GL(nd, IR);
(1.4)
this is a linear real reductive group by (1.3c) and Proposition 1.10. More generally, we have
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255
P r o p o s i t i o n 1.14 Suppose F is a field of dimension d over IR, and G is a linear real reductive group in GL(nd, IR). Then G A GL(n,F) (cf. (1.~)) is a linear real reductive group. This follows from Proposition 1.7, Lemma 1.8, and (1.3c). E x a m p l e 1.15 Suppose p and q are nonnegative integers, and n = p + q. The standard Hermitian form of signature (p, q) on C n is the form Up,q(V,
w)
-- VlW
1 --~- . . . .qt_ V p W p
-
Vp+lWp+
1 . . . . .
VnW n.
The group of complex-linear transformations preserving this form is called U(p, q), the unitary group of signature (p, q). If we identify C n with IR2n, then the real part of Hp,q is the quadratic form B2p,2q of Example 1.13. It is easy to check that a complex-linear transformation preserving the real part of Hp,q must preserve the entire form; so it follows that U(p, q) = O(2p, 2q) N GL(n, C). By Proposition 1.14, this is a linear real reductive group. The proof shows that its maximal compact subgroup is (O(2p) x O(2q))N GL(n, C) = U(p) x U(q).
2
Representations
and operator algebras
Suppose G is a topological group. A unitary representation of G is a pair (Tr, 7-/), with 7-I a complex Hilbert space and 7r: G ~ V(7-/) a homomorphism into the group of unitary operators on 7-/. These are the invertible operators preserving the inner product: the assumption is (v, w e 7-{, g E G).
(2.1a)
Often, it is convenient to formulate this condition as
(v, w E 7-l, g C G),
(2.1b)
256
D. Vogan, Jr.
or simply as 77(g)* = 7 7 ( g - 1 ) . We assume also that 77 is weakly continuous; that is, that the map
G x "H --* ~,
(g, v ) ~ 77(g)v,
(2.1c)
is continuous. An invariant subspace of 7-/ is a closed subspace 7-/0 C that is preserved by all the operators 77(g). In this case, the restricted operators define a unitary representation (770,7-/o) of G. The orthogonal complement 7-ll of 7-/o is a second invariant subspace, and 7 - / - 7-/0 • 7-/1; we write 7r = 770 G 771 accordingly. We say that 77 is irreducible if 7-/~= 0, and the only invariant subspaces of 7-/are 0 and 7-/. Suppose (771,7-/1) and (772,7-/2) are unitary representations of G. An intertwining operator from 771 to 772 is a continuous linear map T from 7-ll to 7-/2 with the property that T771(g) = 772(g)T
(g e G).
The space of all intertwining operators is written Homa(7-/1,7-/2). Forming the adjoint T* defines a conjugate-linear isomorphism Homa(7-/1,7-/2) --~ Homa (7-/2, ~'~1). If some intertwining operator is a unitary isomorphism, then we say that 771 and 772 are equivalent. We write G for the set of equivalence classes of irreducible unitary representations of G. The fundamental problem of abstract harmonic analysis is this: to decompose into irreducible representations an arbitrary unitary representation of G. This problem has a fairly good abstract answer for a large A
class of groups including the real reductive Lie groups. The case of compact groups will be recalled at (2.4). The abstract answer is of little help in finding explicit decompositions for particular interesting representations, however, and this remains an active area of research. A second basic problem in abstract harmonic analysis is the determination of the set G. If this can be accomplished, a question about G may sometimes be analyzed along the following lines. First, the question is made into one about some unitary representation (77,7-l) of G (perhaps on square-integrable functions on a homogeneous space, for example). Next,
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257
the representation 7r is decomposed into irreducible representations. At the same time, one tries to make the original question into a family of questions about the irreducible constituents of ~r. Finally, this family of questions is answered using our knowledge of all irreducible representations. Of course each step of this program is fraught with peril; but it has been carried out with some success in a variety of cases. A striking example quite close to the topic of this chapter is provided by the book [BW], which analyzes the cohomology of cocompact discrete subgroups of real reductive Lie groups. To carry out the program just sketched, one needs to understand not all irreducible unitary representations, rather just those appearing in whatever representation 7r one first constructs. With this in mind, we formulate
Suppose G is a real reductive Lie group. Construct a family of irreducible unitary representations of G sufficiently large to decompose many interesting unitary representations into irreducibles.
P r o b l e m 2.1
Of course this "problem" is not at all well defined, because of the phrase "many interesting." We will make it more precise, and outline what is known about solving it. To explain the approach we will adopt, it is helpful to begin with the case of a finite group G. The group algebra C[G] is the algebra over C with basis {Sg[ g E G}, and multiplication table
696h = ~Sgh
(;)"
(g, h e G).
(2.2a)
This algebra has a conjugate-linear antiautomorphism ,, defined by
gEG
Then, a unitary representation (Tr,7-l) of G defines an associative algebra homomorphism 7r: C[G] --, E n d ( ~ ) ,
7 r ( Z ag6g) = Z
agzr(g),
(2.2c)
satisfying 7r(a*) = 7r(a)*.
(2.2d)
(We say that 7r is a ,-homomorphism.) Conversely, any ,-homomorphism 7r from C[G] to End(T/) arises from a unitary representation of G on 7-/.
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P r o p o s i t i o n 2.2 Suppose G is a finite group. Then the equivalence classes of irreducible unitary representations ~ of G are in one-to-one cor-
respondence with the maximal two-sided ideals Ix in C[G]. This bijection has the following properties. i) If (Tr, ~ x ) is an irreducible unitary representation, then Ix is the kernel of the algebra homomorphism w: C[G] ~ E n d ( ~ x ) of (2.2c). ii) Suppose Ix is a maximal two-sided ideal in C[G]. Choose a maximal left ideal Jr D I~, an d define Vx = C[G]/Jx. Then V~ is isomorphic to ~ x as a module for C[G]. This is well known and not very hard to prove. It is not often used directly to determine the irreducible representations of G, because C[G] is such a complicated algebra. Nevertheless, the proposition suggests a way to approach the representation theory of a group G. First, one should find an associative algebra A(G) the module theory of which is related to the representation theory of G. Next, one should study the ideals in A(G). For a locally compact group G, a natural candidate for the algebra A(G) is the convolution algebra L I(G) (with respect to a left Hear measure dg). Given a unitary representation (w,T/) and f e LI(G), one defines 7r(f) -- fG f(g)Tr(g)dg. Then one can make an excellent correspondence between unitary representations of G and appropriate modules for L I(G). These modules all extend to a certain completion of L 1(G), called C* (G). For type I groups G (which include the real reductive Lie groups), there is a bijection between irreducible unitary representations of G and primitive ideals in C*(G). Now this bijection is a powerful technical tool for example, it is at the heart of the abstract theory of decomposition into irreducible representations. But as a way to describe irreducible representations explicitly, it is (like Proposition 2.2) of little value. The algebra C* (G) is too complicated. We would like instead an algebra the module theory of which is perhaps not quite so perfectly related to the unitary representation theory of G, but with an ideal theory we can hope to study directly. Suppose now that G is a Lie group. Write go = Lie(G),
1~= go ®~ C,
U(g) = universal enveloping algebra of g.
(2.3) One of the central ideas of Lie theory is that these objects can be used to translate problems about Lie groups into linear algebra.
Here is an
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example, along the lines of Proposition 2.2. Recall that a finite dimensional (nonunitary) representation (~, V~) of G is just a continuous homomorphism 7~ of G into GL(V~), the general linear group of some finite dimensional complex vector space V~.
Suppose G is a connected, simply connected Lie group. Then the finite dimensional irreducible representations 7~ of G are in oneto-one correspondence with the two-sided maximal ideals Ir of finite codimansion in U(O). This bijection has the following properties.
P r o p o s i t i o n 2.3
i) Suppose (Tr, V~) is a finite dimensional irreducible representation of G. Write ~: 00 --~ 0[(Vr) for the differential of :r (a Lie algebra representation), a n d ~: U(0) -~ End(V~) for its extension to U(O) (a homomorphism of associative algebras). Then Ir is the kernel of 7r: U(0) ~ End(V~). ii) Suppose I~ is a maximal two-sided ideal of finite codimension in U(O). Choose a maximal left ideal Jr D I~, and define Vr = U(O)/J~. Then the action of Oo on Vr by left multiplication in U(O) is a Lie algebra representation; the corresponding group representation of G is isomorphic to 7r. Like Proposition 2.2, this is an elementary result. Because U(0) is a relatively uncomplicated algebra, Proposition 2.3 is sometimes even directly useful for studying finite dimensional representations. We turn now to the problem of extending Proposition 2.3 to cover unitary representations. A continuous homomorphism between Lie groups is automatically smooth (and even analytic); this is used in Proposition 2.3 to guarantee the existence of the Lie algebra representation ~: 00 --+ 0[(V~). For infinite dimensional representations the situation is more complicated. D e f i n i t i o n 2.4
Suppose G is a Lie group, and (~,:Hr) is a unitary rep-
resentation of G. The vector v C 7-/r is called smooth (rasp., analytic) if the map G ~ ~t~,
g ~ ~(g)v,
is smooth (rasp., analytic). Write T/~ (rasp., 7-/~) for the set of smooth (rasp., analytic) vectors in 7-/~. These are G-invariant linear subspaces of T/r, but they are usually not closed.
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Suppose X E g0. The differential of 7r at X is the linear transformation 7r(X) of ~ defined by
lr(X)v - lim(1/t)(Tr(exp t X ) v - v). t---~O
L e m m a 2.5 The differential of ~ is a Lie algebra representation of $o on Tl~. It therefore defines a homomorphism of associative algebras ~" U(g) ---. End(T/~). All of the resulting operators preserve the subspace ~ C Tl~. It is not very difficult to show that 7-/~ is dense in ~ . It is also true that 7-/~ is dense in T/z; this much deeper result, which we will use in the proof of Proposition 2.12, is due to Nelson. In any case, we can now construct one of the maps of Proposition 2.3 for unitary representations. D e f i n i t i o n 2.6 Suppose G is a Lie group, and (r, 7-/~) is a unitary representation of G. The annihilator of ~ in U(g) is the kernel of the homomorphism of Lemma 2.5: Ann(Tr) = ker(~r: U(g) --. End(T/7) ). This is a two-sided ideal in U(g); it is equal to U(g) if and only if T/~ = 0. As some assurance that this definition is well behaved, here is an elementary lemma. 2.7 In the setting of Definition 2.6, suppose W C 7-l~ is any subspace that is dense in Tl~. Then
Lemma
Ann(Tr) = Ann(W) = {u e U($) l Tr(u)w = O, all w e W } .
So we may compute the annihilator on analytic or (when these are defined) K-finite vectors. W h a t properties can we expect of the annihilator of an irreducible unitary representation? Fairly simple examples show that it need not be a maximal ideal in general. Ring theory has a natural suggestion to offer.
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D e f i n i t i o n 2.8 Suppose R is a ring with unit element. A left R-module M is called simple if it is not zero, and its only submodules are 0 and M . A two-sided ideal I C R is called (left) primitive if there is a simple R-module M such t h a t I = Ann(M)=
{r E R [ r M = 0}.
Suppose G is a connected Lie group and (Tr, ~ ) is an irreducible unitary representation of G. Then Ann(Tr) is a primitive ideal in U(I~). Theorem
2.9 (Dixmier [D])
The reason this is not obvious is that 7-/~ is not a simple U(g) module (unless 7r is finite dimensional). For reductive G, we will see in Theorem 2.17 how Harish-Chandra constructs a simple U(g)-submodule of 7-/~ t h a t is dense in 7-/~. In light of L e m m a 2.7, this proves Theorem 2.9 in the reductive case. Dixmier's proof in general involves several i m p o r t a n t ideas, so we will outline the easiest part of it.
Suppose G is a connected Lie group, (Tr,7-l~) is a unitary representation of G, and V C ~ is a U(l~)-invariant subspace. Then the closure 7-lo of V in TlTr is a G-invariant subspace.
Lemma
2.10
Proof: For any subset S of 7-/~, define
The closure of any subspace S may be characterized as (S±) ±, and so it suffices to show t h a t V ± is G-invariant. For this, fix w E V ± and v E V; we must show t h a t the function f(g) = @(g)w, v) is identically zero. Because 7r is unitary, f(g) = (w, 7r(g-1)v). Because v is assumed to be an analytic vector, the function f is analytic on G. Because G is connected, it therefore suffices to show that all derivatives of f vanish at the identity on G. A typical derivative of f at the identity is (w, 7r(u)v), with u E U(g). Since Y is assumed to be U(g)-invariant, 7r(u)v E V. Since w E V ±, the derivative vanishes, as we wished to show.
71
(I am grateful to P. E. P a r a d a n for showing me this elegant argument.)
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L e m m a 2.11. Suppose G is a connected Lie group and (~, Tt~) is an irreducible unitary representation. Then any nonzero U(~)-invariant subspace V C ~ is dense in TI~. Consequently, A n n ( V ) = Ann(~). (The last assertion uses Lemma 2.7.) P r o p o s i t i o n 2.12. Suppose G us a connected Lie group and (7~,~ ) is an irreducible unitary representation. Then Ann(~) is a prime ideal in U(~).
Proof: Recall that a two-sided ideal I in a (possibly noncommutative) ring R is called prime if whenever J1 and J2 are two-sided ideals with J1 J2 C I, then either J1 C I or J2 C I. So suppose J1 and J2 are ideals in U(g) with J1J2 C Ann(Tr), but J2 ~ Ann(~). Then J27-/~ = V is a nonzero U(g)-invariant subspace of ~ . By Lemma 2.11, A n n ( V ) = Ann(rr). But J1V-
J1 J 2 ~ ~ c
Ann(Tr)7-/~
-
0;
so J1 C Ann(T r), as we wished to show. V1 Dixmier completes the proof of Theorem 2.9 using T h e o r e m 2.13 ([D]) Suppose I c U(g) is a prime ideal. Then I is primitive if and only if the center of the ring of fractions of U ( g ) / I is C. We omit the details. We now have a reasonable analog of Proposition 2.3(i): a map from irreducible unitary representations to primitive ideals in U(~). To get an analog of (ii) (a corresponding parametrization of representations), we need to specialize to real reductive groups. Let us first recall the structure of unitary representations for compact groups. Suppose K is a compact topological group. Then, every irreducible unitary representation of K is finite dimensional. Fix a model (5, Vb) for each equivalence class in K. If (~, 7-/~) is an arbitrary unitary representation of K, define ~-/~ - H o m e (Vb, T/~).
(2.4a)
We make 7-/~ into a Hilbert space as follows. If T and S belong to 7-/~, then
S*T is a map from V5 to V~ commuting with the action of K. By Schur's
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lemma, it is a scalar operator AI; and we define (T, S) - A. An equivalent formulation is (Tv, S w ) ~ . - (T, S)u~ (v, w}v~ (2.4b) for v, w C V£. Now we can form the Hilbert space tensor product ~ ® V£. (Because V5 is finite dimensional, it coincides with the algebraic tensor product.) There is a natural map
~
® V5 ~ ~ ,
T ® v H Tv;
(2.4c)
and (2.4b) guarantees that this map preserves inner products. It is therefore an isomorphism onto its image T/~(5), the 5-isotypic subspace of TI~. This is the largest subspace of ~-/~ on which K acts by a sum of copies of 5. By Schur's lemma again, 7-/~(5) and T/~(5') are orthogonal whenever 5 and 5 t are inequivalent. Consequently, i
A
~. ~ ~~(5)~ h
5EK
0 ~
® V~,
(2.4d)
h
6EK
the direct sums being Hilbert space direct sums. A vector v C ~-/~ is called K-finite if it is contained in a finite dimensional K-invariant subspace. We write ~_/K for the space o f / ( - f i n i t e vectors. Using (2.4d), we find that (2.4e) h
5EK
6CK
the direct sums now being algebraic rather than Hilbert space. The representation 7~ is called admissible for K if all of the spaces ~-/~(5) are finite dimensional; that is, if every irreducible representation of K has finite multiplicity in ~. Suppose now that G is a real reductive Lie group with maximal compact subgroup K. A unitary representation (~, ~-/~) is called admissible if it is admissible for K. Theorem
2.14 (Harish-Chandra [HC1])
Every irreducible unitary repre-
sentation of a real reductive Lie group G is admissible. This is a rather difficult result, relying on a deep study of the adjoint action of K on U(g).
264
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Admissible unitary representations have an excellent algebraic description.
Suppose (Tr,~ ) is an admissible unitary representation of a real reductive Lie group G. Write ~ g for the space of K-finite vectors in ~ .
T h e o r e m 2.15 (Harish-Chandra [HC1])
i) Tl g is a U($)-invariant subspace of the analytic vectors ~ . In particular, ~ g carries representations of the group K and the Lie algebra g. ii) There is a bijection between closed G-invariant subspaces of Tl~ and arbitrary (g, K)-invariant subspaces of ~ g . The correspondence from left to right sends a closed subspace W to W g (the K-finite vectors in W); from right to left it sends Y to Y (the closure of Y in Tl~). To complete this circle of ideas, we will describe formally the algebraic objects arising in Theorem 2.15. (The definition is taken from [L].) D e f i n i t i o n 2.16 Suppose G is a real reductive Lie group with maximal compact subgroup K. A ($, K)-module is a complex vector space V endowed with representations of the Lie algebra $ and the group K, subject to the following conditions. i) The action of K is locally finite and smooth. That is, every v C V belongs to a finite dimensional K-invariant subspace F C V, and the action of K on F is smooth. ii) The differential of the action of K (which makes sense by (i)) is equal to the action of g0 = Lie(K) C g. iii) For k E K, v E V, and X E g, we have k. ( X . v) - A d ( k ) ( X ) . (k-v). For k in the identity component K0 of K, condition (iii) is a consequence of (i) and (ii). We may therefore omit condition (iii) when K (or, equivalently, G) is connected. A ($, K)-submodule of V is a complex subspace W invariant under the representations of $ and g . (By conditions (i) and (ii), invariance under K0 follows from invariance under g. If K is connected, a (~, K)-submodule is therefore just a $-submodule.) We say that V is irreducible if it is not zero,
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and the only submodules are 0 and V. Finally, we say that V is unitary if it is endowed with a positive definite Hermitian form ( , ) satisfying
(k.v, k. w) = (v, w), forv, w E V ,
kEK,
(X.v, w) + (v, X - w ) = 0,
andXEg0.
Theorem 2.15(i) guarantees that the space 7-/g of K-finite vectors in an admissible unitary representation is a unitary (g, K)-module; it is called the Harish-Chandra module ofr~. Theorems 2.14 and 2.15(ii) say that 7-/g is irreducible as a ($, K)-module whenever rr is irreducible. Harish-Chandra's last basic result is a converse. T h e o r e m 2.17 (Harish-Chandra [HC1]) Suppose G is a real reductive Lie group. The map rr ~-+ TI g is a bijection from (equivalence classes of) irreducible unitary representations of G onto (equivalence classes of) irreducible unitary ($, K)-modules. When G is connected, we have seen that an irreducible ($, K)-module is irreducible as a representation of g. This theorem may therefore be viewed as an infinite dimensional analog of Proposition 2.3(ii). It provides a construction of irreducible unitary representations of G from certain (very special) irreducible U(g)-modules. We can now refine slightly Problem 2.1. !
P r o b l e m 2.18 Suppose G is a real reductive Lie group; assume for simplicity that G is connected.
i) Construct a family of interesting primitive ideals I C U($). ii) For each primitive ideal I as in (i), construct a finite set of irreducible unitary representations rr of G, satisfying Ann(Tr)= I (Definition 2.6). This formulation is still imperfect, but it begins to reflect what we will actually do. The constructions in (ii) will generally take place in three steps. First, we will construct some U($)-modules W with annihilator equal to I. This step is usually fairly easy; some possibilities for W will often be suggested by the construction of I in (i). Next, we will construct from each W a (g, K)-module V, still annihilated by I. The principle of this construction (due to Zuckerman) is simple and elegant, but analyzing it in
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detail can be quite difficult. (One minor point is that the annihilator of V may be strictly larger than I.) Finally, we will apply Harish-Chandra's Theorem 2.17 to get a unitary representation of G.
3
P r i m i t i v e ideals a n d D i x m i e r algebras In this section we will consider more carefully part (i) of Problem
2.18: the construction of a family of primitive ideals related to unitary representations. A more detailed account may be found in [V3] and [VT]. The first point is that not every primitive ideal can be the annihilator of a unitary representation. Suppose for a moment that G is a connected noncompact simple Lie group.
Then G has a large family of irreducible
finite dimensional representations (parametrized by a cone in a lattice of dimension equal to the rank of G).
By Proposition 2.3, it follows t h a t
U(~) has a large family of maximal ideals of finite codimension. On the other hand, a unitary finite dimensional representation is a homomorphism 7r: G ---, U(n). Because G is noncompact and simple, such a homomorphism must be trivial. This proves
Suppose G is a connected noncompact simple Lie group, and suppose I C U(g) is a maximal ideal of finite codimension. Then I is the annihilator of a unitary representation if and only if I - gU(~) (the augmentation ideal).
Lamina
3.1
W h a t distinguishes the augmentation ideal among all maximal ideals of finite codimension? If I is such an ideal, then the Wedderburn theorem guarantees that
U(~)/I ~ Mn(C),
(3.1)
the algebra of n x n matrices. Since ~ is semisimple, n = 1 occurs only for the augmentation ideal. So we may ask what distinguishes 1 x 1 matrices from larger ones. One answer is the absence of zero divisors. D e f i n i t i o n 3.2
Suppose I is a two-sided ideal in a ring R. We say t h a t
the ideal I (or the quotient ring R / I ) is completely prime if whenever a and b are elements of R with ab E I, then either a C I or b C I. It is easy to check that a completely prime ideal is prime. (The definition of prime was included in the proof of Proposition 2.12.)
Here is
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some further evidence of the connection between completely prime ideals and unitary representations.
Suppose G is a connected complex reductive Lie group, and 7r C G is an irreducible unitary representation. Then the annihilator In C U(g) (Definition 2.6) is completely prime. P r o p o s i t i o n 3.3 ([V3, Proposition 7.12])
The proof is very easy, requiring no structural information about 7r. Exactly the same result is true for G = GL(n, R); but in this case the proof A
requires a complete and detailed knowledge of G. Any hopes of further generalization are dashed on the rocks of G = SU(2).
This group has
an irreducible unitary representation of each dimension n > 0; and (3.1) guarantees that the corresponding primitive ideal is completely prime only for n = 1.
(More subtle examples are available for noncompact simple
groups as well. Suppose 7r is a holomorphic discrete series representation of Sp(4, R) with Harish-Chandra parameter ~ = (hi, ~2). This means that A1 > ~2 > 0 are positive integers. It turns out that Ann(Tr) is completely prime if and only if/~1 - ~2 = 1.) We can find a way out of this disappointment by looking carefully at the example of G = SU(2). This group acts holomorphically on the Riemann sphere C P 1. The n-dimensional irreducible representation 7rn arises as the space of holomorphic sections of a certain holomorphic line bundle £n ~ CP 1. Define D ( C P 1)n to be the algebra of holomorphic differential operators on sections of /2n. This is an algebra of "twisted differential operators" on C P 1 . (We will return to a more detailed and general discussion of twisted differential operator algebras in Section 9.) The action of G on £~ defines a homomorphism of associative algebras Cn: U(I~) ---+D ( C P 1)n.
(3.2)
Write In for the kernel of Cn.
The ideal In is a completely prime primitive ideal in U(I~), contained in the annihilator Ann(Trn).
Proposition
3.4
S k e t c h o f proof: T h a t In is completely prime follows from the absence of zero divisors in the twisted differential operator algebra D ( C P 1)n. T h a t it is contained in Ann(Trn) follows from the realization of 7rn as holomorphic sections of/:n.
[]
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The lesson to be drawn from this example is that an interesting unitary representation 7r may appear naturally as a module for some completely prime quotient U($)/I, even if Ann(Tr) properly contains I. It is easy to modify Problem 2.18 in accordance with this lesson; we simply weaken the condition in (ii) to Ann(~r) D I. There is a hint here of a second lesson as well. The realization of 71"n on holomorphic sections of £n exhibits 71"n a s a module not only for U(g)/In, but also for the full differential operator algebra D ( C P 1 )n. In this example the homomorphisms ¢~ are all surjective, so that there is no difference between U($)/In and D ( C P 1)n. When we treat general reductive groups, however, we will encounter homomorphisms Oh: U(~O) ~ D(X)~.
(3.3)
Here, X is a "partial flag variety" for $ (a quotient of a complex reductive group Gc by a parabolic subgroup Qc); A is a character of the Lie algebra q; and D(X)~ is a twisted differential operator algebra on X. In this setting, the homomorphism ¢~ is usually but not always surjective. Perhaps the simplest example when ¢~ is not surjective has $ = zp(4, C), X = C P 3 (the variety of lines in the natural four-dimensional representation of 1~), and D(X) ~ the algebra of differential operators on "half forms" on X. (The top exterior power of the cotangent bundle of X has a well defined square root £ --. X in this example; D(X)~ is the algebra of holomorphic differential operators on sections of E.) In any case, we will construct modules for D ( X ) ~ , and not just for U(g)/ker ¢~. In order to accommodate this extra structure in something like Problem 2.18, we need a definition. D e f i n i t i o n 3.5 Suppose G¢ is a complex reductive algebraic group with Lie algebra $. A Dixmier algebra for Gc is a pair (A, ¢) satisfying the following conditions. i) A is an algebra over C, equipped with an algebraic action of Gc by algebra automorphisms Ad(g). ii) The map ¢: U(IJ) --* A is an algebra homomorphism, respecting the adjoint actions of Gc on U($) and A. The differential ad of the adjoint action of Gc on A is the difference of the left and right actions of g defined by ¢: ad(X)(a) = ¢ ( X ) a - a ¢ ( X )
(XE$,
aCA).
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iii) A is finitely generated as a U(g)-module. iv) Each irreducible Go-module occurs at most finitely often in the adjoint action of Gc on A. We say that the Dixmier algebra is completely prime if A is a completely prime algebra. This immediately implies that the kernel I of ¢ is a completely prime ideal in U(g), and one can show (using condition (iv)) that I must also be primitive. If I is a primitive ideal in U(g), then U ( g ) / I is a Dixmier algebra for any connected Go. In the setting of (3.3), D(X)~ is a completely prime Dixmier algebra for Go. The adjoint action arises from the action of Gc on X, by change of variable in the differential operators. It is an idea of Dixmier that for a connected complex algebraic group Gc there should be a close connection between completely prime primitive ideals in U(g) and orbits of Gc on g*. Borho, Joseph, and others found that for reductive Go, this "close connection" cannot be a reasonable bijection. The goal of IV3] was to find a geometric description of all completely prime primitive ideals. That attempt failed, as was shown by work of McGovern. Here is a weaker statement (taken from [VT, Conjecture 2.3]) that appears to be consistent with everything we now understand about Dixmier algebras. C o n j e c t u r e 3.6 Suppose Gc is a connected complex reductive algebraic group. Let (9 C g* be a coadjoint orbit for Go, and let "0 ~ (9 be a connected covering on which Gc acts compatibly. Write R((~) for the algebra of regular functions on (~; this algebra carries a natural algebraic action of Gc by algebra automorphisms. Attached to (~ there should be a completely prime Dixmier algebra (A((~),¢(O)), with the property that A(O) is isomorphic to R((~) as algebraic representations of Go. The "Dixmier
correspondence" (~ ~-~ (A(O), ¢((~)) should be injective. The requirements of the conjecture are very far from specifying A((~) completely; but, fortunately, they are most restrictive precisely in those cases when we know least about how to construct A((~). The following result allows us to relate Conjecture 3.6 to primitive ideals.
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Suppose Gc is a complex reductive algebraic group, and (A, ¢) is a completely prime Dixmier algebra for Go. Then ker ¢ = I(A, ¢) is a completely prime primitive ideal in U(~).
L e m m a 3.7.
Proof: Since A is completely prime, so is its subalgebra U(~)/I(A, ¢). So I(A, ¢) is a completely prime ideal, and therefore prime; we need only show it is primitive. Write Z(fl) for the center of U(g). As a consequence of Theorem 2.13, I(A, ¢) will be primitive if and only if I(A, ¢) N Z(g) is a maximal ideal in Z(fl); that is, if and only if ¢(Z(g)) C C.
(3.4a)
Z(9) = {z e U(g) ]Ad(g)(z)= z, all z e (Go)0}.
(3.4b)
iow~ Let A0 be the subalgebra of A on which (Go)0 acts trivially. By Definition 3.5(iv), A0 is finite dimensional. Since it is also a completely prime algebra over C, we must have A0 = C. Since (3.4b) guarantees that ¢(Z(fl)) C d0, (3.4a) follows. 71 In the setting of Conjecture 3.6, we write
z(¢) =
¢(0)c
(3.5)
for the completely prime primitive ideal provided by the conjecture and Lemma 3.7. The correspondence sending (9 to {I((D) I O a cover of (.O} is (conjecturally) a kind of multivalued Dixmier correspondence from coadjoint orbits to completely prime primitive ideals. From the point of view of primitive ideal theory, the most serious problem with Conjecture 3.6 is that this correspondence is not surjective: Not every completely prime Dixmier algebra is of the form A(0) for a coadjoint orbit cover (0. Even the underlying correspondence (3.5) to completely prime primitive ideals is not surjective. To understand why this is not entirely bad, we recall an example from [J] and [V3]. Suppose Gc is of type G2. There is exactly one coadjoint orbit O8 in g* of dimension 8, and it is simply connected. Joseph found a (unique) completely prime primitive ideal I(O8) with the property that U(g)/I(O8) is isomorphic to R(Os) as representations of Gc. It is therefore reasonable to define A(O8) = U(g)/I(O8) as the Dixmier algebra predicted by Conjecture 3.6.
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Let G be a simply connected split real reductive Lie group of type G2. It turns out that G has exactly one irreducible (g, K)-module V with Ann(V) = I(Os). This (g,K)-module corresponds to an isolated unitary representation 7r; in the classification of G given in [VS], 7r is the unique isolated point among the Langlands quotients of the principal series for the nonlinear group. It is constructed in [V8] as the restriction to G of a ladder representation of SO(4, 3). Certainly ~r is an interesting unitary representation of G, and evidence that the approach of Problem 2.18 will find it is welcome. Joseph found a second completely prime primitive ideal I'(Os) closely related to Os. (Under the adjoint action of Go, U($)/I(Os) is slightly smaller than R(Os).) The Dixmier correspondence of Conjecture 3.6 (or even (3.5)) has no room for I'(Os), however; this ideal is simply omitted. There is exactly one irreducible (g, K)-module V' with A n n ( V ' ) = I'(Os). But it turns out that V' is not unitary; so from the point of view of finding unitary representations, the omission of I ' ( O s ) is harmless (or even desirable). Encouraged by this example, we are going to refine Problem 2.18 to accommodate Dixmier algebras. Here is the setting. D e f i n i t i o n 3.8 Suppose G is a real reductive Lie group. Let Gc be a connected complex reductive algebraic group with Lie algebra g = Lie(G)c. We assume that the adjoint action of G on g factors through a homomorphism
j:G~Gc inducing the identity map on g. (This can always be arranged by an appropriate choice of Gc if G is of "inner type": that is, if every automorphism Ad(g) (for g E G) of g is inner.) In this setting, we say that G is of inner
type Go. Suppose now that (A,¢) is a Dixmier algebra for Go. An (A,K)module is a complex vector space endowed with a module structure for the algebra A and a representation of the group K, subject to the following conditions. (Compare Definition 2.16.) i) The action of K is locally finite and smooth. ii) The differential of the action of K (which makes sense by (i)) is equal to the action of ¢(t~0) C A.
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iii) For k C K, v E V, and a C A, we have
k . (a . v) - [Ad(j(k))(a)] . (k . v). (Just as in Definition 2.16, condition (iii) for k E K0 is a consequence of (i) and (ii).)
A Hermitian transpose on A is a conjugate-linear antiautomorphism • of A of order 2"
(ab)* - b'a*,
(za)* - -2a*
(a, b e A, z e C).
We assume in addition that • is compatible with the usual Hermitian transpose on U(g) defined by the real form g0: [¢(X + iY)]* - ¢ ( - X + i Y )
(X, Y e g0).
Suppose finally that V is an (A, K)-module and that • is a Hermitian transpose on A. We say that V is unitary if it is endowed with a positive definite Hermitian form {, / satisfying {a.v,w} -- {v,a*. w}, forv, w C V ,
kCK,
andaEA.
The map ¢ provides a forgetful functor that makes any (unitary) (A, K)-module into a (unitary) (t~, K)-module. This functor sends (A, K)modules of finite length to (g, K)-modules of finite length, but it need not send irreducibles to irreducibles. T h e o r e m 3.9. In the setting of Definition 3.8, suppose V is a unitary (A, K)-module. Then the Hilbert space completion TI(V) carries a unitary
representation 7r(V) of G. The space 7"l(V) ~ carries a natural action of A. This action preserves the space 7-l(V) K of K-finite vectors (which are automatically smooth), making 7-I(V) K an ( A , K ) - m o d u l e . If V has finite length, then Tl(V) K is equal to V as an ( A , K ) - m o d u l e . This is an immediate consequence of Harish-Chandra's Theorem 2.17.
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Here is a refinement of Problem 2.18. P r o b l e m 3.10 Suppose G is real reductive Lie group of inner type Gc (Definition 3.8). Let (.9 C ~* be a coadjoint orbit for Go, and let (~ ---, (9 be a connected covering on which Gc acts compatibly.
i) Construct a completely prime Dixmier algebra (A((~), ¢ ( 0 ) ) as in Conjecture 3.6. ii) Construct a finite collection {Wi(O) I i - 1 , . . . , r} of modules for A(O).
iii) For ach
on t,'uct from W (O) an (A(O),K)- odul
V (O)
nition 3.8). iv) For each i, construct a Lrermitian transpose • on A(O) (Definition 3.8) and a unitary structure ( , ) on Vi(O) (Definition 3.8). If all these steps are completed, then Theorem 3.9 makes the Hilbert space completion 7-/i(O) of V/(0) into a unitary representation of G. The method of coadjoint orbits suggests roughly how the modules W~(O) should be parametrized. Write g~ for the real dual of the real Lie algebra of G. It is contained in ft* as a real form. L e m m a 3.11 In the setting of Problem 3.10, the intersection O(R) = 0 N g~ is a finite union of orbits of G" 0(]~)
-- 01(]~)
U " " U Os(]~).
The covering map (~ ~ 0 induces (possibly disconnected) G-equivariant coverings having the same degree as O. This is elementary. What the orbit method suggests is that each of the modules in Problem 3.10 should be parametrized by one of the real orbit covers Oi(I~), together with some additional data. (This idea can be made precise and correct for semisimple orbits, but it requires further refinement in general.) In Section 4 we will begin to study real coadjoint orbits, and to see how one might attach representations to them.
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Structure
of c o a d j o i n t o r b i t s
In this section we recall some general structure theory for coadjoint orbits. We work at first with an arbitrary real Lie group G, writing go - Lie(G),
0; - HomR(g0, R).
(4.1a)
We write Go for the identity component of G. Often we write elements of G as lower case R o m a n letters, elements of g0 as upper case R o m a n letters, and elements of g; as lower case Greek letters. The coadjoint action of G on g; is just the transpose of the adjoint action" [Ad* (g)~] (Y) - ~(Ad(g -1)Y)
(~ ~ o;, g e a , Y ~ oo).
(4.1b)
The differential of this action is written [ad* (X){] (Y) - ~ ( a d ( - X ) Y ) - ~([Y, X]).
ad*" go + End(g;),
(4.1c)
The isotropy group for Ad* at ~ is written G~" a~ - {g ~ G IAd*(g)¢ -- ¢}.
(4.1d)
The Lie algebra of G~ is
oa,o - { x c oo lad* ( x ) , f - o} - { x @ oo I ~([Y, x ] ) - o, all Y E Oo }.
(4.1e)
To each ~ E g~ we attach a skew-symmetric bilinear form w~ on g0, defined by
~(x,
Y) - ~([x, Y]) - [~d* (Y)~](x) - [-~d* (X)~](Y).
(4.1f)
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L e m m a 4.1 With notation as in (~.1), the radical of w~ is equal to g~,0. Consequently, w~ descends to a nondegenerate symplectic form (still denoted ~ )
on
go/~,o ~ T~(a. ~) (the tangent space at ~ to the coadjoint orbit through ~). As ~' varies over the orbit W - G . ~, the family of oa~, defines a closed two-form ~ w and, therefore, a G-invariant symplectic structure on W . P r o o f : T h a t the radical of w~ is g~,0 is clear from (4.1e) and (4.1f). It follows that ww is a two-form on W. Each element X E go defines a vector field X w on W. These span the tangent space T W at each point; so to prove that ~ w is closed, it suffices to show that d w w ( X w , Yw, Z w ) - 0 for all X, Y, Z C go. We compute dww ( X w , Yw , Z w ) = X w • ~zw (Yw, Z w ) - Yw • ww ( X w , Z w ) + Z w • ~ w ( X w , Yw) - w w ( [ X w , Yw], Z w ) + w w ( [ X w , Zw], Yw) - ww([Yw, Z w ] , X w ) . Evaluating at the point ~' E W, we get = X w . ~' ([Y, Z]) - Y w " ~' ([X, Z]) + Z w . ~' ([X, Y]) -
~'([[x, Y], z]) + (([IX, z], Y])
-
(([[Y, z], x]).
In the first three terms, we are differentiating the function on W obtained by applying the variable linear functional ~' to a fixed element of go. This amounts to applying - a d * to ~'. We get = - (ad* (X)¢')([Y, Z]) + (ad* (Y)~')([X, Z]) - (ad* (Z)~')([X, Y])
- ~'([[x, Y], z] - [Ix, z], Y] + [Iv, z], x]) = - ¢' ([[Y, Z], X]) + ¢' ([[X, Z], Y]) - ¢' ([[X, Y], Z])
- ~'([[x, Y], z] - [[x, z], Y] + [[Y, z], x]) = - 2{' ([[X, Y], Z] - [IX, Z], Y] + [[Y, Z], X]). The argument of {' vanishes by the Jacobi identity, and so ww is closed. The G-invariance is clear from the definition.
[3
L e m m a 4.1 says that any coadjoint orbit is in a natural way a symplectic homogeneous space. The converse (that a n y symplectic homogeneous
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space is a coadjoint orbit) is not quite true, for two reasons. First, a coadjoint orbit has a slightly stronger structure (which we will describe in a moment). Second, this additional structure lifts to covering spaces. D e f i n i t i o n 4.2. Suppose (W, w) is a symplectic manifold. The symplectic form provides a smooth identification of the tangent bundle of W with the cotangent bundle: To the tangent vector X E Tw(W), we associate the cotangent vector T(X) defined by
(Y this is the contraction of -ww with X. If f is a smooth function on W, then df is a one-form (a smooth section of the cotangent bundle). We may therefore define
Xf - T-l(df), a smooth vector field on W, called the Hamiltonian vector field of f . Using the action of vector fields on functions, we now define the Poisson bracket of the smooth functions f and g by
{ f ,g} = X f . g = d g ( X f ) = w ( X f , X g ) = - X g . f.
The Poisson bracket defines a Lie algebra structure on C a ( W ) . The map f ~ X f is a Lie algebra homomorphism from C a ( W ) to the Lie algebra of vector fields on W. Its kernel consists of the locally constant functions on W .
P r o p o s i t i o n 4.3
Suppose W is a coadjoint orbit in g~, and Y C go. Write f (Y) for the smooth function on W obtained by restricting to W the linear function Y on 9~). Then the corresponding Hamiltonian vector field X f ( y ) is equal to the vector field Yw induced by the action of G on W . The map Y ~ f (Y) is a Lie algebra homomorphism. P r o o f : The assertions about general symplectic manifolds are standard (see, for example, JAr, Chapter 8] or [AM, Chapter 3]; both sources use slightly different sign conventions from ours). For the rest, we compute
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(for Y, Z c 0o) w w ( Z w , Y w ) - f([Z, Y])
(definition of ww) (as at the end of the proof of Lemma 4.1)
= Zw" f(Y) =df(Y)(Zw) = r(Xy(y))(Zw)
(definition of X y ( y ) )
= ww(Zw,Xf(y))
(definition of v).
Because ww is nondegenerate, and the vector fields Z w span each tangent space, it follows that Yw - X f(y). At the same time, we have shown that {f(Z), f(Y)} = X f ( z ) " f ( Y ) - Z w . f ( Y ) - f([Z, Y]), proving the last assertion.
[]
D e f i n i t i o n 4.4 (see [Ko, Section 5]) A Hamiltonian G-space is a symplectic manifold W equipped with a symplectic action of G and a linear map f: g0 ~ C~(W,I~), with the following properties. i) The map f is a Lie algebra homomorphism (for the Poisson bracket Lie algebra structure). ii) The Hamiltonian vector field Xy(y) on W associated to Y E g0 is equal to the vector field Yw obtained by differentiating the action of G in the direction Y. iii) The map f is G-equivariant: f ( A d ( g ) Y ) ( w ) - f ( y ) ( g - 1 , w).
Condition (i) is actually a consequence of (ii) and (iii), but we include it because of its appealing simplicity. For g in the identity component Go of G, condition (iii) is a consequence of (i) and (ii). Condition (ii) also guarantees that the action of Go is symplectic; so the entire definition may be phrased more succinctly for connected G. Suppose (W, f) is a Hamiltonian G-space. The moment map for W is the G-equivariant smooth map p : W -*0~,
u(w)(y)-
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Proposition 4.3 implies that each coadjoint orbit is a Hamiltonian G-space (the requirement in (iii) being easy to verify).
Its moment map is the
identity. It is also easy to see that a G-equivariant covering of a Hamiltonian G-space is again a Hamiltonian G-space. The following result is a partial converse.
Suppose W is a homogeneous Hamiltonian G-space. Then the moment map is a covering of a coadjoint orbit G. ~. Consequently, W ~_ G/G~,I, with G~,I an open subgroup of the isotropy group G~.
Proposition
4.5 ([Ko, Theorem 5.4.1])
P r o o f : Since p is G-equivariant, its image must be a single orbit G. ~. Fix a point w C W with #(w) = ~, and define G~,I to be the isotropy group at w. Obviously this is a subgroup of G~; we need only show it is open. This amounts to showing that the differential of # is one-to-one. Because W is a homogeneous space for G, the tangent space Tw is spanned by the vector fields Yw = X f ( y ) . We must show that Yw vanishes at w if and only if YG.~ vanishes at ~. By the nondegeneracy of the symplectic form on W, X f ( y ) vanishes at w if and only if ¢ o w ( X f ( y ) , X f ( z ) ) -- 0 for all Z C go. By the definition of the Poisson bracket on W, this is the same as the vanishing of all {f(Z), f ( Y ) } ( w ) . By assumption (i) in Definition 4.4 and the definition of p, this is the same as the vanishing of all [Z, Y] (for varying Z) at ~. By (4.1e), this last condition is the same as Y C fl~,0; that is, YG.~ vanishes at ~. U] According to Proposition 4.5, the coverings O appearing in Problem 3.10 are precisely the complex homogeneous Hamiltonian Gc-spaces. This abstract characterization may lend a little respectability to what appears to be an ad hoc setting. We turn now to a preliminary examination of how representations are attached to coadjoint orbits. D e f i n i t i o n 4.6
Suppose G is a Lie group, and { E g~. An integral orbit
datum at ~ is an irreducible unitary representation (r, T/r) of the isotropy group G~, subject to the condition r(exp X ) - e i~(x) • I d ~
(x
(4.2)
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The orbit d a t u m (T, 7-/,) at ~ is equivalent to (~-', 7-/,,) at ~' if there is a g E G so that Ad* (g)~ - ~', and ~- is equivalent to the representation h H 7'(ghg -1) of G~. (The second requirement makes sense because G~, = gG~9-1.) The orbit G . ~ is called integral if it admits an integral orbit datum. To understand this definition, notice first that the linear functional i~: g~,0 ~ iR
(4.3a)
is automatically a Lie algebra homomorphism by (4.1e). If G~,0 is simply connected, we therefore get automatically a unique group homomorphism To" G~,0 --~ U(1),
T0(exp X) - e i~(x),
(4.3b)
on the identity component G~,0 of G~. In general (when G~,0 need not be simply connected), the requirement in (4.3b) still specifies at most one To; the problem is existence. Define (4.3c)
L~,0 - {X ~ fk,0 [exp(X) - e}. Then, To exists if and only if ~(L~,0) C 2~-Z.
(4.3d)
(The necessity of this condition is clear, and sufficiency is not too difficult.) Finally, it is not difficult to show that an integral orbit datum exists if and only if To exists; so (4.3d) is precisely the condition for G . ~ to be integral. When G~ is disconnected, ~- is usually not unique. There is an obvious way to construct a unitary representation of G from an integral orbit datum: by unitary induction from G~ to G. That is, we consider continuous functions
C(G/G~, ~ - ) - {f" G ~ 7-l~ I f ( g h ) - ~-(h)-lf(g)
(g e G, h e G~)}. (4.4a)
Such a function is said to be of compact support modulo G~ if there is a compact subset K of G so that f vanishes outside KG~.
We write
Cc(G/G~, 7-l,) for such compactly supported functions. Suppose fl and f2
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belong to C(G/G~, "H~-). Then we can define a complex-valued function on G by (fl, f2)loc(g) = (fl (g), f2(g))~t~.
(4.4b)
Because T is unitary, (4.4a) implies that (fl, f2)1oc is actually a function on GIGs. If one of the f/ belongs to Cc(G/G~,7-I~-), then (fl,f2)loc is compactly supported on GIGs. In that case we may define
(fl,f2)----:JG
/G~
(fl,f2)loc(x)dx.
(4.4c)
Here the G-invariant measure dx on G/G~ ~_ G. ~ arises naturally from the symplectic structure; the volume form may be taken to be the top exterior power of the symplectic form. In this way, we get a G-invariant positive definite Hermitian form on Cc(G/G~, 7-l~-). Its Hilbert space completion is called L2(G/G~, 7-l~.). This space carries a unitary representation I n d ~ (T) of G, given on C~(G/G~, 7-l~-) by left translation. The difficulty is that this induced representation is almost never irreducible. (An interesting exception occurs when G~ is open in G, that is, when the orbit G . ~ is discrete.) To get an irreducible representation, we would like to make a similar construction on a smaller space; that is, we wish to impose additional conditions on the functions f in (4.4a). One natural idea is to extend the representation T to a larger subgroup H D G~. It turns out to be good to restrict attention to extensions T H still satisfying the analog of (4.2):
TH(expX )
-- e i ~ ( x ) .
Id~
(X E [~0 D ~,0).
(4.5a)
Such an extension of T can exist only if i~ is a Lie algebra homomorphism from ~0 to iR, that is, only if ~([X, Y]) = 0
(X, Y e O0).
(4.5b)
According to (4.1f), this requirement is equivalent to the requirement that the symplectic form w~ vanish on [}0/g~,0 C g0/ft~,0 -~ T~(G. ~). We digress for a moment to recall some linear algebra related to this last condition. Suppose (V,w) is a real symplectic vector space, and Z C V is any subspace. Define
Z ± - {v ~ V lw(v, z ) - O, all z ~ Z}.
(4.6a)
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Then, dim Z + dim Z ± - dim V.
(4.6b)
We say that Z is isotropic if wlz = 0, equivalently, if Z C Z ±. By (4.6b), this implies that dim V - dim Z + dim Z ± _ 2 dim Z
(Z isotropic).
(4.6c)
Dually, Z is coisotropic if Z D Z ±. This implies that dim V = dim Z + dim Z ± < 2 dim Z
(Z coisotropic).
(4.6d)
We say that L is Lagrangian if it is both isotropic and coisotropic. In this case 1 dim L - ~ dim Y (L Lagrangian). (4.6e) Evidently a subspace L is Lagrangian if and only if it is isotropic and dim L - ~1 dim V Lagrangian subspaces always exist; all are conjugate under the action of the group Sp(w). Next, suppose (M, w) is a symplectic manifold. A submanifold A C M is Lagrangian (resp., isotropic or coisotropic) if T~A is a Lagrangian (resp., isotropic or coisotropic) subspace of TAM for every/~ E A. We want to shrink the space (4.4a) as much as possible, to have a good chance of getting an irreducible representation. In the setting of (4.5), this means taking the subgroup H as large as possible. We know already that 00/9~,0 must be isotropic; so it is natural to impose the requirement that ~)0/g~,0 be Lagrangian. Suppose (T, T/v) is an integral orbit datum at ~. An invariant real polarization of T is a closed subgroup H D G~, and an extension T H of T to H , satisfying D e f i n i t i o n 4.7
i) TH(eXpX) = ei~(z). Idu~
(X E ~0).
ii) H is generated by G~ and H0. iii) dimH/G~ = ½dimG/G~. We will show in Section 7 how to construct invariant real polarizations in one large class of examples; but let us consider briefly how one might look for them in general. According to (4.5), the existence of T H forces ~)0/g~,0 to be an isotropic subspace of g0/g~,0 - T~(G. ~). Then (iii) makes
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it a Lagrangian subspace. To construct a polarization, we therefore need first of all a Lagrangian subspace of the symplectic vector space T~ (G. ~). There will be many such subspaces. Each is of the form O0/g~,0, with O0 a subspace of g0 containing g~,0. We will show that each Lagrangian subspace gives rise to at most one invariant real polarization. To correspond to a polarization, O0 must first be invariant under the adjoint action of G~, and it must be a Lie algebra. When these two conditions are satisfied, there is a Lie subgroup H of G with Lie algebra O0, satisfying (ii) and (iii) of Definition 4.7. The requirement that H be closed is not automatically satisfied and is an additional restriction on ~0. Because of (ii), condition (i) (together with the requirement that ~-H extend T) determines TH uniquely; but there is a simple topological obstruction to its existence, which further constrains ~0. Because of these requirements, it may easily happen that no invariant real polarizations exist. When they do exist, there are often many. The only natural notion of "equivalence" of polarizations is conjugation under the group G~. But we have seen that each polarization is determined by a G~invariant Lagrangian subspace; so each equivalence class consists of a single element. For this reason, the problem of relating different polarizations is a difficult one, and we will ignore it entirely. (In the special setting of Section 7, we will find one distinguished polarization; others exist, however.) In the setting of Definition 4.7, we have seen that the tangent space T~ ( H . ~) is a Lagrangian subspace of T~ (G. ~). Since the symplectic structure is G-invariant, it follows that H.~ ~_ H/G~ is a Lagrangian submanifold of G - ~ . A similar argument proves L e m m a 4.8
In the setting of Definition ~. 7, consider the natural projec-
tion 7r: G . ~ ~ G / H ,
.
=
gH.
Then the fibers of ~ are connected Lagrangian submanifolds of G. ~. Let us consider now the construction of a unitary representation of G from an invariant polarization (H, TH) at ~. Certainly, we can define a space of continuous functions
C ( G / H , 7-/~) = {f: G ~ 7-l~- I f(gh) = TH(h)-~ f(g)
(g E G, h e l l ) } . (a.Ta)
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For future reference, notice t h a t we can rewrite at least the s m o o t h functions C ~ ( G / H , ~ - ) as follows. The Lie algebra go acts on s m o o t h functions by differentiation on the right: d
(p(X) f )(g) - -~ f (g exp(tX))[t=o.
(4.7b)
This action makes equally good sense on s m o o t h functions with values in 7-/~. T h e n
C~(G/H,7-L-) = {f e C~(G/G~,7-I,)Ip(X)f--i~(x)f
( x c Do)}
= { f c C ~ ( G , 7-t,-)if(gh) - 7-(h)-lf(g), p ( X ) I - - i ~ ( X ) f (g e G, h e G¢, X e Oo)}.
(4.7c)
Write Cc(G/H, 7-l~) for the subspace of functions of compact s u p p o r t modulo H. Just as in (4.4b), we can define
(fl, f2}loc(g) - (fl (g), f2(g)}7-t.,
(4.7d)
a complex-valued function on G / H . However we cannot imitate (4.4c) with an integral over G/H: Even in very simple examples, this space may not a d m i t a G-invariant measure. To circumvent this problem, we recall the notion of half-density. D e f i n i t i o n 4.9
Suppose V is a finite dimensional real vector space and t
is a real number. A t-density on V is a symbol cldxl t, with c C IR and dx a Lebesgue measure on V. We identify c]dxl t with c']dx'] t if d x ' = j d x and
c -- c~j t. T h e t-densities on V form a one-dimensional real vector space Dt(V). For t = 1, they are just the multiples of Lebesgue measure on V. We have n a t u r a l isomorphisms
Dr(V) ® D~(V) "~ Dt+s(r).
(4.8a)
Suppose t h a t M is a s m o o t h manifold. Define a real line bundle 7Pt on M by Z~t(rn) = Dt(TmM).
We call 7?t the t-density bundle on M; a
section of 7?t is called a t-density on M. If t - 1, sections of
~)1 may be
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identified with densities on M. In particular, if 6 is a compactly supported continuous section of 7)1, there is a natural integral
fM
6(m) E R
(6 E Co(M, 7)1)).
(4.8b)
There are also natural isomorphisms
7Pt ® T)s ~ T)t+s
(4.8c)
as line bundles on M. We return now to the setting of Definition 4.7, and the problem of construction of a unitary representation from ~'H. Write Dt = Dt(go/bo). The adjoint action of H gives rise to a representation ¢ of H on g0/b0, and so to an action Xt of H on Dr. (It is easy to check that xt(h) = I det ¢(h)l-t.) Consider the space of continuous functions
C(GIH, 7lr®Dll2) - { f ' G ~ 7-lr®D112
I f(gh) - (TH®Xll2)(h-1)f(g)}
(4.9a) (What we are doing is twisting the bundle on G / H defined by 7H by the half-density bundle.) The inner product on 7-/~ and (4.8a) provide a sesquilinear pairing
(, >i/2" 7-lr ® D1/2 x ~ . ® D1/2 ~ DCi , ®
®
Given fl and f2 in C(G/H, ~
=
®
(4.9b)
® D1/2), we therefore get
(fl, f2)loc = (fl ('), f2(')}1/2 E C(G/H, D1).
(4.9c)
This is a density on G / H , compactly supported if fl or f2 is. By integrating densities, we therefore get a positive definite G-invariant quadratic form on Cc(G/H, ~ ® D1/2). Its Hilbert space completion L2(G/H, Tlr® Di/2) carries a unitary representation of G, called IndaH(TH). This is often a reasonable unitary representation to attach to the orbit datum (~, T). The most serious shortcoming of the construction is that invariant real polarizations often do not exist.
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285
E x a m p l e 4.10 Suppose G = SL(2, N), so that g0 consists of 2 x 2 real matrices of trace zero. We consider some examples of elements ~ E itS. i) ~ ( ac -ab ) -
~a, 0 ¢ ~ E R. Theelement ~ i s hyperbolic (see
Lemma 5.5). Its isotropy group is
{(x 0) 0
X-1
IXE
"
There are exactly two orbit data at (: both are one-dimensional unitary characters, and T+
(x 0 ) 0
X- 1
- - Ixl iu
T_
(x 0 )=,xli~.sgn(x) 0
X-- 1
Each has an invariant real polarization by the subgroup Hthe characters
T:t=, H
0
are
X-1
,y C ]I(
]XC
trivial on elements
1 0
representations of (4.9) are always irreducible.
y). 1
The unitarily induced
i i ) ~ ( ac -ab ) _ c. The element ~is nilpotent (see Lemma 5.3). The corresponding symplectic form is w~
c
-a
'
c'
-a'
= 2(ca' - ac'),
and the isotropy group is G~-{(
e 0
t e) le--l-1} •
Again the orbit data are two one-dimensional unitary characters T+, defined by e t -1, v_ =e. 7+ 0 e 0 e
( )
Each has an invariant real polarization by the subgroup H of (i). In this case, Inda~(r+,~) is irreducible, but IndaH(T_,H) is a direct sum of two irreducible components.
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286 iii) ~(ac
-ab ) - t / 2 ( b - c ) '
O ~ t E R. This element is elliptic (see
Lemma 5.6). Its isotropy group is G~_{(
cos0 -sin0
sin0) } cos0 IOEIR
WriteZforthematrix
( 0 1 ) so that the general element of G~ is -1 0 ' exp0Z. Since ~(Z) = t, the requirement (4.2) for an integral orbit datum is
Tt(expOZ) = eire. This is a well defined character of G~ if and only if t E Z; so we conclude that ~ is integral if and only if t E Z. The subgroup G~ is maximal in G, and so there is no invariant real polarization of ~-t.
5
( C o ) a d j o i n t o r b i t s for r e d u c t i v e g r o u p s
We turn now to a more detailed study of coadjoint orbits in the case of real reductive groups. We single out three special classes, called hyperbolic, elliptic, and nilpotent. A hyperbolic coadjoint orbit is isomorphic to an affine bundle over a real flag variety. We will eventually attach representations to such orbits by real analysis on the real flag variety. An elliptic coadjoint orbit carries an invariant (indefinite) KShler structure; it is isomorphic to an open orbit in a complex flag variety. We will attach representations to elliptic orbits by complex analysis techniques. A nilpotent coadjoint orbit has (in general) neither real nor complex structure of these kinds; so we do not know how to attach representations to it. In the remainder of this section, we will define the three special classes of orbits and show how to realize a general coadjoint orbit as a combination of them. So suppose G is a real reductive Lie group, with Cartan involution 0, maximal compact subgroup K, and Lie algebra 00. Definition 1.5 allows us to identify 00 with a Lie subalgebra of g[(n, 1~) closed under transpose. On g[(n, 1~) we can define the trace form (X, Y / -
tr XY.
(5.1a)
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This is a symmetric bilinear form, invariant by the adjoint action of GL(n, R): (ad(g)X, Ad(g)Y) = (X, Y) (5.1b) for X, Y E g[(n, N) and g C GL(n, N). By direct computation one finds that ( , ) is positive definite on symmetric matrices and negative definite on skew symmetric matrices, and that it makes these two subspaces orthogonal to each other. One immediate consequence is that ( , ) is nondegenerate. We also deduce L e m m a 5.1 Suppose G is a real reductive Lie group with a homomorphism 7r: G --+ GL(n, I[{) as in Definition 1.5. Then the restriction to go of the trace form on g[(n, N) (cf. (5.1)) has the following properties: i) (Ad(g)X, Ad(g)Y) : (X, Y),
X, Y e go, g e G.
ii) ( , ) is positive definite on P0, negative definite on ~o, and has (P0, [~0) = 0 (notation as in Definition 1.3). iii) ( , ) is nondegenerate on go. Consequently ( , ) defines an isomorphism t~o ~ g~ carrying the adjoint action of G on go to the coadjoint action of G on g~.
Because of this lemma, coadjoint orbits for reductive groups are closely related to adjoint orbits; and adjoint orbits in turn are closely related to conjugacy classes of matrices. We recall some facts from linear algebra. L e m m a 5.2 Suppose X is an n × n real matrix. The following properties of X are equivalent. a) The linear transformation of C n defined by X is diagonalizable. b) The minimal polynomial of X has no repeated factors. c) The conjugacy class
GL(n, N). X
=
{ g X g -1 I g e GL(n, ]t{)}
is closed in g[(n,R). d) There is an X ' = g X g -1 in the conjugacy class of X so that X ' commutes with t X ' .
When these conditions are satisfied, we say that X is semisimple.
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S k e t c h o f proof'. We will sketch the proof that (b) implies (c), which is perhaps one of the least familiar parts of the argument. Let p - P l " " p r be a factorization of the minimal polynomial of X, with pi an irreducible real polynomial. Each pi is either linear or quadratic with no real roots. Let di be the dimension of kerpi(X). We have
]t~n -- ~
kerpi(X),
(5.2a)
i-1
so ~ di = n. Also, G L ( n , R ) . X = {Y e g[(n,R) I dim kerpi(Y) >_ di, all i}.
(5.2b)
(Obviously, GL(n,I~) • X is contained in the right side of (5.2b). For the other containment, notice that for any matrix Z, the sum ~--~i=l kerpi(Z) must be direct, as the Pi are distinct irreducible polynomials. For Y in the right side of (5.2b), it follows by dimension counting that
I~n - ( ~ ker Pi(Y),
(5.2c)
i--1
and that dim ker pi(Y) = di. This implies easily that Y is conjugate to X.) To complete the proof of (c), notice that the right side of (5.2b) is closed in g[(n, R); this follows from the compactness of the Grassmannian manifolds of d-dimensional subspaces of R n. W1
Suppose X is an n × n real matrix. The following properties of X are equivalent.
L e m m a 5.3
a) X is nilpotent. b) The characteristic polynomial of X is t n. c) The closure of the conjugacy class of X contains O. d) The conjugacy class of X contains a multiple r X of X , with r > 0 and e) The conjugacy class of X contains every multiple r X of X , with r > O. f) There is an element A E gi(n, R) with [A, X] = X . When these conditions are satisfied, we say that X is nilpotent.
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P r o o f : We first prove the ascending implications, beginning with (f). Then e x p ( s A ) ( X ) e x p ( - s A ) = e x p ( a d ( s A ) ) ( X ) = e~X, which proves (e). Trivially ( e ) i m p l i e s (d). Assume (d); that is, that g X g -1 = r X for some positive r ~ 1. Possibly replacing g by g - l , we may assume r < 1. Then, g n X g -~ = r n X --~ O, proving (c). Assume (c). The characteristic polynomial d e t ( t I - Z )
is constant on conjugacy classes and depends continuously
on Z; so the characteristic polynomial of X must coincide with t h a t of 0, which is t n. Assume (b). Then X ~ = 0, since every matrix satisfies its characteristic polynomial; so X is nilpotent. To finish, we prove that (a)implies (f). Define H = {r > O l g X g -1 : r X , some g e G L ( n , R ) } . This is a subgroup of the multiplicative group of positive real numbers. If s and s' are any two positive numbers, then s X is conjugate to s ' X if and only if s and s' belong to the same coset of H. Now every multiple of X is nilpotent, and there are only finitely many conjugacy classes of nilpotent n × n matrices. It follows that H has finite index m in R >°. Consequently every m t h power belongs to H, and so H - R >°. Therefore, every positive multiple of X is conjugate to X. It follows that the tangent space at X to the conjugacy class of X contains X. Now the tangent space at Z to the conjugacy class of Z is [g[(n,R), Z]; so (f) follows.
K]
L e m m a 5.4 (Jordan decomposition) Suppose X is an n x n real matrix. Then there are a semisimple matrix Xs and a nilpotent matrix X,, uniquely characterized by the following two properties:
X=X
+Xn.
ii) [Xs, Xn] -- O.
In addition, we have iii) Xs and X n may be expressed as polynomials without constant term in X . iv) A n y matrix commuting with X commutes with Xs and Xn. V) Any subspace of R n preserved by X is also preserved by Xs and Xn. We turn now to an analogous decomposition of a semisimple matrix.
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L e m m a 5.5 Suppose X is an n × n real matrix. The following properties of X are equivalent.
i) X is diagonalizable. ii) The minimal polynomial of X is a product of distinct linear factors. iii) The conjugacy class of X contains a symmetric matrix. W h e n these conditions are satisfied, we say that X is hyperbolic. The proof of the lemma is easy, and we omit it.
Suppose X is an n × n real matrix. The following properties of X are equivalent.
Lemma
5.6
i) The linear transformation of C n defined by X is diagonalizable with purely imaginary eigenvalues. ii) The minimal polynomial of X is a product of distinct factors of the form t 2 + a 2 and t. iii) The conjugacy class of X contains a skew-symmetric matrix. W h e n these conditions are satisfied, we say that X is elliptic. Again we omit the proof of the lemma. Here is a complement to the Jordan decomposition of L e m m a 5.4. L e m m a 5.7 Suppose X is a semisimple n × n real matrix. Then there are a hyperbolic matrix Xh and an elliptic matrix X~ uniquely characterized by
the following two properties:
i) x=xh+x
.
ii) [Xh, X~] = O. In addition, we have iii) Xh and X~ may be expressed as polynomials in X . iv) Any matrix commuting with X commutes with Xh and X~. v) Any subspace of R ~ preserved by X is also preserved by Xh and X¢. vi) If X commutes with t X , then Xh -- ½(X + t X ) and X~ --l(x_tx)" vii) Suppose that X is a derivation of an algebra structure on I~n . Then X~ and Xh are derivations as well. P r o o f : We first establish the existence of the decomposition. Because the definitions of semisimple, hyperbolic, and elliptic are all invariant under conjugation (see Lemmas 5.2, 5.5, and 5.6), we may replace X by a conjugate matrix. By L e m m a 5.2(d), we may therefore assume that X commutes
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with t X. In this case the matrices X h and X~ defined in (vi) obviously commute and have sum X. Also Xh is symmetric, and therefore hyperbolic ( L e m m a 5.5(c)), and X~ is skew-symmetric, and therefore elliptic ( L e m m a 5.6(c)). This proves the existence, as well as (vi). For the uniqueness, regard X, Xh, and X¢ as linear transformations of C n. They commute with each other, and each is diagonalizable; so they are simultaneously diagonalizable (with respect to some basis V l , . . . , Vn of C n). Write zi for the diagonal entries of X as a matrix in this basis, and ai and bi for those of X~ and X h. Then zi - ai + b~, ai is real (Lemma 5.5(a)) and bi is purely imaginary ( L e m m a 5.6(b)). Therefore, ai = Rezi, bi = ~ - l I m z i .
This means
t h a t Xh acts on each eigenspace of X by the real part of the corresponding eigenvalue. Similarly, X~ acts on each eigenspace of X by the imaginary part of the eigenvalue. These descriptions establish the uniqueness of Xh and X~, and properties (iv) and (v) in the lemma follow easily. To prove (iii), we need to know the existence of real polynomials Ph and p~ with the properties t h a t
Ph (Zi) = Re zi,
p~(zi) = v/~-I Im zi.
Because the nonreal zi occur in complex conjugate pairs, this is elementary. Finally, assume that X is a derivation of an algebra structure o. After complexification, this means exactly that if v and v / are eigenvectors of eigenvalues z and z t, then v o v ~ is an eigenvector of eigenvalue z + z ~. The foregoing description of Xh and X~ shows that they inherit this property from X. [El W i t h these facts about matrices in hand, we turn to their generalizations for reductive groups. D e f i n i t i o n 5.8
Suppose G is a real reductive Lie group with 7r: G
GL(n, R) as in Definition 1.5. Recall that the Lie algebra 90 of G is identified with a subalgebra of 9[(n, ]~). An element X E 90 is called semisimple (resp., nilpotent, hyperbolic, or elliptic) if X has the corresponding property as a matrix (Lemma 5.2, 5.3, 5.5, or 5.6). To give more intrinsic characterizations of these properties, we will consider not the action of X on ]~n but rather the adjoint action on 90: a d ( X ) ( Y ) = [X, Y].
(5.3a)
292
D. Vogan, Jr.
The kernel of the adjoint action is the center 5(9o): 3(go) = {Z e go I[Z,Y] = 0, all Y E 9o}.
(5.3b)
Obviously, this is preserved by 0, and so it is the direct sum of its intersections with t~0 and P0:
= 3 (g0) •
p(g0).
(5.3c)
The "image" of the adjoint action is the derived algebra g~ -- [go, go].
(5.3d)
Because of the invariance under ad of the trace form (5.1), the derived algebra is precisely the orthogonal complement of the center for the trace form. Lemma 5.1 therefore implies g0 - g~ ® 3(g0),
(5.3e)
a direct sum of 0-stable ideals. (Notice that although we used structure from matrices to prove (5.3e), the summands are defined in terms of the Lie algebra structure of g0.) It is not difficult to show that the Lie algebra g~) is semisimple, and we will apply to it some standard structural results. T h e o r e m 5.9 In the setting of Definition 5.8 and (5.3), suppose p: 91o End V is a finite dimensional representation of the semisimple Lie algebra g~o, and X ' C 91o. i) If a d ( X ' ) i s semisimple, then p ( X ' ) is semisimple. ii) If ad(X ~) is nilpotent, then p ( X ~) is nilpotent. iii) If T is a derivation of g~o, then there is a unique element X T C gto with T = ad (XT). iv) If X ' - X~ ÷ X~ is the Jordan decomposition of X t as an n × n matrix (Lemma 5.~), then X~s and Xtn belong to g~o. v) I f X ~ is semisimple, then there is a g C G so that Ad(g)(X ~) = gX~ g -~ commutes with O(Ad(g)(X')). Parts (i) and (ii) are proved in [Hu, Theorem 6.5]; part (iii)is [Hu, Theorem 5.3]; part (iv)is [Hu, Theorem 6.4]; and ( v ) i s essentially [W, Lemma 2.3.3].
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L e m m a 5.10 In the setting of Definition 5.8, write X = X ~+ X 3 for the decomposition according to (5.3e).
i) X is sernisirnple if and only if ad(X) is semisimple. ii) X is nilpotent if and only if X 3 = 0 and ad(X) is nilpotent. iii) X is hyperbolic if and only if X 3 C P0 and ad(X) is hyperbolic. iv) X is elliptic if and only if X t C ~o and ad(X) is elliptic. Proof." The matrices X ' and X 3 commute with each other; and a d ( X ) = ad(X1). The matrix X 3 is a sum of commuting symmetric and skew symmetric matrices (cf. (5.3c)), and it is therefore automatically semisimple. We will analyze the bracket action adg[(n,R)(X) of X on all n x n matrices. This space is naturally identified with R n ® (R n)*, and we compute ad~l(~,~) ( X ) ( v ® w) - X v ® w - v ® t X w .
(5.4)
If X is semisimple with eigenvalues {zi}, it follows immediately that ad~l(n,~) (X) is semisimple with eigenvalues { z i - z j }. The implication "only if" in (i) is immediate. For the converse, assume that a d ( X ) = a d ( X ' ) is semisimple. By Theorem 5.9(i), it follows that X ' is semisimple. Therefore, X = X ' + X 3 is a commuting sum of semisimple matrices, and so X is semisimple. For (ii), suppose first that X is nilpotent; say X p = 0. By (5.4), we see that (ad(X)) 2p = 0; so ad(X) is nilpotent. By Theorem 5.9(ii), X ' is nilpotent. The expression X - X ' 4- Xa is a commuting sum of a nilpotent and a semisimple matrix, and so it must be the Jordan decomposition of X. Since X is nilpotent, it follows that X 3 = 0. Conversely, assume that X 3 = 0 and that ad(X) = a d ( X ' ) is nilpotent. By Theorem 5.9(ii), X ' is nilpotent; so since X 3 = 0, X is nilpotent. For (iii) and (iv), assume first that X is hyperbolic; that is, that X is diagonalizable with real eigenvalues. Then (5.4) implies that a d ( X ) = a d ( X ' ) is as well; so ad(X) is hyperbolic.
Similarly, X elliptic implies
a d ( X ) elliptic. To complete the proofs of (iii) and (iv), we will use a lemma. Here is the setting.
Suppose X ' E 1~ is a semisimple element.
According to
L e m m a 5.7, the hyperbolic and elliptic parts Th and T~ of the semisimple derivation a d ( X ' ) of ft~ are also (semisimple) derivations.
By Theorem 5.9(iii), they are given by the adjoint action of unique elements X[h' ] and
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X(~] of g~. On the other hand, the semisimple matrix X t has hyperbolic and elliptic parts X~ and X~.
In the setting just described, X[h1 ] X hI and X[e/ ] particular, the hyperbolic and elliptic parts of X ~ belong to g~o. L e m m a 5.11
-
-
-
-
I X e. In
Proof: Everything here behaves nicely with respect to conjugation by elements of G. According to Theorem 5.9(iv), we may therefore assume that X t commutes with OX ~. In this case, we have Xh_ , (Lemma 5.7(vi)).
-21(X' - OX') ,
1
X~e - -~(X' + OX')
Evidently, these matrices belong t o g~. We therefore
have
ad(X') - a d ( X ~ ) + ad(X~),
a commuting sum of derivations of g~. By what we have already proved, the first term is hyperbolic and the second elliptic as endomorphisms of g~. By the uniqueness in Lemma 5.7, they are the hyperbolic and elliptic parts of ad(X').
C]
We return now to finish the proofs of (iii) and (iv). In general, we will write
for the decomposition of a central element according to (5.3c). Suppose X is hyperbolic. We have already shown that ad(X) = ad(X') is hyperbolic, and so Lemma 5.11 implies that X ~ is hyperbolic. The three matrices X t, X~e , and X3p are therefore hyperbolic, elliptic, and hyperbolic, respectively; and they commute with each other. It follows that X = (X' + X,~p )+ ( X ~ ) exhibits X as a commuting sum of hyperbolic and elliptic matrices. By the uniqueness in Lemma 5.7, the second term is zero; that is, X 3 belongs to P0, as we wished to show. Conversely, assume that ad(X) is hyperbolic and that X~ E P0. Lemma 5.11 shows that X ~ is hyperbolic; so X is a sum of two commuting hyperbolic matrices and is therefore hyperbolic. The proof of (iv) is identical.
[3
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T h e o r e m 5.12 Suppose G is a real reductive Lie group as in Definition 1.5, and X E go. Then there are hyperbolic, elliptic, and nilpotent elements Xh, Xe, and Xn of 90 (Definition 5.8) uniquely characterized by the following two properties: i) X = Xh + X~ + X n . ii) [Xh, Xe] ---~[Xh, Xn] :
[Xe, Xn] : O.
This decomposition has the following additional properties. iii) Any element g E G fixing X (that is, Ad(g)(X) = X ) also fixes Xh, Xe , and Xn. iv) The adjoint representation of 9o preserves the decomposition in (i): ad(Xh) is hyperbolic, a d ( X e ) i s elliptic, and a d ( X ~ ) i s nilpotent.
This is an easy consequence of Theorem 5.9, Lemma 5.10, and Lemma 5.11; we leave the details to the reader.
C o r o l l a r y 5.13 Suppose G is a real reductive Lie group as in Definition 1.5, and X E go is a semisimple element. i) There is a g E G so that Ad(g)(X) commutes with O(Ad(g)(X)). ii) Suppose IX, OX] = O. Then the centralizer G x - {g e G l A d ( g ) ( X ) -
X}
is a real reductive Lie group, with Cartan involution the restriction to G x of O. In addition, Gx=(Gxh)x,
=(Gx,)xh.
Proof." Part (i) is an easy consequence of Theorem 5.9(v) (and (5.3c)). For (ii), the descriptions of G x as iterated centralizers follow from Theorem 5.12(iii). That G x is reductive then follows from Proposition 1.10(ii). [2]
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I n t e r l u d e o n t h e c l a s s i f i c a t i o n of ( c o ) a d j o i n t o r b i t s Corollary 5.13 leads to a systematic procedure for describing all the
coadjoint orbits for a real reductive Lie group. Although we will not really need it, the classification is simple and attractive. The method of coadjoint orbits suggests as well that it should bear a family resemblance to the classification of unitary representations. For these reasons, we will outline the classification here. Most of the proofs are easy; the more difficult results may be found in many places (including [W, Chapter 2]). Suppose G is a real reductive Lie group. A Cartan subspace for G is a maximal abelian subalgebra a0 of P0. Given such a subspace,
D e f i n i t i o n 6.1 we define
M = {k E K ] A d ( k ) ( X ) =
X, all X E a0},
M' = {k E K IAd(k)(ao ) = a0}. The groups M ' and M are compact, and M is an open normal subgroup of M'. The quotient is therefore a finite group
W(G, ao)= M ' / M C Aut(a0), the Weyl group of ao in G. (The action on a0 is by Ad.)
Suppose G is a real reductive group, and ao is a Cartan subspace of po. Then the inclusions
T h e o r e m 6.2
a0 C P0 C go
induce bijections among the following three sets: i) orbits of W = W(G, ao) on ao ii) orbits of K on P0 iii) hyperbolic orbits of G on go Suppose X and G y are equal if and equal. In particular, gation in G) for the
Y belong to ao. Then the group centralizers G x and only if the Weyl group centralizers W x and W y are there are only finitely many possibilities (up to conjucentralizer of a hyperbolic element.
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E x a m p l e 6.3 Suppose G - O(p, q) (Example 1.13) with p _< q. After a simple change of basis, the matrix Ap,q of the quadratic form Bp,q may be replaced by the matrix
1
Ap,q
Ip
0
o
o
o) 0
.
-Ip_q
We write Bl(p,q) for this new form and Ol(p,q) for the corresponding group. If g E GL(p, R), then it is easy to check that G 1 - 0 l(p, q) contains the matrix ,~l(g) __
0 0)
0
t g -1
0
0
0
Ip_q
.
The map /~1 provides an embedding
GL(p, R) C 0 l(p, q) '~ O(p, q), and all maps respect the Cartan involutions. Now the space of diagonal matrices is a Cartan subspace for GL(p, IR), naturally isomorphic to Nv. It is not difficult to check that the image a~ of the diagonal matrices under dk 1 is a Cartan subspace of g~; it consists of diagonal matrices with the first p entries equal to the negatives of the next p, and the last p - q entries equal to zero. The Weyl group of this Cartan subspace acts by permuting and changing the signs of the coordinates of IRP; it is isomorphic to the hyperoctahedral group
w(a, 4 ) - & x (z/2z)p, a semidirect product with the second factor normal. By Theorem 6.2, each hyperbolic orbit has a unique representative
x-
(xl,...,G)
e
X l ~ . . . k X p k O.
We can compute the centralizer G~ as follows. Write p - pr 4-.. "4-pl 4-P0, in such a way that X 1 .....
Xpr > Xpr+l .....
Xpv+...+p I > Xpr+...+pl+ 1 - O .
D. Vogan, Jr.
298 Then,
G x ~- GL(pr, R) × ... × GL(pl, R) × 01 (Po, Po + q - P). (There is a natural embedding of this group in G 1, which is easy to construct using A1.) There is a parallel result for elliptic elements. 6.4 Suppose G is a real reductive Lie group. A Cartan subalgebra for K is a m a x i m a l abelian subalgebra to of ~o. Given such a subspace, we define Definition
T = {k e K l A d ( t ) ( X ) = X, all X e to}, T' = {k e K I Ad(k)(to ) = to}. The group T is called a small Caftan subgroup of K. If K is connected, it is a torus. The group T' is compact, and T is an open normal subgroup of T'. The quotient is therefore a finite group
W(G, to) = T ' / T C Aut(to), the Weyl group of to in G. (The action on to is by Ad.)
Suppose G is a real reductive group, and to is a Cartan subalgebra of ~o. Then the inclusions
Theorem
6.5
to C ~o C $o
induce bijections among the following three sets: i) orbits of W = W ( G , to) on to ii) orbits of K on ~o iii) elliptic orbits of G on 9o Suppose X and Y belong to to. Then the group centralizers G x and G y are equal if and only if 1) the Weyl group centralizers W x and W y are equal; and 2) the elements X and Y annihilate exactly the same weights of to on Po.
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In particular, there are only finitely many possibilities (up to conjugation in G) for the centralizer of a elliptic element. E x a m p l e 6.6 Suppose G = O(p,q) (Example 1.13). Recall that K _~ O(p) x O(q). Write p ' = [/)/2] (the greatest integer in p/2) and q ' = [q/2]. Then there are obvious maps SO(2) p'+q' ~ SO(2) p' x SO(2) q' C O(p) x O(q) C O(p, q). The Lie algebra of SO(2) consists of skew symmetric 2 x 2 matrices, and so may be identified naturally with R. The differential of the first inclusion therefore gives
7-: RP' × Rq' ---~ 1~0. It is easy to check that the image of T is a Cartan subalgebra to of t~0. The Weyl group acts by permutation and sign changes of the first p' and last q' coordinates separately:
(Sp, X (TZ/2%)P')M (Sq, M (TZ/2Z) q'),
W(G,t;o)-
a product of two hyperoctahedral groups. According to Theorem 6.5, every elliptic orbit in go has a unique representative
z - [(z~,..., z 1, ), (Zl~,..., zg, )1 e R~' +~', z l > . . . >_zl, >_o,
Z~l > . . . > zq,2 >o.
We can compute the centralizer G z as follows. Write p' - P'r ÷ " " ÷ P ] ÷ P~ and q' - q~ + . . . q~ + q~o, in such a way that
Z 1 ..... Z~ . . . . .
Z ; , > Z ; , +1 . . . . . 2
2
Z@ > Zq~r+l . . . . .
Z;, + ... +p~+l
=
Z q2 ~+...+q~+l
Z;~+...+p~ > Z;~+...+p[ +1 ~-~ 0, 2 2 -0 Z@+..._Fq; ~> Zq~r-F...+q;+l -- ,
(j-r+l
,
r-1
,''',
1) •
Then,
G z ~ U(P'r,q'r) x ' " x U ( p ~ , q l ) xt
01
( P - - 2 ( P¢r + ' ' ' + P l¢) , q - - 2 ( q ' r + ' ' ' + q l )¢) •
(The idea is that the element Z acts as Z p~r+...+p 1 , l -F1 times a complex y+ structure on a subspace of the form R 2p~+2q~. The centralizer of Z therefore
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acquires a factor consisting of the complex-linear transformations of this subspace that preserve the quadratic form. This is the indefinite unitary group V(p}, q j ) o f Example 1.15.) Here is the procedure for constructing all adjoint orbits for a reductive group G. First, choose a Caftan subspace a0 C P0, and compute its Weyl group. For each element Xh E a0, consider the reductive group Gxh; Theorem 6.2 guarantees that there are only finitely many such groups. (They are described for G = O(p, q) in Example 6.3.) For each such group, choose a Caftan subalgebra txh,0 of t~xh,0, and compute its Weyl group. For each element X~ C txh,0, consider the reductive group Gxh,xe; Theorem 6.5 says that there are still only finitely many such groups. (When Gxh is a product, it suffices to treat each factor separately. The case of O(p, q) is handled in Example 6.6; to treat all of the hyperbolic centralizers for O(p, q), we would need to consider also GL(n,R). The conclusion is that for O(p, q), each group Gxh,x~ is a product of factors of the form GL(a, R), GL(b, C), U(c, d), and O(e, f); there is just one factor of this last form.) To make a list of all semisimple adjoint orbits, we choose one representative Xh in each Weyl group orbit on a0; and then (for each Xh) one representative X~ in each Weyl group orbit on txh,0. The resulting elements
Xs = Xh + X~ provide exactly one representative of each semisimple orbit on 00. This procedure can be continued to give all adjoint orbits: We just need a way to give a representative for each of the (finitely many) nilpotent adjoint orbits for each of the (finitely many) reductive groups Gxh,xe. For the classical groups this problem is not too hard, and for the exceptional groups it has been solved. We refer to [CM] for more information.
7
Hyperbolic elements, real polarizations, and parabolic subgroups We turn now to the problem of attaching representations to orbits for
a reductive group G. Recall from Lemma 5.1 that we can identify go with ft~ by an isomorphism X ~ {x" {x (Y) = (X, Y) = tr XY.
(7.1a)
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Lemma 5.1 allows us to identify the isotropy algebra and group for the coadjoint action at ~x (see (4.1)) as G¢x - {g c G l A d ( g ) X = X } ,
fi~x,O = {Y e fio I[Y,X] = 0}.
(7.1b)
We may sometimes write G x or fix,o accordingly. The symplectic form of (4.1f) is W~x (Y, Z) = {X, [Y, Z]> = - ( Z , [Y, X]);
(7.1c)
notice that this formula is skew symmetric in all three variables X, Y, and Z. We sometimes write simply wx. We call ~x semisimple (or nilpotent, elliptic, or hyperbolic) if X is. P r o p o s i t i o n 7.1 In the setting of (7.1), suppose that ad(X) is diagonalizable with real eigenvalues; this happens in particular if X is hyperbolic (Lemma 5.10). Write fit,o for the t-eigenspace of ad(X), so that fio -- w-"~fit,o,
fix,o -- fio,o.
(7.2a)
tEIR
Define Px - E
gt,o,
nx,o - E
t>O
fit,o,
N x -- exp nx,o.
(7.2b)
t>O
i) The decomposition (7.2a) makes go an R-graded Lie algebra: [g~,0, gt,o] c g~+t,o.
ii) The subspace gt,0 is orthogonal to gs,0 with respect to ~ x unless s = - t . iii) The adjoint action of N x on X defines a diffeomorphism 7(n) = Ad(n)(X).
7: N x ~ X + nx,o,
iv) The coadjoint action of N x on ~x defines a diffeomorphism
Nx ---+
c
I),lpx -
xlpx
},
-y*(n) = Ad*(n)({x).
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v) The group N x is connected, simply connected, and nilpotent. It is normalized by G x , and it meets G x exactly in the identity element. The semidirect product Px = GxNx is a closed subgroup of G; more precisely, this is the Levi decomposition of a parabolic subgroup. vi) Suppose T is an integral orbit datum at ~x (Definition ~. 6). Then ~- has a unique extension Tpx to P X . This extension may be characterized by Tpx (gn) = 7(g)
(g e G x , n e N x ) .
The pair (~'px, Px) is an invariant real polarization of T. Proof: Part (i) is elementary. For (ii), suppose Y E 08,0 and Z C 0t,0. Then (7.1c) implies that x(Y, z ) - s(y, z ) - -t(Y, z ) , and (ii) follows. For (iii), list the positive eigenvalues of ad(X) as tl < t2 < "." < tt. Suppose Y/ and Z~ are elements of 0t~,0. Then we calculate Ad(exp
Y/)(X + E
Zj) -
exp(ad(Y/))(X+ E
J
Zj).
(7.3a)
J
On the right side we are exponentiating the linear transformation a d ( ~ ) of 0o. Now a d ( Y i ) ( X ) - -tiY~, and ad(Yi) carries 0t,o into Ot+t~,o. Consequently, ad(Yi)(0t~,o) C E
0tk,o.
(7.3b)
k>i
Inserting this information in (7.3a) gives i--1
Ad(expYi)(X+E
ZJ) - X + E j
j=l
l
ZJ+(Zi-tiYi)+
E
PJ(~i'Zl'""ZJ)"
j=i+l
(7.3c) Here Pj(Yi, Z 1 , . . . , Zj) C 0t~,0 depends in a polynomial way on Yi and linearly on the various Zk (with k < j).
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We now consider the map from nx,0 to X + nx,0 defined by rr(Y1 + . . - +
Yz) = Ad(exp(Yt).-. exp(Y1))(X).
(7.3d)
We calculate the adjoint action here one factor at a time, using (7.3c). The conclusion is l
~(Y1 + . . . + ~ ) = x + E
( - t j Y j -{- q j ( Y1, . . . Y j - 1 ) ) .
(7.3e)
j=l
Here, qj(Y1,... Yj-1) C l~t~,0 depends in a polynomial way on the various Yk. This description shows that 7r is a diffeomorphism. The map 7r is a composition of 7: n x , o --~ N x ,
r(Y1 + ' - . + I~) : exp(Y1).., exp(Yl),
and the map 3' of (iii) in the proposition. An argument along similar lines to the one just given shows that 7 is surjective. Whenever 3" is smooth, 7 is smooth and surjective, and 3' o 7 is a diffeomorphism, it follows t h a t 3" and 7 are diffeomorphisms. This proves (iii). Part (iv) is just a reformulation of (iii) using the identification of L e m m a 5.1. For (v), N x is connected by definition. The proof of (iii) provided a diffeomorphism v from nx,0 to Nx, and so Nx is simply connected. Nilpotence is immediate from (i). By (iii), only the identity element of N x fixes X, so that G x meets N x exactly in the identity element. The adjoint action of Gx preserves each eigenspace l~t,0 of ad(X), and so it preserves nx,0. It follows that G x normalizes Nx. It follows that Px is a subgroup of G. As a consequence of (iii), we have
Px = {g ~ G I A d ( g ) ( X + nx,0) = (X + nx,0)}. This shows that Px is closed in G. T h a t Px is parabolic means that it contains a minimal parabolic subgroup of G. This is more or less obvious from standard constructions of minimal parabolic subgoups; we omit the details. (Because we do not require G to be in the Harish-Chandra class, there is some question about exactly how a minimal parabolic subgroup should be defined. We define it so that a Levi subgroup is the centralizer
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of a maximal abelian subalgebra consisting of hyperbolic elements. Such a subspace is just a G-conjugate of a Cartan subspace in P0 (Definition 6.1).) For (vi), it follows from (v) that Tpx is well defined. Because distinct eigenspaces of ad(X) are orthogonal with respect to <,>, the linear functional ~x is zero on ~t,0 for t ~- 0. In particular, ~x vanishes on nx; so ~-gx satisfies condition (i) of Definition 4.7. Condition (ii) of the definition follows from (v) of the proposition. By (ii) of the proposition, the nondegeneracy of ~ x forces the pairing it defines between ~t,0 and g-t,0 to be nondegenerate for t ~: 0; so, in particular, these two spaces have the same dimension. Consequently,
dimpx,o/gx,o - dim nx,o - (dim go/gx,o)/2. This is condition (iii) of Definition 4.7, proving that (~-Px, Px) is an invariant real polarization. T h a t 7pz is the only extension of ~- is elementary; we omit the details. V1 Although this construction of polarizations was the main goal of this section, we may as well discuss orbit data in this setting.
In the setting of (7.1), suppose that ~ is a hyperbolic element of ~ . Then the set of integral orbit data at ~ is naturally in oneto-one correspondence with the irreducible representations of the (finite) group of connected components of G~. In particular, this set is nonempty, and so the orbit G. ~ is integral.
Proposition
7.2
Proof: By L e m m a 5.1, we may write ~ - ~x with X a hyperbolic element of 90. By Theorem 6.2, we may (after conjugating by an element of G) assume t h a t X C P0. By Proposition 1.10(ii), it follows that G~ is a real reductive group with Cartan involution 01a ~. Obviously, the linear functional on ~,0 defined by X is just ~lftx,0- W h a t this means is that we have reduced Proposition 7.2 to the case G - G~, which we now assume. This assumption means precisely that X belongs to a~,o - { z c po I G z - G }
= {Z e P0 I a d ( g ) ( Z ) -
0,
Ad(K)(Z)-
Z}.
(7.5a)
Now define Pl,0 - orthogonal complement of al,0 in P0.
(7.5b)
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It is easy to check that gl,0 -- 20 ~- ~1,0 is a Lie subalgebra of g0. It is the orthogonal complement of al,0 in g0. Define G1 = K . exp ~1,0,
A1 = exp al,0.
(7.5c)
If Y E Pl,0 and Z C al,0, then Y and Z commute; so exp(Y + Z) = exp(Y) exp(Z). Now, it follows from the Caftan decomposition (Proposition 1.4) that G1 is a real reductive group, and that
G ---~ G1 ×
A1,
(7.5d)
a direct product. Furthermore, A1 is a vector group, isomorphic to its Lie algebra under the exponential map. The group G1 has compact center. Now the linear functional ~x is given by inner product with X C al,0, and so it is trivial on gl,0. According to Definition 4.6, an integral orbit d a t u m at ~ is an irreducible unitary representation (T, 7-/T) of G, such that ~- is trivial on the identity component G1,0, and T(exp Z) = e i(Z'x) . Idn~
for Z c {11,0. Evidently such representations correspond precisely to the representations of G1/GI,o ~- G/Go, as we wished to show.
[::]
Propositions 7.1 and 7.2, together with the construction of (4.9), provide a finite set of unitary representations of G attached to each hyperbolic coadjoint orbit. It can be shown that these representations are all irreducible. We are left with two (closely related) problems: to interpret this construction as part of a solution of Problem 3.10; and then to extend it to a wider class of coadjoint orbits. In Section 9 we will treat the first of these problems. The Dixmier algebras we need will be algebras of differentim operators. To clarify the formal algebraic construction we will use for these algebras, we examine separately the familiar case of R n .
D. Vogan, Jr.
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Taylor series and differential operators
o n I~n
Suppose f E C°C(IRn). The Taylor series of f at 0 is a formal power series
E
1 oaf (0). c a ( f ) - ~.! Oza
ca(f)xa'
(8.1a)
aEN ~
(Here, we use standard conventions for multi-indices: x a
__ X l a l . . .
Xna~
a! = c~1!-" an!, and so on.) Every formal power series arises as the Taylor series of some smooth function f. We write C [ [ x l , . . . , Xn]] for the space of all formal power series.
Then formation of Taylor series provides a
surjective linear map
T(f)-
T: C ~ ( R n) ~ C [ [ x l , . . . , x ~ ] ] ,
E
ca(f)xa"
(8.1b)
aEN ~
W h a t is essential about the Taylor series is the collection of complex numbers ca; or, what amounts to the same thing, the values of all the derivatives of f at 0. To formalize this point of view, define U -
C[a/OXl,...,a/(OXn]
= algebra of constant coefficent differential operators on Rn.(8.1C) This algebra has as a basis the operators Oa/Ox a. To specify a linear functional on U, we must therefore specify its value at each of these basis elements; and these values can be arbitrary. T h a t is, Home(U, C) "-' families of complex numbers {ta
}.
(8.1d)
We may therefore regard Taylor series as a surjective map
T(f)(u) - (U" f)(0).
7" C ~ (R n ) -o Home(U, C),
(8.1e)
The connection with the formal power series in (8.1a) is 1
0~
(8.1f)
ca(f) - ~T(f)(~x~ ). We could also describe this as an isomorphism
Home(U, C) "-' C[[xl,..., Xn]],
(8.1g) a
,
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307
We want to understand the algebra Dan(R n) of differential operators on R n with analytic coefficients. Such o p e r a t o r s - indeed, even the algebra Df°rm(R n) of differential operators with formal power series coefficients act on C[[Xl,... ,xn]], and therefore on Taylor series. These actions are faithful. (For D an, the reason is that a nonzero differential operator cannot annihilate all analytic functions.) Consequently, U C Dan(R n) C Df°rm(R n) C End(C[[Xl,...,xn]])"~ End(Homc(U,C)).
(8.2)
The problem is to identify the differential operators among all the linear transformations of formal power series. One often thinks of differential operators as characterized by the property of not increasing support. By considering only the action on Taylor series, we have in some sense already taken advantage of that property. What we can study on the level of Taylor series is order of vanishing. For this purpose it is helpful to write
lOLl--- OL1 -~-''"-~- OLn
(OL e 1~ ~).
(8.3a)
Define Up = span of monomials O~/Ox ~ with [a I _< p,
(8.3b)
the constant coefficient operators of order at most p. (We will sometimes be careless about this terminology, saying that any element of Up is of order p even though it might belong to Up_l.) These subspaces form an increasing filtration of U. Similarly, we can define
Dpn(]~n)
-
-
{E
0~
f~(X)~x~
0~ --PDf°rm(]~n) -- { E h(x)-~x~ I~t
(f~ C C~(Rn))},
(8.3c)
(Is
(8.3d)
e
C[[Xl ,. .., Xn]]) },
the differential operators of order at most p with analytic or formal power series coefficients . By the leading coefficients of an operator in -D - pf°rm written as in (8.3d), we will mean the formal power series fa with la] = p. This means that before we can speak of leading coefficients, we must specify the order p we have in mind. If the operator happens to belong to/-)form ~-lp-- 1 then its leading coefficients are all zero. Now define Homc(U, C)p = {# C Homc(U, C) I if(Up) = 0}.
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Under the isomorphism (8.1g), this corresponds to C[[Xl,...
Xn]]p --- { E Cc~Xa
- o,
_< p}.
(8.3f)
P r o p o s i t i o n 8.1 Suppose f C CC°(~ n), and p > O. The following conditions on f are equivalent. i) For every lal _< p, (O"f /Ox~)(O) - O. ii) For every u e Up, (u. f)(0) - 0 (cf. (8.3b)). iii) For every S E Dan(]~n)p, (S" f)(0) -- 0 (cf. (8.3c)). iv) The Taylor series T ( f ) (cf. (8.1b)) belongs to C[[xl,...xn]]p. v) The Taylor series T(f) (cf. (8.1e)) belongs to Home(U, C)p.
This is elementary. When the conditions are satisfied, we say that f vanishes to order p at zero, and we write (8.4)
f e C~(I~n)p.
Because of the proposition, we may refer to (say) Home(U, C)p as the space of Taylor series vanishing to order p. The first property distinguishing differential operators among all endomorphisms of Taylor series is this. P r o p o s i t i o n 8.2 Suppose f E CCX)(]~n)p (cf. (8.~)) and S E Dan(]I(n) (cf. (8.3c)). Then S . f e C ~ (I~n )p-q. Here, we write C ~ (It(n )r = C ~ (l~n) f o r r
Taylor series vanishing to order p -
q. Here, we write Home(U, C)r =
Homc (U, C) for r < O. This is an immediate consequence of the definitions and of the fact that the filtrations in (8.3) respect the algebra structures:
uqu, c Dqn(~ n )Dran(]~n ) C
Dq_~r(I~n),
F)f°rm(]l~n). ]-)form(]~n) F)form(]~n) C ~Jq+r --q wr
(s.5)
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W i t h this property in mind, we say that a linear transformation A of Home(U, C) is weakly of order q if A . Homc(U, C)p c Homc(U,
C)p_q
(8.6)
for all p > 0. (The condition is nonempty only for p _> q.) The most obvious linear transformations weakly of order q are the differential operators of order q with formal power series coefficients. There are more, however. When n - 1, the differential operator A - X~x acts on formal power series by
Ec x ) -
jk0
EJc3x jk0
This action obviously preserves the subspaces C[[x]]p of (S.3f), and so A is weakly of order 0. As a differential operator, however, A is first order. A little reflection shows that the problem is the vanishing at zero of the coefficient of the highest derivative in A. Here is a precise statement. L e m m a 8.3
Suppose 0 OL
S
e
C[[x,,..., Xn]])
IaI<_P
is a differential operator with formal power series coefficients of order at most p. Assume that there is a ~ with I~1 - P and a~(0) ~ 0. (Here, cry(0) means the constant term of the formal power series aZ.)
Then S is not
weakly of order p - 1. P r o o f : The formal power series f = x z vanishes to order p - 1 at 0; but ( S . f)(0) - / 3 ! a~(0) is not zero. So S . f does not vanish to order 0 at 0, and so S cannot be weakly of order p -
1. [~
The lemma says that the notion of weak order for endomorphisms of Taylor series allows us to detect the order of differential operators the leading coefficients of which do not vanish at zero. To continue, we need a way to construct such differential operators from arbitrary ones (with formal power series coefficients). That is, we need a way to reduce the order of vanishing of the leading coefficients. The way to do that is to differentiate
310
D. Vogan, Jr.
those coefficients. The next l e m m a provides a way to differentiate leading coefficients while still thinking of the operators as acting on Taylor series. Recall t h a t a vector field on R '~ with formal power series coefficients is a first order differential operator ~ t h a t annihilates the constant function: n
-- E
i--1
0
fi(X)~X i
(fi E C[[Xl,... ,Xn]]).
(8.7a)
T h e symbol of ~ at 0 is the vector
0"(~)- (fi(O)) E
Cn
(8.7b)
We fix now a set { ~ 1 , . . . , ~N} of vector fields, and assume t h a t the symbols c r ( ~ l ) , . . . , a(~N) span C n. (For the purposes of this section we could just take the n vector fields O/Oxi, but for the applications in Section 9 the more general assumption will be useful.) As a consequence of (8.7c), we can draw the following conclusion. Suppose f is a nonzero formal power series. Then, there is an integer r >_ 0 so t h a t f vanishes to order r - 1 at 0, but f does not vanish to order r. For this value of r, we can find a sequence ( i l , . . . , i r ) C { 1 , . . . , N } r so t h a t (~i, "'" ~i,,f)(O) ¢: O.
(8.7d)
Here is the l e m m a t h a t allows us to apply these ideas to differential operators. Lemma
8.4
Suppose Ooz
s =
C[[x,,... ,Xn]])
I~l
is a differential operator with formal power series coefficients of order at most p; and suppose n i--1
0
c C[[x,,..., =n]])
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is a vector field. Write T = [~, S] = ~ o s -
s o
for the commutator of S and ~. Then T is a differential operator of order at most p with formal power series coefficients: T -
0~ 7~(x)Ox ~
E
(7~ C C[[Xl,... ,Xn]]).
tal_~P
A s s u m e that the leading coefficients of S all vanish to order k at x = O: that is, that
~
e C[[x~,..
,Xn]]~
whenever lal = p.
I f I~1 = P, then T~ -- ~'O'~ e C [ [ X l , . . . , X n ] ] k .
That is, the leading coefficients of T are obtained from those of S by applying the differential operator ~, up to terms vanishing to order k at x = O.
This can be proved by a straightforward computation, which we omit. C o r o l l a r y 8.5
Suppose S is a differential operator of order at most p with
formal power series coefficients. A s s u m e that the leading coefficients of S vanish to order r -
1 at x -- O, but that some leading coefficient does not
vanish to order r. Then we can find a sequence ( i l , . . . , i r ) e { 1 , . . . , N } r so that some leading coefficient of the iterated commutator
T = [ ~ , . . . [ ~ , S]...] has a leading coefficient that does not vanish at x = O.
This is immediate from the lemma and (8.7d). Corollary 8.5 suggests how to refine the definition of weak order q given at (8.6). In the setting of (8.7), we say that an endomorphism A of Homc(U, C) is of order q if for every sequence ( i 1 , . . . , it) e { 1 , . . . , N} r, the iterated commutator [~1 , " " [ ~ , A]...]
(8.8)
is weakly of order q. The definition appears to depend on the choice of the vector fields ~i, but Theorem 8.7 will show (subject to the assumption (8.7c)) that it does not.
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Suppose S is a nonzero differential operator with formal power series coefficients. Then the order of S as a differential operator is equal to its order as an endomorphism of the space Home(U, C) of formal Lemma
8.6
(cf. (s.s)). This follows from Corollary 8.5, Lemma 8.3, and the definitions. Here is the main theorem of this section. 8.7 Suppose T is an endomorphism of Taylor series of order less than or equal to p (cf. (8.8)). Then T is a differential operator with formal power series coefficients of order less than or equal to p. Theorem
S k e t c h o f proof: By an elementary calculation, we can find a differential operator D of order less than or equal to p, with formal power series coefficients, having the property that
D(x ~) = T ( x ~)
(1~1 <_p).
(This amounts to solving a finite system of linear equations in the ring of formal power series. The coefficient matrix is square and upper triangular with nonzero integers on the diagonal; so the system is solvable.) After replacing T by T -
D, we may therefore assume that
T ( x ~) = 0
(lal ~ p).
(8.9a)
On the other hand, if I~1 >- P + 1, then x z vanishes to order at least p at 0, so T ( x ~) vanishes to order at least 0. That is,
T(xZ)(O) = 0
(1~1 >-P + 1).
(8.9b)
We are now trying to show that T -- 0. The first problem is to deduce from the preceding formulas a statement about the effect of T on arbitrary formal power series. For that we use the natural topology on formal power series, in which the various subspaces C [ [ x l , . . . , x~]]p of series vanishing to order p form a neighborhood base at 0. This means that a sequence fm of formal power series converges to f if f -
fm vanishes to order at least
rm, and rm goes to infinity with m. In this topology, a formal power series
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c~x '~ is the limit of its partial sums ~-~]o~[~mCo~Xa" An operator A that is weakly of order p is continuous, and therefore
A (Zco~x'~) -- l i m
E
c~A(x~)"
I~l<m
This means that any fixed coefficient of the formal power series A(~-~ c~x ~) is equal to the corresponding coefficient of }-~l~l_<mc~A(x~) for m sufficiently large. Bearing in mind these remarks, we find that (8.9a) and (8.9b) imply
T(f)(O) - 0
(f e C [ [ x l , . . . , xn]]).
(8.9c)
We are trying to show that T - 0. Because of (8.7d), it suffices to show that if ( i l , . . . , i r ) E
{ 1 , . . . , N } ~, then (~i1"'" ~i~T f)(O) = O.
(8.9d)
We will prove this statement by induction on r, simultaneously with ([~i1' [ ' ' ' [~ir' T]]'" "]f)(O) -- O.
(8.9e)
The case r - 0 is precisely (8.9c); so suppose r _> 1, and that (8.9d) and (8.9e) are known for r - 1. The iterated commutator in (8.9e) may be expanded as a sum of terms of the form i(~iT~jf)(O). Here, I and J are disjoint ordered sets with union { i l , . . . , it}, and ~I is the composition of the vector fields ~i (with i E I) in the order determined by I. Just one of these terms has I I I - r: It is exactly the term appearing in (8.9d). All of the other terms have III < r, and therefore they vanish by inductive hypothesis. It follows that (for a fixed f) (8.9d) and (8.9e) are equivalent. To prove one of these statements, it is enough (by the foregoing remarks about topology) to take f - x ~. If lal _< p, then (8.9d) follows from (8.9a). So suppose lal > p, then x ~ vanishes to order at least p at 0. By hypothesis, the iterated commutator in (8.9e) is weakly of order at most p; so the iterated commutator applied to x ~ vanishes to order at least 0 at 0, as required by (8.9e). This completes the induction and therefore the proof.
[:]
D. Vogan, Jr.
314 9
Twisted differential operator algebras Problem 3.10 suggests that we should construct a representation by
first constructing a Dixmier algebra (A, ¢), and that the action of U(g) on the representation should be given by ¢ and an action of A. The construction given in Section 7, which may be summarized as Indapx (Tpx ), does not proceed in this way; but it can be rearranged to fit better. For the moment we retain the setting of Section 7, although that will soon be modified. The Hilbert space of the induced representation is L2(G/Px, 7-t, ® D1/2). The group G acts by left translation of sections of an equivariant bundle ~ (whose fiber at ePx is 7-/~ ® D1/2). Consequently, elements of the complexified Lie algebra g act by first order real analytic differential operators on 12~; and elements of U(g) act by real analytic differential operators on V,. (Notice that general elements of the Hilbert space of the representation are only L 2 sections of V~, rather than smooth ones; so the differential operatots will not really act on them. But it turns out (since G/Px is compact) that the smooth vectors of the representation (Definition 2.4) are precisely the smooth sections of V~, and differential operators do act on these.) This discussion suggests a candidate for the Dixmier algebra A: It might be the algebra D an of all real analytic differential operators on V~. We have seen that there is a natural algebra homomorphism ¢: U(I~) --~ D an. The action of G on G/Px provides an action Ad of G on D~ n by algebra automorphisms, and this is easily seen to be compatible with ¢ and the adjoint action on U(g). The first problem is that the action of G on D an is not algebraic in any sense. A basic property of algebraic representations is that they are locally finite: Any vector is contained in a finite dimensional G-invariant subspace. The differential operators of order zero include the multiplication operators m I by analytic functions f on G/Px. The adjoint action on m/ is by translation of the function f. If f is not locally constant, then its translates can never span a finite dimensional representation of G. (Proving that statement is a good exercise in elementary representation theory.) One way out of this difficulty is to define D~ lg as the largest subspace of D an on which the adjoint action is algebraic. Since the adjoint action on U(tt) is algebraic, we will have ¢: U(ft) ~ D alg. It is possible to make sense of this idea, but it will be convenient for us to adopt a slightly less direct approach.
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We are now prepared to formulate working hypotheses for the section. Suppose that
Hc C Gc are complex connected algebraic groups. We write b C g for the corresponding Lie algebras. homomorphism
(9.1a)
Fix a Lie algebra
A:b ~ C .
(9.1b)
This amounts to a one-dimensional representation A: ~ ~ End(C~).
(9.1b')
One particular homomorphism plays a special rSle:
5: b ---, C,
1
5 ( X ) - ~ t r ( a d ( X ) on g/b).
(9.1c)
Eventually, we will assume Gc is reductive and Hc is a parabolic subgroup.
(9.1d)
These are all the assumptions and data required for the main construction. To motivate the construction, however, it will sometimes be helpful to assume more. For those purposes, we may sometimes assume G is a real form of Gc, and H = He n G is a real form of Hc;
(9.2a)
and
iA is the differential of a one-dimensional unitary character (A, Ca) of H. (9.2b) We will then write ~0 C g0 for the real Lie algebras. Under these assumptions, the real analytic manifold G/H is a real form of the complex manifold (actually, a complex algebraic variety) Gc/Hc. Then, H acts by the adjoint action on the real vector space g0/O0 (the real tangent space at the base point to G/H); so H acts on the one-dimensional real vector space D1/2(l~o/~o). We get 5 is the differential of the character A of H on D1/2(g0/~}0 ).
(9.2c)
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316
Using this character, we can define /2A -~
line bundle on G / H induced by A ® A,
(9.2d)
C~,A = space of compactly supported smooth sections of/2A. (9.2e) In the setting of (4.9), the Hilbert space of the unitary representation IndaH(A) is constructed as a completion of C 0,A" °o Here is the main result. It is a folk theorem, much harder to attribute correctly than to prove. The main ideas go back at least to Kirillov's work in the 1960s on nilpotent groups, and probably much earlier. The only difficult part of this formulation (that D~ is a Dixmier algebra in the setting of (9.1d)) can be deduced from [CBD]. The paper [BB] demonstrated the particular importance of twisted differential operators on flag varieties, and [BOB1] and [BOB2] provide a thorough treatment.
Suppose we are in the setting of (9.1a, b). Then there is a completely prime algebra D ( G c / H c ) ~ = D~, called a twisted differential operator algebra. This algebra is endowed with an algebraic action Ad of Gc by algebra automorphisms, and with an algebra homomorphism T h e o r e m 9.1
¢~: U(g) --+ Da.
The adjoint actions of Gc on U(g) and D~ are compatible with ¢~. The differential ad of the adjoint action is the difference of the left and right actions of g defined by ¢~ : ad(X)(T) = ¢ ~ ( X ) T - T¢~(X)
(Xcg,
TcD~).
If in addition (9.1d) is satisfied, then (D~,¢~) is a Dixmier algebra for Gc (Definition 3.5). Suppose that the auxiliary conditions (9.2a, b) are satisfied. Then D~ is isomorphic to a subalgebra of the analytic differential operators on sections of 12a (cf. (9.2d, e)). The adjoint action of G arises by change of variables from the action of G on G / H by left translation; and the homomorphism ¢~ arises from the natural action of U(g) by differential operators on C~, a. Proof: The statements in the last paragraph are intended to guide the construction of D~ in general. The problem we face is essentially to find
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a description of the differential operators on £;A that refers only to G, H, and A. To t h a t end, recall that the differential operators are certain endomorphisms of C 0,A" ~ The operators we want are actually going to have analytic coefficients. This suggests the possibility of studying t h e m by means of Taylor series expansions. As a first step, we need to understand Taylor series for sections of £;A. Now, the Taylor series of a function f at a point p can be thought of as a list of all the values at p of derivatives of f. If f is not a function but a section of a bundle, then we should apply differential operators on sections of the bundle, and the values will lie in the fiber of the bundle at p. Here is a description of Taylor series for sections of £A.
In the setting of (9.2), there is a surjective linear map (Taylor series at ell)
L e m m a 9.2
T: C~,a --~ Homv(~)(U(9), Cia+6).
(Here, the Hom is defined using the left action of U(t~) on U(g).) kernel of ~- consists of all sections vanishing to infinite order at ell.
The
S k e t c h o f proof: The fiber of E. at e H is the one-dimensional representation CA ® D1/2(t~o/Oo) of H. Because of (9.2b,c), the Lie algebra ~) acts on this fiber by iA + 5; so (after fixing a choice of a half-density on g0/i~0) we may identify the fiber with Cix+6 • The space C 0~, A may be identified with a certain space of smooth functions on G with values in Ci~+6 (transforming appropriately under H on the right, and satisfying a support condition). The group G acts on this space by left translation, and the Lie algebra go acts by right invariant vector fields. Explicitly, d
(X.cr) (g) - -~ (cr(exp(-tX)g) ) lt: o
(a C C~,A, X C g0, g E G). (9.3a)
This is a Lie algebra representation, and so it extends to an action of the universal enveloping algebra on C~,h. (This is the action mentioned at the end of Theorem 9.1.) The Taylor series map T is defined by
r(a)(u) = (u. a)(1)
(a e C~,A, u ~ U(g)).
(9.3b)
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If Z E b0, then -c(a)(Zu) - (Z . u . a)(1) = d ( u . a)(exp(-tZ))[t=o dt d : d-~[(A ® A)(exp tZ)]. (u. a)(1)lt=0 = [(iI + 6)(Z)]. (u. a(1)) =
[(~ + ~)(z)]. ~(~)(~).
(The third equality follows from the transformation property of u - a
un-
der H on the right.) This shows that 7 maps to the correct Hom space. Surjectivity of 7 is equivalent (after introducing appropriate local coordinates on G / H and using the Poincare-Birkhoff-Witt theorem) to the fact that every formal power series on I~n is the Taylor series at 0 of a smooth function. The assertion about the kernel of ~- is clear from (9.3b). [-3 We want to investigate the space of Taylor series more closely. Notice first that M~ = Homv(~)(V(g), Ci~+~) (9.4a) may be defined in the setting (9.1a-d); the additional structure of (9.2) is not used. The complex vector space M~ carries a U(g)-module structure: (v. #)(u) - #(uv)
(u, v e U(g), # e M~).
(9.4b)
Sometimes it will be convenient to write this action as an algebra homomorphism: ¢~" U(g) ~ End(M~),
(¢~ ( ~ ) , ) ( u ) - ,(u~).
Because we use the right action of U(g) on itself, v . #
(9.4b')
inherits from #
the transformation property under U(I~) on the left, and so it belongs to M~. Next, recall the standard filtration on U(9): Un(g) is the span of all products of less than or equal to n elements of g. In terms of the realization of U(9) as right invariant differential operators on a real form G, the subspace Un (9) corresponds to the differential operators of order at most n. We define the order of vanishing filtration of M~ by
M~,n - {# ~ M~ I~(Un(0)) - 0}.
(9.4c)
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This is a decreasing filtration of Ma" (x)
N M~,n -- O.
M,x D M~,0 D M,x,1 D . . . ,
(9.4d)
n--0
(It is often useful to define U - l ( 0 ) -- 0, and M,x,-1 - M,x.) The action of U(0) is compatible with the filtrations in the following sense:
Up(O). MA,q C M)~,q_p.
(9.4e)
In the setting of (9.2), the Taylor series map 7- of Lemma 9.2 cc (cf. (9.3@) and on Ma (cf. (9.db)) intertwines the action of V(g) on C 0,A The subspace Ma,n is precisely the image under ~ of sections vanishing to order at least n at ell.
L e m m a 9.3
Here, we say that a section er vanishes to order zero at a point p if a(p) = 0; and we say that cr vanishes to order n if (Tcr)(p) = 0 for every differential operator T of order at most n. We omit the simple proof. Now that we have a space of Taylor series, Theorem 8.7 suggests how to define differential operators (with formal power series coefficients). The distinguished vector fields used in (8.8) are provided here by the action of g. D e f i n i t i o n 9.4 In the setting of (9.1) and (9.4), an endomorphism T of Ma is said to be weakly of order q if T(Ma,p) C Mx,p_q for all p >_ 0 (cf. (8.6)). It is of order q if for every sequence (X1, . . . . Xr) of elements of g, the iterated c o m m u t a t o r [¢~(X1), [ ' " [¢x(Xr), T]]...] is weakly of order q. We write/-)form L~A,q for the collection of endomorphisms of order q, and D fOrm _ U
r) fOrm L,A, q •
q
It is not difficult to see that D f°rm is a filtered algebra, isomorphic to the algebra of differential operators in dim g/O variables with formal power series coefficients.
(Consequently, D f°rm is completely prime.) As
a consequence of (9.4e), the map of (9.4b') restricts to a filtered algebra homomorphism ¢,x" U(g) ---+ D f°rm.
(9.5a)
D. Vogan, Jr.
320 We make g act o n D f°rm by
a d ( X ) ( T ) - [¢~ (X), T].
(9.5b)
Because ¢~ is a Lie algebra homomorphism, ad is a Lie algebra representation of g (by derivations) o n D f°rm. It therefore extends to an associative algebra homomorphism ad" U(g) --* End(Df°rm). The algebra D f°rm is close to the requirements of Theorem 9.1. W h a t is missing is the action Ad of Gc by algebra automorphisms. This should be an exponentiated form of the action ad of g. We cannot perform this exponentiation on all of D f°rm. Using an idea of Zuckerman, we will essentially define D~ to be the largest subalgebra of D f°rm o n which the exponentiated action makes sense. Here is a simple version of Zuckerman's idea. D e f i n i t i o n 9.5 (see [V1, Definition 6.2.4]) Suppose Gc is a connected complex algebraic group with Lie algebra g, and V is a representation of g. We will define an algebraic representation FGc(V) = FV of Go. (Recall that this means that every element of FV is to lie in a finite dimensional Gc-invariant subspace, on which the action of Gc is algebraic.) As a vector space, FV will be a subspace of V; and the differential of the representation of Gc is the original action of g. To do this, let Gc be the universal covering group of Gc. There is a short exact sequence
1 --+ Z ---~Gc-oGc--+ 1. Here Z is a discrete central subgroup of (~c. Define FV-
{v ~ V l d i m U ( o ) v < oo}.
This is a 0-stable subspace of V, on which the action of g is locally finite. By the dictionary between finite dimensional Lie group and Lie algebra representations, it follows that FV carries a locally finite representation of (~c, with differential given by the action of g. Set
roY-
{v E FV l z . v -
v, all z E Z}.
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Because Z is normal in Gc, the subspace F0V of FV is (~c-invariant; and the representation of (~c on F0V obviously factors to Go. Finally, define FV = {v c F0V I the function g H g. v from Gc to V is algebraic}. This makes sense because the function takes values in a finite dimensional subspace of V. It is more or less obvious that FV is an invariant subspace of FoV, and that the representation of G¢ on FV is algebraic. We now define D~ - FGc (Df°rm),
(9.6a)
and write Ad for the representation of Gc on Dx. Because g acts by derivations, it is straightforward to check that f'Da is a subalgebra of D f°rm on which (~c acts by algebra automorphisms. It follows easily that Da is a subalgebra on which Gc acts algebraically by algebra automorphisms. Because the adjoint action of Gc on U(g) is algebraic, the image of ¢~ (cf. (9.5a)) is contained in Dx: 4)~: U(g) --~ D~.
(9.6b)
The algebra D~ is completely prime because it is a subalgebra of the completely prime algebra D f°rm.
-
This completes the verification of the statements in the first paragraph of Theorem 9.1. That D~ is a Dixmier algebra under the hypothesis (9.1d) that is, that the finiteness requirements of Definition 3.5 are satisfied is well known, going back at least to [CBD]. (There is an explicit verification in [V7, Corollary 4.17]. One can also find there a discussion of the symbol calculus for D~.) The assertions about the embedding D~ in an algebra of analytic differential operators we leave to the reader; the discussion in Section 8 should help to make them plausible. VI -
E x a m p l e 9.6
Suppose Gc = C n and Hc is trivial, and A = 0. Then
C[[Xl,...,Xn]] (see (8.1g) and (9.4a)). According to Theorem 8.7 and Definition 9.4, Dform ~ Dform(R n),
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the algebra of differential operators on IRn with formal power series coefficients. We also have
u(o) ~ c[o/o~,,..., O/Oxn], and the map Ca from U(0) to D f°rm is the natural embedding. It is therefore easy to compute the adjoint action defined in (9.5)" If T C U(g) is a constant coefficient differential operator, then
ad(r)
0~ ) 0~ f ~~ - E (T f ~ ) Ox ~
E OL
OL
In the setting of Definition 9.5, it follows t h a t f'(D f°rm) consists of those differential operators the coefficients f~ of which have the following property: The space of all derivatives of f~ is finite dimensional. (Here, "derivatives" refers to applying constant coefficient differential operators.) Using linear algebra and calculus, it is not hard to show that these functions are all finite linear combinations of polynomials times exponentials:
fa(x) -- E pa,((x) exp( E ~jxj), ~ECn
with each p~,~ a polynomial in x. The group Gc is simply connected, and so f'(D [°rm) - F0(Df°rm). The adjoint action of Gc is by (complex) translation of the coefficients:
Ad(z) ~ f~~x~ Oz
~(p(z)f~)Ox~, OL
with
(p(z)f,)(x) - f , ( z + z) - ~
p,,~(x + z ) e x p ( ~ ~j(xj + z~)).
EC~ The algebraic functions o n C n are the polynomials. From this, it follows that I ' D f°rm consists of differential operators with polynomial coefficients"
D~, _~ C [ x , , . . . , Xn,
a/Ox,,..., O/OXn].
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To complete this section, we describe the coadjoint orbits to which the Dixmier algebras D ; should correspond in Conjecture 3.6. Assume, then, t h a t we are in the setting (9.1a-d). We follow Section 3 of [V7], which the reader may consult for more details and generalizations. Define E,~,ac/Hc = {~ C g* I ~]~ = A}.
(9.7a)
This is an He-stable affine subspace of it*, of dimension equal to the codimension of O in g. We may therefore use it to construct the fiber product (9.7b)
E A :- G C x Hc EA,Gc/Hc,
an affine bundle over the projective variety G e / H c .
There is a natural
"moment map" #~: Ea + it*,
#a (equivalence class of (g, {)) = Ad*(g)({).
(9.7c)
Obviously, the image of #~ is a union of coadjoint orbits. 9.7 In the setting of (9.1a-d) and (9.7), the moment map #~ is proper and generically finite; its image is the closure of a single coadjoint orbit Oh. More precisely, the parabolic subgroup He has an open orbit O,k,ac/Hc on E)~,Gc/Hc. This defines an open subvariety Proposition
(~A -- GC X Hc OA,Gc/Hc
of E~. The restriction of the moment map It), to ( ~ is a finite covering map onto (9~. The most difficult case of the proposition is ,~ = 0. In t h a t case it is essentially due to Richardson: E~ is the cotangent bundle of the partial flag variety G e / H e , and O is the Richardson nilpotent orbit attached to He. For the general case, we simply refer to [V7]. It is now more or less clear that our constructions for hyperbolic orbits in Section 7 fit into the framework of Problem 3.10. statement.
Here is a specific
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Corollary 9.8
Suppose G is a real reductive group of inner type Gc (Definition 3.8), and that ~ E g~ is a hyperbolic element (Definition 5.8). Define a parabolic subgroup P = L N as in Proposition 7.1, and let Pc C Gc be its complexification. i) The linear functional ~ restricts to a character A(~) of the Lie algebra p. Write D~ = D~(~) for the twisted differential operator algebra on G c / Q c attached to A(~) in (9.6). ii) Suppose 7- is an integral orbit datum at ~ (Definition ~.6 and Proposition 7.2), and (Tr(~), 7-l) is the corresponding unitary representation of G (cf. (~.9)). Then the action of U(g) on the smooth vectors 7-I~ extends naturally to an action of D~. iii) In the notation of (9.7) and Proposition 9.7, E~(~) - (3~(~) ~ (.0h Gc .~. The assertion in (ii) follows from the last part of Theorem 9.1. That in (iii) can be deduced from Proposition 7.1(iv) applied to Gc. We leave the details to the reader, along with such tasks as the construction of a Hermitian transpose on D~. In the setting of (9.1), the correspondence of Conjecture 3.6 should carry the orbit cover (9~ (Proposition 9.7) to the Dixmier algebra D~ (cf. (9.6)). For Gc - GL(n, C), every equivariant orbit cover is of the form ( ~ for some parabolic Hc and character A. (Actually, the covers are necessarily trivial in this case.) The same orbit may arise in several different ways, but Borho has shown in [Bo] that then the various D~ are all isomorphic. (To be precise, one needs in addition to [Bo] the result from [BOB1] that the maps ¢~ are always surjective for GL(n, C).) So there is a well defined Dixmier correspondence for GL(n,C). The injectivity of the correspondeuce is established in [BoJ], proving Conjecture 3.6 for GL(n, C). For most other cases, there are equivariant orbit covers not of the form (9~. (Such covers exist exactly when Gc has a simple factor not of type A, or Gc has disconnected center.) In those cases, the twisted differential operator algebras D~ do not suffice to prove Conjecture 3.6. A discussion of what else is needed may be found in IV7].
10 Elliptic orbits, complex polarizations, and admissibility In Section 7 we gave a construction of unitary representations of a reductive group G attached to hyperbolic coadjoint orbits. This construction
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is very nice as far as it goes, but it does not apply to other coadjoint orbits. In this section we will introduce an analogous construction for elliptic orbits. We begin with some structure theory along the lines of Proposition 7.1.
In the setting of (7.1), suppose that ad(X) is diagonalizable on the complexified Lie algebra g, with purely imaginary eigenvalues. (This happens in particular if X is elliptic (Lemma 5.10).) Write gt for the t-eigenspace of ad(iX), so that
P r o p o s i t i o n 10.1
g - E
fit'
fix - go.
(10.1a)
tCR
Define Px - E
gt,
nx - E
t>O
gt.
(10.1b)
t>O
i) The decomposition (10.1a) makes g an N-graded Lie algebra: [0~, 0 t] C 0 ~+t.
ii) The subspace gt is orthogonal to g8 with respect to cox unless s - - t . iii) Complex conjugation on g with respect to the real form go carries gt onto g-t. In particular,
- P x - E g t,
nx
-Eg
t<0
so that Px + Px = g,
t,
t<0
-
Px N Px
_go
iv) The adjoint action of G x preserves each subspace gt, and so preserves the subalgebra Px. v) Suppose (~,F) is a finite dimensional representation of G x , and that d~(Z) (the differentiated representation applied to the Lie algebra element X ) is a scalar operator. Then d'~ extends uniquely to a representation ¢ of Px. This extension satisfies ¢lnx = O, and ¢(Ad(g)Z) - ~/(g)¢( Z)~y(g -1)
(g E G x , Z C Px ).
( lO.lc)
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D. Vogan, Jr.
Proof: Parts (i) and (ii) are proved just as in Proposition 7.1. (In fact, i X is a hyperbolic element of 9, so (i) and (ii) may be regarded as special cases of Proposition 7.1.) For (iii), use the fact that i X = - i X and the fact that complex conjugation is a Lie algebra automorphism. Part (iv) is obvious. For (v), the property of vanishing on nx is easily seen to define an extension ¢ satisfying (10.1c). For the uniqueness, suppose that ¢ is any representation of Px. Then, if Z C gt, ¢(Z) must carry the ~ eigenspace of ¢ ( i X ) into the ~ + t eigenspace. In our case ¢(iX) is assumed to act by scalars, and so nx must act by zero. [-7 The idea is that the complex Lie algebra P x is something like a polarization at ~x (Definition 4.7). In fact, P x / g x is a complex Lagrangian subspace of the complexified tangent space T~× (G. ~x)¢. There is no subgroup of G with Lie algebra Px; but we observed in (4.7c) that it was possible to write down an interesting representation space for G using only G x and the Lie algebra of the polarization. In the setting of Proposition 10.1(v), the space in question is
= { f e C ~ ( G , F ) lf(gh) - 7 ( h ) - l f ( g ) , p ( Z ) f - - ¢ ( Z ) f
(g~G, h~Gx, Z~px)}.
(lo.2)
That is, we are beginning with sections of the smooth vector bundle on G / G x defined by (~/, F) and imposing a family of first order differential equations corresponding to elements of nx. Proposition 10.1(iii) suggests that these differential equations resemble the Cauchy-Riemann equations. Here is a precise statement. P r o p o s i t i o n 10.2 In the setting of (7.1), suppose that ad(X) is diagonalizable with purely imaginary eigenvalues. Then there is a distinguished
G-invariant complex structure on the coadjoint orbit G . ~x ~- G / G x . It may be characterized by the requirement that P x / g x (cf. (10.1b)) is the antiholomorphic tangent space at the identity coset (corresponding to the point ~x ). In the setting of Proposition 10.1(v), the Lie algebra representation ¢ defines a G-invariant complex structure on the vector bundle jz = G × c x F
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over G. ~x. The space (10.2) may be identified with the space of holomorphic sections of ~. A somewhat more detailed discussion of this result may be found in IV5, Propositions 1.19 and 1.21]. Proposition 10.2 suggests that any generalization of Corollary 9.8 to elliptic coadjoint orbits will need to use complex analysis. To begin, we first consider exactly which (holomorphic) vector bundles on G / G x are relevant. The most obvious possibility is to use the bundle defined by an integral orbit d a t u m (~,7-/~) at ~x (Definition 4.6).
T h a t is, we might
require
d~(Y) = i~x (Y) = i(X, YI
(Y e gx).
But this is not precisely analogous to what we did in the hyperbolic case (cf. (4.9)). There the integral orbit d a t u m was first twisted by the character of the isotropy group on half-densities. Explicitly (for X hyperbolic and g C
Gx), A x ( g ) is the square root of the absolute value of the determinant of the adjoint action of g on (9o/Px,o)*. The differential of A x is a character 5x of 9x,0: 5x(Y) is half the trace of the adjoint action of Y on (9o/Px,o)*. For X elliptic, the quotient 9/Px is only a complex vector space, lacking a G x-invariant real form. There is accordingly no natural analog of half-densities, or of the character A x . But 5x still makes sense: We can compute traces in the complexification of a vector space, and so it is consistent with the hyperbolic case to define a character of 9x by l t r ( a d ( Y ) on ( g / P x ) * ) -
1
~tr(ad(Y)on nx)
This need not be the differential of a character of Gx.
(10.3a)
Nevertheless, a
better analog of (4.9) is to consider bundles on G / G x corresponding to irreducible unitary representations (~/, F~) of Gx with the property that
d (Y) : i¢x (Y) + 6x (Y).
( o.3b)
Such a representation 7 is called an admissible orbit datum at ~. The orbit G.~ is called admissible if there is an admissible orbit d a t u m at ~. (Compare the definition of integral orbit d a t u m in Definition 4.6. If there is a onedimensional character A x of Gx with differential 5x, then tensoring with A x defines a bijection between integral orbit data and admissible orbit data. But, in general, the two notions are simply different.)
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The definition of admissible given here makes sense only for elliptic orbits of reductive groups. There is a more sophisticated notion, due to Duflo, that makes sense for arbitrary coadjoint orbits of arbitrary Lie groups. We refer to [Vb, Definition 10.16] for Duflo's definition; one can also find there a discussion of how it reduces to (10.3) in the elliptic case. According to Proposition 10.2, an admissible orbit d a t u m (~, F~) defines a holomorphic vector bundle ~
over the coadjoint orbit G . ~x. The
unitary structure on the representation V provides a G-invariant Hermitian structure on the vector bundle, and the symplectic structure on the orbit provides a canonical G-invariant measure. Accordingly, there is a natural analog of the unitary representation defined in (4.9) for a real polarization: The Hilbert space is the space L 2 ( G . ~ x , P x ; ~ ) of square-integrable holomorphic sections of the vector bundle (compare (10.2) and (4.7c)). Unfortunately, this space is almost always zero: In fact, 9~ usually has no nontrivial holomorphic sections at all. This is most easily seen when G is compact. Taking, for example, G = SU(2) and ~x any nonzero admissible element of ~ , we find that G . ~x is isomorphic to CP 1, and that the line bundle 9~ is one usually denoted O ( - n -
1), with n a positive integer.
(Essentially, - n arises from the term i~z in (10.35), and - 1 from 5x.) The line bundle O ( - n - 1) has only the zero section. W h a t is interesting is its first cohomology:
HI(G • I x , 9~)
_~ C n ,
the unique irreducible representation of G of dimension n. At first glance, this suggests that we have chosen the wrong complex structure on G . ~x. Indeed, everything in this section works with only trivial changes if we reverse the rbles of Px and Px. With the new complex structure the line bundle ~ becomes O ( n - 1), still with n a positive integer. This bundle
does have holomorphic sections: H°(G • ~x,.T'.r)
,--, C n '
the n-dimensional irreducible representation. There is a price to be paid for such a change, however. If instead we consider the group G = SL(2, R) and take ~x to be a nonzero admissible elliptic element, then G. ~x is the upper half plane with the usual action of
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G by linear fractional transformations. The line bundle ~ is holomorphically trivial, and therefore it has lots of holomorphic sections (and vanishing higher cohomology). When the complex structure is chosen as in Proposition 10.2, there are even square-integrable holomorphic sections; so we get an interesting unitary representation of G (a holomorphic discrete series representation). But if the complex structure is defined instead using the opposite parabolic subalgebra, the only square-integrable section is zero. The conclusion that we draw from these two examples is that one should use the complex structure specified in Proposition 10.2, and look for a unitary representation in some higher cohomology of G - ~ x with coefficients in 9~. The question of exactly which cohomology to consider is illuminated by the following vanishing theorem of Schmid and Wolf. Recall from Theorem 6.5 and (7.1) that any elliptic orbit in ~ has a representative ~x with X C ~0. Theorem
10.3 ([SW])
Suppose G is a real reductive group with maxi-
mal compact subgroup K , and X E ~o. Put on D = G . ~x the complex structure defined by Proposition 10.2. Then Z = K . ~x is a compact complex subvariety of D; write s for its complex dimension. The variety D is (s + 1)-complete in the sense of Andreotti and Grauert. In particular, this means that Z is a compact complex subvariety of maximal dimension in D; and that if ~ is any coherent analytic sheaf on D, then H q ( D , ~ ) = 0 f o r q > s.
In the first example just discussed (with G = SU(2)), we have Z = C P 1 and s = 1. In the second example (with G = SL(2, R)), Z is a point and s---O.
We can now say what representation ought to be attached to an admissible elliptic coadjoint orbit G . ~. By Theorem 6.5, there is no loss of generality in assuming that 0~ = ~. We begin with an admissible orbit datum ? as in (10.3), and form the corresponding holomorphic vector bundle 9~ over the complex manifold G. ~ (Proposition 10.2). Theorem 10.3 says that K . ~ is a compact complex submanifold of dimension s. Form the cohomology group g s(G • ~, . ~ ) . There are at least two important ways to think of this space. (The isomorphism between them is Dolbeault's theorem.) One is as a Dolbeault cohomology group, the quotient of closed (0, s) forms on G . ~ with values in $'~ by exact forms. This shows first of all that the cohomology group carries a natural representation of G (by
330
D. Vogan, Jr.
translation of (0, s) forms). At the same time, it emphasizes the central difficulty in putting a nice topology on the space: It is not obvious that the space of exact forms is closed. (This fact was proved in general by H. Wong in his 1992 Harvard thesis [Wo].) It follows that HS(G.~,Jz~) has a natural complete locally convex Hausdorff topology. A second way to think of H ~(G-~, S'v) is as a (~ech cohomology group with coefficients in the sheaf of germs of holomorphic sections of 9~. From this point of view an element is represented by a family of holomorphic sections of 9c~, each defined over some small open set (say an intersection of s + 1 elements from a covering of G. ~ by Stein open sets). The advantage of this point of view is that every holomorphic differential operator on sections of 9c~ clearly acts on the cohomology. Suppose, for example, that G is of inner type Gc (Definition 3.8). Then, p~ (Proposition 10.1) is the Lie algebra of a parabolic subgroup P~,¢ of Go, and it is not difficult to show that G~ - {g e G I Ad(g ) e P~,c}-
(10.4a)
Consequently, G . ~ ~_ G/G~ is an open submanifold of the flag variety Gc/P~,c. Comparing (10.3) with the definitions of Section 9 (particularly (9.4a)), we find that the twisted differential operator algebra D~ (attached to the character ~ of p~ by (9.6)) may be regarded as an algebra of holomorphic differential operators on 9~. Therefore, D~ acts naturally on H s(G • ~, :F,y).
(10.4b)
The Hermitian structure on F~ gives rise to a natural Hermitian transpose on differential operators on ~'~. This transpose preserves D~, defining a Hermitian transpose of Dixmier algebras in the sense of Definition 3.8. We are now getting close to having the structure required by Problem 3.10. The Dixmier algebra D~ and the group K (or even the larger group G) both act on H ~(G. ~, 9r~). This space fails to be a (04, K)-module only because the action of K is not locally finite. We have already met such a problem in connection with the construction of D~, and we adopt the same solution here: We pass to the subspace of K-finite vectors. Define
V~,~ - HS(G" ~ , ~ ) g = {v ¢ HS(G • ~,Jz~) I d i m ( s p a n ( K , v)) < oc}.
(I0.4c)
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It is not difficult to show that V~,~ is preserved by D~ and K, and that it is a (D~, K)-module in the sense of Definition 3.8. To complete the requirements of Problem 3.10, we need a unitary structure on V~,~. At this point, the analytic ideas that have brought us so far seem to fail: No general construction of such a unitary structure is known. A great deal is known about special cases; two entry points to the literature are [Z] and [RSW]. Briefly, one seeks a unitary inner product defined by integrating certain distinguished Dolbeault cohomology classes. In its simplest form, this program cannot succeed for arbitrary admissible elliptic orbits. To see why, consider elliptic orbits for the group Sp(2n, R) with G~ ~ U(n). Such orbits are parametrized by the nonzero real numbers; the orbit G-~t with parameter t is admissible if and only if 2t + n + 1 is an even integer. (The corresponding character ~ of U(n) is det(2t+n+l)/2.) The number s is equal to zero, and so H 8(G. ~t, ~v) is the space of holomorphic sections of the line bundle on Sp(2n, ]~)//U(n) corresponding to the character ~/. The unitary structure proposed in [RSW] is just integration of holomorphic sections over the orbit. The convergence of these integrals was studied in [HC2]. For t > ( n - 1)//2, all the K-finite sections are squareintegrable; but for 0 < t < ( n - 1 ) / 2 , there are no nonzero square-integrable holomorphic sections. As soon as n is at least two, therefore, we find cases where the inner product does not arise by integration. (It seems likely that the integrals will converge in general for "most" admissible elliptic orbits, as they do in this example. No such convergence has yet been proved, however.) Despite these difficulties, there is a strong positive result.
Suppose G is a real reductive group of inner type Gc, and that ~ E g~ satisfies 0~ - ~ (so that G . ~ is a typical elliptic coadjoint orbit). Assume that G. ~ is admissible, with (~, F~) an admissible orbit datum (cf. (10.3)). Define a (D~,K)-module V~,~ as in (10.~). Then V~,~ carries a natural unitary structure (Definition 3.8). In particular, it is the Harish-Chandra module of a unitary representation ~(~, 7) of G. T h e o r e m 10.4 ([V2])
This result provides unitary representations attached to elliptic coadjoint orbits.
The representation V~,~ turns out to be irreducible as a
(D~,K)-module but not necessarily as a (g,K)-module; so the unitary representation ~(~, ~/) of G may be reducible.
332
D. Vogan, Jr.
We will now outline part of the proof of this theorem. The main point is to give (following Zuckerman) an algebraic construction analogous to the geometric one in (10.4). Roughly speaking, we replace holomorphic functions by formal power series. In the algebraic setting, an invariant Hermitian form can be constructed without much difficulty, and (with a little more difficulty) the positivity of the form can be proved. This is all that is required to attach a unitary representation to the orbit G . ~ (by Theorem 3.9). To complete the proof of Theorem 10.4 as stated, one must also identify the algebraic construction with the geometric one. Roughly speaking, this amounts to proving the convergence of some formal power series solutions of differential equations. The necessary ideas go back to Schmid's 1967 thesis [$1]; the result was proved completely in [Wo]. We begin with the finite dimensional module F~ for G~. By Corollary 5.13, G~ is a real reductive group, with Cartan involution the restriction of 0 and maximal compact subgroup K~ = G~ A K. In particular, K~ acts on F~. At the same time, Proposition 10.1(v) provides a representation of the complex Lie algebra p~ on F.y. The same result ensures that the representations of p~ and K~ enjoy a compatiblity analogous to that required for (g, K)-modules in Definition 2.16; we call F~ a (p~, K~)-module accordingly. Recall now that the differential of 3' is i~ + 5 (cf. (10.3)). In analogy with the definition of M~ in (9.4), we therefore set W~ - Homv(p~)(U(g), F~).
(10.5a)
The Lie algebra g acts on W~ by right multiplication on U(g). The group K~ acts by combining the adjoint action on U(g) with the action on F.~:
(h. w)(u) = h. (w(Ad(h-1)u))
(h E K~, w e W~, u e U(g)). (10.5b)
This action need not be locally finite, and so we define W K~ - {w E W.y I dim(span(K ~ • w)) < oo}.
(10.5c)
Up to this point, there is no need to restrict to the compact subgroup K~; we could just as well have kept track of an action of G~. (The K~-finite vectors turn out automatically to be G~ finite.) But it is the K~ action we will soon need.
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L e m m a 10.5 The space W~ may be identified with the space of formal power series for sections of the vector bundle ?.y at the base point ~ E G. ~. The actions of g and K~ satisfy compatibility conditions analogous to those in Definition 2.16, making W K~ a (g, K~)-module. If G is of inner type Gc, then the Dixmier algebra D~ acts naturally on W~, extending the action of U(g) and preserving the subspace wK~; in this way, W K~ becomes a (D~, K~)-module (Definition 3.8). This elementary result is a holomorphic version of Lemma 9.2. In the setting of Problem 3.10, one can take W K~ as one of the D~-modules
w (c. The next step is to construct from W K~ a (D~, K)-module. We will use a functor F - F ~,K~ e'/< introduced by Zuckerman for passing to the subspace F W K~ of K-finite vectors in W K~. One way to understand this approach is in terms of Theorem 10.3. The "(s+ 1)-completeness" property says roughly that G-( looks like a Stein manifold away from the compact subvariety K . ( . Holomorphic bundles on Stein manifolds have many global sections. This means that (morally) the obstruction to globalizing a formal power series section of 9~ is mostly in the direction of K . ~. Now, it is not difficult to see that a K-finite formal power series section of 9r~ must represent a holomorphic section over K . (. In light of the (s + 1)-completeness, this suggests that elements of F W K~ should represent global holomorphic sections of 9r~. That is, we might expect
FwK~ ~ HO(G . ~,j:.~)g. This turns out to be true, and not too difficult to prove. The problem, as we already observed after (10.3), is that both sides are usually zero; we need analogs not of holomorphic sections but of higher cohomology. Zuckerman's great observation was that the functor F is only left exact, and that it has right derived functors F/. Although it is harder to justify precisely, one might still hope formally that
FiwK~ ~ Hi(G . ~, 9~) g. This statement was proved in [Wo]. Here is the definition of F. (Compare Definition 9.5, which is similar but simpler.)
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D e f i n i t i o n 10.6 ([V1, Definition 6.2.4]) Suppose K is a compact Lie group with complexified Lie algebra t~, and H is a closed subgroup of K. Suppose W is a (t~,g ) - m o d u l e (defined as in Definition 2.16). We want to define a (t~, K)-module p(e,K - ( e , H I W - FW. To begin, let K0 be the universal cover of the identity component of K, so that we have 1 --+ Z ~/2/0 ~ K0 ~ 1. Here Z is a discrete central subgroup of/~0. Define FW-
{w e W ldimU(~)w < ec}.
As in Definition 9.5, F W carries a locally finite representation of/~0; and we can define F0W = {w C F W I z . w = w, all z E Z}. This is a subspace of W carrying a locally finite representation 7r0 of Ko; it is also preserved by the representation ~- of H on W. We may therefore define
r ~ w = {w e r o W l~o(h)w = T(h)w, all h e H N K0}. This subspace is invariant under the representations 7r0 and ~-. Now, define K1 to be the subgroup of K generated by Ko and H. There is a unique representation 7rl of K1 on F~W that extends both 7r0 and 7. Finally, set rw
- Ind/~ r~W,
a locally finite representation of K. The reader may try to understand geometrically each of the steps in the construction of F W when (for example) W is the space of H-finite formal power series sections of a bundle $ on K / H . In this case F W may be identified with the space of K-finite global sections of 8. One interesting step is the last one, of induction from K1 to K. The point there is that the index m of K1 in K is just the number of connected components of the homogeneous space K / H .
Ordinarily, one could not hope to understand
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335
sections on different connected components using Taylor series; but the group action allows us to do just that. The space F1W may be identified with Kl-finite sections of $ over the identity component K 1 / H ~_ K o / ( K o N H). Induction more or less replaces this space by a sum of m copies of it; each copy corresponds to sections supported on one of the components of
K/H. P r o p o s i t i o n 10.7 (Zuckerman; see [V1, Chapter 6])
. The functor F of
Definition 10.6 is a left exact functor from the category of (~,H)-modules to the category of ({~,K)-modules. It has right derived functors F i, which are nonzero exactly for 0 <_ i <_ d i m ~ ( K / H ) . Suppose now that K is the maximal compact subgroup of a reductive group G. If W is a (9, H)-module (defined in analogy with Definition 2.16), then F i W becomes naturally a (9, K)-module. Similarly, if A is a Dixmier algebra and W is an (A,H)-module, then F i W is naturally an ( A , K ) module. The last assertion (about Dixmier algebras) is not part of Zuckerman's original ideas; it is more or less a folk theorem from the early 1980s. It may be proved using the method of [W, Section 6.3]. ~,K f o r T h e o r e m 10.8 (see [V2]) In the setting of (10.5), write F - Fe.K~ the functor of Definition 10.6. Furthermore, write s - d i m c ( K . ~) 1 dim~ ( K / K ~ )
i) r
-o
i¢
ii) F s W ff~ carries a natural nondegenerate (g,K)-invariant Herrnitian f o r?Tt.
iii)
fo m in (ii)
po itiw d¢ nit .
iv) Suppose G is of inner type Gc. Then there is a natural Hermitian transpose on D~ making F e W K~ into a unitary (D~,K)-module. v) F s W ~ ~ is irreducible as a (D~,K)-module. In lieu of a proof, here are some historical remarks.
The vanishing
theorem in (i) is due to Zuckerman for most 7; a proof may be found in IV1]. The proof for all 7 as in (10.3) appears first in IV2]. The form in (ii) was constructed by Zuckerman, but his proof of its invariance under g was incomplete. Repairs were provided in [EW].
D. Vogan, Jr.
336
Part (iii) is (a special case of) the main result of IV2]. The proof there is a reduction to the special case when G~ is a compact Cartan subgroup of G. In that case, Schmid in [$2] had essentially identified F s W K~ with one of the discrete series representations constructed by Harish-Chandra (using deep analytic techniques). Harish-Chandra's discrete series representations are irreducible and unitary; so any nonzero invariant Hermitian form on them must be definite. A purely algebraic proof of (iii) was later found by Wallach (see [W]). Part (iv) is more or less a folk theorem; it can be proved in the same way as (ii). The irreducibility of F s W K~ even as a (g,K)-module was proved by Zuckerman for most -y (see [V1]). The general result in (v) is perhaps an unpublished result of Bernstein; it follows from the generic case using the translation technique in IV7, Corollary 7.14]. Theorem 10.4 follows from Theorem 10.8 and the result of Wong stated before Definition 10.6.
Acknowledgements During the summer of 1994, I lectured on this material at the Nankai Institute of Mathematics; at the Instituto de Matem£ticas at UNAM; at CIMAT in Guanajuato; and finally at the European School of Group Theory. The comments of all of those audiences have been a great help to me in preparing these notes. Henrik Schlichtkrull offered many improvements and corrections to the manuscript. For the flaws that remain, I must of course assume the responsibility.
References
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R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin/Cummings, Reading, Massachusetts, 1978.
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V. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics 60, Springer-Verlag, New York Heidelberg Berlin, 1978.
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A. Beilinson and J. Bernstein, Localisation de g-modules, C. R. Acad. Sci. Paris 292 (1981), 15-18.
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A. Borel and N. Wallach, Continuous Cohomology, Discrete Sub-
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W. Borho, Definition einer Dixmier-Abbildung fiir ~l(n, C), Invent. Math. 40 (1977), 143-169.
[BOB1] W. Borho and J.-L. Brylinski, Differential operators on homogeneous spaces L Invent. Math. 69 (1982), 437-476. [BOB2] ~ , Differential operators on homogeneous spaces III, Invent. Math. 80 (1985), 1-68. [BoJ] W. Borho and J. C. J antzen, Uber primitive Ideale in der EinhiiUenden einer halbeinfachen Lie-Algebra, Invent. Math. 39 (1977), 1-53. [CM]
D. Collingwood and W. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold, New York, 1993.
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T. Enright and N. Wallach, Notes on homological algebra and representations of Lie algebras, Duke Math. J. 47 (1980), 1-15.
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A. Guichardet, Thdorie de Mackey et mdthode des orbites selon M. Duflo, Expo. Math. 3 (1985), 303-346.
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J . E . Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, Berlin Heidelberg New York, 1972.
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A. Joseph, Kostant's problem and Goldie rank, pp. 249-266 in Noncommutative Harmonic Analysis and Lie Groups, (J. Carmona and M. Vergne, eds.), Lecture Notes in Mathematics 880, SpringerVerlag, Berlin Heidelberg New York, 1981.
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A. Knapp and D. Vogan, Cohomological Induction and Unitary Representations, Princeton University Press, Princeton, New Jersey, 1995.
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J. Lepowsky, Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc. 176 (1973), 1-44.
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W. Schmid, Homogeneous complex manifolds and representations of semisimple Lie groups, pp. 223-286 in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, (P. Sally and D. Vogan, eds.), Mathematical Surveys and Monographs 31, Amer. Math. Soc., Providence, Rhode Island, 1989.
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D. Vogan, Representations of Real Reductive Lie Groups, Birkh£user, Boston Basel Stuttgart, 1981.
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R. Zierau, Unitarity of certain Dolbeault cohomology representations, pp. 239-259 in The Penrose Transform and Analytic Cohomology in Representation Theory (M. Eastwood, J. Wolf, and R. Zierau, eds.), Contemporary Mathematics 154, Amer. Math. Soc., Providence, Rhode Island, 1993.
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Index accumulating point, 110 adjoint representation, 10 admissible orbit datum, 327 admissible representation, 263 Airy function, 176 (A, K)-module, 271 algebraic group, 3, 29 amplitude, 168 annihilator, 260 Ann(M), 261 associated variety, 54, 58 asymptotic expansion, 193 Auslander conjecture, 105
Bernstein's theorem, 211 Borel-Weil-Bott theorem, 21 Brauer-Humphreys reciprocity law, 36, 44
c*(a), 258 Calabi-Markus phenomenon, 118 Cartan decomposition, 247, 249 Cartan involution, 246, 248 Cartan subspace, 296 caustic, 169 caustic variety, 192 character, 5, 16 characteristic polynomial, 66 clean intersection, 228
Clifford-Klein form, 100 cohomology group, 20 complete affine manifold, 105 completely prime, 266 coordinate ring, 4 critical point, 169 critical value, 184
D~, 321 derived functor, 17 discontinuous group, 111 Dixmier algebra, 268 dominant weight, 13 dot action, 22 Drinfeld-Jimbo quantization, 83
e-family of symmetric pairs, 146 elliptic, 290 ergodic, 140 exceptional spectrum, 239 Ext, 18
Fa,,x(k),
221 flag manifold, 213 focal points, 191 free action, 109 Fresnel integral, 174 Frobenius kernel, 31 Frobenius reciprocity, 7 341
342
Index
(g, K)-module, 264 Goldie rank, 54 good filtration, 45
linear real reductive group, 248 linkage principle, 45 locally Hausdorff, 113
H° (E), 13 Hamiltonian G-space, 277 Hamiltonian vector field, 276 Harish-Chandra module, 265 Hessian form, 184 Hessian matrix, 169 Hirzebruch's proportionality principle, 128 homogeneous space of reductive type, 123 Hopf algebra, 4 hyperbolic, 290
Mather's theorem, 211 minimal ~-type, 76 moment map, 277 Moore's ergodicity theorem, 140 Morse function, 172, 185 Morse number, 207 Morse polynomial, 208 Morse's lemma, 186, 203
induced representation, 6 induction functor, 41 integral orbit datum, 278 intertwining operator, 256 invariant real polarization, 281 Iwasawa projection, 181 Jordan decomposition, 289 Kempf's vanishing theorem, 37 Kostant's convexity theorem, 219, 227 lacunary principle, 210 Lagrangian subspace, 58, 281 left cell, 63 Leibniz action, 88 light zone, 169, 174
nilpotent, 288
o(v, q), 253 orbital variety, 56 orbit method, 53 oscillatory integral, 168
64 P(R), 63 perfect Morse function, 208 phase, 168 Poincar~ plane, 104 Poisson bracket, 57, 276 Poisson ideal, 57 Prim U(g), 53 prime ideal, 262 primitive ideal, 261 principal spectrum, 239 principle of stationary phase, 171 proper action, 115 properly discontinuous action, 109 property CI, 118 PRV component, 74
Index
quantum group, 39
real reductive group, 249 reductive group, 8 reductive symmetric space, 123 regular representation, 5 relativistic spherical space form, 106 representation, 4 Riemann mapping theorem, 106
SaM's theorem, 184, 202 semisimple, 287 semisimple group, 8 semisimple symmetric space, 123, 145 Serre duality, 25 sheaf cohomology, 18 smooth vector, 259 spectrum, 239 spherical oscillatory integral, 231 Steinberg module, 37 Steinberg variety, 58 submersion, 183 symmetric pair, 145
343
t-density, 283 Taylor series, 306 tilting module, 45 Tougeron's theorem, 211 trigonalizability, 235 twilight zone, 170, 175 twisted differential operator, 316
unfolding, 176 uniform lattice, 139 unitarizable module, 72 unitary representation, 255
v(p, q), 255
V(w), 61
W.~, 332 Wallach set, 79 weight, 10 Weyl's character formula, 39
Zuckerman's functor, 333
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