Real Reductive Groups I
This is Volume 132 in PURE AND APPLIED MATHEMATICS
H. Bass, A. Borel, J. Moser, S.T. Yau, editors Paul A. Smith and Samuel Eilenberg, founding editors A complete list of titles in this series appears at the end of this volume.
Real Reductive Groups I Nolan R. Wallach
Department of Muthemutics Rutgers University New Brunswick, New Jersey
ACADEMIC PRESS, INC.
Harcourt Brace Jovanovich, Publishers Boston San Diego New York Berkeley London Sydney Tokyo Toronto
Copyright 0 1988 by Academic Press, Inc. All rights reserved. N o part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.
ACADEMIC PRESS, INC. 1250 Sixth Avenue, San Diego, CA92101
United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. 2428 Oval Road, London NWI 7DX
Library of Congress CataloginginPublication Data Wallach, Nolan R. Real reductive groups. (Pure and applied mathematics; v. 132 ) Includes index. 1. Lie groups. 2. Representations of groups. I. Title. 11. Title: Reductive groups. 111. Series: Pure and applied mathematics (Academic Press); 132, etc. QA3.P8 vol. 132,etc. 510 s [512’.55] 8632199 IQA3871 ISBN 0127329609 (v. 1: alk. paper) 88899091 9 8 7 6 5 4 3 2 1 Printed in the United States of America
To my mother Puuline Wulluch “For as the sun is daily new and old, So is my love still telling what is told.”
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Contents
xi
Preface
...
Xlll
Introduction Chapter 0. Background Material
1
Introduction Invariant measures on homogeneous spaces 0.1. 0.2. The structure of reductive Lie algebras 0.3. The structure of compact Lie groups The universal enveloping algebra of a Lie algebra 0.4. 0.5. Some basic representation theory 0.6. Modules over the universal enveloping algebra
1 1 4 6 8 10 13
Chapter 1.
17
Elementary Representation Theory
Introduction 1.1. General properties of representations 1.2. Schur's lemma 1.3. Square integrable representations 1.4. Basic representation theory of compact 9 groups 1.5. A class of induced representations 1.6. C" vectors and analytic vectors 1.7. Representations of compact Lie groups 1.8. Further results and comments
17 18 20 22 24 29 31 35 39
vii
viii
Contents
Chapter 2. Real Reductive Groups
41
Introduction The definition of a real reductive group 2.1. 2.2. Parabolic pairs 2.3. Cartan subgroups 2.4. Integration formulas 2.5. The Weyl character formula 2.A. Appendices to Chapter 2 2.A.1. Some linear algebra 2.A.2. Norms on real reductive groups
41 42 48 56 60 65 68 68 70
Chapter 3. The Basic Theory of (9, K)Modules
73
Introduction 3.1. The Chevalley restriction theorem The HarishChandra isomorphism of the center of the 3.2. universal enveloping algebra 3.3. (g,K)modules A basic theorem of HarishChandra 3.4. 3.5. The subquotient theorem 3.6. The spherical principal series 3.7. A Lemma of Osborne 3.8. The subrepresentation theorem 3.9. Notes and further results 3.A. Appendices to Chapter 3 3.A.1, Some associative algebra 3.A.2. A Lemma of HarishChandra
73 74 77 80 82 86 92 95 97 100 103 103 104
Chapter 4. The Asymptotic Behavior of Matrix Coefficients
107
Introduction The Jacquet module of an admissible (9,K)module 4.1. Three applications of the Jacquet module 4.2. Asymptotic behavior of matrix coefficients 4.3. Asymptotic expansions of matrix coefficients 4.4. 4.5. HarishChandra’s Efunction 4.6. Notes and further results 4.A. Appendices to Chapter 4 4.A. 1. Asymptotic expansions 4.A.2. Some inequalities
107 108 112 114 118 125 130 131 131 133
Contents
ix
Chapter 5. The Langlands Classification
137
Introduction 5.1. Tempered (9,K)modules 5.2. The principal series 5.3. The intertwining integrals 5.4. The Langlands classification Some applications of the classification 5.5. 5.6. SL(2, R) 5.7. SL(2, C) 5.8. Notes and further results 5.A. Appendices to Chapter 5 5.A.1. A Lemma of Langlands 5.A.2. An a priori estimate Square integrability and the polar decomposition 5.A.3.
137 138 140 144 149 152 156 159 163 164 164 166 168
Chapter 6. A Construction of the Fundamental Series
173
Introduction Relative Lie algebra cohomology 6.1. A construction of (f, K)modules 6.2. The Zuckerman functors 6.3. Some vanishing theorems 6.4. Blattner type formulas 6.5. Irreducibility 6.6. U nitarizability 6.7. Temperedness and square integrability 6.8. The case of disconnected G 6.9. Notes and further results 6.10. Appendices to Chapter 6 6.A. Some homological algebra 6.A.1. Partition functions 6.A.2. Tensor products with finite dimensional representations 6.A.3. Infinitesimally unitary modules 6.A.4.
173 174 176 179 184 188 193 196 20 1 203 206 207 207 21 1 212 220
Chapter 7. Cusp Forms on G
225
Introduction Some Frechet spaces of functions on G 7.1. The HarishChandra transform 7.2. Orbital integrals on a reductive Lie algebra 7.3.
225 226 230 234
Contents
X
7.4. 7.5. 7.6. 7.7. 7.8. 7.A. 7.A. 1. 7.A.2. 7.A.3. 7.A.4. 7.A.5.
Orbital integral on a reductive Lie group The orbital integrals of cusp forms Harmonic analysis on the space of cusp forms Square integrable representations revisited Notes and further results Appendices to Chapter 7 Some linear algebra Radial components on the Lie algebra Radial components on the Lie group Some harmonic analysis on Tori Fundamental solutions of certain differential operators
243 250 254 259 264 265 265 268 273 277 282
Chapter 8. Character Theory
289
Introduction 8.1. The character of an admissible representation The Kcharacter of a (9, K)module 8.2. 8.3. HarishChandra’s regularity theorem on the Lie algebra 8.4. HarishChandra’s regularity theorem on the Lie group 8.5. Tempered invariant Z(g)finite distributions on G 8.6. HarishChandra’s basic inequality The completeness of the 7c, 8.7. 8.A. Appendices to Chapter 8 8.A.I . Trace class operators 8.A.2. Some operations on distributions 8.A.3. The radial component revisited 8.A.4. The orbit structure on a real reductive Lie algebra Some technical results for HarishChandra’s regularity 8.A.5. theorem
289 290 294 296 31 1 313 320 323 326 326 33 1 337 342
Chapter 9. Unitary Representations and (9, K)Cohomology
353
Introduction 9.1. Tensor products of finite dimensional representations 9.2. Spinors 9.3. The Dirac operator 9.4. (g, K)cohomology Some results of Kumaresan, Parthasarathy, Vogan, 9.5. Zuckerman 9.6. ucohomology A theorem of VoganZuckerman 9.7.
353 354 359 365 368
349
373 38 1 388
xi
Contents
9.8. 9.A. 9.A. 1. 9.A.2.
Further results Appendices to Chapter 9 Weyl groups Spectral sequences
394 396 396 398
Bibliography
403
Index
41 1
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Preface
This book is intended as an introduction to the representation theory of real reductive groups. It is based on courses that the author has given at Rutgers for the past 15 years. It also had its genesis in an attempt of the author to complete a manuscript of the lectures that he gave at the CBMS regional conference at The University of North Carolina at Chapel Hill in June of 1981. When the manuscript for those lectures reached over 300 pages the author realized that the scope of the project involved much more than was expected for a CBMS volume. We apologize to the conference board for not having completed the volume that was expected. We, however, hope that this book will in part fulfill the obligation. Initially, it was our intention to present the subject of representations of real reductive groups from the beginning to recent research, all in one volume. This has also been beyond the ability of the author. We have opted to present the material in two volumes in order to expand upon the original extremely terse exposition and to include recent developments in even the more “classical” aspects of the theory. There are many people that have been helpful in the production of this volume. We thank our students (both former and present) for their patience over the years with the lectures on which this book is based. We especially thank Roberto Miatello for all of the errors that he has found in the various earlier versions of this material and for his many helpful comments. Hans
xiii
xiv
Preface
Duistermaat pointed out a major blunder in our original exposition of HarishChandra’s regularity theorem. His explanation of the method of proof of this theorem that will appear in his forthcoming book with Kolk was very helpful. We also thank Kenneth Gross for having organized the abovementioned CBMS regional conference so well. Finally, we take this opportunity to thank Armand Bore1 for his editorial help, encouragement and patience throughout the preparation of this opus. We also take this opportunity to thank the National Science Foundation for the summer support during the preparation of this volume.
“You d o not understand my philosophy. But that is the way science progresses each generation misunderstands the previous one.” HarishChandra
Introduction
The representation theory of real reductive groups is one of the most beautiful, demanding, useful and active parts of mathematics. Although there have been many important contributors to the field. HarishChandra, through his power and vision, almost singlehandedly changed the field from a backwater of physics to what it is today. For better or for worse HarishChandra, in developing his awesome theory, also established the style of the field. Few disciplines in mathematics put as much emphasis on their technical details. This aspect of the subject makes it an extremely easy part of mathematics to read “line by line” and a very difficult part for those who would just like an “overall’’ picture of the subject. Although this book is a product of the HarishChandra legacy, we have attempted to allow the reader to get a “feel” of the subject without necessarily having understood every line. It is hoped that upon a first reading, the material will be studied by “jumping” from one part, that may seem interesting, to another. We have endeavored to do enough crossreferencing so that a reader could open the book in the middle and understand the material there by following the details backward. A careful reader will find mathematical gems in unlikely places. Kostant’s theorem on ncohomology is in Chapter 9, Zuckerman’s translation principal is in an appendix to Chapter 6, radial component theory is in the appendices to Chapter 7, Kostant’s theorem on nilpotent orbits is in an appendix to Chapter 8. xV
xvi
Introduction
As the title indicates, there is a forthcoming second volume which will contain, in particular, a proof of HarishChandra’s Plancherel theorem. Although both volumes emphasize the analytic aspects of the theory, the material in the volume at hand is more algebraic than the second volume. The reader who is predominantly interested in the algebraic aspects of the theory can read this volume without being too “contaminated” by analysis. Let us now give a “thumbnail tour” of the present volume. Chapter 0 is a compendium of some of the basic results that usually appear in a first course in Lie groups and Lie algebras. It is included to establish notation and references. The purpose of Chapter 1 is to introduce the theory of infinite dimensional representations of Lie groups. The material presupposes no prior knowledge of the reader. Our account is tailored to the needs of the later chapters and since most of representation theory of general Lie groups is unnecessary to the case of real reductive groups, the reader should be aware that this chapter is just the tip of the iceberg. The chapter emphasizes representations on Hilbert spaces. Basic material on smooth, analytic and “Kfinite” vectors is included. A novel aspect of this chapter is the development of the PeterWeyl theory for compact Lie groups as a corollary to the theory of square integrable representations. In Chapter 2, we introduce the class of Lie groups that will be studied throughout the remainder of the book. In particular we make the term “real reductive group” precise. The only prerequisites for this chapter are included in Chapter 0. We develop the theory of parabolic subgroups and Cartan subgroups. We take the more primitive notion to be that of parabolic subgroup and then show how the theory of Cartan subgroups is an outgrowth. Most of the classical groups are introduced in this chapter. We give the Iwasawa, Bruhat and Cartan decompositions for the groups. Integration formulas are given for these decompositions as are various versions of the Weyl integration formula. We also include a proof of the Weyl character formula (the standard one) since a similar proof will be used for the discrete series in Chapter 8. The material of Chapter 3 is the “heart” of the “algebraic” approach to representation theory. It contains various forms of the Chevalley restriction theorem and the HarishChandra homomorphism. The formalism of (9, K)modules is introduced. The critical notion of admissibility is developed. A proof is given of HarishChandra’s theorem that irreducible unitary representations are admissible. The chapter also includes the celebrated “subquotient theorem” of HarishChandra, Lepowsky, Rader and its corollary (in our development), the subrepresentation theorem of Casselman. The latter result is perhaps the most important single theorem to our development. It makes
Introduction
xvii
the theory of the real Jacquet module a viable approach to the representation theory of real reductive groups. Also our proof of this theorem contains ideas that will be critical to later developments in the book. The chapter also includes the basic theory of spherical functions. Most of the material in this chapter is algebraic or at least has algebraic statements. We have, however, given some analytic proofs of theorems that now have completely algebraic proofs. We indicate where thejmore algebraic approach can be found in the literature. Chapter 4 is the core of our approach to the subject. It contains the theory of the real Jacquet module and its consequence (in our exposition) the asymptotic behavior of matrix coefficients.This chapter is strongly influenced by our joint work with Casselman (which was motivated by the padic theory of Jacquet [11)and by HarishChandra’s theory of the constant term. Indeed, as we shall see in Volume 2, this latter theory is a consequence of the material in this chapter. Our approach to the asymptotic expansions is module theoretic. Special cases of the results can also be found in Warner [2]. Also a modern account of HarishChandra’s original approach can be found in Casselman, MiliEic [l]. The critical difference between our results and that of HarishChandra is that we give asymptotic expansions of smooth matrix coefficients rather than just “Kfinite” ones. The point of Chapter 5 is to give a proof of the Langlands quotient theorem (“Langlands classification”). This theorem reduces the classification of irreducible (9,K)modules to the classification of “tempered” (9,K)modules. The elementary aspects of tempered representations and their relationship with square integrable representations is also given. At this point in our development, the critical importance of the irreducible square integrable representations has become manifest. However, in this chapter these representations are described only in the case of SL(2, R). Chapter 6 is devoted to a homologicoalgebraic approach to constructing “admissible” (9,K)modules that is equivalent to that of Zuckerman using derived functors of the “Kfinite functor”. Our approach follows the broad lines of our joint work with Enright. An approach that is closer to Zuckerman’s original ideas can be found in Vogan [2]. Using, what we call Zuckerman’s functors, we construct irreducible unitary representations. These representations had been conjectured to be unitary by Vogan (a generalization of a conjecture of Zuckerman). Vogan gave the first proof of this result, using HarishChandra’s theory of tempered representations. Our proof is elementary, and we use it as a basis for the theory of tempered representations. We single out the families constructed from socalled ‘‘&stable Bore1 subalgebras” and call them the “discrete series”. Using the theory of Jacquet module we
xviii
Introduction
prove that they are square integrable. In Chapter 8 it is shown that these representations exhaust the irreducible square integrable representations. The reader can go directly from this chapter to Chapter 9 which studies the “twisted” (g, K)cohomology with respect to unitary modules. A complete proof (mainly due to Vogan, Zuckerman and Kumaresan) of a conjecture of Zuckerman (that completely calculates this cohomology) is given there using the modules constructed in this chapter. The next step is to prove that the “discrete series” exhausts the irreducible square integrable representations. In our approach, this is where the analysis begins in earnest. The next two chapters are very close to the spirit of HarishChandra’s original approach. In Chapter 7, the basics of HarishChandra’s theory of orbital integrals is given. Our approach differs in one important detail. We do not use the theory of the discrete series to prove that the orbital integrals define tempered distributions. Instead, we use a special case of Kostant’s convexity theorem (essentially due to Thompson [l]). The critical idea in this chapter is HarishChandra’s characterization of the matrix coefficients of the discrete series in terms of the vanishing of certain integral transforms. That is, these matrix coefficients span the space of “cusp forms”. We give HarishChandra’s formula for recovering a cusp form from its orbital integrals. This result implies HarishChandra’s basic theorem that says that irreducible square integrable representations can exist if and only if there is a compact Cartan subgroup. However, the completeness theorem must wait for the results in the next chapter. At this point the reader should have noted a glaring omission in the contents of this book. The only mention of character theory has been in connection with the Weyl character formula. Chapter 8 is devoted to HarishChandra’s theory of characters of admissible representations. These characters are initially defined as distributions on the group (as traces of generalized convolution operators). The main theorem on characters is that they are given as integration against a locally integrable function (HarishChandra’s regularity theorem). Furthermore, on each Cartan subgroup this function has a form reminiscent of the Weyl character formula. With the “local L’theorem” in hand we prove that the Fourier coefficients of orbital integrals of cusp forms are multiples of characters of what we called the discrete series in Chapter 6. The ,completeness theorem is now immediate. As we observed above, Chapter 9 could be read immediately after Chapter 6. This chapter contains a concise introduction to (9,K)cohomology, vanishing theorems due to Kumaresan, Enright, VoganZuckerman and the complete calculation of (9,K)cohomology with respect to a tensor product of a finite dimensional and an irreducible unitary representation (due to Vogan and
Introduction
xix
Zuckerman). The reader should consult Borel, Wallach [l] for an account of the general theory and its applications to discrete groups. We include tables of the vanishing theorems. There are several books whose contents have significant overlaps with this one. Knapp’s recent book (Knapp [l]) approaches the subject through examples. Since this book contains very few worked examples, we recommend that the reader approaching the subject for the first time, study Knapp’s book in conjunction with this one. Since there are important differences in the approaches to the material in these two books, even a more sophisticated (in representation theory) reader would benefit from having read both. Another important reference for the theory is Vogan [l] which covers a good deal of the more algebraic material in this volume. Again, there is a significant difference in emphasis and the student should benefit from a study of both this volume and that of Vogan. There is also a third (very stylish) approach to the subject involving sheaves of differential operators on algebraic varieties. This theory, mainly due to Beilinson, Bernstein and Brylinski, Kashiwara is the subject of a forthcoming book of MiliCic. Other notable books on the subject are Warner [l], [2] and Varadarajan [l]. Both of these works follow HarishChandra’s original methods quite closely. Warner’s treatise in addition contains a very thorough introduction to representation theory (i.e., C“vectors, analytic vectors, induced representations). These books (and Helgason [l]) were valuable aids in the preparation of this work. The literature in the field of reductive groups is vast. We have done our best to give adequate references. However, as is the case in any growing field, there are cases when a result has been proved (partially) by many authors. It would be a project beyond the scope of this book to give the precise history of the genesis of the theorems included in this book. However, in most cases the interested scholar should be able to determine a precise chronology by consulting the citations that we have included. A reader who has mastered the basic graduate curriculum in mathematics should have all the mathematical background necessary to master the material in this volume. However, the serious student should approach this work with an ample supply of paper and pencils. Be patient and it will be yours.
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Background Material
0
Introduction
The purpose of this chapter is to compile some of the background results, terminology and notation that will be used in this book. We recommend that the reader use this chapter basically for reference purposes. However, it might be worthwhile for the reader to skim through it on his first reading to become familiar with some of the notation and definitions. There are almost no proofs in this chapter. Everything covered can be found with adequate explanations in the references that we give, except for the material in Section 6. In Section 6 we give a noncommutative variant of the ArtinRees Lemma of commutative algebra. There is a general ArtinRees Lemma for nilpotent Lie algebras (see McConnell [11, Nouaze, Gabriel [ 13). Lemma 0.6.4appears for the first time in Stafford, Wallach [11. 0.1.
Invariant measures on homogeneous spaces
Let G be a locally compact topological group. Then a left invariant measure on G is a positive measure, dg, on G such that
0.1.1.
I
2
0.
Background Material
for all x E G and all f in (say) C,(G). If G is separable then it is well known (Haar’s theorem) that such a measure exists and that it is unique up to a multiplicative constant. If G is a Lie group with a finite number of components then a left invariant measure on G can be identified with a left invariant nform on G (here dim G = n). If p is a nonzero left invariant nform on G then the identification is implemented by integrating with respect to p using the standard method of differential geometry. If G is compact then we will (unless otherwise specified) use normalized left invariant measure. That is, the total measure is one. If dg is a left invariant measure and if x E G then we can define a new left invariant measure on G, p x , as follows: Px(f)
= j f(gx)dg. G
The uniqueness of left invariant measure implies that Px(f) =
6(x)
J f(g)dg.
G
with 6 a function of x which is usually called the modular function of G. If 6 is identically equal to 1 then we say that G is unimodular. If G is unimodular then we will call a left invariant measure (which is then automatically right invariant) inoariant. It is not hard to see that 6 is a continuous homomorphism of G into the multiplicative group of positive real numbers. This implies that if G is compact then G is unimodular. If G is a Lie group than the modular function of G is given by the following formula: 6 ( x ) = ldet Ad(x)l where Ad is the usual adjoint action of G on its Lie algebra. 0.1.2. Let M be a smooth manifold and let p be a volume form on M . Let G be a Lie group acting on M.Then ( g * p ) , = c ( g , x ) p xfor each g E G, x E M . One checks that c satisfies the cocycle relation
(1)
c ( g h , x )= c ( g , h x ) c ( h , x )
for h , g
E
G, x E M.
We will write J, f ( x )dx for J, fp. The usual change of variables formula implies that
for f (say) in C,(G) and g E G.
0.1. Invariant Measures on Homogeneous Spaces
3
Let H be a closed subgroup of G. We take M to be G / H . We assume that G has a finite number of connected components. A Ginvariant measure, dx, on M is a measure such that
If dx comes from a volume form on M then (3) is the same as saying that Ic(g,x)l = 1 for all g E G, x E M . If M is a smooth manifold then it is well known that either M has a volume form or M has a double covering that admits a volume form. By lifting functions to the double covering (if necessary) one can integrate relative to a volume form on any manifold. Returning to the situation M = G / H , it is not hard to show that M admits a Ginvariant measure if and only if the modular function of G restricted to H is equal to the modular function of H. Under this condition, a Ginvariant measure on M is constructed as follows: let g be the Lie algebra of G and let b be the subalgebra of g corresponding to H. Then we can identify the tangent space at 1H to M with g/b. The adjoint action of H on g induces an action Ad of H on g/b. The above condition says that ldet Ad"(h)l = 1 for all h E H . Thus if H o is the identity component of H (as usual) and if p is a nonzero element of A"(g/Ij)* ( m = dim G / H ) one can translate p to a G invariant volume form on G / H o . Thus by lifting functions from M to G / H o one has a left invariant measure on M. Now Fubini's theorem says that we can normalize d g , dh and dx so that
(4) 0.1.3. Let G be a Lie group with a finite number of connected components. Let H be a closed subgroup of G and let dh be a choice of left invariant measure
on H. The following result is useful in the calculation of measures on homogeneous spaces.
Lemma. I f f is u continuous compactly supported function on H\G (note the
change to right cosets!)then there exists, g , a continuous compactly supported function on G such that f ( H x ) = J g(hx)dh. G
This result is usually proved using a "partition of unity" argument. For details see, for example, Wallach [ l , Chapter 23.
0. Background Material
4
Let G be a Lie group and let A and B be subgroups of G such that A n B is compact and that G = AB. The following result is useful for studying induced representations. 0.1.4.
Lemma. Assume that G is unimodular. If da is a left invariant measure on A and if db is a right invariant measure on B then we can choose an invariant measure, d g , on G such that J f ( g ) d g= J f ( a b ) d a d b AXE
G
forfEC,(G).
For a proof of this result see for example Bourbaki [l]. 0.2. The structure of reductive Lie algebras 0.2.1. Let g be a Lie algebra over C.We use the notation 3(9) for the center of g. Then g is said to be reductive if g = 3(g) 0 [g, g] with [g, g] semisimple. We recall the basic properties of g that will be used in this book with appropriate references. Recall that a subalgebra, lJ, of g is called a Cartan subalgebra if b is maximal subject to the conditions that Ij is abelian and if X E lJ then ad X is semisimple as an endomorphism of g. Here, if X , Y E g then ad X(Y) = [ X , Y] (as usual). Cartan subalgebras always exist and they are conjugate to one another under inner automorphisms (c.f. Jacobson [1. p.2731). If X E g then define the polynomials Djon g by det(t1  ad X ) =
t’Dj(X),
here n = dim g . Let r be the smallest index such that 0,is not identically zero. Set D = D,. X E g is said to be regular if D ( X ) is nonzero.
Lemma. If X is regular then ad X is semisimple. Futhermore, the centralizer in g of a regular element is a Cartan subalgebra of g (Jacobson [ l , p.591). Fix, 6, a Cartan subalgebra of g. If a E b* then we set ga = { X
E
g I [H,X ] = a ( H ) X
for all H
E
lJ>.
If a and ga are nonzero then we call a a root of g with respect to 6, and ga is called the root space corresponding to a. The set of all roots of g with respect to
5
0.2. The Structure of Reductive Lie Algehras
t, will be denoted @(g, 5) and called the root system of g (with respect to 5). We have 9=bO
(1)
0 9,.
aE W g h )
(2)
If
(3)
I f a , B E ~ ( g , b ) t h e n l I 9 , , 9 / 1 1= % t o (Jacobson [1, p. 1 161).
(4)
If a E @(g,6) then the only multiples of are c1 and  c1 (Jacobson [ 1, p. 1 161).
ci
E @(g,b)
then dim(g,)
=
1 (Jacobson [l, p.1111).
c1
in @(g, 6)
0.2.2. Let g be as above. If B is a symmetric bilinear form on g then B is said to be inoariant if B ( [ X , Y],Z)
=

for all X , Y, Z
B( Y, [ X , 21)
E
g.
A nondegenerate invariant form on g always exists. O n [g, g] one takes the Killing form Jacobson [l, p.691 and on j(g) one takes any nondegenerate symmetric form. The direct sum of the two forms is then a nondegenerate invariant form on g. Fix such a form, B. Fix a Cartan subalgebra, b, in g. It is clear that 5 is orthogonal, relative to B,to all of the’root spaces. We therefore see that (1) B restricted to t, is nondegenerate. Thus, if p E b* then we can define H , B(H, H,)
= p(H)
E
5 by for H E.)I
We can then define a nondegenerate symmetric bilinear form ( , ) on b* by b*. One has
( p , z) = B(H,, H , ) for p, z E
(2) (a,a) is a positive real number for c1 E @(g,f)). (Jacobson [l, p.1 lo]) Let bRdenote the real subspace of one has
b spanned by the H, for c1 E @(g,5). Then
(3) B restricted to bRis real valued and positive definite (Jacobson [l, p.1 IS]).
0.2.3. We retain the notation of the previous number. If ci E @(g,b) we denote by s, the reflection about the hyperplane CI = 0 in 6. That is, s,H = H  (2c1(H)/(c(,a))Ha
for H E 6.
0. Background Material
6
sa is called a Weyl rejection. The Weyl reflections have the following properties: (1)
[email protected](g,6) = @(g,6)
(2)
sa6R = b R .
(Jacobson [l, p.1191).
We denote by W(g,9) the group generated by the Weyl reflections. W(g,6) is called the Weyl group of g with respect to 6. Let 6; denote the subset of all H E 6, such that a ( H ) is nonzero for all a E @(g,lj). Let C denote a connected component of 6;. Then C is called a Weyl chamber. (3) W(g,lj) acts simply transitively on the Weyl chambers (Bourbaki [2, p. 1633). A subset P of @(g,6) is called a system of positive roots if @(g, 5) is the disjoint union of P and  P ( = {  a I CI E P}) and if whenever a, p E P and a + p E @(g, 6) then a + p E P. If C is a Weyl chamber then the set of all a E @(g,6) that are positive on C is a system of positive roots. Conversely, if P is a system of positive roots then the subset of I)R consisting of those H such that a ( H ) > 0 for all a E P is a Weyl chamber. Thus specifying a Weyl chamber is the same as specifying a system of positive roots. Fix a system of positive roots, P. Then a E Pis said to be simple if u cannot be written as a sum of two elements of P. The set of all simple roots of P is called a simple system for P or a basis for the root system @(g,$). Let 7t denote the simple system for P. Then z has the following properties (Jacobson [1, p.1201):
0.2.4.
(1)
7t is
a basis for (ljR)*.
(2) If p E P then fl
=
1 n,a
aEn
(3) W ( g ,6) is generated by the s,
with n, E N. for a E n
(Bourbaki [2, p.1551).
0.3. The structure of compact Lie groups
Let G be a compact Lie group with Lie algebra g. Let gc denote the complexification of g. Then gc is a reductive Lie algebra over C. In fact, if ( , ) is any positive nondegenerate symmetric bilinear form on g then we define a new form on g, ( , ), as follows:
0.3.1.
(X, y > = j (Ad(g)X, Ad(g)Y)dg G
for
x,y, E 9.
7
0.3. The Structure of Compact Lie Groups
Here (as usual) d g denotes normalized invariant measure on G. The invariance of dg immediately implies that (Ad(g)X, Ad(g)Y)
=
(X, Y )
for g
E
G and X, Y
E
g,
By differentiating this formula one sees that ( , ) is an invariant form on g. Thus, if u is an ideal of g then the orthogonal complement to u is also an ideal of 9. Hence, dimension considerations imply that g is a direct sum of 1dimensional and simple ideals. This clearly implies that g is reductive. Recall that the Killing form of g, B, is defined by the following formula:
B ( X , Y ) = t r ad X ad Y for X, Y E g. Since ad X is skew adjoint relative to ( , ) for X E g it is clear that B ( X , X) I 0 for X E 9. Also, B(X, X ) = 0 if and only if ad X = 0. Thus, g is semisimple if and only if B is negative definite. The converse is also true.
Theorem. ff g is a Lie algebra over R with negative dejinite Killing form then any connected Lie group with Lie algebra 9 i s compact. This theorem is known as Weyl’s theorem. For a proof see, for example, Helgason [1, Theorem 6.9, p. 1 331.
0.3.2. In this book a commutative compact, connected Lie group will be called a torus. Let T be a torus with Lie algebra t. If we look upon t as a Lie group under addition then exp is a covering homomorphism of t onto T. The kernel of exp is a lattice, L , in t. That is, L is a free Z module of rank equal to dim t. Let TAdenote the set of all continuous homomorphisms of T into the circle. If p E TAthen the differential of p (which we will also denote by p ) is a linear map of t into iR such that p ( L ) c 2niZ. If p is a linear map of t into iR such that p ( L ) c 2niZ then p is called integral. If p is an integral linear form on t then we define for t = exp(X), t w = exp(p(X)).This sets up an identification of integral linear forms on t and characters of T. 0.3.3. Let G be a compact, connected Lie group. Then a maximal torus of G is (as the name implies) a torus contained in G but not properly contained in any subtorus of G. Fix a maximal torus, T, of G. Then t, is a Cartan subalgebra of 9., The elements of @(gc,),t are integral on t and thus define elements of T ” . Thus, we will look upon roots as characters of T. We now list some properties of maximal tori that will be used in this book.
8
0. Background Material
(1) A maximal torus of G is a maximal abelian subgroup of G (Helgason C1, p.2871). (2) If T and S are maximal tori of G then there exists an element g E G such that S = gTg’ (Helgason [l, p.2481). (3) Every element of G is contained in a maximal torus of G. That is, the exponential map of G is surjective. (Helgason [l, p.1351.) (4)If T is a maximal torus of G then G / T is simply connected. (This follows from say Helgason [l, Cor.2.8, p.2871.)
Let T be a maximal torus of G. Let N ( T )denote the normalizer of T in G (the elements g of G such that gTg’ = 7’). Let W(G,T ) denote the group N ( T ) / T .Then W(C,T ) is called the Weyl group of G with respect to T. If g E s E W(G,T ) then we set sH = Ad(g)H for H E t. This defines an action of W(G,T ) on t. ( 5 ) Under this action W(G,T ) = W(g,,tc) (Helgason [l, Cor.2.13, p.2891).
0.3.4. Let g be a semisimple Lie algebra over C. Then a real form of g, u, will be called a compact form if 11 has a negative definite Killing form. The following result is due to Weyl. Combined with Theorem 0.3.1 it is the basis of what he called the “unitarian trick”.
Theorem. If b is a Cartan subalgebra of g then there exists a compact form, u, of g such that u n is maximal abelian in u. (Jacobson, [l, p.1471.) 0.4. The universal enveloping algebra of a Lie algebra
Let g be a Lie algebra over a field F which we will think of as R or C . Then a universal enveloping algebra for g is a pair ( A , j ) of an associative algebra with unit, 1, over F, A , and a Lie algebra homomorphism, j , of g into A (here an associative algebra is looked upon as a Lie algebra using the usual commutator bracket, [ X , Y] = X Y  Y X ) with the following universal mapping property: If B is an associative algebra with unit and if a is a Lie algebra homomorphism of g into B then there exists a unique associative algebra homomorphism CT of A into B such that a(X) = o “ ( j ( X ) ) . It is easy to see that if ( A , j ) and (B, i) are universal enveloping algebras of g then there exists an isomorphism, T, of A onto B such that Tj = i. Thus, if a universal enveloping algebra exists then it is unique up to isomorphism. The usual construction of a universal enveloping algebra of g is given as follows: Let T ( g )denote the free associative algebra over F generated by the
0.4.1.
9
0.4. The Universal Enveloping Algebra of a Lie Algebra
vector space g. That is, T(g)is the tensor algebra over the vector space g. Let I ( g ) denote the two sided ideal of T ( g ) generated by the elements XY YX  [X, Y] for X, Y E g. Set U ( g ) = T(g)/Z(g). Let i denote the natural map of g into T(g).Let p denote the natural projection of T(g) into U ( g ) . Set j = pi. Then it is easy to see that ( U ( g ) , j )is a universal enveloping algebra for g. The basic result on universal enveloping algebras is the PoincareBirkoffWitt Theorem (PBW for short):
Theorem. Let X , , . . . ,X,, be a basis of g. Then the monomials j ( X, )" . . . j(Xn)",
form a basis of U ( g )(Jacobson [l, p.1591). 0.4.2. In light of the uniqueness of universal enveloping algebras and PBW we will use the notation U ( g ) for the universal enveloping algebra of g and think of g as a Lie subalgebra of U ( g ) .Thus,j will be looked upon as the canonical inclusion. Let U m ( g denote ) the subspace of U ( g )spanned by the products of m or less elements of g. Then U m ( g )c U r n +'(9) defines a filtration of U ( g ) . This filtration is called the canonical jiltration of U ( g ) .With this filtration U ( g )is a filtered algebra (that is, U p ( g ) U 4 ( gc ) U p + q ( g ) ) . Let G r U ( g ) denote the corresponding graded algebra. g generates U ( g )and the elements XY  YX are in U ' ( g ) for X, Y E g. Hence Gr U ( g )is a commutative algebra over F. Let S(g) denote the symmetric algebra generated by the vector space g. Then there is a natural homomorphism, p, of S(g) onto Gr U(g).PBW implies that this homomorphism is an isomorphism. If X I , . . . ,X, are in g then set
symm (X, . . . X,)
= ( 1/ k !)
1X u
. . . Xu,
U
the sum over all permutations o of k letters. Then symm extends to a linear map of S(g) to U(g).Let q be the projection of Um(g)into G r U(g).If x E S(g) is homogeneous of degree k, then it is easily checked that q(symm(x)) = x. Hence symm defines a linear isomorphism of S(g) onto U(g).In particular, if X E g then symm(X") = X" (the multiplication on the left hand side is in S(g) on the right hand side it is in U ( g ) ) .symm defines a linear isomorphism of S(g) onto U ( g )which is called the symmetrization mapping. We note that if a the Lie algebra (0) then U(a) = F . Let E be the Lie algebra homomorphism of g onto a given by E(X)= 0. Then E extends to a homomorphism of U ( g )onto F which we also denote by E (rather than 6 " ) . E is called the augmentation homomorphism.
0. Background Material
10
We denote by goPpthe Lie algebra whose underlying vector space is g with bracket operation { X , Y } = [ Y , X ] . Then U(goPp)= U(g)Opp(the opposite algebra). The correspondence X H  X defines a homomorphism of g onto gOPP whose extension to U ( g )will be denoted x T . We note that the linear map x H x T is defined by the following three properties: (1)
l T = 1.
(2)
XT=X
(3)
( ~ y=)y T~x T
for X E g. for x, y E U(g).
0.4.3. Let b be a subalgebra of g. PBW implies that the canonical map of U(b) into U ( g ) is injective. We can thus identify U(b) with the associative subalgebra of U ( g )generated by 1 and b. Let V be a subspace of g such that g = b @ V. Then PBW implies that the linear map U(b) 0 S ( V )
+
U(g)
Given by b 0 u H b symm(u) for b E U(b), v E S ( V ) , is a surjective linear isomorphism. Hence U ( g )is the free module on the generators symm(S(V ) )as a U(b) module under left multiplication. Similarly, U ( g )is the free right U(b) module generated by symm(S(V ) )under right multiplication by U(b).
0.5. Some basic representation theory 0.5.1. One of the most useful elementary results in representation theory is Schur’s Lemma. There is a Schur’s Lemma for most representation theoretic contexts (algebraic, unitary, Banach, etc.) In this book there will be several such Lemmas. We begin this section with a particularly useful one (usually called Dixmier’s Lemma). It is based on the following result: Lemma. Let V be a countable dimensional vector space ouer C . If T is an endomorphism of V then there exists a scalar c such that T  c l is not invertible on V. Suppose that T  cl is invertible for all scalars, c. Then P(T) is invertible on V for all nonzero polynomials P i n one variable. Thus if R = P / Q is a rational function with P and Q polynomials then we can define R ( T )to by the formula P(T)(Q(T)’). This rule defines a linear map of the rational functions in one variable, C(x),into End(V). If u E V is nonzero and if R E C ( x ) is nonzero with R = P / Q as above then R(T)u = 0 only if P(T)u = 0. Thus the map of
0.5. Some Basic Representation Theory
C(x) into V given by R
11
R(T)u is injective. Since C(x)is of uncountable dimension over C this is a contradiction. H
0.5.2. We now come to Dixmier’s Lemma. Let V be a vector space over C. Let S be a subset of End( V ) .Then S is said to act irreducibly if whenever W is a subspace of V such that SW W then W = V or W = (0).
Lemma. Suppose that I/ is countable dimensional and that S c End(V) acts irreducibly, If T E End( V ) commutes with every element of S then T is a scalar multiple of the identity operator.
By 0.5.1 there exists c E C such that T  c l is not invertible on V. Since the elements of S preserve Ker(T  c l ) and Im(T  c l ) and since at least one of the two spaces must be proper, we see that T = c l . 0.5.3. Let g be a Lie algebra over F = R or C . Then a representation of g is a pair (a,V ) with V a vector space over C and a a homomorphism of g into
End(V). The universal mapping property of U ( g )implies that it extends to a representation of U(g).We will write (T rather than a for this extension. If (T is understood we will usually use module notation for representations of Lie algebras (and their extensions to enveloping algebras). That is, we will write xu for (T(x)u.We will then call V a gmodule or a U(g)module (which, of course, it is in the usual associative algebra sense). If V and W are gmodules we denote by Hom,(V, W ) the space of all gmodule homomorphisms (or intertwining operators) from V to W. That is, the space of all linear maps, T, of V to W such that TXu = XTu for X E g and u E I/. We say that V and W are equivalent if there exists an invertible element in Horn,( V, W ) . Let V be a gmodule. Then a subspace, W,of V is said to be inuariant if X W is contained in W for all X E g. V is said to be irreducible if the only invariant subspaces of V are V are (0). In this context Schur’s Lemma says: Lemma.
If V is an irreducible gmodule then Horn,( V, V ) = CZ.
Let u be a nonzero element of V. Then U(g)u is an invariant nonzero subspace of V. Hence U(g)u = V. PBW (0.4.1) implies that U ( g )is countable dimensional. Thus V is a countable dimensional. The result now follows from Lemma 0.5.2. 0.5.4.
We now concentrate on a particularly important class of Lie algebras. A Lie algebra 5 over C is called a three dimensional simple Lie algebra (TDS for
0. Background Material
12
short) if it has a basis H , X , Y with commutation relations [ X , Y ] = H , [ H , X ] = 2 X , [ H , Y ] =  2Y. A concrete example of a TDS is eI(2, C)the Lie algebra of 2 by 2 trace zero matrices. Here one takes
x=["
'1,
Y = [ l0 0 O].
.=[;
3
0 0
We therefore see that if 5 is a TDS and if u is the real subalgebra of 5 with basis X  Y, i(X Y ) ,iH then u is isomorphic with the Lie algebra of SU(2) (the group of 2 by 2 unitary matrices of determinant 1). Let (a, V )be a finite dimensional representation of 5 (that is, dim V is finite). Since SU(2) is simply connected, there is a Lie homomorphism a of SU(2) into GL( V )(the group of invertible elements of End(V)) whose differential is a restricted to u. Let du be normalized invariant measure on SU(2). Fix ( , ) a positive nondegenerate Hermitian form (inner product for short) on V. Then we define a new inner product ( , ) on V as follows:
+
(u, w ) =
J
SU(2)
(a(u)u, a " ( u ) w ) du
for u, w E V.
Then ( ~  ( U ) V , ( T " ( U ) W ) = ( u , w ) for u E S U ( 2 ) and u, w E V. Differentiating this relation gives ( X u , w ) =  ( u , X w ) for X E u and u, w E V. Thus if W is a 5invariant subspace of V then so is the orthogonal complement of W. We have proved:
Lemma. I f V is a ,finite dimensional smodule then V splits into a direct sum of irreducible 5submodules. The proof we have just used is a special instance of the celebrated "unitarian trick". This trick was also used in 0.3.1. 0.5.5. Thus to describe finite dimensional 5modules it is enough to describe irreducible ones. To do this we will use the following commutation relation in U(5):
(1)
[ X , y"]= nY"'(H  n
+ 1)
for n
=
1,2,
Let V be a finite dimensional irreducible smodule. Then H has an eigenvalue on V of maximal real part, c. Let u be a nonzero eigenvector for H with eigenvalue c. By the commutation relations defining a TDS we see that HXu = (c + 2)Xu. Thus Xu = 0. O n the other hand, (2)
HY"u = (c  2n)Y"u
and X Y " u = n(c  n
+ l)Y"'u
by (1). We therefore see that there must be a nonnegative integer, m, such that
13
0.6. Modules Over the Universal Enveloping Algebra
Y"v is nonzero but Y m + l v= 0. Set vo = u and u, = Ynvfor n = 1,2,. ... Then ( 2 ) implies that v o , . . ., v, is a basis for a nonzero invariant subspace of V. Since V is irreducible, this implies that v o , . . . , v, is a basis of V. (2)now implies that tr H = ( m + l)(c  m) on I/. Since [ X , Y ] = H we must have tr H = 0 on V. Thus c = m. If W is an m + 1 dimensional vector space over C with basis w o , . . . , w,. We define the endomorphisms x, y and h of W by the following formulas:
( 3 ) xwo = 0, yw,=wntl hw,
xw,
= n(m  n
+ l ) ~ ,  ~ for n = 1,..., m;
f o r n = O , ..., m  1
= ( m  2n)w,
for n
and
yw,=O;
= 0,. . ., m.
Then it is not hard to show that x, y, h satisfy the commutation relations of a TDS. Putting all of this together we have proved: Lemma. Let B be a TDS with standard basis X , Y, H. Then for every strictly positive integer m + 1 there exists up to equivalence exactly one irreducible m 1 dimensional irreducible smodule, W. Furthermore, W has a basis wo,..., w, such that X , Y, H correspond to the elements x , y, h in ( 3 ) respectively.
+
0.6. Modules over the universal enveloping algebra Let A be an associative algebra over C . Then A is said to be (left) Noetherian if whenever I , c . . . c 1, c ... is a chain of left ideals in A then there exists, m, such that 1, = 1, for all k > m. Let g be a Lie algebra over C .
0.6.1.
Lemma.
U ( g ) is Noetherian.
If I is a subspace of U ( g )set
Here the notation is as in 0.4.2.If 1 is a left ideal of U ( g )then Gr(l)is easily seen to be an ideal in Gr U ( g ) .Gr U ( g )is isomorphic with S(g). The Hilbert basis theorem implies that S(g)is Noetherian (Atiyah, Macdonald [l, p.811). Hence we conclude that there is m such that G r 1, = Gr 1, for all k > m. But then I,,,= lk for all k > m.
0. Background Material
14
0.6.2. If A is an algebra with unit over C then an Amodule, M , is said to be finitely generated if there exist elements m,, . . ., m, of M such that M = E Amj.
Lemma. Let A be Noetherian and let M be a finitely generated Amodule. I f M I c . . . c M,, c . . . is a chain of submodules of M then there exists m such that M,,, = Mk for all k > m. This is proved by induction on the number of generators and is left to the reader (cf Atiyah, Macdonald [ 1, p.751).
0.6.3. Let A be as in the previous Lemma. Let I be a twosided ideal of A. We set I kequal to the ideal in A generated by the products of k elements of I. Then I is said to have the ArtinRees property (AR property for short) if whenever M is a finitely generated Amodule and N is a submodule of M there is a nonnegative integer k such that ( I k f j M )n
(1)
N
=
I j ( l k MnN )
for all j > 0.
If t is an indeterminate set A r t ] = A 0 C [ t ] .That is, A [ t ] is the algebra of all polynomials in t with coefficients in A. If I is a two sided ideal in A then we set I * = A + t l + t Z I 2+ ... + t k l k+ ... in A [ t ] .
Lemma.
I has the A R property if I* is a Noetherian algebra.
Let M be a finitely generated Amodule. Set M*
=
M
+ t l M + t 2 1 Z M + ....
Then M * is a finitely generated I*module. Let N be a submodule of M . Put N,
=N
Nk = N
+ t ( l M n N ) + t Z I ( I Mn N ) + ... + t'l'(1M
nN )
+ ...
+ t ( l M n N ) + . . . + t k ( l k Mn N ) + t k + ' l ( l k Mn N ) + ...
Then N , c N , c . . . is a chain of I *submodules of M *. There is thus a k such that N k + j= Nk for all j > 0. This is the AR property. If n is a Lie algebra over a field then we set n, = [ n , n ] and n,, = [n,, n ] for m = 1, 2 , . . . . n is said to be nilpotent if there exists k such that nk = 0. Let g be a Lie algebra over C. Let n be a nilpotent Lie subalgebra of U ( g ) such that if X is in g then [ X , 111 c 11. Let I = nU(g). Then I is a two sided ideal in U ( g ) . 0.6.4.
0.6. Modules Over the Universal Enteloping Algehra
15
Proposition. I has the A R p r o p e r t y in U ( g ) . Setg" = g + t n + t 2 n , +t'ilt, + . . . i n U ( g ) [ t ] . S i n c e i i j = O f o r j > > O , g A i s a finite dimensional Lie algebra over C. Thus if i is the natural inclusion of g A into U ( g ) [ t ]then we have the extension i to U ( g A )I.t is easy to check that i " ( U ( g " ) = I*. Thus since U ( g " )is Noetherian, I* is also. Thus Lemma 0.6.3 implies the result. 0.6.5. We conclude this section with a particularly important construction of U(g)modules. Let b be a Lie subalgebra of g. Let M be a U (b)module. Let U ( g )act on U(g) 0M by left translation in the first factor. Let V, be the U ( g ) submodule of U ( g ) @ M generated by the elements h 0m  10 bm for m E M and b E U (b). Then we sct U h ) @ M = ( U ( g )0W l V M U(b)
We now collect some properties of this construction. Let N be a U ( g ) module and let T be a U (6)module homomorphism of M into N , then (1) Then there exists a unique U(!J)module homomorphism of U ( g )@, into N , T" such that T"(1 0m ) = Tm.
M
Indeed, put T  ( g 0m) = yT(m). Then Ker T  contains V,. Hence T induces a U(g)module homomorphism T Aof U ( g )@,,, M into N . The rest is equally clear. (2) Let 0 + A 5 B
k C + 0 be a U (b)module exact sequence. Then
is a U(g)module exact sequence Let I/ be a subspace of g such that LJ = b 0 V. U ( g )= S(V )0 U ( 6) as a right U(6)module under right multiplication (0.4.3).Thus we can look upon the modules U ( g )@,, D as S ( V )0D for D = A , B,C . Under this identification, a" = l o x
the result is now clear.
and
=lob,
This Page Intentionally Left Blank
1
Elementary Representation Theory
Introduction In this chapter we develop most of the general representation theory that will be needed in this book. We have attempted to make the material as elementary as possible. The infinite dimensional representation theory of Lie groups is a vast subject that has been studied by many authors in that last 40 years. Thus, a short chapter such as this one can only “scrape the surface” of the material. A much more encyclopedic account can be found in Chapters 4 and 5 of Warner [11.The more general theory is not really necessary to our book, since we will be studying mainly reductive groups. We now give a description of this chapter. The first section is canonical except for the introduction of the conjugate dual to a Hilbert representation. This notion is of great importance to the representation theory of reductive groups. In the second section we give a variant of Schur’s Lemma. As we indicated in Section 0.5 there are many variants of this Lemma. The one that we give for irreducible unitary representations is sufficient for our purposes. Section 3 is devoted to the most elementary properties of square integrable representations. As we will see in the later chapters, these representations are
27
18
1.
Elementary Representation Theory
the basic ingredients in the harmonic analysis of real reductive groups. Section 4 contains the PeterWeyl theory of representations of compact groups. It also contains the critical (for our purposes) notion of isotypic component. In Section 5 we study a very special class of induced representations. A good exposition of the general theory of induced representations can be found in Warner [l, Chap. 51. Included in this section is Frobenius reciprocity for compact groups. In Section 6 we introduce just enough of the theory of smooth and analytic vectors to do the representation theory of the later chapters. Again, the serious reader can consult Warner [l, Chap. 41 for a much more comprehensive account. Section 7 is devoted to giving the CartanWeyl classification of irreducible representations of connected compact Lie groups. We give some details of these wellknown results, since the proof we use involves concepts that will be needed in later chapters. 1.1.
General properties of representations
1.1.1. Let G be a separable, locally compact group with left invariant measure, d g (0.1.1).Let V be a topological vector space over C. We denote by, End(V), the space of continuous endomorphisms of V and by G L ( V ) the group of all invertible elements of End( V ) .Then a representation of G on V is a homomorphism, n, of G into GL( V ) such that the map G x V + V given by g, u H n ( g ) v is continuous. That is, the homomorphism, n, is strongly continuous. We will say that (n, V )is a representation of G. Let (n,V )be a representation of G. Then a closed subspace, W, of I/ will be said to be invariant if n(g)W is a subspace of W for all g E G. (n, V )will be said to be irreducible if the only invariant subspaces of V are (0)and V. If (n,V )and (a,W )are representations of G then a continuous linear map, T, of V to W such that T n ( g ) = a(g)T for all g E G is called an intertwining operator or Ghomomorphism. We use the notation HornG(V, W )for the space of intertwining operators. We say that (n,V )and (a,W )are equivalent if there exists a bijective element, T, in Hom,(V, W )such that T  l is in HomG(W,V ) . If G is a Lie group and if V is a Frechet space (c.f. Reed, Simon [l, p. 1323) then a representation (n, V ) of G is said to be smooth if the maps of G to V given by g H n(g)v are C" for all u E V. 1.1.2. In this book the most important class of representations that we will study will be representations (n,H ) where H is a (separable) Hilbert space. Such a representation will be called a Hilbert representation. If ( n ,H) is a Hilbert representation and if n(g)is a unitary operator for all g E G then we call (n,H ) a unitary representation.
1.1.
19
General Properties of Representations
Let (n,H)be a Hilbert representation of G. Let I...[ denote the operator norm on End(H). The principle of uniform boundedness (c.f. Reed, Simon [1,111.9, p.811) implies: (1) If R is a compact subset of G then there is a constant, C,, such that In(g)(s C,
for all g E R.
The definition of a representation also implies:
(2) If u, w
E
H then the map g H ( n ( g ) u ,w ) is continuous on G.
1.1.3. Lemma. Let H he a Hilhert space and let n be a homomorphism of G into G L ( H ) .I f (n,H ) satisjes ( 1 ) and ( 2 ) above then (n,H ) is a representation of G . If f
E
C,(G) then we define for u, w E H the sesquilinear form pf(u,w ) by
s
[email protected]>w ) = f ( g ) ( 7 m 4 w > dg. c;
Let supp f be contained in a compact subset, R, of G. Then I P f ( ~ P ) I5 C,U ll
*
Iwlll.flll(llflll is the L' norm of f).
Hence there is an operator n ( f )in End(H) such that In(f)l 5 C,llflll
w ) = (nn(f)u,w >
and
for 0, w E H.
If f is a function on G we set L ( g ) f ( x )= f ( g  ' x ) for g, x n ( L ( x ) f )= n ( x ) n ( f )
(1)
for f
E
CJG), g
E
E
G. Then
G.
If U is an open subset of G such that Cl(U) is compact then we will use the notation L ' ( U ) for the space of all f E L'(G) such that supp f is a subset of U. The above considerations imply that n extends to a bounded linear map of L'( U ) into End(H) and that (1) is satisfied. Assume that 1 E U. If V is an open subset of U containing 1 and having the properties that V V is contained in U and that if u E V then up' E V then the map V x Li(V) to L ' ( U ) given b y x, f H L ( x ) f is continuous. Thus the map of V to H given by x H n(x)n(.f)vis continuous for f E L ' ( V ) and u E H. Let vj be a decreasing sequence of open relatively compact subsets of G such that V, = (1). Let { u j } be a sequence of nonnegative, continuous, functions on G such that supp uj is contained in and
0
j uj(g)dg = 1.
G
20
1.
Elementary Representation Theory
Then one shows easily (using uniform continuity) that (2)
for u, w E H .
lim (n(uj)u,w ) = ( u , w )
j m
Let H , denote the subspace of all u in H such that the map g H n(g)uis continuous on G. Then 1.1.2 ( 1 ) implies that H , is closed in H. Now (2) implies that H , is weakly dense in H. Thus H , = H. If u, w E H , and if x, y are in a compactum R contained in G then l I 4 x ) u  n(Y)4I + CQllU  WII.
I l 4 x ) u  74Y)Wll
This completes the proof of strong continuity. Note. The part of the proof using H , is taken from Warner [ l , p.2381. In that reference it is shown (using the Theorem of Krein and Smulian) that only condition (2) is needed. 1.1.4. If (n,H ) is a Hilbert representation of G then we set z*(g) = (n(g)l)*. Then the conditions ( 1 ) and (2) of 1.1.2 are clearly satisfied by n*. Hence, (K*,H ) is a representation of G which is called the conjugate dual representation of (71, H ) . Clearly, one has
(n(g)u,n * ( g ) w )
= (u, w )
for u, w E H , g
E
G.
1.2. Schur's lemma
Let G be a topological group. In this section we study variants of Schur's lemma that apply to unitary representations of G. The first and simplest form is:
1.2.1.
Lemma. Let ( n , H ) he an irreducible unitary representation of G. Then Hom,(H,H) = CI.
This result is easily proven using the spectral theorem. If T E Hom,(H, H ) then T* is also. Since
T
=(T+
T * ) / 2 + i ( T  T*)/2i,
it is clearly enough to prove that a selfadjoint intertwining operator is a scalar. We thus assume that T is selfadjoint. Let {P,} be the family of spectral projections corresponding to T (Reed, Simon [ l , p.2341). Since n(g)Tn(g)' = T for all g E G, the uniqueness of the spectral family for T implies that each
1.2. Schur’s Lemma
21
Hom,(H, H). This implies that P,H = H or (0) for each Bore1 set in R. It follows that there is a closed interval J = [  a , a ] such that P ’ = I. If we bisect J then one of the two halves, say, J , will have spectral measure I. Continuing to bisect in this way we find a nested sequence J1 =) J , =I ... of intervals each having spectral measure 1. Since .Ikis a point, { p } , we see that P is supported on { p } . Hence T = p l .
P,
E
0
1.2.2. We now give a useful refinement of the above result. For this we need some notation. Let H be a Hilbert space. If B is a subset of End(H) then set B’
=
{XE End(H) I TX
for all T EB } .
=XT
Let B be a subalgebra of End(H) such that I Then Von Neumann’s observation is:
E
B and if T E B then T * E B.
If u E H then ( B ’ ) ’ u c Cl(Bu).
(1)
Indeed, since T* E B if T E B, the orthogonal complement to Cl(Bu) is Binvariant. Thus, if P is the orthogonal projection of H onto CI(Bu) then P E B‘. Hence, if T E (B’)’ then TP = P T . Thus, TCl(Bu) is a subspace of CI(Bv).(1) now follows since u E CI(Bu). We can now give a refinement of Schur’s lemma. Proposition. Let ( x , H) be an irreducible unitary representation of G. Let D be a dense subspace of H that is G invariant. Let T be a linear map of D into H (there is no topology on D ) such that Tx(g)u = n(g)Tu for all g E G, u E D . Assume also that there exists a dense subspace D‘ of H and S a linear map of D’ into H such that
(To, w )
=
( u , Sw)
for v E D, w E D‘.
Then T is a scalar multiple of 1 restricted to D.
Let A denote the subalgebra of End(H) spanned by the operators x ( y ) for g E G. If X E A then, clearly, X* E A. Since n(1)= I , I E A . We also note
(2) I f x , y E H, X that
E
End(H) and if 6 > 0 is given then there exists U
IIUx  Xxll < 6
and
+ Yy,Z x + Wy)
A such
llUy  Xyll < 6.
Indeed, set I/ = H 0 H with the direct sum inner product. Let B U E A ) . Then B’ is the space of operators of the form U ( x ,y ) = ( X x
E
=
with X , Y, 2, W E A’.
{U 0 U I
22
1. Elementary Representation Theory
Now Lemma 1.2.1 implies that (A')' = End(H). Thus it is easy to see that (B')' is the space of all operators of the form with Z
U ( x ,y ) = ( Z x ,Z y )
E
End(H).
(1) now implies that if Z E End(H) then ( Z x ,Z y ) E CI(B(x,y)). This clearly implies (2). Let T be as in the statement of the result we are proving. Assume that u E H and that u and To are linearly independent. (2) implies that there exists a sequence { U j } in A such that
lim
q u =u
lim ~ T = vu.
and
Now, if w E D' then ( u , w)
= lim ( ~
T uw), = lim ( T q u , w)
= lim
(uju, Sw) = ( u , S w )
= (Tu, w).
Since D' is dense in H this implies that Tu = u. Since this is ridiculous, we conclude that if u E D then u and Tu are linearly dependent. This easily implies that T is a scalar multiple of I on D.
1.3. Square integrable representations Let G be a locally compact, separable group. Fix, dg, a right invariant measure on G. Let L z ( G )denote the space of all square integrable functions with respect to dg. If f E L2(G)and if x E G define R(x)f by 1.3.1.
R ( x ) f ( g )= f ( 9 4
for 9 E G.
Since dg is right invariant R(x)is a unitary operator for all x E G . Furthermore,
( W u , 0)
=
J u(gx)o(g)dg,
G
which is easily seen to be a continuous function of x . Lemma 1.1.3 implies that ( R , L2(G))is a unitary representation of G , called the right regular representation of G. 1.3.2. If (n,H ) is a Hilbert representation of G and if u and ware in H then we use the notation c " , for ~ the function
9 H (n(g)u,w). The functions c " , are ~ called coeficients or matrix coeficients of n. Let (qH ) be an irreducible unitary representation of G. Then we say that ( n , H ) is square integrable if it has a nonzero, square integrable matrix coefficient.
23
1.3. Square Integrable Representations
Lemma. Zf (n,H) is a square integrable representation of G then euery matrix coejicient of n is square integrable. Furthermore, there exists a unitary operator, T
E
Hom,(H, L2(G)),
with closed range such that
T ( H )is a subspace of C(G) n L2(G).
Fix w‘ and u‘, unit vectors in H such that c,.,,. is square integrable. Set D‘ = { u E H I c,. E L2(G)}. We note that D’ contains span{n(g)w’I g E G). Thus D‘ is dense in H. Also, if v E D‘ then n(g)u E D’. We define a map T from D‘ to L2(G)by Tu = c,,,.. Then Tn(x)u= R ( x ) T L ~ for u E D’. Define on D‘ the inner product ( (0, W)
, ) given by
= (11, W )
+ (Tu,T w ) .
(1) D‘ is complete relative to ( , ).
Indeed, let { uj} be a Cauchy sequence in D’. Then { uj> is Cauchy in H and Tuj is Cauchy in L2(G).Thus, uj converges to u E H and Tujconverges to u E L2(G). In particular, a subsequence of Tujconverges pointwise, almost everywhere to u. But c,,,.,. converges uniformly to c,,,.,. Thus, u = c,,,. almost everywhere. This implies that u E D’. Let S denote the canonical inclusion of D’ into H. Then S is clearly a bounded linear mapping of D’ into H. Let S* denote the adjoint map from H to D‘. Then S* satisfies the hypotheses of Proposition 1.2.2 with D = H . Thus S* = aZ with a E R.So D‘ = H. It also implies that there exists b > 0 such that
(Tu,Tw)
(2)
= b(v,w)
for u, w E H .
The lemma now follows using the map h’”T in light of (2) and the already observed fact that D’ = H. 1.3.3. The above proof has as an immediate consequence the Schur orthogonality relations: Proposition. (1) If
Let (n,H ) and (a. V ) he square integrable representations of G.
and a are not equivalent then
J ( n ( g ) x , y >conj((a(g)z,w>) d g = 0.
G
for all x , y E H, z, w E V.
24
1. Elementary Representation Theory
Define the operators T and S by T(u)(g)= ( n ( g ) u , y ) and S(u)(g) = (a(g)u, w ) . Then the proof of the preceding result implies that there exist t > 0 and s > 0 such that ( l / t ) T and ( l / s ) Sare unitary intertwining operators from H and V, respectively, to L2(G).It follows that there exists a positive constant C such that I(T(u),S(u))l I Cllullllull
for all u E H , u E V.
Hence, for each u E V there exists a unique A(u) E H such that (T(u),S ( u ) ) = ( u , A(u))
for all u E H .
It is easy to see that there exists a positive constant, a, such that ( l / a ) Ais a bijective unitary intertwining operator. This proves ( 1 ) . We now assume that n = (r. Then the above argument implies that
J (n(s)x,Y > conj((n(g)z, w>)dg = a
G
h Y ) ( x ,z>
= b(x,Z K W ,
Y).
Thus a(w,y) = ( l / d ) ( w , y ) with d > 0. This completes the proof of (2). 1.3.4.
If (n,H ) is a square integrable representation of G then the number
d(x) in 1.3.3(2) is called the formal degree of n. d(n)has an interpretation as a generalized dimension in the theory of VonNeumann algebras (Dixmier [l, p.2811). If G is compact then we will see (in the next section) that d(x) = dim H < co. 1.4.
Basic representation theory of compact groups.
1.4.1. Let H j , j IN , N 2 00 be Hilbert spaces then the symbol OjHjwill mean the Hilbert space completion of the algebraic direct sum of the Hj with the inner product
(1 1 w,) = 1 ( v j , wj>, uj,
uj, wj E Hj, j I N .
Let G be a topological group. Let (n,,Hi) be unitary representations of G for j I N . Let H = OiHj. Then the representation of G, n, on H given by the extension to H of n(g)(I:uj) = X xj(g)ujis called the direct sum of (nj,Hj). Let G be a separable, locally compact, unimodular, group with invariant measure, dg. A sequence { u j } of nonnegative continuous functions on G is
25
I .4. Basic Representation Theory of Compact Groups
called a delta sequence if the following three conditions are satisfied: (1)
supp uj+ is contained in supp uj and
(2) (3)
Uj(X)
=
{ l},
= uj(x'),
J uj(g)dg=
G
n supp uj
1
for all j .
The following result is due to Gelfand, Graev, PiatetskiShapiro (the proof we give is due to Langlands): Proposition. Let (n,H ) be a unitary representation of G. If there exists a delta sequence uj on G such that each n(u,) (1.1.3)is a compact operator (c.f. 8.A.l.l) on H then there exist unitary irreducible representations ( n j Hj), , j < N , N 5 co, such that (n,H ) is equivalent with the direct sum of the (nj,Hi). Furthermore, for each i there are only a finite number of ( n j ,H j ) equivalent with (xi,Hi). Let S be the set of all collections of closed, invariant, mutually orthogonal, irreducible subspaces of H. We order S by inclusion. Zorn's lemma implies that there is a maximal element, T, of S. Let V be the Hilbert space direct sum of the elements of T. Let X be the orthogonal complement to I/. Then X is a closed, invariant subspace of H . Suppose that X is nonzero. Let v be a unit vector in X . Since lim n(uj)u = u we see that there exists i such that if u = ui then n(u)vis nonzero. Now, (2) implies that if Q = ~ ( urestricted ) to X then Q is nonzero and selfadjoint on X. Also, by assumption Q is compact. Let 2 be an eigenspace for a nonzero eigenvalue for Q on X (such exist by the spectral theorem for compact selfadjoint operators c.f. Lemma 8.A.1.2). Then 2 is finite dimensional. Let R be a nonzero subspace of Z of minimal dimension subject to the condition that R = W n 2 for some closed invariant subspace, W,of X . Let Y be the intersection of all invariant subspaces of X containing R. If Y were reducible then Y could be written as an orthogonal direct sum A + B with A and B closed invariant subspaces of Y. Since Q leaves invariant any invariant subspace of X , we see that R must be completely contained in A or in B. But this contradicts the definition of Y. Hence Y is irreducible. We have now contradicted the definition of T. Hence X = 0 so V = H. The last assertion follows from the fact that the nonzero eigenvalues of each n(uj)have finite multiplicities. 1.4.2. For the rest of this section we will assume that G is compact. If f E C ( G )and if u E L 2 ( G )then
R ( f ) u ( x )= J u(xg)f(g)ds G
=
1u(g)f(xls)ds.
G
26
Elementary Representation Theory
1.
Hence R(f)is the integral operator on L 2 ( G )with kernel K ( x , y ) = f ( x  ' y ) . Since G is compact we see that R ( f ) is a HilbertSchmidt operator. Hence R ( f )is compact. The previous proposition therefore applies to ( R , L2(G)).We now derive some consequences of that result.
Proposition. Let (n,H ) be an irreducible unitary representation of G. Then dim H < co. Since G is compact and the matrix coefficients of n are continuous, (n,H ) is square integrable. Thus Lemma 1.3.2 implies that n is equivalent to an irreducible closed subspace of L 2 ( G ) which is also contained in C(G).The result now follows from: Scholium. Let (X, p ) be a measure space with total measure 1. If V is a closed subspuce of L2(X)contained in L"(X) then dim I/ < co. Let 11.. denote the Lznorm and let 11. . clear that
llfll 5 Ilfll,
(*I
.llm
for f
denote the Lconorm.Then it is
E
L"(X).
Let Q be the inclusion of V into L2(X).Let W be the closure of V in L"(X). Then (*) implies that Q extends to a bounded operator from W to V. Hence W = V. The closed graph theorem now implies that there exists a positive constant such that
(**I
Ilfll,
5
cllfll
for f
E
V.
Let f l , . . . , fd be anprthonormal set in V. If p i E C for i
=
1,. . . , d then
IC pifi(x)I 5 IIC p i f i I l m 5 c l l z pi.fiII = ~ (Ipi12)1'2. 1 Choose p i = conj(f.(x)). Then we have
1 Ifi(x)12 I c ( C lfi(~)1~)'/'
for a.e. x E X.
This implies that
C Ifi(x)lz I c z
for a.e. x E X.
Integrating this inequality over X yields d I c2. This proves the result. 1.4.3. As we have observed, if (n,H ) is an irreducible unitary representation of G then n is square integrable.
1.4.
Basic Representation Theory of Compact Groups
27
Lemma. Let (n,H ) be an irreducible unitary representation of G. Then the formal degree of n is equal to dim H .
Let d be the formal degree of n. Let u l , . . ., u, be an orthonormal basis of H . Set f;i = c " , , ~ Then ,. the matrix [,fij(x)] is unitary. Hence
1 IAj(x)12= n
for all x E G.
If we integrate both sides of this equation over G then 1.3.3(2)implies that ( l / d ) n 2= n. Hence d = n as asserted. 1.4.4. Let G" denote the set of all equivalence classes of irreducible unitary representations of G. If y E G" we denote by L2(G)(y)the sum of all invariant, irreducible subspaces of L 2 ( G ) that are in the class y. The material in 1.4.2 implies that
(1)
dim L2(G)(y) < a
and
L2(G)= @L2(C)(y).
Let y E G" and let (n,H ) E y. We set d(y) = d ( n ) = dim H ( < co). We put for g E G x,(d = x,(d = tr 4s). 1.4.5.
Then xy is called the character of y . Lemma.
I f y, p are in G" then
f x&)
G
conj(x,(s)) dg
=
dy,p.
This is an immediate consequence of 1.3.3(2)and 1.4.3. 1.4.6. Let for y E G " , ay = d(y)conj(xy).Let P denote the orthogonal projection of L 2 ( G )onto L2(G)(y).
Lemma.
Py = R(a,).
This result is also a direct consequence of 1.3.3(2)and 1.4.3. Corollary.
If y
E
G" then dim L 2 ( G ) ( y )= d(y)'.
By the above lemma dim L z ( C ) ( y = ) tr R(a,). This is easily seen (using the material in 1.4.2)to be equal to d(y) conj(a,(x'x))dx G
= d ( y ) conj(x,(l)) =
28
1. Elementary Representation Theory
1.4.7. Let (n,H ) be a unitary representation of G. If y E G” then we set H ( y ) equal to the closure of the sum of all the closed, invariant subspaces of H that are in the class y. H ( y ) is called the yisotypic component of H.
Lemma (1)
H(y) = n(gy)H.
(2)
H is the Hilbert space direct sum of the H(y).
If u, w E H then R(cly)c,,, = cU,, with u = n(cl,)u.If u E H ( y )then c , , , is a sum of matrix coefficients of y. Hence (n(cc,)u,w) = (u, w) for all u E H(y), w E H. This implies that H ( y ) is contained in n(ay)H. We now prove the reverse inclusion. If u E n(cty)Hthen R(ay)c,,, = cU,,for all w E H. Hence c,,,, E L2(G)(y)for all w E H. Let Z = span{n(g)u 1g E G}. Then dim spanfc;,, I z E Z , w E H } I d(y)’. This implies that dim Z < co. Hence, 2 splits into a finite direct sum of irreducible invariant subspaces each in the class of y . This completes the proof of (1). We now prove (2). We note that if u E H(y), w E H ( p ) with y and p distinct, then c,,, E L 2 ( G ) ( yn ) L z ( G ) ( p )= ( 0 ) .This implies that ( H ( y ) , H ( p ) ) = 0. We must therefore only show that the sum of the H(y)is dense in H. We label G” as y l , y2,. . . If y = y j then we set cly = c l j . If u, w E H then lim
1
N+m j < N
R(Nj)cu,w
= cv.u
in L 2 ( G ) .Thus if w E H is orthogonal to the algebraic sum of the H ( y ) then c,,, = 0 for all u E H. Hence w = 0. (2) now follows. 1.4.8.
We conclude this section with a useful variant of the “unitarian trick”.
Lemma. Let (n,H ) be a Hilbert representation of G (still assumed to be compact). Then there exists an inner product ( , ) on H that gives the original topology on H and is suck that relative to ( , ), n is unitary. Define ( , ) as follows: ( u , w) = j
c
(n(g)u,n ( g ) w ) dg
for u, w E H.
There is a positive constant C such that In(g)l I C for all g E G (1.1.2(1)).Since
29
A Class of Induced Representations
1.5.
n(g)n(g')= I we also see that 11n(g)ull 2 C'llull for all g E G. Hence
for all u E H .
C'(u,u) I (u,u) I C(u,u)
so ( , )defines the same topology as ( , ). The rest of the argument goes as usual (0.3.I , 0.5.4). 1.5.
A class of induced representations
1.5.1. Let G be a unimodular, locally compact group. Let K and P be closed subgroups of G such that K is compact and such that G = P K . Let 6 denote the modular function of P (0.1.1). Let d p denote left invariant measure on P and let dk denote normalized invariant measure on K . Then we can choose invariant measure on G so that
J f(g)dg
G
=
j f(pk)dpdk
PXK
for f
E
C,(G)
(Lemma 0.1.4).
We extend 6 to G by setting 6 ( p k ) = d(p) for p E P, k E K . This makes sense since 6 ( p ) = 1 for p E K n P. If .f is a function on K such that f ( p k ) = f ( k ) for p E K n P then we extend f to G by setting f ( p k ) = f ( k ) for p E P and k E K .
( I ) If f is integrable on K and if f ( p k ) = , f ( k ) for all p
j f ( k g ) W g )dk
K
=
E
P n K then
j ,f(k)dk.
K
Indeed, there exists, g E C,(G) such that (see 0.1.3)
1g ( p k ) d p = , f ( k )
for all k
E
K.
P
For this g we also have
If x E G we set x = p ( x ) k ( x )with p ( x ) E P, k ( x ) E K . This decomposition is not necessarily unique, but the ambiguity will be irrelevant to our argument. We have for x E G
J f ( k ) dk = Gj g(u)du = Gj g(ux)du = P xj K g ( p k x ) d p d k
K
=
J G(kX)f(kx)dk.
K
since G ( p ( x ) ) = 6(x). This proves (1).
30
1.
Elementary Representation Theory
1.5.2. Let (a, W )be a Hilbert representation of P. In light of Lemma 1.4.8,we assume that the restriction of a to K n P is unitary. Let ( H " ) , be the space of continuous functions, u, from G to W such that
u ( p g ) = cVp)"2a(p)u(g)
(1)
If u, IJ
E
for p
E
P, g E G .
( H " ) , then we set (u, I J > =
(2)
j ( u ( k ) ,4 k ) ) dk.
K
Let H" denote the Hilbert space completion of ( H " ) , relative to ( , ). If u E ( H " ) , and if g E G we set
(3) Clearly q,(g)u
n,(g)u(x) = u(xg)
for all x
E
G.
E (H " ) , .
1.5.3. Lemma. (1) If g E G then n,(g) extends to u bounded operator on H". (2) (n",H " ) is a Hilbert representation of G which is unitary if a is a unitary representation of P. As above we write g = p ( g ) k ( g )with p ( g ) E P, k(y) E K . Since the ambiguity in the definition of p ( g ) is in the compact set P n K , it follows that if R is a compact subset of G then there exists a compact subset, R', of P such that p(R) is contained in 0'. Let u E H and let R be a compact subset of G. If g E R then
By the above p ( k g ) E (KR)'. Hence 1.1.2(1) implies that there is a constant E , such that la(p(kg))l < E , for g E R. Put Dn equal to the supremum of 6''' on K , . Then (i) implies that
This proves (1). Set C, = D,C,.
Using (ii) it is easy to see that if u, u, z
Since it is clear that the functions c , , , ~for u, u E
E
H and if g E R, then
are continuous, the above
31
1.6. C" Vectors and Analytic Vectors
inequalities imply (see Lemma 1.1.3) that (no,H " ) is a Hilbert representation of G. (i) combined with 1.5.1(1) implies that if a is unitary then 71, is unitary. The representation (q,, H " ) constructed above is a special case of an induced representation. We will not have any use for a more general definition of induced representation. Thus, in this book, induced representation will mean the above construction. We will also use the notation 1.5.4.
Indi(a)
for (n",H").
1.5.5. We now look at the special case when G is compact. Let P be a closed subgroup of G. We may take G = K in the above construction. Let (a, W )be a finite dimensional unitary representation of P. We study (q,, H"). Let y E G" then (with notation as in 1.4.7)
(1)
H"(y) is contained in (H"),.
Indeed, if u E C(G), and if u E H then 7c,(u)u E (H"),. Thus (1) follows from Lemma 1.4.7( 1). Fix (p,V ) , a finite dimensional unitary representation of G. Let T E Horn,( V,H"). Then (1) implies that T ( V )E ( H " ) , . Thus we can define T" (u) = T(u)(l)for u E H. It is clear that T " is in Hom,(V, W ) .We have (2) The map TH T" defines a linear isomorphism of Homc(V,H") onto Horn,( V, W ) .
Indeed, if S E Hom,(V, W )define S  ( u ) ( g ) = S ( p ( g ) u ) .Then it is easy to see that S E Hom,(V, H"). It is also clear that (T")" = T and (S)" = S. (2) is usually called Frobenius reciprocity. It immediately implies
(3) 1.6.
dim H"(y) = d ( y ) dim Horn,( V, W )
for y E G", ( p , V ) E y.
C" vectors and analytic vectors
1.6.1. For the rest of this chapter we will be studying representations of Lie groups. Let G be a Lie group with a finite number of connected components. We fix a left invariant measure, dg, on G. Let (71,H) be a Hilbert representation of G. If u E H is such that the function @(g) = n(g)uis of class C" from G to H then u is called a Cmuectoror smooth vector for (n,H ) . The following result was first observed by Girding in order to prove Theorem 1.6.2.
32
1.
Elementary Representation Theory
Lemma. If f E CF(G) and if u E H then n ( f ) u is a smooth vector for (n,H ) . Let U be a relatively compact open subset of G containing 1. Let L'(U) be the space of all L' functions on G with support in U. Let V be an open subset of U such that if x E V then xl E V and VV c U . Then
(1) Let f class C".
E
Cp(V) then the map of I/ to L'(G) given by F(x) = L(x)f is of
Indeed, if X E g (the Lie algebra of G,as usual) then we set L(X)f(g) = d/dt,,,(f(exp(  tX)g). Taylor's theorem implies that there is E > 0 and E a bounded function of t , g for It/ < E, such that f(exp(tX)g) = f(g) tL(X)f(g) + t 2 E ( t ,9) for 1 t I < E and g E I/. This implies that
+
with Can appropriate constant for It\ < E. This implies that F(x) is of class C ' . This argument can be iterated to prove (1). We have seen in the proof of Lemma 1.1.3 that the correspondence f to n(f)u is a bounded linear map of L ' ( U ) into H. Thus the map of I/ to H given by x H z(L(x)f)u is a C" map. Since n(L(x)f) = n(x)n(f). We see that the map of I/ into H given by x H z ( x ) n ( f ) uis of class C". The lemma now follows since n(x) is a bounded linear operator on H hence it is of class C". 1.6.2. Theorem.
The space of C" uectors of H is dense in H.
As is well known, there exists a delta sequence u j (1.4.1) consisting of C" functions on G.Since lim n(uj)u = u for u E H . The result follows from the previous Lemma. 1.6.3. Let H " denote the space of all C " vectors for n. If u E H m, and if X then we set
Eg
n(X)u = d/dt,=, n(exp(tX))u. Then n(X) maps H" into H" and it is not hard to show (using Taylor's Theorem) that
(1)
n([X, Y]) = n(X)n(Y)  n(Y)n(X) on H
for all X, Y E g.
33
1.6. C" Vectors and Analytic Vectors
Hence (n,H ") defines a representation of g. The universal mapping property of U ( g )implies that n extends to U ( g ) . If D
E
U ( g ) then we set p D ( u ) = Iln(D)ull
for u E H".
We give H" the topology induced by the seminorms p D for D E U ( g ) . Lemma. (1) H" is a FrPchet space. ( 2 )(71, H " ) is a smooth representation of G(l.l.l). 1.6.4.
Since U ( g ) is countable dimensional it is enough to show that H" is sequentially complete to prove (1). Let ( v j ) be a Cauchy sequence in H". If X E g then { u j } and {Xuj} are Cauchy sequences in H. Thus there exist v, u E H such that lim uj = 11
and
lim Xuj = u.
We note that lim Xn(exp(tX))uj= n(exp(tX))u and that d/dt(exp(tX)u = z(X)x(exp(tX))u. Hence uj
1
+ j n(exp(sX))n(X)ujds= n(exp(tX)uj. 0
If we take the limit of this expression in j we have t
u
+ j0 n(exp(sX))uds= n(exp(tX))u.
This implies that the map t H n(exp(tX)u is of class C' with derivative equal to n(exp(tX))u. Hence g H n ( g ) u is of class C'. This argument can be iterated to show that u is a smooth vector. Hence H" is complete. We now prove (2). We first observe that
(i) The map U j ( g )0 H"
f
H
given by g, u H n ( g ) u is continuous.
We also have (ii)
n(g)n(X)u = n(Ad(g)X)n(g)u
Hence if D E U ( g ) ,g E G and if pD(n(g)u

1, E
for g E G , X E g
H" then
u, = Iln(g)n(Ad(g')D)u n(D)ull.
and
uE
H".
34
1.
Elementary Representation Theory
In light of (i), we have shown that (n,H " ) is a representation of G. Now the argument that we used to prove (1) completes the proof of (2). 1.6.5. Let ZG(gc) denote the subalgebra of U(g,) consisting of those g E U(g,) such that Ad(x)g = g for all X E G. If G is connected then Z,(g,) = Z(g,) the center of U(g,).
Lemma. Let (n,H ) be an irreducible unitary representation of G . Then each acts by a scalar multiple of I on H".
z E Z,(g,)
If X E gc we will use the notation, conj(X), for complex conjugation of X relative to g. That is, if X = X, + i X , with X,, X, E g then conj(X) = XI iX,. We define a conjugate linear antihomomorphism of U(g,) onto U(g,), XHX* as follows: (1)
(2)
(3)
1* = 1,
X*
=
conj(X)
(xy)* = y*x*
for X
Eg ,
for x,y E U(g,).
It is clear that (ZG(gc))*= Z,(g,). If we take D = D' = H" and T = ~ ( z ) , S = n(z*)then the lemma follows from Proposition 1.2.2. to C If (n,H ) is a representation of G and if x is a homomorphism of ZG(g) such that n(z)u = x(z)ufor z E &(g) and u E H then x is called the infinitesimal character of n. 1.6.6. Let (n,H ) be a Hilbert representation of G. Then we say that u E H is an analytic vector for (n,H ) if the function 9 ++
(n(gb9 w>
is real analytic for all w E H . This agrees with the standard terminology (Warner [1, p.2781) since weak analyticity implies strong analyticity. However, we will only need this notion of analyticity in this book. We use the notation H" for the space of analytic vectors of H . It is clear that if u E H then n(g)u E H and n ( X ) uE H for g E G and X in g. Hence, H" is a representation of g. The main reason for the introduction of analytic vectors is the following result: Proposition. Let G be connected. If V is a ginvariant subspace of H", then C 1( V )(in H ) is a Ginvariant subspace of H .
35
1.7. Representations of Compact Lie Groups
If W is a subspace of H we denote by W1 the orthogonal complement of W in H. Then it is easy to see that C l ( W ) is equal to (W')". Let X E g, let u E V and let w be in V'. Then there exists E > 0 such that if It1 < E then (n(exp(tX))zi, w )
=
1(t"/n!)(n(X")v, w)
and the series converges absolutely. Since w E I/' it follows that (n(exp(tX)o, w )
=0
for It( < E.
The real analyticity of t H (n(exp(tX))u, w ) now implies that (n(exp X)v, w )
=0
for all u E V, w E V' and X E g.
This implies that V1 is invariant under the operators n*(exp X ) for X E g (see 1.1.4for n*). Since exp(g) generates G as a group, we see that V' is an invariant space for n*. Hence (V')' is an invariant subspace for n.
1.7. Representations of compact Lie groups Let g be a reductive Lie algebra over C.We will use the notation of section 0.2. Fix 6, a Cartan subalgebra of g. Fix B, an invariant nondegenerate form as in 0.2.2. Set @(g, 6) = 0.Fix P, a system of positive roots for @. Let A = { a l , .. . ,a,} be the simple roots in P. 1.7.1.
(1)
If X E ga and if Y E g, then [ X , Y]
=
B ( X , Y)H,.
Indeed, [g,,g,] is a subspace of 6. If HE^ then B([X,Y],H) B(X, [H, Y]) = c((H)B(X,Y ) . So (1) follows from the definition of Ha.
=
1.7.2. Lemma. Let (a,V )be an irreducible jinite dimensional representation of g. Then the elements oft, act semisimply on V.
Let for a E P, X E ga, Y E g P a be nonzero. If H = (2/(a,a))H, then X , Y, H span a TDS (0.5.4),5,. Hence Lemmas 0.5.4 and 0.5.5 imply that each Ha, a E P, acts semisimply on V. Schur's Lemma implies that the elements of j(g) act by scalars on V. Since the span of the Ha,a E P and 3(g) is 6 the lemma follows.
1.7.3. We note that the argument in the proof of the above Lemma actually proves (1) Let (6,V ) be a finite dimensional representation of g. 6 acts semisimply on V if j(g) does.
36
1. Elementary Representation Theory
Let (a, V )be a (not necessarily finite dimensional) representation of g such that lj acts semisimply on I/. If p E b* then we set V, = { v E 1/: hu = p(h)u for all h E 6). Then V, is called the pweight space of V and if p E b* and V, is nonzero then p is called a weight of V. We now assume that V is finite dimensional. We partially order the weights of V by saying that p 2 y if p  y is a sum of elements of P. Let A be a weight of V that is maximal relative to the partial order. '
(2)
2(A, cr)/(a,a) is a nonnegative integer for c1 E P.
Indeed, let s, X , Y be as in the proof of Lemma 1.7.2. Then XV, c V,+,. Thus X V , = (0). The result now follows from 0.5.4 and 0.5.5. If A is an element of b* satisfying ( 2 )then we say that A is dominant integral.
(3) If p is a weight of V then 2(p, @)/(a,a) is an integer for all a E (D. This also follows from TDS theory. (4) If p is a weight of V then so is sup for all c1 E @. Indeed, let 5, be as above. Let u be a nonzero element of V,. Then there exists r > 0 such that X'v is nonzero but X'+Iu = 0. By TDS theory ( p + 2rcr)(H) = m, a nonnegative integer. Also, TDS theory implies that if w = X'u then YJw is nonzero for j = 0,. . ., m. Thus, the forms p + 2(r  j)cr are weights of V f o r j = 0,. . . , m. Since sup is on this list of weights (4)follows. 1.7.4. We are now ready to give the CartanWeyl classification of irreducible finite dimensional representations of g.
Theorem. (1) If V is an irreducible, jinite dimensional gmodule then I/ has a unique highest weight (i.e.,maximal weight), which we write as A,. Furthermore, the A, weight space is one dimensional. ( 2 ) I f V and W are irreducible jinite dimensional gmodules then V and W are equivalent if and only if Av = A w . ( 3 ) If A is a dominant integral linear form on b then there exists an irreducible jinite dimensional gmodule, V, such that A, = A. We set n + implies that
C a E Pga and n 
=
X a t P g,. Then g = n  0 t, 0 n + . PBW
U(n) = U(n)U(b)U(n').
(i) v
=
We now prove (1). Let p be a maximal weight of V. Fix a nonzero element V,. Then U ( g ) v = U ( n  ) u by (i). Since V is irreducible, this implies that
E
1.7.
37
Representations of Compact Lie Groups
V = U ( n  ) u . Since the weights of 1) on U ( n  ) are of the form  C njaj with nj nonnegative integers, and the 0 weight space consists of the scalar multiples of 1. (1) now follows. Before we begin the proof of (2) we will introduce a concept that will be useful in the later chapters. Let b = 1) 0n'. b is usually called a Bore/ subalgebra of g. If p E l ~ *we denote by C, the ldimensional bmodule C with lj acting by p and n+ acting by 0. We set (0.6.5) M ( p ) = U ( g )@ C, (0.6.5).
(ii)
U(b)
M ( p ) is usually called a Verma module. By the first part of this proof l j acts semisimply on M ( p ) and the weights of M ( p ) are the linear forms p  C njuj with nj nonnegative integers. Furthermore, the pweight space is spanned by 10 1. Let N be the sum of the submodules of M ( p ) that do not intersect C 1 0 1. Then it is easy to see that N is the unique proper maximal submodule of M ( p ) . Hence, M ( p ) has a unique, nonzero irreducible quotient which we denote L(p). Let V be an irreducible, finite dimensional gmodule with highest weight A. Then we have seen above that 11 'V , = 0. Hence there is a surjective gmodule homomorphism of M(A) onto V (0.6.5( 1)). But then V is equivalent to L(A). This implies (2). To prove (3) we need only show that if p is dominant integral then L ( p )is finitedimensional. So, assume that p is dominant integral. Let c( be simple root in P and let 5, = 5 be the corresponding TDS. Set m = p ( H ) + 1(X, Y, H are as above). Then
(iii) Ym(l0 1 ) E N (the maximal proper submodule of M ( p ) ) . Indeed, set u = Ym(l0 1). If p E A is not equal to c1 then [gs, Y] = 0 by the definition of simple root. Also 0.5.5(1) implies that X u = 0. Since the simple root vectors generate n + as a Lie algebra (0.2.1(3)),see that n'u = 0. Now (i) implies that U ( g ) uE N . This proves (iii). (iv) If c1 is a simple root in P and if u E L ( p )then U(5,)u is finite dimensional. Indeed, this is true if u is the image of 1 0 1 in L ( p ) . Let us call that element w. Set 5 = 5,. Let Z = U (5 )w. Clearly, the union of the spaces U'(g)Z is L(p). Since each of these spaces is finite dimensional and 5 invariant (iv) follows.
(v) If
CJ
is a weight of L ( p )and if s E W(g,6) then so is a weight of L(p).
This follows from (iv) using the argument proving 1.7.3(4) and 0.2.4(3). (vi) If
G
and y are weights of L ( p )agreeing on I),then o
=
y.
38
1. Elementary Representation Theory
This is clear since j(g) acts on L ( p )by scalars. (vii) L ( p ) has only a finite number of weights. We set W = W(g,b).If a is a weight of L ( p ) then n is integral. 0.2.3(3) implies that there is s E W such that sa is dominant integral. Thus in light of (iii)we need only show that there are only a finite number of dominant integral weights. We may (in light of (vi)) assume that 3(9) = 0. But then the integral forms are in a lattice in b*. If a is a dominant weight then (T = p  Q with Q a sum of elements of P. Thus (0, a> =
(P

Q, a> 5 = ( P , P  Q> 5
( K P>.
Thus the dominant weights are contained in the intersection of a discrete set and a compact set. This proves our assertion. It is not hard to show that the weight spaces of M ( p ) are all finite dimensional. (One must show that the weight spaces of b on U(n) are finite dimensional.) Hence (vii) completes the proof of (3). 1.7.5. Let G be a compact Lie group with maximal torus T. For the rest of this section we will use the notation g for the complexification of the Lie algebra of G. We will also write b for t,. Then g is a reductive Lie algebra over C and b is a Cartan subalgebra of g. We may thus continue with the notation of the previous paragraphs. Let (n,H) be an irreducible (unitary) representation of G. Then an isotypic component for T (1.4.7) is a weight space for 5. We will thus use the notation H ( p ) for the p weight space and also think of p as a character of T (0.3.2).In particular we will look upon the highest weight of H as a character of T. We now assume that G is connected. Let G" be the simply connected covering group of G. Let p be a dominant integral functional on b that is also 7'integral (0.3.2). Then there is a representation n of G on L ( p ) whose differential gives the action of g. Let Z denote the kernel of the covering homomorphism of G onto G. We assert that 2 is contained in Kern. Assuming this for the moment, we have
Theorem. Let p be a dominant integral, Tintegral form on 6. Then there exists an irreducible unitary representation (n,,,F") of G whose differential is equivalent to the gmodule L(p). Let y,, denote the equivalence class of rc,,. Then G" = {y,,: p dominant integral and Tintegral}.
We must show that 2 is contained in Kern,,. Let p be the covering projection of G" onto G. Set T" = p  ' ( T ) .Then C / T  is a covering space
39
1.8. Further Results and Comments
of G / T . Since G / T is simply connected (0.3.3(4)), this implies that T“ is connected. Since Z is a subgroup of T” we see that p ( Z ) = 1. Z is easily seen to be central, so Schur’s Lemma completes the proof.
1.8.
Further results and comments
This section contains some results that are related to the material of this chapter. Some of them will be referenced to the literature and others will be left as exercises to the reader. They will not be used in the body of this book. 1.8.1.
The material in Section 1.3 is strongly influenced by the material in Bore1 [l] on irreducible square integrable representations. We note that there is a slightly more general notion of square integrability which we will now discuss (we use the notation of Section 1.3). Let Z be the center of G. Let d ( Z g ) be a right invariant measure on Z\G. If x E ZAthen we write L’(G; x) for the space of all measurable complex valued functions on G such that 1.8.2.
,f(zg) = X ( Z ) f ( S )
Ilfll’
=
for z
E
z,9 E G
and
j If(zu)12md< a.
%\G
We set ( n , ( g ) f ) ( x )= f ( x g ) for x, g E G and f E L’(G;z). Then (n,, L’(G; x) is a unitary representation of G. Let ( n , H ) be an irreducible unitary representation of G. Then Schur’s lemma implies that there exists x E ZAsuch that n(z)= x ( z ) I for z E 2.We say that (n,H ) is square integrable modulo, the center with central character x, if there exist u, w E H  (0) such that cU+ E L’(G;x). The analogue of Lemma 1.3.3 is true in this context. The orthogonality relations (Proposition 1.3.3)also have an analogue. Here (71, H ) and (a, V ) should be taken to have the same central character and the integration should be over Z\G. The proofs are essentially the same as those of Section 1.3.
1.8.3. As we indicated in Section 1.5, the notion of induced representation that we introduced is a special case of a more general theory. The interested reader should consult Warner [l], Chapter 5 for a comprehensive account of induced representations of Lie groups and for a complete set of references to the vast literature.
40
1.
Elementary Representation Theory
1.8.4. We now use the notation of Section 1.6. DixmierMalliavan [l] have proved that if (x,H) is (say) a Hilbert representation of G then H" is the span of the spaces x ( f ) H with f a smooth compactly supported function on G . This result allows one to give a simple proof of the following result.
Theorem. Let P and K be closed subgroups of G with K compact. Let ( 0 , W )be a Hilbert representation of P and let (H,)" denote the space of all smooth elements of with the topology of uniform convergence on compacta with all derivatives. Then (xo,(H,)m) is a smooth Frechet representation of G that is equivalent to ( T IH , "). The Verma modules (1.7.4(ii)) will be studied in more detail in Chapters 4,6 and 9. There is a vast literature on this subject. The best reference is Dixmier [I], Chapter 7. 1.8.5.
2
Real Reductive Groups
Introduction In this chapter we introduce the class of real Lie groups that we will be studying throughout this book. The definition of a real reductive group that we give in Section 1 can be shown to be the same as that in Borel, Wallach [l, 0.3.11 if we add the condition of inner type. We have opted to give the more cumbersome definition since it allows an extremely elementary entry into the fine structure of these groups. We hope that the experts will not become too impatient with our presentation of the material. To the less expert reader we wish to issue a warning about some of the proofs in this chapter. Although, at first sight, they seem to be complete (indeed, possibly overdetailed) there are many points that have been left to the reader. Also the examples in this chapter should really be looked upon as exercises. The first section of this chapter gives the definition and basic structure of real reductive groups. It contains the Cartan and Iwasawa decompositions of these groups. The second section is, perhaps, the most important section of this chapter. It introduces the notion of parabolic subgroup and of parabolic pair. The theory of parabolic subgroups makes the harmonic analysis on real reductive groups tractable, since it reduces many problems on a real reductive group to corresponding problems on the Levi factors of these subgroups. In
41
42
2.
Real Reductive Groups
Section 3 we show how the theory of parabolic subgroups can be used to study Cartan subgroups of real reductive groups. The relationship between Cartan subgroups and cuspidal parabolic subgroups is one of the basic ingredients in HarishChandra's Plancherel formula. Section 4 contains integration formulas associated to various decompositions of real reductive groups that are consequences of the results in earlier sections. In the final section of this chapter we show how to use the Weyl integral formula to derive the Weyl character formula. We include this material since it contains many of the ideas that will be used in our exposition of the theory of square integrable representations of real reductive groups.
2.1. The definition of a real reductive group Let F = R or C. Let (as usual) M,(F) denote the space of all n x n matrices over F. Let GL(n,F) denote (as usual) the group of all invertible elements of M,(F). Let fi,.. . ,f, be complex polynomials on M J C ) such that each is real valued on M,(R) and such that the set of simultaneous zeros of the in GL(n,C) is a subgroup, G,, of GL(n,C). Then Gc is called an afine algebraic group dejned ouer R. The subgroup, GR = G, n GL(n,R)is called the group of real points. If in addition, g* E G, for g E G, then G, is called a symmetric subgroup of G L ( n , C ) . We define an automorphism 8 of GR by =W 1 ) * . Let C, be a symmetric subgroup of GL(n,C) with real points G R . By a real reductive group we will mean a finite covering, G, of an open subgroup Go of GR. Thus the statement "G is a real reductive group" carries with it all of the above data. We will also write p for the covering homomorphism from G onto Go. We will identify the Lie algebra of G with that of GR.Thus we can define on g, the Lie algebra of G , an involutive automorphism, 8, given by 8 ( X ) =  X * . This automorphism is usually called a Cartan involution. 2.1.1.
2.1.2. Examples 1. GL(n,R).GL(n,R) is clearly a real reductive group. 2. SL(n, R). Let SL(n, F ) be the subgroup of GL(n,F ) consisting of all g with det(g) = 1. Then all of the hypotheses are satisfied by SL(n, R). 3. GL(n,C). Here we look upon C"as R2"and multiplication by idenoted by J . Then GL(n,C) is the subgroup of GL(2n,R) given by the equations g J Jg = 0. We can choose the identification of C" with R2" so that J * =  J . Thus the conditions in the definition are satisfied.
43
2.1. The Definition of a Real Reductive Croup
4. S L ( n , C ) . S L ( n , C ) = ( g E M J C ) ldet g = I}. We leave it to the reader to
show that SL(n, C ) is a real reductive group. 5. O ( p ,q). Let p and q be nonnegative integers with p q = n > 0. We look upon R“ as the direct sum of R Pand Rq.Let Ip,¶ be the operator on R ” given by I on R P and  I on R4.Then O ( p , q )is given by the equations glp,sg* = I p , ¶ . Clearly, O(n,0) = O(0, n) is compact. We write O(n) for O(n,0). 6. SO( p , q). SO( p, q) = O( p , 4) n SL(n, R). We write
[email protected]) for SO(n,0). 7. U ( p ,q). We look upon C P + ¶ as the direct sum of C p and C4. These in turn we identify with real vector spaces of twice the dimension as in Example 3. Then U ( p ,q) = GL(n,C)n O(2p,2q)(n = p q). We write U ( n )for U ( n ,0). 8. S U ( p , q ) .S U ( p , q ) = U ( p , q ) n S L ( n , C )We . write SU(n)for SU(n,O). 9. Sp(n,R). We take J on Rz”as in Example 3. Then Sp(n, R) is given by the equations g J g *  J = 0.
+
+
The above list just gives some of the socalled classical groups over R. We now give a general “example”. In the proof of the next Lemma we will use several standard concepts that we have not yet defined. The point of this lemma is to reassure the experts that our concept of real reductive group is the “usual one”.
2.1.3.
Lemma. A connected semisimple Lie group with .finite center is a real reductive group. Let G be as in the statement. Let g be the Lie algebra of G. Let H be a Cartan involution of g. Then if B is the Killing form of g, the form ( X , Y ) =  B ( X , O Y ) is an inner product. Let X , , . . . , X, be an orthonormal basis of g relative to ( , ). We use this basis to look upon g as R“. Let Gc be the automorphism group of gc. If g E Gc then g* = 0 conj y  ’ conj
0
(here conj is complex conjugation in gc relative to g). Also GR = Aut(g). Set Go = (GR)’. Then Ad is a covering homomorphism of G onto Go. Thus all the conditions are satisfied. Let G be a real reductive group with Lie algebra g. We assume all of the data in 2.1.1. Let B ( X , Y ) = tr X Y for X , Y in g. If X E g then 0 ( X ) =  X * E g. ( X , Y ) =  B ( X , O Y ) defines an inner product on g. Hence B is nondegenerate. Set f equal to the + 1 eigenspace for 0 in g and set p equal to the  1 eigenspace of H in g. Then the decomposition g = f 0p is called a Cartan decomposition of g. One has: 2.1.4.
(1)
f is the Lie algebra of a compact subgroup of G.
2. Real Reductive Groups
44
Indeed, € is the Lie algebra of GR intersected with the orthogonal group >. of
0} and OG {g E G f det(g)2 = l}. (2) Let G = GL(n,C).G = G', S is as in (l), OG = {g E G Idet(g)l = 1). For all of the other examples in 2.1.2, G example.
=
G+
= OG.
=
We give one more
(3) G = GSp(n,R). Let J E GL(2n,R) be as in 2.1.2 Example 3. Then GSp(n, R) is the subgroup of all g E G1(2n, R)such that g J g * is a scalar multiple of J . We leave it to the reader to check that GSp(n, R)is a real reductive group. Then S is as in the above examples. G = G'. OG = {g E G I gJg* = * J } . 2.2.4. Let a, Q, = @(g, a), etc. be as in 2.1.6. Let rn = { X E g [ X , a] = O}. Set M equal to the set of all g E G such that Ad(g) is I on a. This is clearly an algebraic condition that is invariant under taking adjoints. Thus M is a real reductive group. The standard split component of M is A since m = a 0O n t (2.1.6). It is also clear that OM = M n K . 2.2.5. Let t be a maximal abelian subalgebra of Om. Set Ijo equal to the complexification of (1)
=t
0a. Set b
Ij is a Cartan subalgebra (0.2.1)of g r .
If X X =U
E
g and if X commutes the elements of Ij then so does OX. Hence V with U E €, V E p and both U and V commute with the elements
+
50
2. Real Reductive Groups
of 6. But then V must be in a and U must be in Om.Hence V must be in t. We have therefore shown that b is maximal abelian in g. If X is in t or a then X acts semisimply on gc. Thus the elements of b act semisimply on gc. So b is a Cartan subalgebra of gc. Let @(g,, 6) be (as usual) the root system of gc relative to 6. It is obvious that (2)
6)
@D(g,a)
= @(gc>
la 
(0).
Since the elements of @(gc, 6) are real valued on a and take pure imaginary values on t it follows that (see 0.2.2)
(3)
bR
= (it
+ a)
[gC,
Let H , be an element of a‘ n bR.Let H , , . . . , H, be a basis of bR. We order @(gc, b) lexicographically relative to this basis. Let R denote the corresponding positive root system (0.2.4). Let R , be the set of all p E @(g, a) such that p ( H , ) > 0. Then R, is a system of positive roots for @(g,a) (2.1.10).Then it is clear that RI,  (0) = R,. Let A (resp. A,) be the corresponding system of simple roots for R(resp. R , ) (0.2.4, 2.1.10). Set F, = { a E A I ala = O}. Then
(4)
(A

Fo) la = A,.
Indeed, if p E A. and if ct E R restricts to p then c( = p1 + ... + 8, with pj simple in R. Only one of the pj can have a nonzero restriction to a since p is simple in R,. This implies (4). (5)
A,, is a linearly independent subset of a*.
Indeed, if p E b* set conj(p)(H) = conj(p(conj(H)))for H E 6 (here conj(X) is conjugation in gc relative to 9). If ct E R then its restriction to a is given by (a conj(a))/2. Let A  F, = { a , , . . . , a,}. Then there is a permutation j H j ’ of 1,. . ., r such that conj(ctj) = aj. + C a o F on,a. ( 5 ) follows easily from this.
+
2.2.6. Let F be a subset of A,. We set a, = { H E a I p ( H ) = 0 for p E F } . Set m, = {X E g I [ X , a,] = {O}}. Put MF = {g E G I Ad(g)H = H for H E a}, and A , = exp aF. Then MF is a real reductive group with (M,)’ = M , = O(M,). Also relative to 8 the split component of MF is A,. Let R, be the subset of those roots in R, whose restriction to a, is nonzero. Set nF
=
@ g”.
PER
Let NF denote the connected subgroup of G with Lie algebra n,
51
2.2. Parabolic Pairs
Lemma. (1) nF is a nilpotent Lie subalyebra of g. (2) I f X E nF then ad X is nilpotent on 8.
Let H E aF be such that p ( H ) > 1 for all p E A,  F. Set n = nF. Put n, = [n,n] and n j+, = [nj,nj]. Then, recall that n is nilpotent if nk = ( 0 ) for k large. Since ad H has all of its eigenvalues greater than or equal to j on nj there must be an index such that nj = ( 0 ) . Let c be the lowest eigenvalue of ad H on g. Then (ad X ) m g is contained in the sum of the eigenspaces of ad H with eigenvalue at least c m. This implies (2).
+
2.2.7. Set PF = MFNF. Then PF is called a standard parabolic subgroup of G. The word standard has to do with the choices of a and R,. The pair (PF,AF) will be called aparabolic pair (ppair for short). Lemma 2.2.2 implies that under the multiplication mapping MF is isomorphic with A , x OMF. We have Lemma. ( 1 ) The map MF x NF 4 PF given by m, n H mn is a surjective difleomorphism. (2) The map OMF x A , x NF + PF given by m, a, n H man is a surjective diffeomorphism.
It is enough to prove (1) since (1) combined with Lemma 2.2.2 implies (2). It is an easy calculation to see that the differential of the map in (1) is everywhere regular. Thus (1) will follow if we show that the map is injective. So suppose that m, m , E MF and n, n1 E NF and that mn = m l n l . Then m,m' = n(n,)'. Hence we must show that MF n NF = (1). Let H be as in the proof of Lemma 2.2.6 and set a, = exp t H . Then lim a,na_, = 1 ,a,
for all n E NF. Since the a, are central in MF, this clearly implies that MF n NF = { l}. So the lemma follows. The decomposition in (2) is called a Langlands decomposition of PF. PB is called a minimal parabolic subgroup of G. The PF are standard relative to PD.
2.2.8. We say that a real reductive group is of inner type if Ad(G) is a subgroup of Int(g,). Lemma.
Let G be a real reductive group of inner type. Let ( P F ,AF) be as above.
(1)
MF is a real reductive group of inner type.
(2)
K o P F = G.
52
2.
Real Reductive Groups
Set K F = K n M,. Then Theorem 2.1.8 implies that MF = KF(MF)O.Thus it is enough to show that Ad(K,) is contained in Int((mF)c).Set * a F = a n Om,. Let k E KF. Then there exists u E (K,)Osuch that Ad(u)*a, = Ad(k)*a,. Thus we may assume that Ad(k) stabilizes *aF. But then Proposition 2.1.10 implies that we may assume that Ad(k) restricted to *aF is the identity. That is, we may assume that k E OM. Let t be as in 2.2.5. Then if we argue as above we may assume that Ad(k) is the identity on t . Let g, = € @ i p . Then the connected subgroup, G,, of Int(g,) corresponding to g, is compact by Theorem 0.3.1. t 0 ia is the Lie algebra of a maximal torus, T. of G,. Hence, 0.3.3(2) implies that Ad(k) E T, which is a subgroup of Int(g,). This proves (1). In the first part of this proof we have shown that K = OMDKo. Thus KoPD = G which proves (2).
2.2.9. GL(n, R). g = M,(R). We take a to be the diagonal matrices. Let be the matrix with 1 in the i, j position and 0's everywhere else. If H is the diagonal matrix with h i , .. . , h, on the main diagonal we set E ~ ( H=) hj. Then @(g,a)=(EiEj i # j ) . We take R o = { c i  E j I i < j } . A 0 = q  c z , c ; ~  E ~ , . . . , E ,  ~  E , . If m , , ..., m, are positive integers adding up to n then we set P ( m , , ..., m,) equal to the subgroup of all matrices in the following block form: First we write every matrix in the form [ A , ,j ] with A i , an mi by mj matrix. Then the form of the elements of P ( m l ,. . ., m p ) is A,, = 0 for i > j . This describes all standard parabolic subgroups of GL(n,R).We leave it to the reader to find which subset of A. corresponds to m , , . . . , m,. 2.2.10.
We now give a proof of an important Theorem that is usually known as the Bruhat Lemma. This result was first proved by Bruhat for the classical groups. The general result is due to HarishChandra [4]. We will follow HarishChandra's original argument. Fix ( P D ,A,) = (P,A ) , a minimal ppair. Let N,(A) = { g E G I Ad(g)a = a}. Set W(G,A ) = N , ( A ) / M . We look upon W ( G , A ) as a group of linear automorphisms of a. We leave it to the reader to prove that
(1) If G is of inner type then W(G,A ) = W(g,a). If s E W(G,A ) then we can choose s* E K such that k choice for each s E W(G,A ) .
E
s. We fix such a
Theorem. Assume that G is of inner type. Then G is the disjoint union of the sets Ps*P, s E W(G,A ) .
2.2.
Parabolic Pairs
53
We have seen that G = K P. Thus to prove that G is the union of the asserted subsets of G, it is enough to prove that if k E K then k E Ps*P for an appropriate s E W = W(G,A ) . Fix k E K . (2) p (the Lie algebra of P ) is the sum of Ad(k)p n p and n. Recall that ( X , Y) =  B ( X , UY) defines an Ad(K)invariant inner product on g. Relative to this inner product p' = 8n. Thus (p + Ad(k)p)' = O(n n Ad(k)n) = O(n n (ad(k)p n p)). It is also clear that dim(p
+ Ad(k)p) + dim(p + Ad(k)p)'
= dim
9.
We therefore have dim((p n Ad(k)p)
+ n)
+ dim It dim((Ad(k)p n p) n n) = dim(p n Ad(k)p) + dim 11 dim(p + Ad(k)p)' = dim(p n Ad(k)p) + dim 11 dim g + dim(p + Ad(k)p) = dim p + dim Ad(k)p + dim n dim g = 2 dim p + dim n dim 9. Since dim n = dim On and dim p + dim On = dim 9, the above equations imply that dim((Ad(k)pn p) + 11) = dim p. Ad(k)p n p is a subspace of p thus =
dim(p n Ad(k)p)



(2) follows.
(3) If X X E 3(g)
E
p and if ad X has real eigenvalues (as an endomorphism of 9)then
+ a + n.
Let H and a, be as in the proof of Lemma 2.2.6 for F = 0. Then Ad(a,)Y = Y for Y E m and Iim,+ I Ad(u,)X = 0 for X E It. If x E g then ad(Ad(a,)x) = Ad(u,)ad x Ad(u,)'. So ad x and ad(Ad(a,)x) have the same eigenvalues. Assume that X E p and that ad X has real eigenvalues. Then X = Y + 2 with Y E rn and Z E 11. If we take the limit to  a3 of Ad(u,)X then we see that Y has real eigenvalues. Now Y = U + h with U E O m and h E a. Since ad h has real eigenvalues, this implies that ad U has real eigenvalues. The elements of ad(Onr) have purely imaginary eigenvalues. Hence ad U = 0. This proves (3). ~
(4)
If h E a' then Ad(N)h = h
+ n.
If X E tt then eadXh= h + C j , o (ad X ) j h / j ! E h + n. If X E n then set d(X) = Ad(exp X ) h  h. Then d & ( X ) = [ X , h] for X E n. This implies that
54
2. Real Reductive Groups
there is an open neighborhood of 0 in n such that 6 ( U ) is an open neighborhood of 0 in n. We now prove (4). Let X E n then there exists t > 0 such that Ad(a_,)X E 6 ( U ) (see the proof of (3)). Thus X E Ad(at)6(U)= 6(Ad(at)U). Hence 6 is surjective, which is the content of (4). Let h E a'. Then (2) implies that there exists X E n such that h X E Ad(k)p n p. (4) says that there exists n E N such that h X = Ad(+ This implies that there exists y E p such that Ad(n)h = Ad(k)y. In particular this equation implies that ad y has real eigenvalues. Thus, ( 3 ) implies that y = z + h, + u with z E 3(9), h , E a and u ~ n As. above ad(y) has the same eigenvalues as ad@,). Thus h, E a'. So ( 3 ) implies that y = Ad(n,)(z h,) for some n , E N. Recall that g is identified with the Lie algebra of G R .Thus if g E G and if x E g then x and Ad(g)x have the same eigenvalues, set g = n  l k n , . Then Ad(g)(z + h , ) = h. Thus (if we compare eigenvalues) z E a. Thus we may use the notation h , for z + h , . m = ker(ad h) = Ker(ad h,). Thus Ad(g)m = m. Since Ad g preserves eigenvalues, Ad(g)a = a. Thus there exists s E W such that g E s*M. But then k E Ns*P. To complete the proof we must show that if t, s E W and if p, p1 E P and if ps* = t*p, then s = t. Let h E a then Ad(p,)h = h X, Ad(p)sh = sh X, with X, X , e n . Thus t h + Ad(t*)X, = sh + X . Ad(t*)X, = U V with U E n and V E On. Thus sh X ,  U = th V. Thus sh = th. Since h is arbitrary in a, this implies that s = t.
+
+
+
+
+
+
+
+
2.2.11. We will now apply this result to prove the socalled GelfandNaimark decomposition (which first was proved for general groups and minimal parabolics by Moore [l]). Let F be a subset of A. and let ( P F , A F be ) the corresponding ppair. Let PF = MFNFas usual.
Corollary. Assume that G = G t . The map of O(N,) x PF to G given by x, p H xp defines a difleomorphism onto an open subset oj G whose complement has measure 0 relative to dg. We first prove the result in the case when F = 0. We use the notation of the last number. We observe that W is a finite set. Indeed, the elements of W permute the roots and are completely determined by the corresponding permutation. If s E W then Ad(s*)n = C a z Og"". Thus Ad(s*)n = (Ad(s*)n)n n (Ad(s*)n)n On. Let Us(resp. V,) be the connected subgroup of G with Lie algebra (Ad(s*)n)n n (resp. (Ad(s*)n)n en).
+
(1)
s*N(s*)'
=
v,us.
2.2.
55
Parabolic Pairs
Let y(u,u) = uu for u E V,, u E U,. Then dy,,,(X, Y) = X + Y for X E v,, Y E U , . Thus the image of y contains an open neighborhood of 1 in s * N ( s * )  ' . Fix H E a such that s a ( H ) < 0 for all positive roots, a. Set a, = exp t H . If x E s*N(s*)' then lim,+ma,xu, = I . Since, u,V,u, = V, and a,U,a, = Usour usual argument now implies (I).
(2) If x E ON, p subset of G.
E
P set p(x, p ) = x p . Then fi is a diffeomorphism onto an open
+
We first assume that G = Go. Then d p J X , Y ) = x X p xpY for X E On, Y E p. If this expression is 0 then X = pYp'. The right hand side of this equation is in p and the left hand side is in On. Since these two spaces have 0 intersection, this implies that /? is everywhere regular. If p(x, p ) = p(xl, p , ) then (xl)lx = pIp'. Let H, a, be as above for s = 1. If Y E ON and lim,+m a,ya_, exists then it is easy to see that y = 1. O n the other hand, it is easy to see that lim,+ u,qa, exists if q E P. So 1 = (xl)Ix = plp'. Thus x = xl and p = p l . So fi defines a diffeomorphism onto an open subset of Go. Let G now be arbitrary (subject to our hypotheses). Let p also denote the corresponding mapping for G. Let 4 be the covering homomorphism of G onto Go. Then qp is everywhere regular by the above. Since the center, Z , of G is contained in P and Z n O N = @, it is not hard to see that p is a diffeomorphism onto an open subset of G. Fix t E W such that ta is negative for all positive roots a. Then G is the disjoint union of the sets (t*)l
P(rs)*P,
sE
w.
Now P(ts)*P = N ( t s ) * P = ( t s ) * ( ( t s ) * )  ' ) N ( t s ) * P= (ts)*V,P (by (1)). (2) implies that V,,(ts)*P is a submanifold of G of dimension dim dim P. Thus, if is not equal to ON then ( t * )  ' P ( t s ) * P is a submanifold of G of lower dimension. Hence up to a set of measure 0, G is the union of the sets
v, +
v,
(t*)'(ts)*(ON)P,
V,, = ON.
If V, = ON then tf's preserves the Weyl chamber. Thus t's = I. So s = t . Thus if V,, = ON then s = 1. The corollary now follows in this case. Now let F be arbitrary. It is clear that O(NF)PF 3 O ( N , ) P, . Hence, 8(NF)pFhas total measure in G. Let p F ( x ,p ) = x p for x E O(NF),p E P F . Since g = O(n) 0 p, the argument used to prove (2) shows, in this case, that PF is everywhere regular. If we use H E aF such that a ( H ) > 0 for a E @(PF,A F ) and argue as we did for we find that is injective.
56
2.
Real Reductive Groups
2.3. Cartan subgroups
Let G be a real reductive group. Then a Cartan subalgebra g is a Lie subalgebra, 6 such that I), is a Cartan subalgebra of gc. We define the polynomials Dj on g by
2.3.1.
det(t1  ad X ) =
1t J D j ( X ) .
Let 1 be the dimension of a Cartan subalgebra of gc. Then using the theory of complex reductive Lie algebras (0.2.1) one sees that Dk = 0 for k < 1. We set D = D,. Then D is a nonzero polynomial function on g. Set g’ = { X E g D ( X ) # 0). Then g’ is open and dense in g. Let Int(g) denote the group of automorphisms of g generated by the automorphisms of the form exp(ad X ) for X E g. As is well known Int(g) = Aut(g)’. If X E g is such that ad X is asemisimple endomorphism of gc then we say that X is semisimple. Lemma. (1) 1 j X E g ’ then X is semisimple and C,(X) = { Y E g I [A’, Y ] = 0} is a Cartan subalgebra of 8. ( 2 ) I f X is a semisimple element of g then C , ( X ) is a reductive subalgebra of g that contains a Cartan subalgebra. (1) is an immediate consequence of 0.2.1. We now prove (2). Let u = C,(X). Let V, be the sum of the eigenspaces with nonzero eigenvalue for ad X in gc. Set V = V, n g. Then g = u @ V. Let q be the map of V x II to g given by 4 ( y , x) = exp(ad y ) x . Then d q o , z ( y , x )= ad Y Z + x.
Thus dq,,, is surjective. The inverse function theorem implies that there are open neighborhoods V of X in u and W of 0 in V such that 4(W,V) is open in g. Hence, if D is identically 0 on u then D is zero on g. Since this is contrary to our assumptions we see that g’ n u is nonempty. Hence (1) implies that u contains a Cartan subalgebra b of g. Let 6 be the complexification of bo. Set @ = @(gc,6). Let @, = { a E @ a ( X ) = O}. Then it is clear that ’lc =
0
0kc),. ‘I€@
Let a be an abelian ideal in u,. Then, in particular, a is invariant under ad Let Q be the set of all roots, a, such that (gc), is contained in a. One has a=
5.
b n a O @(gc),. asQ
If
c1
E
Q then, in particular, a ( X ) = 0. Thus, since a is an ideal in uc, it is a
2.3. Cartan Subgroups
simple matter to see that  a E Q. Since a is abelian, this implies that Q Thus a is central in uc. This implies that u is reductive.
57 = @.
2.3.2. Fix a Cartan involution 0, of y. Let B be a nondegenerate 8 and g invariant form on g such that ( X , Y ) =  B ( X , 8 Y ) defines an inner product on g. We say that 8 is associated with B. Lemma. If 8, is another Cartan involution of g that is associated with B then there exists x E Int(y) so that x8xI = 0,.
Set N = 88,. Then our assumptions imply that (NX, Y) = (X,NY) for all X , Y E g. Thus N 2 = exp W with W a selfadjoint endomorphism of g. The condition that exp W is an automorphism of g is a polynomial condition. Thus, since exp(mW) is an automorphism for all integral m, exp tW is an automorphism for all t E R (2.A.1.2). But then W is a derivation of g. Hence W = ad X for some X E g. Since 8,NO, = N', 2.A.1.2 implies that 8, exp t X 8 , = exp(  t X ) for all t E R . We therefore see that if x = exp((1/4)ad X ) then x0xl commutes with 8,. Since both 8 and 8, are Cartan involutions associated with B this implies that 8, = x 8 x p ' . The above argument is due to Mostow [l]. 2.3.3. Lemma. Let lj be a Cartan subalgebra of g. Then there exists x Znt(g) such that xlj is 8invariant.
E
We may assume that g is semisimple since every Cartan subalgebra contains the center. Let u be a compact form of gc such that u n h c is a maximal abelian subalgebra of u (0.3.4). Let y denote conjugation on gc relative to u and let a denote conjugation on gc relative to g. Let B denote the Killing form of gc. Set ( X , Y) =  B ( X , y Y ) for X , Y E gc. Then ( , ) is an inner product on gc. Set N = ay.Then ( X , NY) = (NX, Y) for all X , YE gc. Thus if we argue as in the proof of Lemma 2.3.2 we see that N 2 = exp(ad X ) with X E iu. We note that a N a = N  I and yNy = N  ' . Hence the usual argument shows that y exp t ad Xy = exp(  t ad X ) and a exp t ad Xa = exp(  t ad X ) . From this it is easy to deduce that if y = exp((l/4)ad X ) then p = yyy' commutes with 0.The restriction to g of p is a Cartan involution of g, 8,, associated with B. Lemma 2.3.2 implies that there exists z E Int(g) such that 8, = z ~ z  ' .Then x = zy is the desired element of Int(g). 2.3.4. Let lj be a &stable Cartan subalgebra of g. Then we say that lj is a maximally split Cartan subalgebra of g if lj n p is maximal abelian in p. We say that lj is fundamental if lj n f is maximal abelian in f .
2.
58
Real Reductive Groups
Lemma. Fundamental and maximally split Cartan subalgebras exist. Furthermore, any two fundamental (resp. maximally split) Cartan subalgebras are conjugate under Int(g).
The Cartan subalgebra in 2.2.5 is clearly maximally split. Let t be a maximal abelian subalgebra of f. Let a, be maximal abelian in p subject to the condition that [t,al] = 0. Set t), = t + a , . We may argue as in 2.2.5 we see that 5, is a Cartan subalgebra of g which is clearly fundamental. Let K , denote Ad(K'). Let bj, j = 1, 2, be maximally split Cartan subalgebras of g. Let aj = bj n p , j = 1,2. Then Lemma 2.1.9 implies that there exists k E K , so that ka, = a,. Thus we may assume a, = a, = a. Let t j = f n bj, j = 1, 2. Then t j is maximal abelian in 'm (2.1.6),j = 1, 2. Thus there exists m E Ad('M0) such that mt, = t, (0.3.3 (1)). This completes the proof in the case of maximally split Cartan subalgebras. Let Q, j = 1, 2 be fundamental Cartan subalgebras for g. Set t j = f n bj for j = 1, 2. Then there exists k E K , such that k t , = t,. We may thus assume that t , = t , = t. Let u = C,(t). Then u is reductive (Lemma 2.3.1) and @invariant. Let aj = p n bj for j = 1, 2. Then each a j is maximal abelian in p n u. Hence Lemma 2.3.1 implies that there exists u E Int(u) n K , with ua, = a,. The result now follows.
2.3.5. Let real root of Lemma.
b be a Cartan subalgebra of b if CL is real valued on b.
9.If a E @(gc,bc) then a is called a
b is fundamental if and only if it has no real roots.
Let t, be a Cartan subalgebra of g which we assume (as we may) is 0invariant. Let CI be a real root for IJ and let 4 be the corresponding TDS in gc. Then 5' = s n g is a @stablesubalgebra of g isomorphic with sl(2,R). We can clearly choose a standard basis X , Y, H of so such that H E b n p and OX =  Y. Since CI is real, it follows that a is 0 on t = bnf. Thus R(X  Y) t is an abelian subalgebra of f. This shows that if t, is fundamental then has no real roots. We prove the converse by induction on the dimension of g. If dim g = 0 the result is obvious. Assume the result for all reductive Lie algebras of smaller dimension. Suppose that b is &stable and that IJ has no real roots. Set t = n f. Let u = C,(t). If t is nonzero then u is reductive, 0stable and of lower dimension. Thus IJ n u is fundamental in u. But u clearly contains a fundamental Cartan subalgebra of g. Hence b is fundamental in g. If t = (0) then
+
2.3. Cartan Subgroups
59
all of the roots are real. But then 9 is abelian and so the result is also true in this case. Let a be maximal abelian in p. Fix G = A N K an Iwasawa decomposition of G. Then a standard ppair, (PF,AF),is said to be cuspidal if OmFhas a Cartan subalgebra, ,t completely contained in f. Set 6, = t F aF.
+
Proposition. Let t) be a Cartan subalgebra of g. Then there exists a standard, cuspidal, ppair, ( P F , A F )and , x E Int(g) such that xt) = bF.
We may assume that t) is &stable and that 4 n p is contained in a. Let Qo denote the set of roots of a that are nonzero on t) n p = a,. Let H E a, be such that a ( H )is nonzero for all a E Q,, . There is s E W(g,a) so that a ( s H ) > 0 for all a E P (the positive system corresponding to the choice of n, Lemma 2.1.10(2)). Let k E s. We replace t) by Ad(k)b. Let F be the set of all a E A0(2.2.5) that vanish on H. Then, n p is contained in aF. The result now follows. 2.3.6. Let t) be a Cartan subalgebra of g. Then a subgroup of the form C,(b) = { g E G I Ad(g)Ih = I } will be called a Cartan subgroup of G. A standard ppair, ( P F , A F )is, cuspidal if and only if OMF has a compact Cartan subgroup, TF. In this case H F = TFAF is a Cartan subgroup. Proposition 2.3.5 immediately implies: Proposition. If H is a Cartan subgroup of G then there exists a standard cuspidal ppair, (PF,AF), and g E Go such that gHg' = HF. If H is a Cartan subgroup of G then we call H fundamental (resp. maximally split) if t) is fundamental (resp. maximally split).
2.3.7. A parabolic subgroup of G is said to be maximal if it is proper and is not properly contained in any parabolic subgroup of G. The maximal parabolic subgroups of G are conjugate to the subgroups, PF, with F of the form A,,  { a } with a a simple root. Proposition 2.3.6 implies that if His a noncompact Cartan subgroup of G and if G = ' G then there is a maximal parabolic subgroup, PF, of G such that H is Int(g) conjugate to a Cartan subgroup of M F . This gives an inductive technique for finding all Cartan subgroups up to conjugacy. Let us give some examples. 1. SL(n,R). Let us denote by P ( m , , ..., m k ) the intersection with SL(n, R) = G of the groups so designated in 2.2.9. Then if k = 2, P ( m , , m 2 ) is
2.
60
Real Reductive Groups
maximal. If n > 2 then G has no compact Cartan subgroups. The cuspidal parabolics correspond to the cases when mi = 1 or 2 for j = 1,. . . , k. 2. S U ( p , q ) . We assume that p > q > 1. We choose a to be the space of all matrices h(t,, . . ., t,), t j E R,j = 1,. . ., q, given by
Set Ej(h(tl,.. .,t,)) = tj for j = 1,. . .,q. @(g, a) consists of E~ k E~ for i # j, * 2 e j for j = 1,. . ., q and if p > q, k~~for j = 1 , . . . , q. Choose the Weyl chamber corresponding to t , > t , > ... > t,.
If p > q (resp. p = q) then the simple roots are  E~ ,..., E ,  ~  E,, E, (resp. 28,). Set Hj = h(t,, .. .) with ti = 1 for i < j and ti = 0 for i > j . Then the m’s for maximal standard parabolics are of the form C,(Hj),j = 1,. .. ,q. We leave it to the reader to describe the Cartan subalgebras of 9. 2.4. Integration formulas 2.4.1. Let G be a real reductive group. Fix 8, a Cartan involution, and G = N A K , an Iwasawa decomposition of G. Let ( P F , A F )be a standard ppair. If p E (aF)* and if H E aF we write a” = exp p ( H ) if a = exp H. We define p F E (aF)* by p F ( H )= (i)tr(ad H
lnF).
Lemma. Let dn, da, dm be respectively invariant measures on NF, A,, OMF.Let dk be the normalized invariant measure on K . Then we can choose an invariant measure dg on G such that
j f ( 9 )4 = N F
G
for f
E
X
s
AF X ’ M F X
KF
f (namk)a2PF dn da dm dk,
C,(G). Also if u E C ( K ) then
j u(k)dk
K
=
s
KXKF
u(kFk(kg))a(kg)2PF dk, dk
2.4.
Intergration Formulas
here
if g E G and if g
61 =
nak, n E N , a E A, k
E
K then a(g) = a and k ( g ) = k.
Let d p denote a left invariant measure on P,. Then we can choose an invariant measure, dg, on G such that
j f ( s ) 4= PI; j K f ( P k ) d P d k
G
x
by Lemma 0.1.4. Thus we must show that up to scalar multiple d p = a2pr.dn da dm. Lemma 2.2.7 implies that d p = h(n, a, m)dn da dm with h a smooth function on NF x A, x OM. By left invariance h is independent of n. By definition of OM, the modular function, 6, of PF is 1 on OM,. Thus d p is right invariant under OMF. Hence h is a function of a alone. The Jacobian of the action n H m a  ' is det(Ad(a)I,) = a2pFfor a E A,. Thus a2pFdn da dm is left A,invariant. We now prove the second assertion of the Lemma. According to Lemma 0.1.3 there exists a continuous compactly supported function f on G such that kE K.
j f ( p k ) d p = J u(k,k)dk,, KF
PF
Thus we have
J f ( x ) d x = J u(k)dk.
G
K
Now,
We write kg = na(kg)k(kg)as above. d p transforms by 6 under right multiplication by elements of PF.Since G(na(kg))= a(kg)2pF,we have
j u(k)dk
K
=
J a ( k g ) 2 p F fpk(ky)) ( d p dk
PF x K
=
J u(k,k(kg)) dk, dk.
KFXK
As was to be shown. 2.4.2. For our next integration formula we assume that G is of inner type. Let R be the system of positive roots for @(g, a) corresponding to the choice of n. Set a' equal to the Weyl chamber corresponding to R (2.1.10). Set A + = exp(a+).If a E A, a = exp H , we set y(a) = naGR sinh(u(H)).
Lemma. d g can be normalized so that
62
2.
Real Reductive Groups
For simplicity of notation we will write M for OM. Let p: a' x K I M + p be defined by p ( H , k M ) = Ad(k)H. Let p' denote the range of p. Since Ad(K)a = p, Ad(K)a+ = Ad(K)a' and Ad(K)(a  a') is a finite union of submanifolds of lower dimension, p' is open, dense and has a complement of measure 0 in p. It is easy to check that /?is a diffeomorphism onto p' (Proposition 2.1.10). Let p: K x A + x K I M +G be defined by p ( k , a , x M ) = kxax'. The above remarks and Theorem 2.1.8 imply that p is a diffeomorphism onto an open subset of G that has a complement of measure 0. This implies that there is a smooth function h such that
J f(g) dg = K x AJ KIM h(k, a, x ) f ( k x a x  ' ) dk da d ( x M ) .
G
X
Since dg is left and right invariant it is easy to see that h is a function of only a. Let X, be a basis of n such that ad HXj,= ocj(H)X, for all j and H E a. Set rj = X, + OX,. Let 2, be a basis of rn and let H, be a basis of a. We may look upon the 5 as a basis of the tangent space at 1 to K I M . A direct calculation yields
~ p 1 , a , 1 ( ~ r n ~= O ~( ZOr)n l a , d ~ .a, 1
H, 0) = ( H j ) a 7
dpi,,,i(O,O,
9
= ((1  Ad
al)q)a.
It is now easily seen that the Jacobian of p at 1, a, 1 is
nj(aaJ 
1.
Hence h(a) is a constant multiple of ?(a). Since we are using normalized measure on K , M and K I M , we may replace the integration over K I M by integration over K . Since dk is invariant d(kx) = dk and d ( k  ' ) = dk. The result now follows. We continue our assumptions of 2.4.2. Proposition 2.3.6 implies that there exist &stable Cartan subalgebras bl,. . . ,b, that are mutually nonconjugate and such that every Cartan subalgebra of cj is conjugate to one of them (here conjugation is relative to Ad(G)). Let I l l , . .. , H, be the corresponding Cartan subgroups. Set 4= { g E G I Ad(y)bj = bj}. Then 4 contains Hi = and it is easily seen that Wj = N,.JHj is a finite group. u(H) Let 5 be a system of positive roots for @ ( g c , ( t ) j ) c ) . Set nj(H)= nmEp for H E bj. Let D be is as in 2.3.1. Then 1D(H)I = Inj(H)IZ.Since G and each
2.4.3.
63
lntergration Formulas
2.4.
Hj are unimodular, each coset space G / H j has a Ginvariant measure, dxj (0.1.2). Proposition. There exist positive constants c j , j of Lebesgue measure on g and the bi such that
=
1,. . . , r and normalizations
For the moment, fix j, and set 1ij = t), etc. Let p : G / H x 5' g be defined by p(gH, h) = Ad(g)h (here 5' = g' n 5). We may identify the complex tangent space at 1H to G / H with nc + I T  . Translating by the elements of G allows us to identify the tangent space at gH with this space. A direct calculation yields dpgH,h(X,
for X
E 11'
+ n,
Z
E
z,= Ad(g)(ad X h + z,
5. This implies
(1) The Jacobian of p at gH, h is D(h),up to sign. This implies that p is everywhere regular. The remarks preceding the statement we are proving now imply that pis a [W]fold covering of its range. Lemma 2.3.1 implies that g' is the disjoint union of the open subsets Ad(G)(5j)c. The result now follows from (1). The above result is sometimes called the Weyl integral formula for g. We now derive the Weyl integral .formula for G. We define real analytic functions dj on G by 2.4.4.
det(tZ  (Ad g

I)) =
tjdj(g).
Here n = dim G. Set d = dj for j = rank(g,). We set G' = { g E G I d(g) # 0). Then G' is open, dense with complement of measure 0 in G. We retain the notation of 2.4.3. Proposition. There exist positive constants mj so that if d g and dhj are respectively invariant measure on G and Hj then G
We fix j and for the moment drop the index j. Let o : G / H x H' defined by o(gH,h) = ghg' (here H ' = H n G').We have d'gH,h(X,Z)
= (Ad(g)((Ad(h')

+ Z))o(gH,h),

G be
64
2.
Real Reductive Groups
for X E n+ 0 n, 2 E b. The rest of the proof is now almost identical to that of Proposition 2.4.3 and we leave it to the reader. We now derive some integration formulas that are related to the GelfandNaimark decomposition. We will use the notation of 2.4.1. We set V, = ON,. Fix invariant measures dn, dm, da, dv respectively on N F , OM,, A , and V,. 2.4.5.
Lemma.
The invariant measure, dg, can be normalized so that
s f ( 9 )ds
=
G
s
a 2pFf(nmav)dn dm da dv ~
N F x ' M F x AFx V F
for f E C,(G). I f u E C ( K )then
j u(k)dk
K
s
=
KFXV
U(v)2PFu(k,k(U)) d k , dv.
Let p: N, x OMF x A , x V, + G be defined by p(n,m, a, u) = nmav. We have seen (2.2.11) that p is a diffeomorphism onto an open subset of G whose complement has measure zero in G . Thus there exists a smooth function, h, on N, x OM, x A , x V, such that
s f ( 9 )d g G
s
=
N p X O M Fx A F x V F
f (nmax)h(n,m, a, v) dn dm da dv.
As usual, the biinvariance of d g implies that h is a function only of a. We may
now argue as in the proof of Lemma 2.4.1 to complete the proof of the first integration formula. We now prove the second one. We may replace u by the function u(k) = J u(k,k)dk, K
and therefore assume that u(k,k) that
= u(k)for k , E K , .
Let c1 E C,(PF/K,) be such
jP 4 P ) d P = 1. Put h(pk) = a ( p ) u ( k )for p E P,, k
s f ( k )d k K
=
j h(g)d g
G
= PF
=
1 X
{
VF
PF x V F
As was to be proved.
=
NF x
E
K . Then
s
OMF
x
h(po)dpdv =
x VF
PF x V F
a2pFh(nman) dn dm da dv
h(pa(v)k(u))dpdv
s
a(o)2pFh(pk(u))dpdv = a(v)2pFu(k(u))do. V
65
2.5. The Weyl Character Formula
2.5.
The Weyl character formula
The purpose of this section is to show how to use the Weyl integral formula to prove the Weyl character formula. Let G be a compact Lie group.
2.5.1.
Lemma.
G is a real reductive group.
Since G" is countable (Theorem 1.7.5) we may write G" as {yl,yz, ...}. Let (nj,V,) E y j . We set Hj = V, with the direct sum inner product. Let p j be the direct sum representation. Let G j be the kernel of p j . Then Gj conGj = { 1) (1.4.4(1)).This implies that for some index, k, the tains Gj+ and Lie algebra of Gk is 0. Hence Gk is finite. Hence there is an index k' such that Gk, = { 1). Let pk,= p, Hk' = H. We look upon H as C" with the usual inner product, ( , ). We then look upon C" as Rz" in the usual way and take ( , ) = Re( , ). We identify G with its image in GL(2n,R). Let I be the set of all real valued polynomials on M,,(R) that vanish on G. Let P be the algebra of all real valued polynomials on M,,(R). Let M be the set of zeros in GL(24R) of I . If f E I then f(X*) = g ( X )defines g E P which is clearly in 1. Since M is the Zariski closure (c.f. Mumford [l, p.11) of G, M is an algebraic group. Thus A4 is a real reductive group. Let Pc be the algebra of complex valued polynomials on M,,(R). The StoneWeierstrass theorem implies that the restriction of P , to M is uniformly dense in C ( M ) . Let M act on Pc by mf(X)= f(Xrn)for f E Pc, X E M,,(R) and m E M. Since the space of homogeneous polynomials of a fixed degree is invariant under the action of M and is finite dimensional we see by 1.4.4(1) and 1.3.2 that the restriction of P, to M is precisely the algebraic sum of the isotypic components of L 2 ( M ) . But then the space of Ginvariants in P, restricted to M is uniformly dense in the space of G invariants in C ( M )under the right regular action. Now the condition that a polynomial be Ginvariant is itself a polynomial condition. Thus every Ginvariant polynomial is Minvariant. But then the Ginvariant continuous functions are Minvariant. This implies that C ( M / G ) consists of the constants. Hence G = M.
Oksj
n
Note. This lemma is an important part of the Tannaka duality theorem.
We now assume that G is connected. Let T be a maximal torus of G. Let t, = t,. Fix R a system of positive roots for @(gc,$). Let 6 E *)I be half the sum of the elements of R. Fix ( , ) an Ad(G)invariant inner product
2.5.2.
2. Real Reductive Groups
66
on g (0.3.1). Denote by ( , ) the induced symmetric nondegenerate form on fJ*.Let A be the simple root system of R. (1)
2(6, a)/(a,a) = 1
for
GI
E
A.
If a E A let s, be the corresponding Weyl reflection (0.2.3). Let fl E R  { a } then s,fl E R by 0.2.4(2) and 0.2.1(4). Thus s,R = ( R  { a } )u {  a } . This implies that sad = 6  a. (1) now follows. In particular, (1) implies that 6 is dominant integral. Hence there is a finite covering G" of G so that if T" is the corresponding maximal torus then 6 is T" integral (see 1.7.5). Define on T" by A ( t ) = t6lIaER(1 t"). Then IA(t)I2 = Id(t)l (2.4.4).0.3.3(1)says that, up to conjugacy, T is the only Cartan subgroup of G. If we carefully follow the argument in 2.4.4 one finds that if all the measures are normalized measures then (m1)' = [W(G, T ) ]= w. We therefore have Proposition. Let dg and dt be normalized invariant measure on G and T respectively. Then
2.5.3. We assume for the remainder of this section that G = G". If p E T" we set A ( p ) ( t )= C s e Wdet(s)tsp(W= W(C, T ) ) . We say that p is regular if s p # p for s E W  { 1). It is easy to see that A ( p ) = 0 if p is not regular. If p is regular than there exists s E W such that s p is dominant integral. Let p and fl be integral, dominant integral and regular then
JT A b ) ( t )conj(A(/j))(t)dt= ~ 6 , , ~ .
(1)
This is an immediate consequence of Lemma 1.4.5.
Lemma. A
=
A(6).
Let a be a simple root then using the material in the proof of 2.5.2(1) we see that A(s,t) =  A ( t ) for t E T. Now A is a sum of characters of T with coefficients +1. The coefficient of t6 is 1. The other characters that come into the expansion are of the form 6  q with q a sum of distinct elements of R. Thus A = C c,A(6  q ) the sum over all q that are sums of distinct elements of R and the coefficients cq are integers. We assert that if A(6  q) is nonzero then A(6  q) = fA ( 6 ) (here q is a sum of distinct elements of R). Indeed, if s E W then s(6  q ) = 6  q' with q' a sum of distinct elements
67
2.5. The Weyl Character Formula
of R . Thus we may assume that 6  q is dominant and regular. This implies that 2(6  q, a)/(a,a) is a positive integer for all simple a. Hence 2.5.2(1) implies that 2(q, a)/(@,a) 2 0 for all simple roots a. Thus (4,a) I 0 for all a E R. But then (4, q) I 0. So q = 0 as asserted. We therefore conclude that A = cA(6). Proposition 2.5.2 implies that
S 1AI2c2t = W . T
So (1) above implies that c
2.5.4.
=
1.
We now come to the Weyl character formula.
Theorem. Let y E G” and let A be the highest weight of y relative to R (Theorem 1.7.5).Let xy be the character of 1’. Then
A ( 6 ) z y= A ( A
+ 6).
We order the weights of L(A) (1.7.4) by saying that p 2 (T if p  0 is a sum of elements of R . If p is a weight of L ( A )and if s p 2 A for some s E W then s p = A. This implies that A(6)xy= A ( A 6) f with f = C c,A(A 6  q) where 4 is a sum of elements of R and A 6  q is dominant integral and regular. Applying 2.5.2( 1) we have
+ + +
+
JT A(s)(t)x,(t)(conj(A(6)(t)Xy(t))dt = w + jT I f ( t )I
dt.
Lemma 1.4.5 combined with Proposition 2.5.2 now imply that
1T If ( t ) l ’ d t = 0.
so f
= 0.
2.5.5. We now show how one uses the Weyl character formula to derive the Weyl dimension formula.
Theorem. Let y E GA have highest weight A relative to R . Then
Clearly, ~ ~ (=1 d(y). ) Hence d(y) = lim Xy(exp(itHs))= lim A ( A + 6)(exp(itH,))/A(d)(exp(itH,)) f+O
f0
= lim A(G)(exp(itH,,+ h))/A(S)(exp(itH,)). t0
68
2.A. 2.A.1.
2. Real Reductive Groups
Appendices to Chapter 2 Some linear algebra
2.A.l.l. We put the usual inner product, ( , ), on C". If X E M,(C) then we denote by X * the conjugate transpose of X . Then X * is the adjoint operator to X relative to ( , ). If X = X * then we say that X is selfadjoint. If X is selfadjoint and if ( X u ,v) > 0 for all nonzero u E C" then X is called positive nondegenerate (or positive definite). If X E M,(C) then we write exp X for the usual power series
1( l / m ! ) x m . Then exp defines a complex analytic mapping of MJC) into G L ( n , C ) . As is well known, if X is selfadjoint then there is a unitary operator, u, on C" such that u X U  ' is diagonal with real entries. Hence it is clear that (1) If X is selfadjoint then exp X is positive nondegenerate. (2) If A is positive nondegenerate then there is a selfadjoint matrix, X , such that A = exp X .
We may assume that A is diagonal with positive diagonal entries, a l , . . . , a,. Take X to be the diagonal matrix with diagonal entries log(a,), . . ., log(a,).
2.A.1.2. The following lemma is due to Chevalley. It will be used several times in this chapter. Lemma. Let f be a real or complex valued polynomial function on M,,(C). Suppose that Y is selfadjoint and that ,f(exp m Y ) = 0 for all m = 1,2,. . . . Then f (exp t Y ) = 0 for all real t. Let u be a unitary matrix such that uYu' is diagonal. If we replace f by the polynomial g ( 2 ) = f ( u  ' Z u ) we may assume that Y is diagonal with real diagonal entries a,, . . . , a,. We restrict f to the diagonal matrices. Our assumption now says that f(exp(ma,), . . ., exp(ma,)) = 0 for m = 1,2,.. . .
2.A.1.
69
Some Linear Algebra
Set p ( t ) = .f'(exp(ru,),. . ., exp(ta,)). If p is not identically zero then
b, exp(/A,,)
p(t) =
with A , > ... > A ,
with b , nonzero. Thus if s is real and sufficiently large then J b , exp sA,I >
1 b, exp s A ,
.
1m> 1
Thus p(m) is nonzero for sufficiently large integers, rn. Since this is contrary to our hypothesis, we must have p ( l ) = 0 for all t. 2.A.1.3. If X power series
E
M,(C) we define ( I
1( Lemma.
Let X , Y be
E
d exp,( Y)

I)"( l/(m
exp(X))/X to be the sum of the
+ l)!)Xm.
M,(C) then =
exp X ( ( I  exp(  ad X))/ad X) Y.
This result can be proved directly by manipulating power series. See (e.g., Wallach [ 13). 2.A.1.4.
Let p, denote the space of selfadjoint elements of M,(C).
Lemma. The mup U ( n ) x p, surjective difleomorphism.
+ GL(n,C )
given by u, Y H u exp X deJnes a
Let g E GL(n,C). Set A = y*g. Then A is positive nondegenerate. So A = exp Y with Y E p,. Set X = (3)Y and p = exp X . Then p2 = A. It is easy to check that yp' E V ( n ) .Thus the map is surjective. Suppose that g = u exp X = u' exp X'. Then exp 2X = exp 2X'. This implies that exp 2mX' commutes with X for all m = 1,2,. . . . Lemma 2.A.2 implies that exp t X ' commutes with X for all real r. Hence X ' commutes with X. Thus X and X ' can be simultaneously diagonalized using a unitary matrix. Since exp 2 X = exp 2 X ' this implies that X = X ' . Thus u = u'. So the map is injective. Let f denote the map we are studying. The Lie algebra of U ( n )can be identified with the Lie algebra of all skewadjoint matrices ( Y * =  Y). If X * = X, if Y, Z E pn and if u E U ( n ) then dfu,,(X, Y)
= u(X exp
Z
+ d / d r , = , exp(Z + rY)).
Thus if df,*,(X, Y ) = 0 then X exp Z must be selfadjoint. But then X exp 2 =  X exp Z. So (exp Z)X(exp(  Z ) ) = X. After an orthonormal change of
70
2. Real Reductive Groups
basis we may assume that Z is diagonal with real diagonal entries a,, . . .,a,. Thus the eigenvalues of T H (exp Z )T(exp(  Z ) )are of the form exp(aj  ak) which are all positive so X = 0. Lemma 2.A.3 implies that
((I

exp(  ad Z))/ad 2 )Y
= 0.
But (ad Z ) 2 k + Y ' is skewadjoint and (ad Z ) Z k Yis selfadjoint. Thus we see that V Y = 0 with
V
=
C (ad Z ) 2 k / ( 2 k+ l)!.
The eigenvalues of V are of the form
1(ai

aj)"/(2k
+ l)!,
which are positive. Thus Y = 0. So f is everywhere regular and bijective. Hence
f is a diffeomorphism. 2.A.1.5. Let F = R or C . If X E M,(F) then Xis said to be nilpotent if X k = 0 for some k. If g E G L ( F )then g is said to be unipotent if g  I is nilpotent. Lemma. If Y is nilpotent then exp Y is unipotent. If g is unipotent then g = exp Y with Y nilpotent.
It is clear that exp Y
=I
+ YZ with
[ Y , Z ] = 0. Thus ((exp Y) 
=
Y k Z kThus . if Y is nilpotent exp Y is unipotent. Let g be unipotent. Set Z =
g  I . Put log(g) = C , , Zm/rn. Since Z is nilpotent this series is actually finite. The obvious formal manipulation of power series gives exp(log(g)) = g (it is rigorous since all series are finite). Since log(g) = ZW, log(g) is nilpotent so the lemma follows.
2.A.2.
Norms on real reductive groups
2.A.2.1. Let G be a real reductive group. Then as in 2.1.1 there exists G, a symmetric algebraic subgroup of GL(n,R ) (for appropriate n) and, p , a finite covering homomorphism of G onto an open subgroup of GR.Furthermore, we can choose a Cartan involution 8 of G such that p ( 8 ( g ) ) = p ( g  ' ) * . On R2",which we look upon as R" R",we put the standard inner product. If g E G L ( n , R ) then we set llgll = 119 0 (g')*ll where 11 11 is the operator norm. If g E G then we set llgll = IIp(g)ll. Let K be the maximal compact subgroup of G corresponding to 8. Let g = f 0 p be the corresponding Cartan decomposition of g. Then 11. . .I[ has the following properties:
+
2.A.2.
Norms on Real Reductive Groups
{g E G I llgll I r } is compact
(3) (4)
71
Ilk, exp(tX)k,ll
=
llexp XII'
for all r < co.
for all k,, k , E K , X and all t E R, t 2 0.
E
p
These properties are easy to prove and are left to the reader.
2.A.2.2. Lemma. Let (n,H ) be a Hilbert representation of G. Then there exist constants C > 0, r > 0 such that Iln(g)ll I Cllgll'for all g E G. (Here llAll denotes the operator norm of A , ) We note that llgll 2 1 for all y E G. We set a(g) = log(llgll). Then a(x) 2 0, + o(y) and a(x') = a(x). Set p(g) = log IIn(g)llfor g E G. Then AXY) I Ax) + AY) for x, Y E G. Put B, = {g E G o ( g ) I r}. o(xy) I o(x)
(1) There exists a positive constant, C, such that p(x) I C for x
E
B,.
This follows from (3) above and 1.1.1 (1). (2)
e"ln(x)l I In(kx)I I e'In(.x)l
for x E G, k
E
K.
This follows from (1) since K is contained in B , . Let X E p . Then o(exp t X ) = to(exp X ) for t 2 0. Let j be a nonnegative integer such that j < o(exp X ) I j 1. Then o(exp(X/(j 1)) < 1. Hence p(exp(X/(j 1)) I C. This implies that
+
+
p(exp X ) I (.i
+
+ 1)C IC(1 + o(exp X ) ) .
Thus, if k E K then p(k exp X ) < C
+ C( 1 + o(exp X ) ) = C(2 + o(exp X ) .
Theorem 2.1.8(1) now implies that if 9 E G then Iln(g)ll I e2cIIgIIc. This completes the proof. The above result will play an important role in our study of matrix coefficients of representations. The method in the above proof was suggested by the proof of Warner [1,4.4.5.9]. 2.A.2.3. We will call any function, 11. ..I\, on G with values in [I, m) satisfying (l),(2), (3),(4)of 2.A.2.1 a norm on G. We note that the proof of 2.A.2.2 implies
72
that if )I Ill and q > 0 such that
2.
Real Reductive Groups
11 1 , are norms on G then there exist constants C > 0 and
(1)
llsll, I CllSlll?
for all 9 E G.
We fix an Iwasawa decomposition, G = NAK, with a contained in p . We assume (for the sake of simplicity) that G has compact center. Let ' 0 be the set of positive roots of 0(g, a) corresponding to N. Let { a l , .. ., a,} be the simple roots in 0 ' . By our assumption, the simple roots span a*. We define H , , . . ., H, E a by aj(Hk) = 8 j . k . Let A + be as in 2.4.2.
Lemma. Let 11. . 1 1 be a norm on G. Then there exist p, j E a*, with p ( H j ) > 0 for all j , and positive constants C , , C , such that C l a pI llall I C2aP, ,for all a E Cl(A').
In light of (1) we may assume that [I...(( is given as in 2.A.2.1. Let Z denote the weights of a on R2"corresponding to the representation p ( g ) 0 (p(g)')* for g E G . We partially order C by p 2 j if p ( H j ) 2 j(Hj) for j = 1 , . . .,r. Let pl,. . . , p d be the maximal elements of C. Then llall is the maximum of the ap',j = 1 , . . . , d , for a E Cl(A+).Set y = pl + ... + pd. 2.A.2.1 (3) implies that y ( H j )> 0 for all j = 1,. . . , d. Hence it is clear that I llall I a7
2.A.2.4. Lemma.
Let 11..
for a E Cl(A+).
be a norm on G. Then there exists d > 0 such that
j llgllddg < 03.
G
Let y be as in 2.4.2. Then y(a) I a Z p for a E Cl(A'). Let p be as in Lemma 2.A.2.3. Let d be so large that dp(Hj)> 2p(Hj) for j = 1,..., r . The result is now a direct consequence of 2.4.2.
3
The Basic Theory of (g, K)Modules
Introduction In this chapter we begin the representation theory of real reductive groups. The theory of (9, K)modules (first introduced by HarishChandra for connected K and later defined in general by Lepowsky) is the connecting link between the algebraic results of Chevalley and HarishChandra and group representation theory. The main results of this chapter are HarishChandra’s theorem that implies that irreducible unitary representations are admissible (Section 3.4), the subquotient theorem of HarishChandra, Lepowsky, Rader (Section 3.5) and its important refinement due to Casselman (Section 3.8). Section 1 contains the theorem of Chevalley that relates the polynomial K invariants to the invariants of the Weyl group. This theorem is one of the main ingredients in HarishChandra’s proof of the isomorphism between the center of the universal enveloping algebra and the Weyl group invariants on a Cartan subalgebra. This result and HarishChandra’s determination of all “infinitesimal characters” is the content of Section 2. In Section 3, Lepowsky’s definition of (9, K)modules is introduced. The most important example is the space of Kfinite, smooth vectors of a Hilbert representation. The main result
73
74
3. The Basic Theory of (9, K)Modules
in Section 4 is Theorem 3.4.1 which asserts that the isotypic components of a finitely generated (g, K)module are finitely generated as modules for the center of the universal enveloping algebra. This theorem combined with Schur's Lemma implies the above mentioned theorem of HarishChandra on irreducible unitary representations. In Section 5 we give Lepowsky's proof of the subquotient theorem. It also contains preliminary results on the algebraic structure of (9, K)modules. Section 6 is devoted to an exposition of some of HarishChandra's theory of the spherical principal series. The main result of this section is the exact sequence in 3.6.6. However, the estimate in 3.6.7 will be fundamental in later developments. The material in Section 7 will be useful in the theory of the Jacquet module. Section 8 is devoted to a new proof of the subrepresentation theorem of Casselman. Although this theorem appears to be only slightly stronger then the subquotient theorem, we will see in the next chapter that the difference between the two theorems is significant.
3.1. The Chevalley restriction theorem 3.1.1. Let G be a real reductive group. Let 0 be a Cartan involution for G and let g = f 0 p be the corresponding Cartan decomposition. Let K be as in 2.1.8. If V is a real vector space then we denote by P ( V )the space of complex valued polynomial functions on V. Let K act on P(p) by kf(X) = f(Ad(k')X) for k E K, X E p and f E P ( p ) . We denote by P(p)" the space of all f E P ( p ) such that k f = f for all k E K . Let a be as in 2.1.6. Let W = W(g,a) be as in 2.1.10. Let W act on P(a) by sf(H) = f(s'H) for s E W , H E a, .f E P(a). Let P(a)" denote the set of all f E P(a) such that sf = f for all s E W. If V is a real vector space and if W is a real subspace of V then we define for f E P( V ) ,Res,/,(f) to be the restriction of f to w.
3.1.2. Theorem. Assume that G is of inner type (2.2.8). Then Res,,, is an algebra isomorphism of P(p)" onto P(a)W. As we have seen in 2.1.10, W = {Ad(k)I, I k E K,Ad(k)a = a}. Thus
(1)
Res,/,(P(p)") is contained in P(a)W.
(2)
Res,/, is injective on P(p)".
This follows from Lemma 2.1.9. (3) Let Hj E a, j = 1, 2. If there exists k there exists s E W such that sH, = H , .
E
K such that Ad(k)H, = H , then
75
3.1. The Chevalley Restriction Theorem
Clearly, a and Ad(k)a are maximal abelian in C,(H,) n p . Since C,(H,) is real reductive and &stable (2.3.1(2)),there exists k, E ( K n C,(H,))' such that Ad(k,)(Ad(k)a) = a. Takes = k , k l , . (4) If Hj E a, j = 1, 2 and if WH, n WH, = 0then there exists a continuous function f on p such that f(Ad(k)X) = f ( X ) for all k E K and X E p and f W 1 ) = 0, f(H2) = 1.
By (3), Ad(K)H, n Ad(K)H, = 0. Thus there is a continuous function, h, on p such that h i s identically 0 in Ad(K)H, and identically 1 on Ad(K)H,. Set f ( X ) = J h(Ad(k)X)dk. K
( 5 ) Let H j , j = 1, 2 be as in (4). Then there exists p pW,) z P(H2).
E
P ( P ) such ~ that
Set C = Ad(K)H, u Ad(K)H,. Then C is a compact subset of p . Let f be as in (4).The StoneWeierstrass theorem implies that there is a polynomial q on p such that for x E C . 1q(X)  f ( X ) (< 4
sK
Then p ( X ) = q(Ad(k)X)dk defines the desired polynomial. Let F denote the quotient field of P(a). Let L be the quotient field of J = Res,,o(P(p)K). Let Dj be as in 2.3.1. Set f ( z ) = C z j Res,,,Dj. Then the roots of .f are the elements of @( 9, a). If p E a* and if p vanishes on 'a (2.2.2) then p E J . Thus we see that F is a normal extension of L(see any book on Galois theory). So L = { f E F I af = f for all a E Gal(F/L)}, here Gal(F/L) is the group of all automorphisms of F that are equal to I on L. By the above, if a E Gal(F/L) then a(a*) = a*. Hence, aP(a) is contained in P(a) for all a E Gal(F/L). Denote by U the group of all automorphisms of P(a) that are equal to I on J . Then we have shown that J = { f E P(a) I of = f for all a E V } .If a E U and if H E a then S ( f ) = af(H) defines a homorphism of P(a) into C. Hence the nullstellensatz (c.f. Mumford [l, p.31) implies that there exists H , such that S ( f ) = f ( H , ) for all f E P(a). Now, af = f for f E J , so ( 5 ) implies that there exists s E W such that H, = sH. We have therefore shown that if f E P(a)" then af = f for all a E U . Hence P(a)" is contained in J . Now (1) implies the result. Note. The above Theorem is the celebrated Chevalley restriction Theorem. We note that if G is not necessarily of inner type and if we define N,(A) = {k E K I Ad(k)a = a} and W = NK(A)/'M then the conclusion of the above theorem is still true (with the same proof).
76
3. The Basic Theory of (g, K)Modules
3.1.3.
We now derive a corollary to Theorem 3.1.2 which is also called the Chevalley restriction theorem. Let g be a reductive Lie algebra over C. Let P ( g ) denote the space of all complex polynomials on g. We define an action of g on P ( g ) by X f ( Y ) = d/dt,=,f(exp(t ad X ) Y ) . Set I ( g ) = { f P (~g ) IXf = 0 for all X E g}. Let $ be a Cartan subalgebra of g. Let W = W(g,$).We let W act on P ( b ) by s f ( H ) = f ( s  ' H ) . Let I ( b ) denote the Winvariants in P ( 6 ) .
Theorem. ResSibis an isomorphism of I ( g ) onto I ( I)). Since the center of g is contained in b, we may assume that g is semisimple. Let g, be a compact form of g such that g, n b is maximal abelian in gu (0.3.4). Set G = Int(g) which we look upon as a real reductive group. Let 8 denote conjugation on g relative to g,. Then 6 is a Cartan involution of g (looked upon as a real Lie algebra). If we set € = 9, and p = ig, then g = € 0 p is the corresponding Cartan decomposition. Since g, is a real form of g, Res,,p is an isomorphism of I ( g ) onto P ( P ) ~Set . a = b n p . Then Reshiais an isomorphism of I(b) onto P(a)W.The result is now an immediate consequence of Theorem 3.1.2.
Example. We look at the case when g = M J C ) (the Lie algebra of GL(n,C)).We take for $ the space of diagonal matrices. If H E and if H has diagonal entries h , , . . ., h,,, then define E ~ ( H=) h j . Then O(g, $) is the set of all & j  &k for distinct j , k. We take @+ to be the set of all & j  &k for j < k. Then if a = c j  & k , then s,H has diagonal entries, h g l ,..., h,,, with a the permutation ( j , k ) . We therefore see that W is the set of all permutations of the diagonal entries. Thus, the fundamental theorem of invariant theory for the symmetric group (Weyl [l, pp.37, 381) implies that P ( € J )is~ equal to C[al,. . .,a,,], where aj is the jth elementary symmetric function in the diagonal entries of H . Recall that these functions are defined by 3.1.4.
n
1< j < n
Define for X
E
(t
+ hj) = 1 t"'aj(H).
M,,(C)the polynomials pj by
det(tI
+ X ) = 1t "  J p j ( X ) .
Then it is clear that Resgibpj= aj. Theorem 3.1.3 now implies that P ( g ) gis the polynomial algebra in p l , . . ., p,, .
3.2. HarishChandra Isomorphism of Center of the Universal Enveloping Algebra
77
3.2. The HarishChandra isomorphism of the center of the universal enveloping algebra Let g be a reductive Lie algebra over C. Let Z ( g ) be the center of U(g)(0.4.1).In this section we will give HarishChandra's determination of the homomorphisms of Z ( g ) into C . In order to carry this out we will use the HarishChandra isomorphism. In Section 6 we will give a related (but different) mapping that is called the HarishChandra homomorphism. Let Ijbe a Cartan subalgebra of g (0.2.1).Fix R a system of positive roots for @(g,b). Let n + (resp. n) be the sum of the ga (resp. g P m )for c1 E R. Then
3.2.1.
g = n+
0I] 011
PBW (Theorem 0.4.1) implies that
U ( g ) = U ( b )0(tIU(g)
+ U(g)n+).
Let q denote the projection of U(g) onto U (6) corresponding to this direct sum decomposition. Let be the set of all x E U ( g )that commute with every element of 6. Lemma.
q is an algebra homomorphism of' U(g)h into U ( Ij).
We enumerate @+ as {a1,..., a,,).Let X j , j = 1,. . .,d, be a basis of n+ with X j € g , , . Let 3 be a basis of 11 with YEY,,. Let Hk be a basis of 6, k = 1,..., 1. If n E N d then set X"
= (X,)"'
'
. (Xd)"", '
Y" = ( Y,)"' . * ( Yd)n". '
If k
E
N ' then set Hk = (Hl)k'...(HL)kL.
Then PBW implies that the elements Y m H k X "form a basis of U ( g ) . (1) U(gIhn (nU(g)
+ W g ) n + )= U(gIhn n U ( g )= U(dhn u(g)n+.
Y m H k X "with the sum over all m, k, n such that If x E U ( g ) hthen x = C C mjaj = E njaj. Which clearly implies (1).
Let ui E
U ( g ) hf o r j =
1,2.
Then u I u 2 = u,q(u,)(mod U(g)n+).
78
3. The Basic Theory of (g, K)Modules
Hence (1) implies that
This is the content of the Lemma. 3.2.2. Fix an invariant from, B, on g as in 0.2.2. We define a mapping X H X # of g onto g* by B( Y , X ) = X # ( Y )for Y E g. Then X H X # induces an algebra isomorphism of S(g) onto P(g).ad induces an action of g on S(g) as derivations. Under X H X # this corresponds to the action of g on P ( g ) in 3.1.3. We may thus identify S(g) and P(g)as gmodules. Let p E b* be half the sum of the elements of R . We define an isomorphism, p, of S(9) given by p ( H ) = H  p ( H ) on b and extended to S(b) by the universal mapping property. Since b is abelian, U ( b )is isomorphic with S(t)). Thus we will use S(b) and U ( b )interchangeably. We define a homomorphism, y, of Z ( g ) into U ( b )by y = p q. (Note that Z(g)is contained in U(g)').) Under our identification, the ginvariants in S(g),S(g)g,correspond to P ( g ) g . We also have an action of W = W(g,b)on S(b). The Winvariants in S(b) correspond to P( b)w. (Here we have replaced g by b in the above discussion.) We write U ( 6)" for the Winvariants in U (I)) ( = S(b)). We can now give the HarishChandra isomorphism for the center of the enveloping algebra. 0
3.2.3. Theorem. y ( Z ( g ) )is contained in U ( b ) w .The map y de$nes an algebra ) ~ ( b ) ~ . isomorphism of ~ ( gonto We first note that the result follows from
(*I
y ( Z ( g ) )is contained in U(b)".
We use the standard filtration of U ( g )(0.4.2).Then G r U ( g ) = S(g). We can therefore consider Gr q: S(g) + S(t)).The direct sum decomposition g = [) 0(n+ 0TIC)
is B orthogonal. Thus, under our identifications, it is an easy matter to see that G r q = Resgih.Thus, if we compare the filtration to the grade and apply Theorem 3.1.3 combined with (*) the result follows. We are thus left with proving (*). Let m be a simple root in R . Let m' = b + g m + gCz.Set naequal to the sum of the root spaces corresponding to the elements of R  { a } .Set fi" equal to the sum of the root spaces corresponding to the roots  p for p E R  {m}. Then g = ma 0f i a 0nn.
3.2. HarishChandra Isomorphism of Center of the Universal Enveloping Algebra
79
PBW implies that
+
U ( g ) = U(trt")0(iiaU(g) U(g)na). Let q" be the linear projection of U ( g )onto U(ma)corresponding to this direct sum decomposition. Define p a in (ma)* by p"(X) = ($)tr(ad XI,,=)for X E ma. Define a homomorphism, (T of U(nt") to itself by a(X) = X  p"(X) for X E ma.If we argue as in the proof of Lemma 3.2.1 we see that (T q restricted to Z ( g ) is a homomorphism into Z(m"). Let y" be the HarishChandra homomorphism associated with tit'. Then y a 0 q = y. Thus if we show that y"(Z(ma))is contained in the s, invariants of U ( b ) then (*) will follow from 0.2.4(3). We therefore can assume that g = ma.That is, R = { a } . We are reduced to the case when [g, g ] = g1 is a TDS (0.5.4).Let X, Y, H be a standard basis for gl. Set C = H 2 2(XY YX). Then a simple computation shows that C is in Z(g).Let c E S ( g ) be the element given by the same formula in S ( g ) (which we have identified with P(g)). Then Resgib(c)= H 2 . Now W = { I , s,} and s, restricted to j(g) is I , s,H =  H . Thus Theorem 3.1.3 implies that S(g)g = S(j(g))C[c]. If we compare the standard filtration of Z ( g ) with G r Z ( g ) we see that Z(g) = U(j(g))C[CJ. But it is clear that y(C) = H 2  1. Thus (*) is true in this case. This completes the proof. 0
0
+
0
+
3.2.4. We now show how one uses the HarishChandra isomorphism to derive HarishChandra's formula for infinitesimal characters. If p E b* then set xr = p y ( p extends to a homomorphism of U ( b ) to C by the universal mapping property of U ( 6 ) ) . 0
Theorem. Let x be a nonzero homomorphism of Z(g) to C . Then there exists p E b* such that x = I,,. Furthermore, if p, p' E b* then xr = xp. if and only if there exists s E W such that s p = p'. Let Djbe as in 2.3.1. Set pi = Resqib(Dj).Set f ( t ) = X t'pj. If CI E (D(g,b) then ~ ( c I=) 0. This implies that U ( b )(which is identified with S(b) which is in turn identified with P ( b ) ) is integral over U ( t ) ) w(cf. Zariski, Samuel [ l ] ) . Hence every nonzero homomorphism of P ( ~ Jinto ) ~ C is given by point evaluation ([op. cit.]). In light of our identifications, this implies the first assertion. The second assertion follows from the observation that if h, h' E b and if f ( h ) = f ( h ' )for all f E P ( b)w then there exists s E W such that h' = sh. (cf. the proof of 3.1.2).
3.2.5. We now look at what these results say for g = M,(C). If A is an associative algebra over C and if [ u ~ , is~ ]an n by n matrix over A we set
80
3. The Basic Theory of (9, K)Modules
d e t ( [ ~ ~ ,= ~ ]C) sgn(o) nj.= aoj.j, the sum over all permutations of nletters. We take Ej,k to be the standard basis of M,,(C) and look upon these elements as being in U ( g ) . Let t be an indeterminate and set aj,Jt) = Ej,k + ( j  1 t ) d j , k . Write d e t ( [ ~ ~ . ~ ( t=) ]C) t n  j u j . Then the content of the classical Cappelli identities (Weyl [l, p.421) is that uj E Z ( g ) . One computes that y(uj) = oj (3.1.5).
+
3.3. (9, K)modules 3.3.1. Let G be a real Lie group with Lie algebra, g. Let K be a compact subgroup of G. We recall Lepowsky's definition of a (9,K)module. Let V be a gmodule that is also a module for K (for the moment we ignore the topology of K ) . Then V is called a (9,K)module if the following three conditions are satisfied : (1)

k X . v = Ad(k)X  k  u
for v E V, k
EK,
X
E g.
(2) If u E V then Ku spans a finite dimensional vector subspace of V, W,, such that the action of K on W, is continuous.
(3)
If Y E f and if u E V then d / d t , = , exp(tY)u = Yv.
If V and Ware (9, K)modules then we denote by Horn,,,( V, W) the space of all ghomomorphisms that are also K homomorphisms of V to W . V and W are said to be equivalent if there is an invertible element in Hom,,,(V, W). We denote by C(g, K ) the category of all (g, K)modules with Hom in this category given by Horn,,,( V, W).
3.3.2. A (9,K)module, V, is said to be Jinitely generated if it is finitely generated as a U(g)module. V is said to be irreducible if the only g and K invariant subspaces of V are V and (0). In this context we have the following variant of Schur's Lemma.
Lemma.
Let V be an irreducible (9, K)module. Then Horn,,,( V, W) = CI.
Let c be a nonzero element of V. Let W, be as in 3.3.1(2). Then U(g)W, is a g and a Kinvariant subspace of V. Hence, V = U(g)W,. In particular, this implies that V is countable dimensional. The result now follows from Lemma 0.5.2. 3.3.3. Let V be a (g, K)module. Let y E K A .Then we set V ( y )equal to the sum of all the Kinvariant, finite dimensional, subspaces of V that are in the class of y. Lemmas 1.4.7 and 1.4.8 immediately imply
3.3. (a, K)Modules
81
Lemma. As a Kmodule, V direct sum.
=
ByeV(y).Here the direct sum is the algebraic
If y E K" then we call V(y)the yisotypic component of V. We say that V is admissible if dim V(y)< cc for all y E K " . 3.3.4. Lemma. Let V be a (g, K)module. Then V is admissible if and only if dim Horn,( W, V ) < cc for all jinite dimensional Kmodules, W.
Let W be a finite dimensional Kmodule. Let T be a Khomomorphism of W into V. Then T ( W ) is a direct sum of irreducible Ksubmodules of V (Lemma 3.3.3). Since W has only a finite number of inequivalent irreducible quotients, there exists, F, a finite subset of K" depending only on W,such that T (W ) is contained in V(y).The lemma now follows.
BYSF
3.3.5. Let ( T I ,H ) be a Hilbert representation of G . Then according to Lemma 1.4.7, H is the Hilbert space direct sum of the H ( y ) for y E K " . Here we are assuming, as we may, that TI IK is unitary (Lemma 1.4.8).Lemma 1.4.7(1)implies that H ( y )n H" (1.6.3) is dense in H ( y ) for all y E K " . We set HK equal to the algebraic direct sum of the H(y)n H" for y E K". By the above, it is clear that HK is a dense subspace of H (resp. H"). Lemma. HK is a ginvariant subspace of H". With this structure of g and K modules, HK is a (g, K)module. We note that HKis the space of all C"vectors, u, of H such that z ( K ) vspans a finite dimensional subspace of H. 1.6.4(ii) says that if X E g, g E G and u E H" then n(g)n(X)v= n(Ad(g)X)n(g)u.Thus, if u E H K , if X E g and if W, is the span of T I ( K ) U then W, is contained in HK and T I ( X ) U E n(g)W, a finite dimensional Kinvariant subspace of H c5'. The result now follows. HK is called the space of C", K,finite vectors of H or the underlying (9, K ) module of H. We say that H is admissible if H K is admissible. H is said to be infinitesimally irreducible if HK is irreducible as a (g, K)module. If ( T I ,H ) and ( 0 , V ) are Hilbert representations of G then TI is injinitesimally equivalent with o if the (g, K)modules HK and VK are equivalent.


3.3.6. Let V E C(g, K ) . If p E V * then we write X p (resp. k p) for the functional X p ( v ) =  p ( X u ) (resp. k p(v) = p(k'v). Then relative to these actions V* is a g and a Kmodule that satisfies the compatibility condition 3.3.1(1). We set V" = { p E V * I K p spans a finite dimensional subspace). We may argue as we did above to see that V" is a g and a Ksubmodule of V * . Hence V" is a (9, K)module. V  is called the (g, K)dual module of V.

82
3. The Basic Theory of (9, K)Modules
We set V # equal to the space of all conjugatelinear functionals on V with g and K acting on V # as above. We set V = { p E V # I K p spans a finite dimensional subspace). Then as above V is a (9,K)module that is called the conjugate dual (g, K)module of V. A basic theorem of HarishChandra
3.4.
Let G be a real reductive group. We return to the notation of 3.1.1. Let of U ( g ) consisting of those elements u E U ( g ) such that Ad(g)u = u for all g E G.Notice that if G is of inner type (3.1.1) then z&) = Z ( g ) . The purpose of this section is to prove several important theorems of HarishChandra [l] the first is:
3.4.1.
z&) denote the subalgebra
Theorem. Let V be a finitely generated (3.3.2) (9,K)module. If y V(y)is jinitely generated as a Z,(g)module.
E
KA then
The proof of this result involves several steps which we now begin. We fix V a finitely generated (9,K)module. Let W be a finite dimensional Kinvariant subspace of V such that V = U(g)W. In light of the material in 0.4.3, we see that V = symm(S(p))W. We define V, = W and y + l= p Q + 5 for j = 0, I , . . . . Then each 5 is Kinvariant, p 5 is contained in V,, the union of the 5 is V. Set Gr(V) equal to the direct sum of the spaces (V,/y here V = (0). Then Gr( V ) is equivalent with V as a Kmodule. Let pi be the natural projection of into y/V, If X E p, u, w E V, and if pj(u) = pj(w)then pi+ l(Xu  X w ) = 0. We may thus define an action of each X E p on Gr(V) by Xpj(u) = p j + l ( X u )for u E V,. 3.4.2. (1)
We define a new Lie algebra structure on f 0 p as follows:
If X , Y E f or if X
E
f, Y E p then [ X , Y ] has the same meaning as it did
in g. If X, Y E p then [X, Y]
(2)
= 0.
We denote by gC the Lie algebra f 0p with commutation relations given as in (l), (2). We form a Lie group GCwith total space K x p and multiplication given by:
(3)
(k,X)(u,Y)
= (ku,Ad(u')X
+ Y),
Then GCis a Lie group with Lie algebra gC.
k, u E K , X , Y E p.
3.4.
83
A Basic Theorem of HarishChandra
Lemma.
Gr( V ) is a finitely generated (gc, K)module.
Let and X, YEP. Then XYpj(i;)=pj+,(XYu)=pj+,(YXu+[X, Y ] u ) = since P~+,~(YX V ) [ X , Y ] E € .Thus X Y o = Y X o for all uEGr(V) and X , Y E P . It I S therefore clear that Gr(V) is a g"module. Conditions 3.3.1(1), (2), (3) are all assertions for K and they follow from the fact that V is a ( g , K ) module. Let Gr( V)j = F/? Then p Gr(V ) , = Gr( V)j+ I . Thus Gr(V ) is finitely generated.
3.4.3. We may look upon p as an abelian normal subalgebra of gc. Then S ( p ) is the universal enveloping algebra of p. Clearly, S ( P )is~contained in the center of U ( g c ) . Lemma.
If y E K A then Gr( V ) ( y )is Jinitely generated as a S(p)Kmodule.
Let y E KA and let ( p , X ) E y. We look upon Hom,(X, Gr(V)) as an S ( p ) and a Kmodule with the actions (uT)(u)= u(Tu) and (kT)(u)= k(T((k')u)) for u E S ( p ) , k E K and u E Gr(V). As a S(p)module HomJX, Gr(V)) = X* O G r ( V ) with S(p) acting on the right factor. Thus under this action Hom,(X, Gr( V ) ) is finitely generated as a S(p)module. Also Hom,(X, Gr(V)) is the space of Kinvariants in Hom,(X, Gr(V)). Set L = S(p) Hom,(X, Gr(V)). Since S(p) is Noetherian (0.6.1) there exist elements TI,.. ., in Hom,(X, Gr( V ) ) such that L = X S(p)T,(0.6.2). If T E Hom,(X,Gr(V)) then T = C p j T , with p j € S ( p ) . Hence T=kT=C(Ad(k)pj)Tj for all k E K . Hence if we set for p E S ( p ) , po
then T = (*)
1(pj)'Tj.Since p o
E
=
S Ad(k)pdk
K
S(p), for all p E S(p), we have proved:
Hom,(X, Gr(V ) )is finitely generated as a S(p)Kmodule.
Let u: Hom,(X, Gr(V ) )0X + Gr( V ) ( y ) be defined by u(T 0 x) = Tx. are as above then Then u is surjective. Thus we see that if TI,..., Gr(V)(y) = C S ( P ) ~ T ( X This ) . completes the proof of the lemma.
3.4.4. Let P(g)' be the algebra of all polynomials on g, f , such that f Ad(g) = f for all g E G. 0
Lemma.
P(p)" is ,finitely generated us a Res,,,(P(g)')module.
84
3. The Basic Theory of (9, K)Modules
Let a be maximal abelian in p. We use the conventions in 2.1.1. In particular we identify the Lie algebra of G with that of GR. We define polynomials, q j , on 9 by det(t1
+ X ) = c tJqj(X)
for X
E g.
Let C denote the set of all weights of a on R". Then clearly, C spans a*. If jE C then C PjResg,,(qj) = 0. Since qn = 1 this implies that (*)
P(a) is finitely generated as a Resg,a(P(g)G)module.
, the result follows 2.1.9 implies that Res,/, is injective on P ( P ) ~ so from (*).
3.4.5. We define a linear map, 6, of U ( g )to S(p) by h(symm(p)k) = E(k)p, for p E S(p) and k E V(f), here we are using 0.4.3 and I: is defined as in 0.4.4. If u E U j ( g )(0.4.2)we set hj(u)equal to the jth homogeneous component of 6(u). We note that (1)
If u E U j ( g ) ,u E
4then Pj+k(m) = hj(u)pk(u).
Fix, B, an invariant nondegenerate form on g. As in 3.2.2, we identify P(g) (resp. P(p)) with S(g) (resp. S(p)). Set I(p) = Resgi,(P(g)G).We look upon I(p) as a subalgebra of S(p). Then S ( P )is~ finitely generated as an I(p)module (Lemma 3.4.4). We also note that (2) If u E S(g)' then symm(u) E ZG(g)and G(symm(u))= Res,,,(u). 3.4.6. We are finally ready to complete the proof of Theorem 3.4.1. Lemma 3.4.3 now implies that if 1' E K A then Gr(V)(y) is finitely generated as an I(p)module. Let U1,.. . , Ed be homogeneous generators with 5 homogeneous of degree k j . Let uj E 4, project onto 4.3.4.5(1) and (2) now imply that @Pk((c
zG(g)uj)
4)=
I(p)q = Gr(v)(y),
Hence C ZG(g)uj= V(y), which was to be proved. 3.4.7. We now derive some consequences of Theorem 3.4.1. The first is immediate. Corollary. Let V he a Jinitely generated (g,K)module such that if V then dim ZG(g)u < 00. Then V is admissible.
U E
3.4.8. Corollary. Let V be an irreducible (9,K)module then V is admissible.
3.4. A Basic Theorem of HarishChandra
85
Lemma 3.3.2 implies that the elements of Z&) act on V by scalars. The result now follows from 3.4.7. Before we can give the next application we must introduce some notation and results. Let C E ZJg) be the Casimir operator of G corresponding to B. That is, if XI,. . ., X , is a basis of g and if X', ..., X" are defined by ~ ( x , .X , k ) = d j , k then C = Z x J x j .
3.4.9.
Theorem. Let ( n , H ) be a Hilbert representation of G. Suppose that if u E HK (3.3.5) then dim C[C]u < m. Then HK is a subspace of the space of analytic vectors f o r 71 (1.6.6). Let CK be defined for ( f , K ) in the same way as C was defined for ( g , G ) . We note that If u E HK then dim C[C, CK]u < co.
(1) Set A (2)
=
C  2cK then (1) implies that If u E HK then dim C[A]u < co.
Fix u E HK and let w E H. Set f = cL,+ (1.3.2). We look upon U ( g ) as the space of all left invariant differential operators on G (as usual). Then (2) implies that there is a monic polynomial, p , such that p ( A ) f = O . Let X I , . . . ,X , be an orthonormal basis of 9 relative to the inner product, ( , ), given by (X, Y ) = B(X,BY). Then A = C(T.)'.Thus, in local analytic coordinates, p(A) is an analytic elliptic operator. Analytic elliptic regularity (Nirenberg [ l , p.1581) implies that f is real analytic. The following result is the basic theorem in the title of this section. Theorem. Let (71, H ) be an irreducible unitary representation of G. Then (n,H ) is admissible.
3.4.10.
In light of Lemma 1.6.5, and the previous theorem HK consists of analytic vectors. Let u E H K be nonzero and set V = U ( g ) span(n(K)v). Then 1.6.5 combined with Corollary 3.4.7 implies that V is an admissible ( 9 , K ) submodule of H K . Now G = KG', so Proposition 1.6.6 implies that Cl(V) is a Ginvariant subspace of H. Hence C1(V ) = H . Since Cl(V ) ( y )= CI( V ( y ) ) for all y E K " , this implies that HK = V. 3.4.11. Theorem. Let (n,H ) be a unitary representation of G. Then (71, H ) is irreducible if and only if it is injinitesimdy irreducible (3.3.5). I f (n,H) and
86
3. The Basic Theory of (5, K)Modules
(0, V ) are irreducible unitary representations of G then n and equivalent if and only if they are injinitesimully equivalent,
(T
are unitarily
Suppose that (n,H ) is irreducible. Then, as we have seen in the preceding proof, if W is a nonzero (9,K)submodule of HK then W = H K . Suppose that HK is irreducible. If H is reducible then H = H I 0H , unitary direct 0(H2)K. sum of closed, nonzero, Ginvariant subspaces. Thus HK = This contradiction implies the first part of the result. We now prove the second assertion. Let A be an invertible element of Horn,,,(H,, VK).Then A maps (HK)(y) to (V,)(y)for all y E K A . We may thus define A* E Horng,'(VK,HK) by (A*v, w ) = ( v , A w ) for v E V(y)and w E H(y) (here we have used the admissibility of V, and HK). Then A*A E Homg,K(VK, V'). Thus the first part of this theorem and Lemma 3.3.2 imply that A*A = cI with c > 0. Set T = cP1"A. Then T extends to a unitary operator from H onto V which is clearly a Kintertwining operator. It is easy to see that if X E g then Tn(exp(X)) = n(exp(X))T on H K . Since G = KG', this implies that T defines a unitary equivalence. 3.4.12. Theorem. Let (n,H ) be an admissible Hilbert representation of G. Then (n,H ) is irreducible if and only if it is infinitesimally irreducible.
If (n,H ) is reducible then there exists a closed, proper, nonzero, Ginvariant subspace V of H . Since V is admissible it is clear that V, is proper. If H K is reducible then H is reducible by the argument in the first part of the proof of 3.4.10.
3.5. The subquotient theorem 3.5.1. The purpose of this section is to give a proof of the celebrated subquotient theorem of HarishChandra [3], Lepowsky [13 and Rader. We first must establish some generalities about (g, K)modules. We return to the notation in 3.3.1. In this section U ( g ) will denote the universal enveloping algebra of gc.
Lemma.
U ( g ) Kis a Noetherian algebra over C .
Let I be a left ideal in U ( g ) KThen . U ( g ) I is a left ideal in U ( g ) .Since U ( g )is Noetherian, there exist x j E I , j = 1 , . . . , d such that U ( g ) l = C U ( g ) x j . Hence, if y E I then y = C u j x j with uj in U ( g ) .If k E K then Ad(k)y = y and
87
3.5. The Subquotient Theorem
Ad(k)xj = xj. Thus we may replace uj by its projection in U ( S ) ~This . implies that I = C U(g)Kxj.
3.5.2. For simplicity, we now assume that K is connected. If y E K" then we fix V, E y. Set I , = { x E U ( € )x acts by 0 on Vy}.If y, o E K" then set UY." = {x E U ( g ) I,x c U(g)I,,}. We note that Schur's Lemma implies that U(f)/l, is isomorphic with End(Vy).We look upon End(Vy)as a Kmodule under left multiplication. If we apply the material in 0.4.3 and 0.6 it follows that
U ( g ) / U ( g ) l ,is U(g)module isomorphic with U ( g )@,, End(?).
(1)
The latter module can be considered to be a (g,K)module if we use the Kaction, k ( g 0 T ) = Ad(k)g 0 kT. Thus, in light of (l), we may look upon U ( g ) / U ( g ) Z ,as a (9,K)module.
Lemma.
(U(g)/U(g)Z,)(o)= U ( g ) " ~ y / ( U ( g ) "nq U(g)I,) Y
for ally, CT E K A .
If V is a (g, K)module then (since K is assumed to be connected) V ( y )=
{ u E Vl1,v = 0). Let V = U ( g ) / U ( g ) I and , set q equal to the natural projection of U ( g ) onto V. If g E U(g)"Y then I,q(g) = 0. Also, if g E U ( g ) and if
I,q(g) = 0 then g E U(g)"*,. The result now follows.
E K" and let X be a U ( g ) Kand U(f)invariantsubspace of W(y).Then ( U ( g ) X ) ( y )= X .
3.5.3. Lemma. Let W be an admissible (g, K)module. Let y
We first observe that (1)
U ( g ) y 3IW(,) y
=
U(dKU(f) IW(?).
Indeed, let A denote the left hand side of (1). Let B denote U ( € I)W ( y ) .Then B is isomorphic to End(Vy).Thus in particular, B is a finite dimensional simple algebra over C. This implies that A = B'B where B' is the commutant of B in A (for this case this result is implicitly proved in 1.2.2). It is easy to see that B' = U ( g ) K I W ( y ) . We now prove the Lemma. U ( g ) X = ( U ( g ) / U ( g ) l , ) Xwhich is the direct sum of the spaces U(g)"VyX.So ( U ( g ) X ) ( y = ) U ( g ) y , y X= X by (1).
3.5.4. The following result is true for general real reductive groups of inner type. We will give the necessary modifications of the proof below in Section 3.9.
88
3. The Basic Theory of (9, K)Modules
Proposition. Let V be an irreducible (9,K)module. Let y E K " . Then Horn,( V,, V(y)) is an irreducible U(g)'rnodule under left multiplication. Furthermore, if W is an irreducible (9, K)module with W(y)nonzero and if Horn,( (recall our identification). It is clear that Ad(k)X(V) = X(V) for k E K . Assume that G is semisimple. Let K , be the subgroup of Int(g,) generated by exp(ad &,). If x E pc then we say that x is nilpotent if ad x is nilpotent as an endomorphism of gC. Let .,q(p,) be the set of all nilpotent elements in pc. Assume that V is irreducible. Then results of Kostant, Rallis [ l ] imply that X(V) is contained in , t"(p,). Also in the above mentioned paper it is proved that K , has only a finite number of orbits on .h"(p,). One can show that the degrec of the Hilbert polynomial is equal to max{dim K , x I x E X ( V ) } which we write as Dim V. One can show that Dim V is equal to the GelfandKirillov dimension of V (Gelfand, Kirillov [11). The above constructs deserve further study.

3.9.7. In 3.5.24 we gave we developed some results of Lepowsky, McCollum [ l ] that culminated in the proof of Theorem 3.5.5. We now
102
3. The Basic Theory of (5, K)Modules
show how one can extend these results to the case of (possibly) disconnected real reductive groups. So let G be a real reductive group of inner type and let K be as usual. Let H ( K ) denote the space of all Kfinite functions on K , under the left (hence also the right) regular action of K . Then H ( K ) is a representation of K under both L and R ( L ( k ) f ( x )= f ( k  ' x ) , R ( k ) f ( x )= f ( x k ) ) . Set H = H ( g , K ) = U(g,) H ( K ) with the tensor product taken with respect to the action, L, on H ( K ) . O n H we define a multiplication as follows
&,,
A direct calculation shows that this multiplication is well defined on H and makes H into an associative algebra over C. If V is a (9, K)module then we let H ( K ) act on V by f . ~ J= f(k)k*udk. K

 .
We write (g 0f ) u = g f u. We leave it to the reader to show that this defines an Hmodule structure on I/. We have thus canonically assigned to each (g,K)module, V, an Hmodule such that H V = I/. Such an H module is called faithful. One can show that the above correspondence defines an equivalence of categories between C ( g , K ) and the category of all faithful Hmodules. In particular, an irreducible (g, K)module defines an irreducible Hmodule and viceversa. We note that if we identify H ( K ) with 1 0 H ( K ) then H ( K ) as an algebra under convolution (fi * f 2 = L(f1)f2) is a subalgebra of H. Let j be the We put on natural mapping of U ( g , ) @ H ( K ) onto U(g,)&,,H(K). U(g# 0H ( K ) the tensor product algebra structure. Then j defines an algebra homomorphism of U ( g , ) K 0H ( K ) into H. We make H into a (9, K ) module by letting g act by left multiplication and by setting k (g 0f )= A W g 0L(k1.f. If y E K" then set H ( K ) , = {f E H ( K ) f V, = 0 ) . Then the material in 1.5.4 implies that H ( K ) / H ( K ) , is isomorphic with End( 0 such that n k + ' V n V, = nk(n'Vn Vn,)for r 2 0. Thus V, = nV,. Hence 4.1.5(1) implies that V, = (0). This implies that Ker T = (0). Now, Ker & 3 Ker G, Hence, Ker & = (0)for k sufficiently large, since I/ has finite length (4.2.2). This completes the proof. The following result is due to Casselman, however his original proof was much more complicated.
0,
0
0
4.2.4. Corollary. Let V E 2 then there exists a Hilbert representation of G , (r,H ) , such that V is equivalent to H K .
Let a be as in 4.2.3 and let T be an injective element of Horn,,,( V,X u ) . X " is contained in (H")" (3.4.9) since X " is admissible, hence CI(T(V)) is a Ginvariant subspace of H". Since X u is admissible, it is also clear that CI(T(V))K= T(V). Take H = Cl(T(V))and r the induced action of G.
If V E 2 and if (q2) is an admissible Hilbert representation of G such that HK is equivalent to V as a (g,K)module then we call ( r , H ) a realization of I/. The content of 4.2.4 is that every V E 2 has a realization.
4.2.5.
4.2.6. Our next application of the Jacquet module is a technically useful criterion for admissibility due to Stafford and the author.
114
4. The Asymptotic Behavior of Matrix Coefficients
Theorem. Let V be a (9, K)module that is jinitely generated as a U(n)module. Then V is admissible. Let Vj.[n] = V j . be as in the proof of Lemma 4.1.4. Then Vj. = (V/njV)*. Hence dim V j . < 00. The argument at the end of the proof of Lemma 4.1.4 implies that V*[n] is a direct sum of generalized weight spaces and each generalized weight space is finite dimensional. Let x be a homomorphism of Z(g) to C.Let p E (V*[n])x. Suppose that p E V j . but that p is not an element of Vj.Then there exist elements X,, E n, k = 1,. ..,j  1 such that Xl...Xjlp is a nonzero element of VT. This implies that if (V*[n])x is nonzero then so is (V*[n])x n VT. Now 4.1.3(2) implies that V*[n] E 7T Let V, = { u E VIp(u) = 0 for all p E V*[n]}. Then V, = njV. Thus Proposition 0.6.4 implies (see the proof of 4.2.3) that nV, = V,. Now V, is a gsubmodule of V (Lemma 4.1.4), hence V, is a finitely generated (9, KO)module. Theorem 3.8.3 now implies that V, = 0. The proof of Theorem 4.2.1 only uses the following properties of V : it is finitely generated as a U(n)module and V, = 0. That argument therefore proves that V has finite length. Since an irreducible (g, K)module is admissible (3.4.8), V is admissible. 4.2.7. Corollary. If V is a jinitely generated, admissible (g, K)module,then V is jinitely generated and admissible as a (g, KO)module. 3.7.2 implies that V is finitely generated as a U(tt)module. The result now follows from the previous theorem.
4.3. Asymptotic behavior of matrix coefficients 4.3.1. Let G be a real reductive group. We will assume throughout this section that G o = ‘(GO). We retain the notation of the previous sections. Let A. be the set of simple roots of @(P,A ) . Let F be a subset of A. and let ( P F ,A F ) be the corresponding standard ppair. Lemma. Let V E H . Then V/nFV is an admissible jinitely generated (mF,K n P,)module. Let *nF = n n m F .Then *nF is the “n” for a minimal parabolic subgroup of M F . 3.7.2 implies that V is finitely generated as a U(n)module. Hence V/n,V is finitely generated as a U(*n,)module (n = *nF0 nF). The result now follows from Theorem 4.2.6.
4.3. Asymptotic Behavior of Matrix Coefficients
115
Let V be an admissible (g, K)module. Then we denote by V" the space of all elements, p , of V * such that K p spans a finite dimensional subspace of V * (here kp(u) = ~ ( K ' uas) ,usual).
4.3.2.
Lemma.
If V E 2 then V"
E
H.
Let V ; c V , c ... be an increasing chain of submodules of V " . Set V, = ( u E V [ V,:(u) = O}. Then V, 3 V, 3 ... is a decreasing chain of sub
modules of V. Now V is of finite length (4.2.1). Hence there exists k such that V, = V, for all j > k . Since V is admissible, V l = ( p E V" [ p( 5) = 0 ) . We therefore see that VJ: = V L for j > k . Thus V" is finitely generated. Since V  is clearly admissible (V' = 0 V ( y ) * ) ,V" E H. 4.3.3. Let (n,H ) be a Hilbert representation of G. Let (H")' be the space of all continuous linear functionals on H a .If g E G (resp. X E g) and if p E ( H m)' then we define gp (resp. Xp) by g p ( u ) = p ( n ( y  ' ) u ) (resp. X p ( u ) =  p ( n ( X ) u ) )
for u E (H")'. Then 1.6.4(ii)implies that
(1)
gXp
= (Ad(g)X)gp
for y
E
G, X
E
g and p E (H")'.
Set ( H " ) ; equal to the space of all p E ( H " ) ' such that K p spans a finite dimensional space. Then (1) implies that ( H " ) ; is a (g, K)module. Let for u E H , CJ(U) E H ' be defined by a(u)(w) = (w, u ) for w E H. Then a is a conjugate linear continuous isomorphism of H onto H'. Lemma. I f (n,H ) is udmissihle (H");, = ( H K ) " .
then ( H " ) ;
=
o(HK).Furthermore,
This is clear, since dim H ( y ) < m for each y E K" Let V be an admissible finitely generated (g,K)module. Let F be a subset of A,, and let ( P F , A F )be the corresponding standard ppair. Then V/nFV is an admissible finitely generated (m,,K,) module (here K , = MF n K ) . Since aF is contained in Z,,,(ni,), this implies that
4.3.4.
(1)
V / n F V = 0 ( V / n FV ) , the sum over p E
Furthermore, there exists d such that ( H E ((aF)C)*and H E aF. Set E ( P F , V , = (I* ((aF)C)*i ( V / n F V ) , f O}.
p

p ( H ) d ( V / n , V ) ,= 0 for all
116
4. The Asymptotic Behavior of Matrix Coefficients
Indeed, V/n V = (V/n, V)/*n,( V/n, V ) . If 6 E E(P,, V ) then 3.8.3 implies . now follows. that * t i F (V/n, V)a # (V/n, V ) &(2) . . , a r ] . Define H,, . . . , H, Let A. = {al,. then we define A,, E a * by
4.3.5.
E
a by ai(Hj) = 6 i , j .If V E A?
Av(Hj)= max{ Re p(Hj) I p E E(P, V  ) } Fix a norm 11...11 on G (2.A.2.3). The following Theorem generalizes an unpublished result of HarishChandra. Theorem. Let (n,H) be a jinitely generated, admissible, Hilbert representation of G. Set V = HK and A = A". There is a positive constant d such that if p € ( H r n ) ; ( then there exists a continuous seminorm, q,, on H" with the property that
I(P(n(a))v)lI ( 1 + log Ilall)daAap(v) forvEH"andaECl(A+). Let p E (H");. 4.3.3 implies that p = o ( w ) with w E HK. Lemmas 2.A.2.2 and 2.A.2.3 imply that there exists 6 E a* and C > 0 such that if x , y E H then
This clearly implies that if p E (H"); then there exists, 01, a continuous seminorm on H" such that (1)
I(p(n(a))u)lI a'o;(v)
for u E H" and a E CI(A+).
The idea of the proof is to show that if 6(Hj) > A(Hj)then we can replace 6 in (1) by 6  maj with m = min(l/2,p(Hj)  A(Hj)} at the cost of possibly changing the seminorm 01 and putting in a term (1 log Ilall)d. Let c1 E A o . Set F = A.  {LX}.If LX = aj then set H = Hj. Then aF = RH. Set a, = exp(tH). If a E CI(A+) then a can be written uniquely in the form a = a'al with a = exp(C XkHk), xk 2 0, xj = 0 and t 2 0. Let q be the canonical projection of V" onto V"/n, V  .
+
(2) If q ( p ) = 0 then there exists a continuous seminorm T; on H Ip(n(a)v)lI aa"y;(u), for a E C I ( A + )and u E H".
such that
Let X , , . . . , X,, be a basis of nFconsisting of root vectors for a corresponding to the roots PI,. . .,P, respectively. Our assumption implies that p = C X k p k
4.3.
Asymptotic Behavior of Matrix Coefficients
117
with pk E V  . Hence I(pu(n(a)u)l = lC Xkpk(n(a)')l
=
I c pk(n(xk)n(a)u)I
'
Ipk(n(a)n(Ad(a~')Xk)u)l = 2
aBIpk(n(a)n(Xk)U)I
a"'a;(n(Xk)U)).
(2) now follows from 3.8.6(1). Let for z E C, ( V  / n F ) V  ) =denote the generalized eigenspace for H with eigenvalue z. Let P, be the projection of (V/n,V) onto (V/n,V), corresponding to the Hweight space decomposition. Let p E V  . Then 4 ( p ) = C P&). Let p z E V" be such that q ( p , ) = P24(p). Then p  C p z E n,V". We now estimate p,(n(a)u) for each z. Set p z = v. Let V1,. . ., V,, be a basis for U(a,)q(v). We assume that Vl = 4(v). Let v k E V" be such that q(vk)= & for k 2 2. Now Hfk and B
=
=
bknfn
[bkn]has the property that
(3)
(B ~ 1 ) '= 0.
We also note that = HVk
(4)
Let a'
E

1
bknVn E
nFV
Cl(A+)be such that ( a ' ) a= 1. We set
and G(t,a'; U) =
Then ( g ) F ( f , a'; U)
(5)
=  B F ( t , a'; U)

G(t,a'; u).
This implies that
(6)
r
F(t, a'; u ) = exp(  tB)F(O,a'; u )  exp(  f B ) f exp(sB)G(s,a ' ; U) ds. 0
118
4. The Asymptotic Behavior of Matrix Coefficients
We now estimate the terms in (6).(1) implies that (7) IIF(0,a';u)ll I (a')'p(o)
with
b a continuous seminorm on H " .
(2) implies that IIG(t,a'; u)ll 5 exp((G(H) l)t)(a')'b'(u) continuous seminorm on H ffi.
(8)
with
b' a
+
d I)exp(sB)III C(1 Isl)PeSRez for s E R.Here p I (see the beginning of the proof).
(9)
This follows immediately from (3). These estimates imply that if t 2 0 then
for some continuous seminorm p on H" and some positive constant C. We observe that (1 + s)Pe&' is bounded by a constant C for E > 0 and s 2 0. We therefore have
+
+
(10) ~ ~ ~ ( t , a ' I ; u~) l (~ 1t)peCtRe2(a')'p(u) C(1 for t > 0.
+ t)pe'(6(a)2i3) (0")
Here C is a positive constant and fi is a continuous seminorm on H". There are now two cases. Case I: 6 ( H ) 
I A ( H ) .Then there is a continuous seminorm, p, on H",
such that IIF(t,a';u)ll I ( 1
+ t)pe'A'H)(a')'b(o),
for t 2 0.
6 ( H )  3 > A(H). Then in (1) we may replace 6 by 6  (+)a (after having argued as above for all a). We may clearly iterate the argument leading to (10).After a finite number of steps we will be in Case I. Case IZ:
If we apply this argument to all simple roots, the desired estimate follows.
4.4.
Asymptotic expansions of matrix coefficients
In this section we show how the technique of the last section can be refined to prove asymptotic expansions of certain matrix coefficients of an admissible finitely generated Hilbert representation. We retain the notation and assumptions of the previous section. Let F be a subset of Ao, then we have the corresponding standard ppair (PF,A F ) .
4.4.1.
119
4.4. Asymptotic Expansions of Matrix Coefficients
Let (n,H ) be an admissible finitely generated Hilbert representation of G. Set V = H K . As in 4.3.3 we identify ( H " ) ; with V  . Set K , = K n M,. Lemma 4.3.1 implies that V/(II,)~ V" is an admissible finitely generated (m,,K,)module. Since a, is a subspace of Z,(m,), we have (1) (nF)kV"/(nF)kV"splits into the direct sum of finitely many generalized weight spaces for a,. Let Ek denote the corresponding weights. = V.
Here we write (n,)'V"
Let S,: nF 0( ( n , ) k V v " / ( n F ) kV+"l ) + Sk(X 0(u
( r 1 , ) k + 1 1 /  / ( i t , ) k + 2 V " be
defined by
+ (nF)k+lV)= xu + (n,)k+*v.
Then Sk is a surjective a,module homomorphism. Since the weights of a, on n, are precisely the elements of @(P,, A,), the Lemma follows. 4.4.3.
Set E
=
E c{p
(1)
u
+ CI
Ek.Then p E E(PF, V  ) ,
ci
a sum of elements of @(P,, A F ) } .
Let S = { j I cij E F } (here we are using the notation of 4.3.5). Let L+ = ( Z j G snjaj I nj E N}. In this notation (1) implies
(2)  E c { p  c i i p E E(PF, V " ) , CI E L'}. Furthermore, if 6 E  Ek then 6 = p  ci with p E E(PF, V " ) and c( = Z j S snjaj with C nj 2 k. If p, 6 E (aF),* then we say that p 2 6 if p  6 E L + . Let E o be the set of all maximal elements of E(they are clearly contained in  E o =  E(PF,
v")).
Set *AF=A n OM,. Then A
=
*AFA, and Cl(A+)= ( * A Fn CI(A+))C1((AF))'
Let d and Av be as in 4.3.5.
Theorem. Let o E V  . If p E Eo, Q E L+ and u E H" then there exists a polynomial of degree at most d on a, P , , , ~ ( HCT, ; u ) such that (i) The map a, 0H"
+
C , h, u
H p,.a(h;
r ~ u) , is continuous and linear in v.
120
4. The Asymptotic Behavior of Matrix Coefficients
(ii) If H
E
(aF)+then o(n(exp tH)u) is asymptotic to
as t + + co.(4.A.1.1.) (iii) If a E * A Fn CI(A+)then with A
= A",
and B is a continuous seminorm on H" (depending on p and Q).
Fix H as above. Set a, = exp tH. Put ( p ( H )I p E  E } = { z j } with Re z1 2 Re z2 2 .... Let for each j , kj be defined to be such that if Q E L+ and Q= n p , with C n, = kj then Re zj > (A  Q)(H). Let q k denote the natural projection of V" onto V  / ( n , ) ' V  . Let N be a gap in the sequence { z j } (4.A.1.1).Set k = k,.
xqEF
, on (I) If q k ( p )= 0 then there exists E > 0 and a continuous seminorm j H" such that I(Ann(ara)u)I 5 ex~(t(Re ZN
 &))(I+ log IIaII)'aABdv) for t 2 1, a E CI(A+)and u E H " .
Here we have used Theorem 4.3.5. The last inequality clearly implies (I). Suppose that qk(p) is nonzero. Let ,!il= qk(p),... , p,, be a basis of U(aF)qk(p).If x E aF then x p j = Z bjr(x)pr with bjr E ((aF)C)*. Let B ( x ) = [bj,(x)]for x E aF. Let p = p l , .. .,p,,E V" be such that qk(pj) = p j . Then xpj
=
1bjr(x)pr +
rj
with yj = yj(x)E (n,)"V". Let a be as in the statement of the Theorem, then we set F(t,a'; u) =
[
]
PI (n(a,a')u) pp(nc(ara')u)
4.4.
Asymptotic Expansions of Matrix Coefficients
and G(t,a';u)=
[
121
1.
yl(74ala')u) y,(n(a,a')u)
Then as in 4.3.5 we have
(IJ)
($)F(t, a; 0 ) =

B(h)F(t,a; u)  G(t,a; u).
This implies that (111) F ( t , a; u) = exp(  tB(h))F(O,a; u )  exp(  tB(h))
1
exp(sB(h))G(s,a; u) ds. 0
Let Q be the projection of C p onto the direct sum of the generalized eigenspaces for  B ( h ) with eigenvalue whose real part is less than Re z N . Then if we argue as in the proof of 4.3.5 we find that if t 2 1(B = B(h))
(IV)
I
IIQ(eIBF(O,a; u)  eIB eSBG(s,a; u)ds)II 0
exp(t(Re ZN

+
~ ) ) ( 1 log Il~ll)'~''\B(u)
with \B a continuous seminorm on H". As in the proof of 4.3.5 we find that if R (V)
=
I  Q and if t 2 1 then
+
([RefB G(t,a;o)ll I C E f ( l log Ilall)'a''B(u)
with p a continuous seminorm on H" (V) implies that
7 R(ehBG(s,
a; u ) ) ds
0
converges absolutely. Set 30
1
F o ( t , a; u ) = elBRF(O,u ; u )  elB R(e"G(s, a; v)) ds. 0
Then RF(t, a; u)  F o ( t , u ; u)
=

7
e  t B R(e"G(s, a; u)) ds. I
122
4. The Asymptotic Behavior of Matrix Coefficients
A straightforward estimation shows that there exists E' > 0 such that IleIB
a,
1 R(e"G(s,a; u))dsII I (1 + log Ilall)da"exp(t(Re zN

E))~(u)
I
for t 2 1 with p a continuous seminorm on H". Set fN(t, a; u) equal to the first component of F o ( t ,a; u). Then fN(t, a; u ) = C j 5 Nexp(tzj)uj,,(t, a; u), with u ~ , ~ a; ( u), a polynomial in t of degree at most d. If t 2 1 then the above inequalities imply that
with fl a continuous seminorm on H . If M is a gap of the sequence { zj} and M > N then the above estimates imply that uj,N = uj,Mif j I N . We set p,(t; u) = pj,N(t, 1; u ) for N > j. We have at this point shown that
(VII) p(n(u,)u) is asymptotic to the exponential polynomial series C exp(zjt)pZJ(t;u ) as t + + 00. We now refine the above argument to prove the Theorem. Let for E > 0, SF,&= { h E aF I llhll = 1 and ~ ( h>) E for all tl E @(PF,A,)). If h is a nonzero element of aF then set a(h) = h/l\hl\. (VIII)
If HI, H ,
E
SF,€and if t, s > 0 then o(tH,
+ sH,) E SF,€.
This is an easy consequence of the triangle inequality. Set I ( & ) = max{A(h) h E S,,,}. (A = Av.) (IX)
If h E S,,,, p E  E k then Re p ( h ) I I ( & )  kE.
This follows from 4.4.3(2). Set 'E equal to the set of weights of a, on V/(n,)kV". Set Fk = vj, k j E . Then (IX) implies
(XI
If p E Fk then Re p ( h ) < I ( & )  kE
for h E SF,€.
Put Ek,&= { p E k E I Re p(h) > I ( & )  ke for h E S,,,}. Since SF,&is compact it is easy to see that (XI) There exists 6 > 0 such that if h E SF,&and if p E Ek,e then Re p(h) > Y ( E )  kE  6. Let p E V  . Fix k > 0. Let jil, . . .,ji,, pl = 1.1,. .. ,pp and yl,. . ., yp, and B(h) be as above. Then the eigenvalues of B(h) are the 8(h)with 8 E kE.Let P = pk be the projection of C ponto the sum of the generalized eigenspaces for  B(.) that
4.4.
123
Asymptotic Expansions of Matrix C'oefficients
are elements of Ek,'. We set
and
for u E H " . Then (as usual), (i)F(n(exp th)u) =  B(h)F(n(expth)u)  G(n(exp th)u). This implies that
(XII)
t
F(n(exp th)u) = e'B'h)F(u) e('s)B'h'G(n(expsh)u)ds. 0
Set Q
= Qk =
I

P. The standard estimates yield
IIQF(n(exp th)u)ll I exp(r(e)  ke)t)(l L)'~B(u) for t 2 1, h E SF,e and B a continuous seminorm on H".
+
We also note that (in light of (XI)) if s > 0 then IlesBB'h)G(n(exp
sh)u)(lI ( 1
+
S)2de(ker(&)6)s
eS ( r ( E )  k d P
W
Here we have used the obvious estimates in order, and /Iis a continuous seminorm on H".This implies that the integral
7 e"B'h'PG(n(expsh))ds 0
converges absolutely and uniformly for h E SF,e.We set a,
Fk,h(U) =
PF(0) 
0
esBB'h)PG(.rr(exp Sh)U) dS
for u E H" and h E SF,e.The above estimates imply that
(XIII) IJF(n(expth)u)  e  ' B B ' h ) F k , h ( u ) I 1 I (1 + t)2dexp(t(r(s)  k ~ ) ) B ( u ) , for h E SF,,,u E H" and /? is a continuous seminorm on H ".
124
4. The Asymptotic Behavior of Matrix Coefficients
This implies that
It follows that (XV)
Fk,h(n(expth)) = etB(h)Fk,h(u) for h E SF,&and u E H".
We now assume that k has been taken so large that k  T ( E ) > 0. Let 6 be as in (XI). Let 0 < c < 1 be such that k  I ( & )  (+)6 = (ks  T(E))c. Let H , and H , E SF,&be such that ( H , , H , ) > c. (We note that ( H , H ' ) > 0 for H, H' E aF.) It is easily checked that if t , s > 0 then IItH, sH,II > cs t . We leave it to the reader to show (using the above) that
+
IleXP(f~(H2))(Fk,Hl(~(eXP(tHZ)D) PF(n(exp tH,)ull 5 (1
for t 2 1 and fl is a continuous seminorm on H". This implies that limt+ + " etB(H2)Fk,H1(n(eXP tH,)u) lim
lim
=
+
+ t)2dest'2fl(u)
Fk,H,(U).Hence
(n(exp(tHl + s H 2 ) ) u ) = Fk,H1(').
eB(tH~+sH2)~
s++m t'frn
We therefore see that Fk.H2(n(eXP
= exp(tB(Hl))Fk,H,(u)'
If we interchange the roles of H, and H , , this implies that lim
eXP(t~(H2))Fk.H,(n(exp(tHZ))u) = Fk,HI
(').
t'+W
We have (finally) shown that
&HI
= Fk,H2. Since
equal to the common this implies that F k , h is independent of h E SF,&.Set Fk,E value of the Fk,h. If we combine all of the above we have
for t 2 1, h E SF,&and fl is (you guessed it!) a continuous seminorm on H". We now note that if we choose a smaller E then Fk.& will not be changed. We may therefore denote Fk,& by Fk.If we now combine (XVI) with (VII) the theorem now follows.
125
4.5. HarishChandra's 3function
4.5.
HarishChandra's =function
4.5.1. We retain the notation of the previous sections. Let V be an admissible (9, K)module. Let V  be as in 4.3.2. Then V" is also an admissible (g, K)module. The next result gives a characterization of V " .
Lemma. Let W be a (g,K)module. Suppose that there exists a complex bilinear mapping b : V x W + C such that (1) (2)
b(Xu, W ) =  b(u,X W ) ,b(ku,kw) = b ( ~W,) for u E V, w E W, X E g and k E K . If b(u, W ) = 0 then u
=0
and if b(V,w) = 0 then w
=0
(ie., b is nondegenerate). Then W is (9,K)isomorphic with V" If w E W then set T(w)(u)= b(u,w). Then T defines a g and Kmodule homomorphism of W into V*. Thus T ( W ) is contained in V  . The nondegeneracy of b implies that T is injective. If y E K" let y* denote the class of the dual representation of any representative of y. Then (2)combined with the Kinvariance of b implies that dim W ( y * )= dim V(y).Hence T is surjective. 4.5.2. to 0.
Let (no,p,H",') be as in 3.5.5. Let
0"
denote the dual representation
Lemma. (H"F~))K" is isomorphic with (H""."),. Let f E (H'P)Kand g E ( H ' " *  P ) K .We set
( f , g > = j ( f ( k ) , g ( k ) dk. ) K
Then Lemma 2.4.1 implies that ( , ) satisfies 4.5.1(1). We leave it to the reader to prove that ( , ) satisfies 4.5.1(2). 4.5.3. We are now ready to study the Zfunction. Let Ep be defined as in 3.6.1. We set E = Eo. We have followed HarishChandra in giving this zonal spherical function a special name. Lemma 3.6.7 indicates its special role. Also the function E will be used in the definition of HarishChandra's Schwartz space (7.1).
126
Theorem.
4.
The Asymptotic Behavior of Matrix Coefficients
There exist positive constants C and d such that ap 5
+ log Ilall)d
Z(a) Ic a  q 1
for a E Cl(A+).
Let (n,H) denote ( n O , H o (see ) 3.6.1). Under the pairing ( , ), H K = ( H K ) ” . Let 1, be as in 3.6.1. Then Z(g) = (n(g)lo, lo). Set V equal to the (g,K)submodule of HK generated by I,. Then under ( , ), V” = V. Suppose that p p is a weight of a on V/nK Let (i be an OMtype of the p + p weight space of V/nV. Then Lemma 3.8.2 implies that there is a nonzero element of Horn,,,( V,( H P ) K )Frobenius . reciprocity implies that (i must be the trivial OMtype. Now this implies that ZP = Eo. Theorem 3.6.6 now implies that p = SO for some element in W(G,A ) . Hence p = 0. We therefore conclude that (in the notation of 4.3.5) A, =  p . The upper inequality now follows from Theorem 4.3.5. We now prove the lower inequality. Formula (*) in 3.6.7 says that
+
1
S(U)= up ~ ( i i ) ~ ~ ( ~  ’ E ~ ) ” d i i . N
We make the change of variables 6 H aria'. Then we have Z ( U )=
1(a(aEa’)a(E)‘)”a(6)2pdE. N
Lemmas 2.4.5 and 3.A.2.3 now imply the first inequality. 4.5.4.
We now show how HarishChandra used the above result to prove the convergence of two important integrals. These results will be used in the next chapter to prove the conversion of the intertwining integrals of KunzeStein, KnappStein and HarishChandra. Our exposition follows that of HarishChandra [8]. Theorem. Let d be as in Theorem 4.5.3. If
E
> 0 and if F is a subset of A. then
1 U ( E ) ~ (~ p(1og a ( 6 ) ) )  d  E d n< co. NF
Let h E Cl(a+).Set a, = exp th. Then 4.5.3 implies that there is a positive constant C such that (1)
(a,)PE(a,)I C(l
+ t)d
for t 2 0.
We have seen in 4.5.3 that (2)
S
(u,)”Z(UJ= ~ ( E ) ~ a ( a , E’)” a ; dE. N
127
4.5. HarishChandra's =function
We now choose h to be the element such that a(h) = Ofor CI E F and or@) = 1 for CI E A.  F. Then mF = C,(h). Set *iiF=mF n ii. Then t i = *tiF@ tiF.2.4.5 implies that we can normalize the invariant measure on *& such that
J
a(*E,)2d*E, = 1.
"r;
4.A.2.1 and 4.A.2.2 imply that we can normalize the invariant measure on NF such that i f f E CJN) then
1f ( E ) d E =
(4)
N
j *NFX N F
f(*EFn,)d*6,dE,.
We assert that (a,)%(a,) =
j
NF
a ( i i ) P a ( ~ , E' ~ ) p; dE.
Let Z(t) denote the right hand side of (5). Since a,xa_, = x for x E *NF and x E N a ( x ) k ( x )the obvious manipulation of (2) using (4) yields I(t)=
*NF
j
a(* fi,)2Pa(k( * E,)E,)Pa(k(* e~)CZ~fia,)~ d * EF d E,. NF
Now k(*N,) is contained in KF which commutes pointwise with the a, for R . Also, a(ka,Ea,) = a ( a , k E k ~ ' a  , for ) k E K , , t E R and Ti E &. Since K , is compact, dkE,k' = dE, on N, for k E K,. The obvious calculation now yields tE
(3) now implies (5). In particular ( 5 ) implies
(6)
J a ( n ~ ) ~ a ( a , % , a dn, _ , ) ~5 c(1 + t ) d
for t 2 0.
NF
We now use the notation in 4.A.2.4 (with the "F" there equal to @). Then we have for t 2 0 a(a,nFaf)'
Now
IIo(arnFar)uO
II
=
\\o(a[Eat)'uol[.
128
4. The Asymptotic Behavior of Matrix Coefficients
We have proved the following inequality
(7)
U(U,~~,U_,) 2
+ e'a(iiF)p2P)'i2
(1
If r > 0 then we set (&)r
= {ii E
for t 2 0 and ii,
E
flF.
NF I a(%)2 r } . Then 4.A.2.3(2)implies
(8) (NF)ris compact for all r > 0. In (7) we take t =  2 log r for 0 < r < 1. Then ( 6 ) implies that if Ti E (&)r then a ( ~ , E a  2 ~ )2''2. We therefore find that C(1
+ t)d 2
J
a(Ti)Pa(a,Tia_,)PdTi2 2  l / *
(NFL
j a(n)Pdn (NF)*
which implies (9) We now take r p = exp(  2 p )for p = 0, 1,. . . . With this notation (9) implies that
j a(Ti)Pdn I C'(1 + 2P+l)d IC"2Pd.
(NFbp
If ii E (N,)rp+, (&)rp we have rp 2 a(ii)P 2 rp+ Hence on this same set we have 1 + 2 p I 1  p(log a(%))I 1 + 2 p + ' . This implies that if E > 0 then
j (NF)rp
I
a(E)P(l

p(l0g a(n))d&dn
(NF)rp
I C"(1 + 2p)dE2pd I C"'2&P.
If we sum over p > 0 we find that
j_ a ( n ) ~ (1 (log a(ii)))d"n
N F ( N F ~ , ,
c"'C 2pEp< 00.
I
This implies the theorem since (NF)ris compact. 4.5.5.
We retain the notation of the previous paragraph. If g E G then we can write g = nFmF(g)aF(g)kF(g) with nF E NF, m,(g) E OM,, a&) E A , and k&) E K . We leave it to the reader to check that aF(g) is determined uniquely by g but that m&) and k F ( g ) can be replaced by m,(g)k and k'k,(g) for k E K , . Fix a norm 11...11 on G (2.A.2). Let EF be the ''2'function for OM,. We extend Z, to G by setting =.,(namk) =E,(m) for n E N F , a e A F , m E o M F and k E K . The above considerations imply that this extension is well defined.

I29
4.5. HarishChandra’skfunction
4.5.6.
Theorem. If r > 0 and if q > d
+ r then
j a(fi)PFsF(n)(l + log IlmF(6)ll)d(l p(l0g a(n))‘dfi
of F such that the elements of M , have block diagonal form A2
0
O and the elements of NF have block form
Ad
I
130
4. The Asymptotic Behavior of Matrix Coefficients
Now, if k E K F and if ii
E
NF then
kii = knmF(ii)aF(E)kF(ii) = knk'km,(ii)k'a,(ii)kk,(n)
with n E NF. Thus kmF(ii)= m,(kiik') and aF(kii)= +(ii). This implies that I
J
=
NF x
KF
J
= NF
x KF
a(kiik')PFa(mF(kiik'))(l  p(1og a(ii))'qdiidk ~ ( i i ) ~ "( p(1og l a(kiik'))'qdiidk
I J a(ii)P(l  p(l0g a(ii))'qdiidk NF
< 00
by Theorem 4.5.4.
4.6.
Notes and further results
The theory of the real Jacquet module is an outgrowth of work of Casselman and of Casselman and the author to introduce a functor on the category X with the same exactness properties as the Jacquet module (Jacquet [l]) in the case of padic groups. With this notion in hand many arguments for the "real case" are proved in a manner quite analogous to the way they are proved in the "padic case". Indeed, the material in this chapter is more strongly influenced by HarishChandra's work on padic groups than it is by that on real groups. A more complete exposition of the theory of the Jacquet module can be found in Wallach [2] and Wallach [3]. The category V,introduced in Section 1, is essentially the same as what some authors call 0'. This category is an extension of the category 0, which was introduced by Bernstein, Gelfand, Gelfand [l] to study the structure of Verma modules. Further results on Verma modules will be proved in Chapter 6. The best reference for the theory of Verma modules is Dixmier [2, Chapter 71. 4.6.1.
We use the notation of Section 4.3. If in Theorem 4.4.3 the space H a , is replaced by H K then the expansions in Theorem 4.4.3 can be found in Casselman, MiliEic [l] (these results sharpen earlier work of HarishChandra). Their proof uses the theory of regular singularities as generalized in Deligne [l]. If u E HK their results imply that the expansions actually converge to the matrix entry. However, one must still prove that their expansions are asymptotic in our sense (c.f. Borel, Wallach [l, Chapter 31). We will see in Volume I1 that HarishChandra's theory of the constant term is a fairly direct consequence of Theorem 4.4.3 (in light of Lemma 7.7.5).
4.6.2.
4.A.1.
131
Asymptotic Expansions
4.6.3. If we combine Theorem 4.5.3 with Lemma 3.6.7 then, in the notation of 3.6.7 we have Proposition.
If v E (ac)* then IZu(a)II alRev+p(l + log Ilall)d
for a E CI(A+).
The number d that appears in 4.5.3 can be taken to be I W(G,A ) [  1 since one can show that dim(Ho),/n(Ho),
=
I W(G,A ) [ .
4.A. Appendices to Chapter 4 4.A.1.
Asymptotic expansions
4.A.l.l. By a formal exponential polynomial series we will mean a formal sum of the form
where pj,, is a polynomial in t for each j, n. The point here is that we do not care if the series converges. Fix such a formal series. Then we may rearrange it in the following way:
with uj E {zk  n 1 < k < p , n > 0, n E N}, Re u1 2 Re u2 2 ..., and pu, is the sum of the pk," with zk  n = u,. We will call N a gap of the series if uN > uN + 1. Iff is a function on R then we say that f is asymptotic as t , cc to the formal polynomial series given as in ( 1 ) if for each gap, N , there exist positive constants (depending on N ) C and E such that
+
(3)
I f ( t )
1 exp(ujt)pu(t)l5 C exp((Re uN  E ) t )
for t 2 1 .
j O}. We set +a* = { p p = C xjaj with x j > O}. Let (x,H ) be an admissible Hilbert representation of G. Then we say that ( q H ) satisfies the weak inequality if there exists a nonnegative constant, d, such that if W E H , , and U E H" (1.6.1) then I ( x ( g ) u , w ) l I o ( u ) ( l log Ilyll)'E(g) for all g E G and CJ is a continuous seminorm on H" depending only on w. Here E is as in 4.5.3. We say that ( n , H ) satisfies the strong inequality if for each d > 0 and w E HK, u E H" then I(x(g)u,w)l Igd(u)(1 + log Ilyll)dE(g) for all y E G. Here, 0, is a continuous seminorm on H" depending only on d and w . These definitions are provisional, unitary representations satisfying the weak inequality will be called tempered later in this chapter. We will also see that if (x,H ) is irreducible and unitary then (x,H ) satisfies the strong inequality if and only if it is square integrable. Let V be an admissible finitely generated (9,K)module. Let A, be as in 4.3.5. Then we say that V is tempered if A, p E Cl(+a*). We say that V is rapidly decreasing if A, + p E +(a*).
+
+
5.1.2. Proposition. Let ( n , H ) be a Hilbert representation of G . If HK is tempered then (n,H ) satisjies the weak inequality. lf HK is rapidly decreasing then (n,H ) satisfies the strong inequality.
Let V = HK. Theorem 4.3.5 implies that there exists d > 0 such that if w E V then there exists a continuous seminorm, C J ~depending , on w such that I(x(a)u,w ) l I a,(u)(l + log Ilall)'a" (A = A") for all a E Cl(A+). Let w l , . . ., w p be a basis of the span of K w . Let a(u) = supksKC a,(ku). x(k)w = I:gj(k)wjwith each gj a continuous function on K. (x(k,ak,)u, w ) = ( n ( a k 2 ) u , x ( k , ) pw1 ) = C conj(gj((k,)'))(n(ak,)u,w ) . It follows that (1) If w E V and u E H" then l{x(k,ak2)u, w ) l I o(u)(l
for a E CI(A+)and k,, k ,
E
+ log Ilall)da"
K.
We now prove the result. Suppose that V is tempered. Then aA I app for all a E CI(A+). Now, I =(a) for all a E CI(A+) (Theorem 4.5.3).
5.1.
139
Tempered (9, K)Modules
Since Z ( k , g k , ) = E(g) for all k , , k , E K, the first assertion now follows immediately from (1). If p E +a* then for each r > 0 there exists a positive constant C, such that, a' I Cr(l log Ilall)' for a E CI(At). Hence, the second assertion is also a direct consequence of (1).
+
5.1.3.
Proposition. Let V be an admissible finitely generated (9,K)module.
If V is rapidly decreasing then V splits into a direct sum V = @ V, with
4
irreducible. Furthermore, there exist ( n j ,H j ) irreducible (unitary) square integrable representations of G such that is equivalent to Before we prove this result we must prove a lemma which will be useful in the later chapters.
Lemma.
There exists a positive constant r such that
Let y(a) be as in 2.4.2. Then y(a) Ca2Pfor a E CI(A+). We now apply Lemma 2.4.2 (using the left and right Kinvariance of E and 11...11)
I C
l
a2PZ(a)2(1+ log Ilall)'da
A+
< C' J (1 + log I(all)drda.

A+
Here we have used Theorem 4.5.3. Since the last integral is finite for r sufficiently large the result follows. 5.1.4. We now prove the above proposition. Let (n,H ) be a realization of V (4.2.5). Let (.*,El) be the conjugate dual representation. Let V" be the underlying (9,K)module of 7t*. Then V  is admissible and finitely generated (see 4.3.2). Let u l , . . ., up be a set of generators for V  as a (9,K)module. Then the set {n*(g)vjlg E G, j = 1,. .., p } spans a dense subspace of H . If u, w E V then we put (V,W) =
Ci G1( n ( y ) v ? v j )conj((n(g)w, vj>)dg*
The above integral converges absolutely by Lemma 4.5.3. The choice of the uj implies that (u,v) > 0 for nonzero u. Since dg is right invariant it
5. The Langlands Classification
140
follows that (1)
( X u ,w ) =  ( u , X w ) = (u,w)
(ku,kw)
and for X
E
g, k
E
K, and u,w E V.
This implies that if W is a (9,K)submodule of V then W', its orthogonal complement relative to ( , ), is also a (9,K)submodule. Since V has finite length as a (g, K)module (4.2.1), it is clear that V splits into a direct sum of irreducible (g, K)modules. Since V  * j ( V ) and V  V" are exact functors (Theorem 4.1.5) we see that each summand of V is rapidly decreasing. Thus to complete the proof of the proposition we may assume that V is irreducible. Fix w a nonzero element of V. Let T(u)(g)= (n(g)u,w) (( , ) is the original inner product on H ) for u E V. We have shown that T(u)E L 2 ( G ) for all u E V. If x E U ( g ) then xT(u).= T(xu).Thus T ( V )consists of smooth vectors (1.6.1)for L 2 ( G ) .The argument in the proof of Theorem 3.4.9 implies that T (V ) is contained in the space of analytic vectors for L2(G).Thus if we set H , = CI(T(V)) then H , is an R(G)invariant subspace of L 2 ( G ) (here R(g)is right translation by g and we have used Proposition 1.6.6). Set nnl(g) equal to the restriction of R(g) to H , . Then it is clear that ( H I ) K= T ( V ) and that T is a (9,K)module isomorphism of V onto T ( V ) .The result now follows from Proposition 1.3.3(2). 5.2. The principal series 5.2.1. We retain the assumptions and the notation of the previous section. Let F be a subset of A. and let (PF,A F )be the corresponding ppair (2.2.7). Let ( a , H , ) be a Hilbert representation of OMF which is unitary when restricted to K, = K n PF. Let p E (a>)c. We define mHPF*u*M to be the space of all smooth functions J':G + (H,)"such that f(namg) = a p + p a ( mf) ( g ) for n E N F , a E A,, m E OMF and g E G. We define for f , g E mHPF.u*fl (f>S> =
1 ( f ( k ) , g ( k ) )dk.
K
Let H p F , u + denote the Hilbert space completion of mHPF,a,a.Then 1.5.3 implies that if we define nPF,,,,(g).f(x) = f ( x g ) for g, x E G then ('PF.6.F HPF+u+) is a Hilbert representation of G. We denote by I,,,,,, the underlying (9,K)module of (n,,,,,,, HPF*u+ 1. 5.2.2. Lemma. If (a,H,) is admissible and finitely generated then Ip,.,.p i s an admissible (9,K)module. Furthermore, IPF,,,, is the space of all f E mHPF.a*p
5.2. The Principal Series
141
such that (1) f ( K ) c W c (H,)K with W a ,finite dimensional subspuce depending only on f . ( 2 ) f is right Kjnite.
Let f E I,,,,,,. Then in particular, 1' is a smooth vector for T C ~ , , , Also ~ ~ ~ . f is Kfinite, which means that there exist f ,,.. . , f nE I p F , g , , such that n,,,,,,(k)f = E uj(k)& for k E K . Here aj is a smooth function on K. Thus f ( k ) = C uj(k)&(l).Now, if k E K , then f ( k ) = o ( k ) f ( l ) .Thus f satisfies (1). (2) is an immediate consequence of the definition of Kfinite vector. The converse is equally easy and left to the reader. We now assume that (o,H,) is admissible and finitely generated. Let *PF = P n OM,. Then 4.2.2 implies that (in the notion of 4.2) is equivaY y a finite dimensional representation of lent to a submodule of O M F Xwith *PF(here the subOM, indicates that we have replaced G by OMF).Hence, as a K,module (Hn)K is equivalent to a subrepresentation of Ind(y l o M) . This implies that I,,,,,, is equivalent as a Kmodule to a subrepresentation of Indf,(IndtA(y = IndfM(y Frobenius reciprocity now implies that IPF,n,, is admissible.
IOM).
loM)
5.2.3. We now give another variant of Frobenius reciprocity which seems to have been first observed by Casselman. We retain the notation of the last paragraph. Let V be a (g, K)module. If T E Hom,,,(V, I,,,.,) then we set T A(u) = T(u)(l).Since T(u)(n)= T ( u ) ( l )for n E NF we see that T"(n,u) = 0 for u E V. If X E mF then T ( X u ) ( l )= (XT(u))(l)= d/dtI,=,T(v)(exp tX) = X ( T ( u ) ( l ) ) Here . the action is on the module ( H , , J K which is with a, acting by p + p F . We therefore see that T A defines an element of H o m ~ ~ , K ( V / n F V,( H o , p ) K ) . We have Lemma. The map T H T" defines a bijection between Hom,*,(l/; I,,,,,,) H0rnInF..K( V/n, V,( H , , , ) K ) .
and
The proof is exactly the same as that of 3.8.2. Let W be an admissible, finitely generated (Om,, K,)module. Let H,) be a realization of W.Then the (g, K)module, I,,,,,, depends only on W for each p E We write I p F , w , , for this (g, K)module.
5.2.4. (0,
is equivalent with IPFIW, @.
Lemma.
If f
E
I,,,,.,
and if g E
~
,then we set
142
5. The Langlands Classification
Here, ( , ) denotes the natural pairing of Wand W " . The result now follows from 2.4.1 and 4.5.1. Proposition Let V be an irreducible, tempered, (g, K)module. Then there exists a standard ppair, (P,, A,), and an irreducible unitary representation, (a,H,,), of OM, such that (H,,)Kis rapidly decreasing and p E (aF)*such that V is isomorphic to a summand of IPF,a,ip. 5.2.5.
Set E( V )=  E(P,, V " ) (4.3.4). If ,u E E ( V ) then set F ( p ) = { j Re(p + p, pj) < O } . Let A. E E ( V ) be such that F(Ao) has the minimal number of elements. Set F = { c t j j E F(Ao)}.Let p denote the restriction of A. to a,. Then X = ( V " / n F V " )  pis nonzero, admissible, finitely generated ( m FKF), module. By the definition of F, Re(/\, + p, pj) = 0 for j $ F(Ao). Thus p pF = iv with v E (aF)*.Let W be an irreducible (nonzero) quotient of X . Lemma 5.2.3 now implies that V  is isomorphic with a submodule of I p F , w , i v . Suppose that (X/*n,X), is nonzero for some [ E (*a,):. Then 4' + p E E ( V ) . By the above Re( [ p p, Pj) = Re(i * p F , pj) for all j . Thus, the definition of F implies that Re( [ + * p F , pi) < 0 for all j E F(Ao). We conclude that W  is a rapidly decreasing (Om,, K,)module. 5.2.4 implies that V is equivalent with a quotient of IPF,W,iv.5.1.3 implies that W" is the underlying (Om,, K,)module of an irreducible, square integrable representation, (6,IT,,), since ( z ~ ~ . , HP.U.iV , . ~ ~ ,) is unitary (1.5.3).The result now follows.
+
+ +
+
5.2.6. Corollary. Let V be an irreducible, tempered (g,K)module then V is equivalent t o the underlying (9, K)module of an irreducible unitary
representation.
This follows directly from the last part of the proof of the preceding theorem. Fix a subset, F , of Ao. Let ((T,H,,) be an admissible, finitely generated, Hilbert representation of OM,. Let p E (a,):. Let ( 7 ~ ~ ~ , , ,H, ~P, F s a 3be p ) as above. Since PF will be fixed in this number, we will drop the P, in our notation. Let (H,,)" be endowed with the usual topology (1.6.3).We set ( H " 9 p ) ,equal to the space of a smooth functions from G to (H,,)" that are in m H O ~ If p. x E U(g) and if 6 is one of the seminorms defining the topology on (H"."), we set S,( f )= supkEKh ( ~ , , ~ f( x(k)) ) for f E (H"+), . Then it is easy to see that (H"*p)m defines a smooth Frechet representation of G, that IpF,o,p is a dense subspace and that (H"."), is contained in (H".')". It can be shown (c.f. Borel, Wallach [l, III,7.9] that (IT"+), is equivalent to (H",p)" as a smooth Frechet module. 5.2.7.
143
5.2. The Principal Series
5.2.8.
The following result will be used in Section 4.
Here we have used Ilrn,(g)ll I C‘llg[l)d’. To see this, we choose ( B , W )a finite dimensional irreducible representation of G that is unitary for K and is such that if W, = { w E W I a(n)w = w, n E N F } then the representation of OM, on W, has compact kernel. If a E A , then a(a) = a’Z on W,. Thus if g = n&)aF(g)m,(g)kF(g) and if w E W, is a unit vector then I[a(g‘)w\\= aF(g)’\ 1 o(rnF(g))p wI 1. Hence 11 o ( m F ( g ) ) 11 = a F ( g ) ’ l l B(gp‘ I w 11 I cllg 11‘ for some q. We observe that sup{Ila(m)wll;w E W,, IIwJJ= l} is a norm on M F . We now continue the argument. The last expression above is equal to
NOWa(m,(g))aF(g) = a(g) and a,(kg) = a,(g) for k E K , . Thus
=
C(l
+ log lIgl[)‘’KJ a(kg)Rej’+pFdk
=
C(l
+ log
~ ~ ~ ~ ~ ) “ Z R e j ’ ( ~ ) ~
The last inequality in the statement follows from Lemma 3.6.7.
5. The Langlands Classification
144
5.3. The intertwining integrals We retain the notation of the previous sections. Let F be a subset of A. and let (PF,A F )be the corresponding standard ppair. Fix (a,H,) a representation of OMF that satisfies the weak inequality. We set (as usual), NF = @NF) and KF = K n M F .Let O ( P F , A Fdenote ) the set of roots of aF on i t F .
5.3.1.
Lemma. Let p E (aF);be such that Re(p, a) > 0 for d l (1) I f f E ( H U s f l ) and , if w E then
j I(f@),
CI E
@(PF,AF).
w>l dii < a.
NF
Furthermore the map
f H J (f(n),w> dii NF
is continuous on ( H " 3 p ) ., (2) If w E ( H g ) Kis nonzero then there exists f E IPF,,,p such that
J
NF
(fm,w>dn
is nonzero
We will use the notation of 4.5.5. We first prove (1). If ii E NF then f ( i i ) = f(nm,(ii)aF(ii)kF(ii))with n E N F .Thus f(n) = ~ F ( n ) ~ ' ~ ~ a ( m F ( i i ) ) f ( k This F(n)). implies that
J
NF
I(f(n),w>I
d~ =
J
NF
I j
aF(n)ReP+P~(a(mF(n)f(kF(n)),w>~dii
+ log IImF(n)II)r~:F(mF(ii))dii
fl(f(kF(~))aF(~)'+~(l
NF
with p a continuous seminorm on (H,)". Here we have used the weak inequality. Set y ( f ) = supkEK fi(f(k)). Now a F ( i i ) R eI p Cq(l  p(1og for all q > 0 (4.A.2.3). Thus the integrand is dominated by 
Cqy(f)a,(n)a,(m,(n))(
p(lOga F ( i i ) )  4 .
(1) now follows from 4.5.6. We now prove the second assertion. Let h E C,(N,) be such that
J
h(ii)dE = 1.
NF
Set f(nrnaii) = afl+Pa(m)h(ii)w for ii
E
N,, rn E OMF,a E A , and n E N F . Extend
5.3.
145
The Intertwining Integrals
f to G by 0. Then f
E
( H g g p ) and , ( f ( i i ) , w ) dn = ( w , w )
J
> 0.
NF
Since I,,,,.,
is dense in (H“.’),, the continuity assertion in (1) now implies (2).
5.3.2.
We retain the above assumptions on o and p.
Lemma.
Let f E IpF,a,p then there exists a ,finite dimensional subspace, V ( j ) , such that j ( f ( i i ) , w ) dn = 0
of
NF
for all w E Let w
J
E
orthogonal to V (.f ). If k
E
( f ( i i k ) , w ) dn
K , then = NF
NF
=
(a(k)f’(k‘iik),w ) dii =
J
( o ( k ) f ( i i ) w) , dii
NF
( f ( i i ) , a ( k ‘ ) w ) d i i .
J NF
Here we have used the invariance of dii on NF under conjugation by K , . Let S be the set of all elements, y , of K A such the projection o f f into the yisotypic component of ZPF,a,pis nonzero. Let V ( f ) be the sum of the 6isotypic components of H, with 6 a constituent of some y E S restricted to K,. Then since S is finite, V ( f )is finite dimensional. The above formulas now imply the Lemma. of 5.3.3. The preceding Lemma implies that there exists a linear map, b,F,u,ll, IPF*,,,L to ( H u ) K , such that
J ( f mw>dn =
(bPF,o,p(.f)9
w>
Nr
E I p F . n , p ,w E ( H o ) K .The calculations in the proof of 5.3.2 imply that flPF,,,, is a K,module homomorphism.
for all f
Lemma. ~pF,,,,(fi,lp,,,p) = 0. Let of l P ~ , o , p / f i F I P ~ . ointo , p ( H n ) K . Then @PF,,,p E
CL,~.,,~
Hom,,.dl,,
(see 5.2.3 for notation). Let f E lpF30,p and let X
E TI,.
vJf)
be the corresponding linear map
, , , ~ ~ / ~ ~ l , ~ . , , , pp ,r (MH u , p
If w E (H,,)Kthen set =
j
N I‘
( f ( f i ) , w> dn.
146
5. The Langlands Classification
Lemma 5.3.1 implies that yw is a continuous functional on (H".,)),. Thus y,(Xf) = d/dtI,=,y,(n,,,(exp tX)f) = 0 by the right invariance of dii on &. If X E OmF then y,(Xf) = d/dtIr=oyW(nu,,(exp tX)f) =
1) J (f(E dt NF t=o
exp tX), w ) dii (o(exp tX)f(exp(  tX)ii exp tX), w ) dii
=
dt t = o NJF (a(exp tX)f(E), w ) dii.
The last equation follows from the invariance of dE on NF under conjugation by elements of OMF.We leave it to the reader to see that the estimates in 5.3.1 justify the interchange of differentiation and integration. We have thus shown that BpF,a,r(fiFI~F,u,r) =o and that p~,,~,,(xf) = x f l ~ F . ~ . u ( f ) for X E OmF. If h E aF then we may argue in exactly the same way (taking into account d(aEa') = a2pFdii on N,) to find that BpF.,,,(hf) = ( p  pF)(h)BPF,u,p(f). This completes the proof. 5.3.4. The above lemma combined with 5.2.3 implies that there exists a (9,K)module homomorphism jPF,n.P of lpF,u,P into IpF,o,P such that jPF,,,,(f)(l)= a P F I U , , ( fAlso ) . 5.3.1 implies that j P F , , , ,is nonzero. (This is a critical point for later applications). We now give an important interpretation of the above integrals that is due to Langlands (in this generality).
Theorem. We maintain the above assumptions. Let f E ( H ~ ~ ' ) ,and let g E lpF,o,p. Let h E aF be such that a(h) > 0 for all a E @(PF,A F ) .Then lim I'fm
Here n
e t ( ~ M h ) ~
(n(exp th)f, s> =
j (f(ag(1)) d3.
NF
= TC~,,,,~.
Since g is Kfinite, the span of g ( K ) is finite dimensional. Also, our hypothesis on f implies that f(g) E (H,)"for each g E G. Set a, = exp th. Then
by Lemma 2.4.5.
5.3. The Intertwining Integrals
147
NOW 6 = nmF(ii)aF(ii)kF(ii)with n This implies that (n(aOf,g>
=
J
E
NF. Hence k F ( i i )E NF(mF(n)aF(6))’n.
aF(fi)2PaF(fi)np 0.
The justification for the interchange of limit and integration is then a consequence of Vitali’s convergence theorem (c.f. Dunford, Schwartz [ 11). We are left with proving (1). The transformation rule of f implies that I,(E) =
J u ~ ( u ~ ~ u ~ ~ ) ”  ” ~ ~ ( E ) ” ’ ” ( ~,)‘mF(E))f(E), ( ~ ~ ( u , ~ ~ u g(k(alEa,))dii.
E
The integrand is dominated by a constant times ~ F ( a ~ i i u  ~ )EF(m,(ulEa P~~P
1 ‘mF(ii))~F(ii)PR e ”( 1 + log 11 mp(i~) 11 Id. +
We now analyze this expression. We first observe that EF(x’y) =
S ~ ( k x ) ” ~ ( dk, ky)~
for X , y E OMF.
Kp.
Indeed,
J
KF
since k x Now
a(kx’y)dk
=
J
a ( k ( k x ) . ~ C ’ y ) a ( kdk x )= ~ ~J a ( k y ) P a ( k x ) P d k KF
KF
= n(kx)a(kx)k(kx).
S u(kx)Pa(kx)Pdk =
KF
J
a(k(i7)x)”a(k(ii)y)Pa(fi)’P d7i.
*Np.
If we now use the fact that k(ii) E *AJ,a(n)’ii we have
(3)
5. The Langlands Classification
148
(3) implies that 1r(E) 5 a,(a,iia,)PRe~u( *nmF(ariia,))Pa( *fimF(ii))”u(n)aF(n)p+Re P dii d*ii
S
EX*NF
+
with o ( i i ) = ( 1 log Iliill)d. Now a(m,(g))a,(g) = a(g) and *%n,(a,6a,) set u ( Z ) = (1 log I l ~ l l ) then ~,
+
l,(E) 5
J
=
j a( *iia,iia,)P Re ”a( * i i i i ) R e p + p u ( i i ) d*E dii
=
a(k(*ii)a,iia,)PRe”a(k( *ii)ii)Re’ ‘%(ii)a(* i i ) 2 Pd*E d n
1 E a(a,k( * ii)iik(* f i )
‘NF
hence, if we
*NF x E
*NF x E
=
= m,(alfEna,),
‘ a r ) P
k( * ii)Ek(* n )  l ) p ’
 R e ”a(
Re
%(E)a(* ii)2P d * ii dE
X
j a(a,na,)PRe”a(n)p+Re”(l
+ log J J f i ( ( ) d d i i .
E
Let 0 < E < 1 be such that (Re p a(a,fia JP
 Re’a(n)P
 a(a,6ap,)~ W
Re

EP,, a )
> 0 for a E @(PF,AF).Then
”
F ~ ( ~ , ~ ~ _ , )  w( dRa ~( qPR e p
 & P F ~ ( ~+)WPF
 @)P
0 there exists C, > 0 such that U(ii)&PF+Q
I C,a(E)P(l

log l l u ( i i ) l p .
Set u(ii) = a(ii)P+EpF(l + log Iliill)d. Then u is integrable on N and we have just shown that I , @ ) I j u(ii)dii. E
This completes the proof of the theorem. 5.3.5. We will see in the next section that this result is one of the main ingredients in Langlands’ classification of irreducible (9,K)modules. The above proof is due to HarishChandra [15]. Special cases of this Theorem had been proved earlier in Helgason [3] and Knapp, Stein [l]. We should point out that in the literature just cited f is also taken to be Kfinite. Since we do not need this condition, some of our later arguments will be simpler than the originals.
5.4. The Langlands Classification
5.4.
149
The Langlands classification
We retain the notation and assumptions of 5.1. Let F be a subset of A. and let (PF,A F )be the corresponding ppair. Let (CJ,H,) be an irreducible unitary representation of OMF such that (H,)K is tempered (5.1.1). Let p E (aF): be such that Re(p,ci) > 0 for ci E @(PF,AF).We call such a triple, ( P F , g , p ) , Langlands data. (We allow PF = G, that is to say F = Ao.) Set P F = MFNF. The following theorem is a combination of a basic result of Langlands [l] and a refinement of the result by Militic [l].
5.4.1.
Theorem. Assume that (PF,CJ, p) is Langlands data (1) If f E I,,,,,, and if jp,,,J f ) is nonzero then f generates 1,,,,,, as a (9,K)module. (2) jPF~a,,(IPF,o,S) is the unique nonzero irreducible (9,K)submodule of IPF,,,, which is also the unique irreducible quotient of IPF,,,,. W e denote this module by JPFsU,S'
(3) If (PF,CJ,p) and (PF,ts',p') are Lunglands data and if JPF,,,, is equivalent to J p F , o g , p f then F = F', p = p' and CJ is unitarily equivalent to CJ'. We first show that (1) implies (2). Let Z be a proper (g,K)submodule of I,,,,,,. Then (1) implies that j p F , u , p (= Z )0. Since jPF,a,pis a nonzero module homomorphism (5.3.1), this implies that Ker j,,,,,, is the unique maximal, proper (g,K)submodule of I,,,,,,. We therefore see that JPF,,,, is irreducible. Now, IF,,,,^) = I,,,,,, (5.2.4) and (P,,cJ", p) is Langlands data if we replace A. with  A o . The above now implies that I ~ , , ,  ,  , has a unique nonzero irreducible quotient (g, K)module, hence IF,,,,, has a unique nonzero irreducible (9,K)submodule. This completes the proof of (2) assuming (1). We now prove 5.4.1(1). Let ,f be as in the statement of (1) above. Let Z = U ( g )span {Kf}. Then Cl(Z) is a Ginvariant subspace of H = HU3'. If 2 is a proper subspace of HK then Cl(2) is also proper in H. Hence there exists a nonzero element g E HK such that (g,Cl(Z)) = 0. Let W = span{Kg). Then ( W , C l ( Z ) ) = 0. Since kg(1) = g(k), we may therefore assume that g(1) is nonzero. Now, j p F , , . , ( f ) ( k )is nonzero for some k E K . If we replace f by kf we may assume that j p F , u , S ( f ) ( l=) / ? p , , u , p ( f ) is nonzero. With all of this in place we are ready to derive a contradiction.
5.4.2.
150
5. The Langlands Classification
Let h E aF be such that ~ ( h>) 0 for all c1 E @(PF,AF).Set a, = exp th. Let m E OM,. Then Theorem 5.3.4 implies that
0 = lim
et(Pt4(h)
f++m
(x(atm)f, s>= (fmP(f), dl))?
here x = xu,, and p = pPF,,,,. Since CT is irreducible this implies that g( 1) = 0, which is the desired contradiction. 5.4.3.
We now prove 5.4.1(3) (we use the notation therein). Let V Z J , , , ~ , # Z
Jp,,,r,p..We choose a realization (x7H ) of V. Lemma 5.2.8 implies that there is a constant d > 0 such that if u E V, u” E V  and if a E CI(A+)then lcu,u(a)l 0 for all then we have lim
aPRepw
lcu,u(a)l< C ’ a R e p ’  p +log (l tl E @(&,A,,).
C ~ , ~  ( U =~0)
ll~\l)~.
If we set a, = exp th
for all E > 0.
t++m
+
This implies (here we use 5.3.4 and 5.3.1) that (Re p ~ p ) ( h> ) Re p’(h) for all E > 0. If we take the limit to E = 0 then we have Re p(h) 2 Re p’(h) for all such h. This in turn implies that Re(p,bj) 2 Re(p’,pj) for all j not in F‘. Hence F is contained in F‘. If we interchange the rolls of F‘ and F we find that F = F’ and Re p = Re p’. Let h be as above. Then a , ~) ( uu,  ) ~ lim a f ~ p c u , u  ( = t++m
lim a f ~ ~ ’ c , , ,  ( a ,=) p(u, u), t+m
both exist and the bilinear forms c1 and fi are both nonzero. We may thus choose u, u“ so that ct(u,u) is nonzero. Then lim,,,, a:”’ = p(u, v)/c((u, u“). But Re(p  p ’ ) ( h )= 0. Hence p = p’. We are left with proving that CT z 0’.Let S(resp. U ) be a (g,K)module homomorphism of I,,,,,, (resp. I,,.,,,,) onto V. Let u E V and let f E I,,,,,, (resp. g E be such that Sf = u (resp. U ( g )= u). Let k E K , and let m E U(Om,). Set f l = mkf and g1 = mkg. Then S(fl) = mku = U(gl). Let h be as above. Then (n,,,(at)f17f) = (7cg.,,(a,)gl,g) for all t. If we replace f by k’f (if necessary) we may assume that f(1) is nonzero. Theorem 5.3.4 implies that (mkBP,,,,,(f), f(1)) = (mkpPF,g,,,(g)7g(l)). Since k E KF and m E U(’m,) are arbitrary, this implies that CT z 0’.The proof of the theorem is now complete.
5.4.
The Langlands Classification
151
5.4.4. We are now ready to state the celebrated Langlands classification of irreducible (9,K)modules.
Theorem. Let V be an irreducible (9,K)module. Then there exists Langlands data ( P , , c , p ) such that V is (9,K)isomorphic with JpF,b,ll. In light of the uniqueness statement in 5.4.1 the above Theorem reduces the classification of irreducible (9,K)modules to the classification of irreducible, tempered (9,K)modules. 5.2.5 reduces this question to the classification of irreducible “rapidly decreasing” unitary representations of the OMF and the determination of the constituents of the unitarily induced representations in 5.2.5. We will see that the “rapidly decreasing” representations are the “discrete series” which we will parameterize in the next three chapters. The full determination of the tempered, irreducible (9,K)modules has been carried out in Knapp, Zuckerman [11. 5.4.5. We now begin the proof of 5.4.4. Let V be an irreducible ( 9 , K ) module. Set E ( V ) = {  p p I p E E(P,, V ) } .We use the partial order and notation in 5.A.1. Let A E E ( V ) be such that (Re A)o is a maximal element among the (finite set of) (Re A’)o, A‘ E E ( V ) . Put F = F(Re A). We will identify F with the corresponding subset of Ao. Set p = Ale. Then Re p = (Re A)o. Set W = ( V “ / n F I / “ )   / , + pThen . W is a nonzero finitely generated, admissible (m,, KF)module (3.7.2, 4.2.6). Let 6 E (*aF): be such that ( W / T ~ , W ) ~is nonzero. ~+~ Then 6  p p E E(P,, V ) .We relabel { 1,. . . ,r } so that F = { 1,..., t } . Then *aF is the linear span of { Ha , r . . ., HNt}.Let ,..., be the corresponding ‘‘Pi’ for *aF and (crl ,..., cr,}. Set 1 = Re(6 *pF). Then 5.A.1.3 implies that there is, a subset, F‘, of { 1 ,..., t } such that
+
+
p,
Dt +
with ti > 0 for j > t. Now pj > 0 for j = 1,..., t (5.A.1.1(1)). We assert that F‘ = { 1,. . . , t } . If not then A + Re p >  X j e F , y j a j + X j , t xjPj. Hence 5.A. 1.3 implies that (A + Re p ) o > X tjPj = Re p = (Re P ) ~ .Since A was chosen such that (Re A)o is maximal we have a contradiction. If we “unwind” the minus signs we have shown (1)
W“ is a tempered
(Om,,
&)module.
5. The Langlands Classification
152
The exactness of the Jacquet module (4.1.5) now implies that if Z is an irreducible, nonzero quotient of W then Z" is tempered. Let (a,H,,) be an irreducible unitary representation of OMF such that ( H , , ) K = Z" (5.2.6). Lemma 5.2.3 implies that V" is equivalent with a submodule of IPF,,,,p. Hence V is equivalent with a quotient of ZPF,,,,8. Since (P', a,p) is Langlands data, Theorem 5.4.1 implies that V is (9, K)isomorphic with JpF,,,,p.This completes the proof.
5.5. Some applications of the classification
5.5.1. In this section we will use the results of the last section to derive some results that refine the growth conditions of Section 4.3. We will also drop the provisional definitions of Section 5.1. We begin with the following direct application of 5.4.4 and 5.4.1. Theorem. Let (71, H ) be an admissible Hilbert representation of G that satisfies the weak inequality. Then HK is tempered. (See 5.1.1 for the definitions.) If (n,H ) satisfies the weak inequality then every subquotient of (n,H ) does also. The exactness of the Jacquet module implies that HK is tempered if and only if every irreducible subquotient of HK is tempered. Thus to prove the Theorem we may assume that (71, H ) is irreducible. According to 5.4.4 there exist Langlands data (PF,a,p)such that HK is equivalent to JPp,,,,p.If PF is proper then 5.4.1 combined with 5.3.4 implies that (n,H ) cannot satisfy the weak inequality. If PF = G then HK is tempered by the definition of Langlands data.
5.5.2. In light of the above result, we will use the term tempered to describe the weak inequality as well as the definition in 5.1.1. The next result uses an idea due to MiliEic [l]. Theorem. Let V be an admissible finitely generated (9,K)module. Let (n,H ) be a realization of V. I f p E a* is such that if a E Cl(A+) then I(x(a)v,w>l
CaV
+ 1%
Ilall)d
for v, w E HK ( = V ) for some constants C and d (possibly depending on v, w). Then p 2 Av (see 4.3.5). Let E be a finite dimensional irreducible (9,K)module with highest weight A relative to @(P,A). Then a acts on E"/nE" by A. Clearly,
153
5.5. Some Applications of the Classification
(V"/nV") 0 (E"/nE") is a quotient of (V" 0E")/n(V" 0E"). Let C E E(P, V"). Then a acts on ( V " / ~ I V " ) ~0( E " / n E " ) by the generalized eigenvalue [  2.. This implies that V" 0E"/n(V" 0 E " ) has a nonzero (m, OM)module quotient of the form HO.[ I with g an irreducible finite dimensional representation of OM. Thus there exists a nonzero (9, K)homomorphism, T, of V" 0E" into lp,u,cd,,. By duality, there exists a nonzero (9, K ) homomorphism of I p , O  , d + p  iinto V @ E. Let 1, be so large that (P, D , I p  [) is Langlands data. Let a+ = { h E alci(h) > 0 for c( E 0 then D,O D  , imbeds in Ibvpand the corresponding quotient is isomorphic with F,,  1. (b) If p < 0 then F,+ is the unique irreducible submodule of Iu,, and Iu,,JFw+l is isomorphic with D,,0 D,. (c) If p = 0 then IJ = E and lc,ois isomorphic with D+,o 0D,o. Suppose that k > 0 and that D, imbeds in lu,,. Then IJ = ck+' and 5.6.1(5) and the calculation of the eigenvalue of C on V  k  l imply that p = & k . If p =  k then (1) implies that FkPlimbeds in lu,,and since (P,o, k ) is Langlands data, we would have the contradiction D, z F,  We have shown
'.
(3) If k > 0 then Dk imbeds in lu,wwith other lo,,.
IJ = ck+'
and p
=k
and in no
5.7.
SL(2,C)
159
Similarly, we have (4) If k < 0 then D, imbeds in Iu,p with other Iu,u.
IT
=E
~
and + ~p
=
k and in no
Fix for the moment k > 0, IT = 6 , ” . Then D, and D  , are both isomorphic with submodules of I u , k . Both of these (9, K)modules are irreducible and since they are inequivalent (even as Kmodules). It follows that the direct sum D, 0D  , is isomorphic with a submodule of la,,. As a Kmodule, D, 0Dk 0Fk is isomorphic with lo,,. (2) now implies (a). (b) follows from (a) and I , , % Iu, ,, Dk % D and FL % Fk 1. If p = 0 then as above, the only place that D ,o can imbed is in Also as a Kmodule is isomorphic with D+.o 0D  , o . So (c) follows as above. 
5.6.4.
,
In light of the above results and the Langlands classification, we have
with Re p > 0 and (1) The nontempered representations consist of the p $ Z o r p ~ Z a n d o # ~ ” + ‘ a nt hde F , , k r O ( = J p , , k , , + , ) . Lemma 5.6.3 implies that if V = Dk or D  , with k > 0 then Av = k  1. Thus Dk and D  , are the underlying (g,K)modules of irreducible square integrable representations. We can now give the list of irreducible representations of SL(2,R).
(I) The square integrable representations D,, k E Z, Ikl > 0. These are usually called the discrete series. (11) The unitary spherical principal series, Il,i,,p E R. The irreducible unitary nonspherical principal series, Ie.ipwith p E R  {O). (111) D+.o and D  3 0 , the constituents of the reducible unitary principal series. These are sometimes called limits of discrefe series. (IV) The finite dimensional representations Fk, k E N. with Re p > 0 and p not an integer or p E N and CJ # E ” ’ ~ . (V) The
5.7.
SL(2,C)
In this section we use elementary methods to give the classification of irreducible (g,K)modules for G = SL(2,C). We look upon G as a real reductive group. Thus the Lie algebra of G , g, is looked upon as a real Lie algebra. On the other hand, g = 5 4 2 , C), which also has the structure of a Lie algebra over C. We will write J for multiplication by ion g and look upon J as a real endomorphism.
5.7.1.
5. The Langlands Classification
160
We choose K = SU(2) and P to be the group of upper triangular elements of G. Let H be as in 5.6.1. We set a = R H . If t E R then we set a, = exp t H and we take A = { a , ; t E R ) . We set
Then 'A4 = (m(8)I 8 E R}. We note that in this case, 'A4 = T, a maximal torus of K . If k E Z then we define E T" by ak(m(8)= eikO.Then T" = (0, I k E Z}. We look upon a, as C by identifying p with p ( H ) . With this identification, p = 2. From the representation theory of SU(2), we know that K " = { y k k E N} with dim y k = k 1. We will use the following tensor product formula repeatedly.
+
The easiest way to prove (1) is to use characters. We leave this as an exercise to the reader who has not seen this formula before. 5.7.2. If V is a (g,K)module then we write V ( k ) for V ( y k ) . We note that g = t 0 J € and that (Ad, t,) E yz. This combined with (1) above implies (1) If V is a (g,K)module then g V ( k ) c V ( k we set V ( j )= 0 if j < 0.
+ 2) 0 V ( k )0 V ( k

2). Here,
We write Ik,pfor Ip,gk,pfor k E Z and p E C.We note that the multiplicity of oj in yk is 1 if ljl I k and k + j is even and it is 0 otherwise. Thus Frobenius reciprocity implies that
(2)
=
@ Ik,p(2j+ lkl)
with dim I k & ? j
+ lkl) = 2 j + Ikl + 1.
jr0
The subrepresentation theorem now implies that (3) If V is an irreducible (9, K)module then dim V (j ) < j
+ 1.
If V is an irreducible (9, K)module then we set k ( V ) = min{k V ( k )is nonzero). Then k( V ) is called the minimal K  t y p e of V. (4) If V is an irreducible (g, K)module with minimal Ktype, k
there are two possibilities: (i) V is finite dimensional. (ii) V ( k 2 j ) is nonzero for all j 2 0.
+
= k( V ) ,then
SL(2,C)
5.7.
161
Indeed, if V is infinite dimensional and if V ( k + 2 j ) = 0 for some j > 0 then C,, V ( k + 2v) is ginvariant by (1). This is a contradiction. ( 4 ) implies
(5) If I k , p is reducible then it must have either a finite dimensional submodule or a finite dimensional quotient module. is reducible. Then 5.5.5 implies that I has an Indeed, assume that I = lk,p irreducible, nonzero submodule, V. If V is finite dimensional then we are done. If V is infinite dimensional then V ( k (V ) + 2 j ) = I ( k ( V ) + 2 j ) for j >> 0. Hence, 1/V is finite dimensional. (5) reduces the study of the reducibility of the lk,p to the determination of the imbeddings of the finite dimensional representations in the principal series.
5.7.3. We note that [ J x , y ] = J[x,y ] = [ x , J y ] for x, y E 9. This implies that u = {x E gc; J x = ix} and U = {x E 9,; J x = ix} are commuting ideals in gc such that gc = u 0U. Let X be as in 5.6.1. Then X, JX is a basis of it,. Also, b = CH 0 C J H is a Cartan subalgebra of 9,. Clearly, ad(H)X = 2X and ad(JH)X = 2 J X . Define a l , a, E b* by a,(H) = a,(H) = 2 and a l ( J H ) = 2i, a,(JH) = 2i. Then {oc,,~,} is a system of positive roots for @(gc,b). Let Hj, j = 1, 2 be defined by ctj(Hk)= 26j.k. Then CH, and CH, are respectively Cartan subalgebras for u and U. We have (1)
H
= HI
+ H,
and
JH
=
i ( H ,  H,).
We can now apply the theorem of the highest weight to see that the finite dimensional irreducible (9,K)modules are parameterized by pairs of nonnegative integers. We write F j , k for a representative. We leave it to the reader to check that
(2)
As a Kmodule Fi,k = y j 0 y k .
Now b acts on F j , k l ~ , F j * by k the lowest weight of F j , k . Thus H, acts by  j and H, acts by  k . We recall that p = 2. We have therefore proved ( 3 ) F j * k imbeds in I k  j ,  j p kand  2 it imbeds in no other principal series representation. The conjugate dual representation of FJskis F k , j . Thus we have
(4) Fk3’is a quotient of I k pjj+,k + 2 and it is a quotient of no other principal series representation.
162
5.
The Langlands Classification
5.7.2(5) now implies
(5) The only reducible principal series representations are I k  j ,  j  k  2 and I k  j , j + k + Z forj, k nonnegative integers. The first type has F j * k as a submodule, the second type has F k * jas a quotient module.
+ j + 2 > 0 for j , k nonnegative integers, F k . j = Jp,
Since k Zk ~
(6)
j, + +
j , k + j + 2 .
Let
be the maximal proper submodule of 1,  j , j + k + 2 .
z k j , j + k +
is irreducible.
Indeed, if it were reducible then it would contain a finite dimensional subquotient module, F. Now F would have the same infinitesimal character as F k , j . This implies that F is isomorphic with F k , j . This contradicts 5.5.3. To complete the classification we need only identify the modules zk
j , j + k
+2.
5.7.4. Let y E u be such that ad Hly = 2y. Then ad H2y = 0. Let y y1 iy, with yj E g for j = 1,2. If f E Cm(G) then we set
+
=
L(Y)f(d = d/dtl*=o(f(exP(tYl)s)+ if(exp(ty2)g)). (1) Suppose that k E Z, p E C and that 4(k + p ) =  p with p > 0 and p E Z. Then L( y)’(Ik,@)is a submodule of 1, + + 2 p = I  @ ,  k . Furthermore, L( y)” is a nonzero (g,K)homomorphismof I k , @ into
We note that
[LI,y] = 0. If x E LI n n,
CL(X),L(Y)’I
=
and if [x,y]
PL(Y)Pl(L(H,)
= H,
then
+ P  1).
The asserted intertwining properties now easily follow. We leave the details to the reader. Since, H 2 p l contains ~ Cp(N)the last assertion is also clear. We are now ready to identify the &  j , j + k + 2 .
(2) Let j , k be nonnegative integers. Then I  j  k  2 , j  k isomorphic with Z k  j , j + k f 2 as a (9, K)module.
+
+
Indeed, +(( j  k  2) ( j  k ) ) =  ( k 1). I  j  k 2 , j  k is irreducible. Hence, (1) implies (2).
is irreducible and
5.7.3(5) implies that
For the classification we will need one more observation which follows immediately from 5.7.3(5).
(3) If k
E
Z, p
E
R then Ik,ip is irreducible.
Here is the classification:
I. The tempered representations consist of the each is irreducible.
with k
E Z,
p
ER
and
5.8.
163
Notes and Further Results
11. The finite dimensional irreducible (g, K)modules. 111. The Ik,pwith R e p > 0 and at least one of i ( p k ) or +(p  k ) is not
+
a strictly positive integer.
5.8. Notes and further results The results in Section 5.6 are originally due to Bargmann [l]. In a very real sense, this work of Bargmann is the first to use the “infinitesimal method” to study representations of semisimple Lie groups. It contains the pivotal ideas of expanding in terms of isotypic components and the use of the Casimir operator. It seems that Bargmann did this work on the suggestion of Pauli.
5.8.1.
5.8.2. The results of Section 5.7 are originally due to Gelfand, Naimark [11. In this paper the methods are of a more global nature. The point being that every irreducible unitary representation is either the trivial onedimensional representation or is infinitesimally equivalent to an irreducible principal series representation (i.e., either unitary principal series or complementary series).
Proposition 5.2.5 is essentially (that is after the material in 5.5 is taken into account) a result of HarishChandra, Langlands [13 and Trombi [11.
5.8.3.
5.8.4. The intertwining operators as studied in Section 5.3 are due to HarishChandra. The motivation for these operators comes from the earlier work of Kunze, Stein [l], [2], who studied these operators in the case of minimal parabolic subgroups. See also Knapp, Stein [ 11. The main point in the earlier papers was to give a meromorphic continuation of the operators jp,o,vof 5.3.4 to allow v to be purely imaginary. This analytic continuation will be implemented in Volume 2 of this book.
5.8.5. As was indicated in the body of this chapter the Langlands classification is due to Langlands [l]. The formulation given involves some ideas of MiliEic, and it follows the broad lines given in Chapter 4 of Borel, Wallach [11. To complete the classification of irreducible admissible (g, K)modules, it is necessary to classify the irreducible tempered representations. In light of 5.2.5 and 5.5.4 it is enough to determine the irreducible square integrable representations (this will be completed in 8.7) and to find the equivalences between the irreducible components of the representations Zp,o,iv for o irreducible and square integrable, v real. The latter part has been done by Knapp, Zuckerman [ 11. In that paper, an unambiguous parametrization of the irreducible tempered representations is also given.
164
5. The Langlands Classification
5.8.6. Theorem 5.5.6 is usually proven using the theory of characters, in particular HarishChandra’s regularity theorem (8.4).
5.A. Appendices to Chapter 5 5.A.1.
A Lemma of Langlands
5.A.l.l.
Let V be a real vector space with inner product ( , ). Let {al,. . . , a,} be a basis of V such that ( a j , ak) I 0 for j # k . Let /Ik, k = 1,. . . , r be defined by (P,, ak) = 6 j . k for j , k = 1,. . . , r. We define a partial order on V, x 2 y if x  y = Z uj aj with uj 2 0. (1)
Pj
2 0,
j = 1, ..., r.
Let y j be the GramSchmidt orthonormalization of the tlk. Then our hypothesis on the c(k implies that y j 2 0 for all j = 1,. . ., r. The definition of the Pj now implies that (P,, Y k ) 2 0 for all j , k. (1) now follows. Let C = {x E V l ( x , a j ) 2 0 for all j = 1,. . .,I } . Then C is a closed convex cone containing no line through the origin. If x E V let C, = { y E C I y 2 x). Then it is clear that C, is a closed, nonempty, convex and Co = C. (2) Let x E I/. There exists a unique element xo E C, such that llxoll Illyll for all y E C,. Let z E C,. Then T = { y E C, I llyll I l l z I I } is compact. Hence 11...11 achieves a minimum on T at (say) xo. If u E C, is such that llull = llxoll then tu ( 1  t ) C, ~ ~for ~ all 01t11. ~~tu+(1t)xo(~2=t2~~u~~2+2t(lt)(u,xo)+ (1  t)211x0112I (tllull (1  t)llxoll)* = llxo112 with equality if and only if u = C X ~ Thus . u = x0.
+
+
5.A.1.2.
We note that If G c { 1,. . . , r } then the set { zj I z j is a basis of V.
(1)
If x
(2)
E
= aj, j E
G;zj
=
aj,j $ G }
V then xo = C ujPj. Clearly, uj 2 0 for all j . We set F ( x ) = { j  ; u j= 0).
If j $ F ( x ) then
(
~
0
9P
j)
= (x,
Pj).
Since xo E C,, it is clear that if j $ F ( x ) and if ( x ~ , / ? ~#) ( x , P j ) then < x O , b j ) > (x,Pj). We thus assume this inequality. Let E > 0. If k E F ( x ) then (xo

Eaj, ak) =  & ( a j, a k) 2
0.
5.A.1.
A Lemma of Langlands
165
(3) with zj I 0 for j E F(x). (1) implies that x = CjcF(,, zjaj C j & F ( wjpj. x) If j is not in F ( x ) then (x, j j ) = (x,, j j ) . If we now observe that
+
det([(bj,
b k ) j.k $ F ( x ) ] ) / O ?
it follows that wj = uj for j 4 F ( x ) . Since xo E C,, x for j E F(x). (4)

xo I 0. Thus zj I 0
If x, y E I/ and if x 2 y then xo 2 y o .
We first show that if y E C, then y 2 xo. Let X be the linear span of the aj f o r j E F(x) and let Y be the linear span of the pi f o r j $ F(x). Then I/ = X + Y an orthogonal direct sum. Let P be the corresponding orthogonal projection onto Y. If j 4 F ( x ) then
This easily implies that xk,j 2 0. We have thus shown that if z 2 0 then Pz 2 0. We also note that if z E C, then Pz E C,. Indeed, Pz 2 Px by the above and if j E F ( x ) , ( z , P a j ) 2 ( z , a j ) . Finally, if z E C, one sees easily that z 2 Pz. Thus, if z E C, then z 2 Pz 2 P x = xo. We now prove (4).Let x 2 y. Let z E C,. Then z 2 x so z 2 y. Hence z E C,. But then, xo E C,. Thus, the observations above imply that xo 2 y o .
(5) If G is a subset of { 1,. . .,r } and if x=
1sjaj + 1 tjpj .i E G
with sj 2 0, j E G and tj > 0, j 4 G then G
i4G
=
F(x).
166
5. The Langlands Classification
Set y = C j g c tjPj. Then Y E C,. If j is not an element of G then ( x , P j ) = ( y , P j ) 2 ( x , , P j ) by (4). But xo E C x SO < X , , P j ) 2 < X , P j > . Hence, ( x o , j j ) = ( y , P j ) for j 4 G. It is now clear that x o 2 y (use the argument in (4)). Hence (4) implies that y = x o . This completes the proof. 5.A.1.3. We now apply the above results to root systems. We use the notation in 5.1.1. If we replace V by a * and ( , ) by ( , ) then we have proved.
Lemma. Let p E a*. Then there exists a unique subset F ( p ) of { 1,. . . , r ) such that
with yj 2 0 and x j > 0. Set p o Po 2
=
Cj4F(p)xjPj. If cr, p
E
a* and if p 2 cr then
00.
A similar proof of this lemma has been given by Carmona [l]. An alternate constructive proof can be found in Borel, Wallach [I, Ch. 4, Appendix].
5.A.2.
An a priori estimate
5.A.2.1. If x E R" then we denote, as usual, the coordinates of x by xl,. . . , x,. Set (R')" equal to the set of all x with x j > 0 for j = 1,. . . , n. If S is a subset of { 1,. . . , n} then we set x s equal to the element of R" with ( x ' ) ~= 0 if j 4 S and ( x ' ) ~= x j if j E S. Thus x 0 = 0 and x ( ~ * . , , = * "x). If x E (R')" we set R ( x ) equal to the convex hull of the x s . Then R ( x ) is a rectangle whose interior is contained in (R')". We will use standard multiindex notation. Thus, if I = (il,. . . , in) with ij N then = x i 1l x i z2. . . x $ , a1 = a i l a i l . . . a$ (with aj equal to partial differentiation in the jth coordinate) and 111 = i , + ... + i n . (We realize that there is an overlap in notation, so multiindices will be denoted by I , J , K and subsets of { 1 , . . . , n} will be denoted by S , T.) We say that I I J if i, 5 j, for k = 1,. . ., n. We fix K = (1,. . . , 1). The "fundamental theorem of calculus" implies (1) R(x)
a K f ( x ) d x= (  1)"
( l)lsf(xs)
IS[ denotes the cardinality of S.
for f
E
Cm(R")and x
E (R')".
5.A.2.
167
An a priori Estimate
5.A.2.2. If H is a subset of R" and if y E R" then we write y { y x x E H}.We also write H S = {xsI x E H ) .
+
+H
=
Lemma. Let S be a nonempty subset of { 1,. . . , n}. Let E > 0 be given. Then there exists a positive constant Ct,s such that if xo E Cl((R+)") and if x E ((R')"))' with xj  ( x ~ >) E~ ,for all j E S then if f E C"(R") is such that a"j' E L'((R+)") f o r all I IK then
It is enough to prove the result for xo = 0, since we can translate f by xo. So assume that xo = 0. If h E Can(Rn)and if h(xS)= 0 for S # T and h(xT) = 1 then (1) above implies
The Leibniz formula applied to (1) yields
Here C, is a constant depending only on I and n and (1.. .IIR(x),m is the sup norm on R(x). Let S be fixed as in the statement of the Lemma. If x E (R')", then we set u(x) equal to the element given by U(X)~ = xj for j E S and U(X)~ = 211x11 for j $ S. Then u ( x ) E (R')" and I ~ U ( X ) ~> I I '411x11' for T # S, Let a E C,(R) be such that a(t) = 0 for It1 2 2, a ( t ) = 1 for It\ I 2 and 0I a(t) I 1 for all t. Let fl E C"'(R) be such that 0 I P(t) I 1 for t E R and P(t) = 0 for It\ I +,P ( t ) = 1 for It1 2 1. Set
a.
If xj > E for j E S then ~ ( u ( x ) = ~ )0 for T # S. Since U ( X )= ~ x for ((R')")', (2) gives an estimate for If(x)l (use u(x) in place of x and T = S in (2)). We must therefore show that Ila'hllR(u(x,,,mis bounded by a constant depending only on E and S for each I I K . Leibniz's rule implies that it is enough to estimate
x
E
for J I I I K . This expression is 0 if { k l j , > 0)
=
( J ) is not contained in S .
168
5. The Langlands Classification
Otherwise it is equal to
1 ~ (1 If z E R(u(x)) then 1 1 ~ 1 < estimate.
+ 4(n  lS1))11~11~.This
implies the desired
5.A.2.3. This result has as an immediate consequence the following fact, which will be used in Section 5.5. Corollary.
Let the notation be as in the previous result. Let x E ((R')")' then
f S # @
lim f(tx) = 0. f++C€
5.A.3.
Square integrability and the polar decomposition
5.A.3.1. We maintain the notation of Section 5.1. Let y(a) be defined as in 2.4.2. If f E C " ( G ) then we say that f is Kfinite if R(K)L(K)f'spansa finite dimensional space. Lemma. Let f E Cm(G)be Kfinite. Then f is square integrable fi and only ij(1)
j+y(a)lf(k,ak,)12 da ( 00
A
for all k , , k ,
E
K.
We first prove that (1) for all k , , k , E K implies that f is square integrable. Let u l , . . . , u,, be a basis for span{R(K)L(K)f}. Our hypothesis implies that each ui is square integrable on A + . Now f(k,ak,) = R(k,)L(k,)f(a) = C hj(k,, k,)uj(a), with hj E C"(K x K ) . Thus, there exists a positive constant, C, such that If(klakz)l I C C luj(a)l for k , , k , E K and a E A + . Thus f is square integrable by 2.4.2. Suppose that f is square integrable. Then Lemma 2.4.2 combined with Fubini's theorem implies that (1) is true for almost every k , , k , E K . Let S be the set of all ( k , , k , ) such that (1) is true. Then K x K  S has measure 0 and if ( k , , k , ) E S then y " 2 L ( k l ) R ( k 2 ) f E L2(A'). Since S is dense in K x K it is easy to see, using Kfiniteness, that y '"span(L(K)R(K)f) is contained in L2(A'). Hence S = K x K .
5.A.3.
Square Integrability and the Polar Decomposition
5.A.3.2. Lemma.
Let f
C " ( A ) then
E
H
j r(a)lHf(a)I' du < m,
A+
if and only
169
E
u(a),
if f
A+
a2p1Hf(a)12da ( a,
H
E
U(a).
Since ?(a) I CaZPfor a E CI(Af) the sufficiency of the above condition is clear. We will use the following result to prove the necessity. (Notation as in the previous appendix.) Scholium. Let e l , .. .,en be the standard basis of R". Let a,, . . . , aP E (R")* (0) be such that ai(ej)2 0 for ail i, j and ai(ej)= 6i,j for 1 I i, j 2 p . Set = a1 + ... + ap. There exists a constant C such that
1
j e"'"'lf(x)12dx I C
(1) f o r all f
p
E
( n sinh ai(x))la"j(x)12dx
j
I l l s p (it+)''
(R+ )"
C"(R").
We first prove that there exists a C > 0 so that (1) is true for f n then the result follows from (sin hO = 0)
E
C,(R"). If
=
d
cc
f sinh x  If(x)l' dx
0
dx
m x
=

j cosh xlf(x)12dx. 0
+
Since ld/dxlf(x)I'l i l f ( x ) I 2 I d / d x f ( x ) 1 2and cosh x 2 e"/2 for x 2 0. So the result is obvious for p = n. Assume that the result is true for p  1 2 n. We prove it for p . If we reorder the coordinates on R" we may assume a p ( e l )> 0. Then
+ 1orj(el) j
f cosh 3(i(x)
a,(e,) f cosh a p ( x ) (R+ )"
n sinh
i#j
(R+)"
fl
isp 1
ai(x)l.f(x)12 dx.
sinh ai(x)lf(x)12dx
170
5.
The Langlands Classification
Thus
j cosh ap(x) (R+)"
n
sinh ai(x)lf(x)12 dx
isp 1
with C, = 2/a,(e,). If we replace f by exp(crp/2)f. The inductive hypothesis for p  1 implies the result for p . We have thus proved the existence of C such that (1) is true for f E C:(R"). We now prove the result using this C . Suppose that f E Cm(R").If the right hand side of (1) is infinite there is nothing to do. So assume that it is finite. Let u E Cm(R)be such that u(x) = 1 for 1x1 I 1 and u(x) = 0 for 1x1 2 4. Set for t > 0, h,(x) = u(llxl12/t2).Then h, is smooth, h,(x) = 0 for llxll 2 2t and u,(x) = 1 for llxll I t. If t 2 1 then la'h,(x)l I C1 for all x E R" (the important point is that C1 is independent of r.) Indeed, (a/axi)h,(x) = uyllxl12/t2)2xi/t2
and IxiJ I 2t when u'(llxllz/t2)is nonzero. We now leave it to the reader to prove the inequality for all I . Now lim alhrf(x) = a'f(x) r+m
and the preceding remarks imply that la1h,f(x)i2 I
D,
1
iaJf(X)12
for x
E
R"
IJ l < l I l
with Dl depending only on 1. Thus
by dominated convergence. The Scholium now follows. We now prove the Lemma. It is clear that we may assume that O(G0)= GO. Take q,.. ., a, to be the simple roots in (D(P,A). Let @ ( P , A )= {al,..., a,} (here each root is counted dim g" times). Take ej E a to be the elements defined
5.A.3.
171
Square Integrability and the Polar Decompositon
Use this basis to identify R" with a. The Lemma is now an by ai(ej) = easy consequence of the Scholium. 5.A.3.3. Lemma. Let f x E U ( g ) if and only if
E
C"(G) be Kfinite then xf
E
L z ( G ) for all
j a2Plxj(k,ak2)12da< co
A+
for all k , , k ,
E
K and all x
E
U(g).
This is an easy consequence of 5.A.3.1 and 5.A.3.2. 5.A.3.4. Proposition. Let f E C'(G) be Kfinite and such that xf E L 2 ( G ) for all x E U ( g ) . I f h E a(0) is such that sc(h) 2 0 for all c1 E @(P,A ) then
lim e'p'h'f(expth) = 0 1+a
Set g(a) = a2Plf(a)12.5.A.3.3 implies that hg result now follows from 5.A.2.3.
E
L ' ( A + )for all h E U(a). The
This Page Intentionally Left Blank
6
A Construction of the Fundamental Series
Introduction As we have seen in the last chapter, the tempered representations (in particular the square integrable representations) are the basic “building blocks” to construct all irreducible admissible representations (up to infinitesimal equivalence) of real reductive groups. Except for the simple case of SL(2,R)we gave no indication of how one might construct irreducible square integrable representations. In this chapter we use a method that is equivalent (see 6.10)to Zuckerman’s derived functors to construct (9,K)modules (our method is based on the results in Enright, Wallach [2]). An exhaustive account of Zuckerman’s functors can be found in Vogan [2]. The key new ingredient in our presentation is the a priori proof of the unitarity of the fundamental series. This combined with our theory of the real Jacquet module leads to a proof that the fundamental series is tempered and square integrable when the parameters are regular and there is a compact Cartan subgroup. In Chapter 8 we will show that the square integrable representations constructed in this chapter (which we call the discrete series) give all of the irreducible square integrable representations of real reductive groups. We also derive many of the algebraic properties of the derived functor construction. In particular, we prove generalizations of Blattner’s formula for the Kmultiplicities. In our development Blattner’s conjecture (a theorem of
I73
174
6.
A Construction of the Fundamental Series
Schmid [2] and HechtSchmid [l]) is proven before the characters of the discrete series have even been defined. In Section 10 we discuss the relationship between the material in this chapter and the corresponding results in the literature. There are four appendices at the end of this chapter. Two of them (3 and 4) contain basic results of the theory. Appendix 3 is an exposition (based on the Jacquet module) of some of the results in Zuckerman [l] on "coherent continuation". The technique is based on unpublished joint work with Casselman. Appendix 4 contains the theorem of HarishChandra [11 which asserts that an admissible finitely generated infinitesimally unitary (g, K ) module is the underlying (g,K)module of a unitary representation of the group. Chapter 9 is independent of the material in the next two chapters. Thus a reader interested in the applications of the results of this chapter to ( g , K ) cohomology can go directly to Chapter 9. 6.1. Relative Lie algebra cohomology 6.1.1. Let G be a real reductive group and let K be a maximal compact subgroup of G. Let M be a closed subgroup of K such that det Ad(@ = det Ad(m)l,. Let C ( g , M ) be the category of all (g,M)modules (see 3.3.1). If V is a (9, M)module then we define Cj(g,M ; V )to be Hom,(Aj(g/m), V ) .We define for j? E Cj(g,M ; V ) ,
+ rc< s ( 
l)""([Xr,Xs],
x o , . . .)
x,,.. , is,. . .) Xj). 


.
Here, Xj E g/m and Xj is a representative in g. It is standard that ( C * ( g , M ) d, ) is a complex. The cohomology of this complex is denoted by H j ( g , M ; V ) . A complete discussion of this cohomology can be found in Borel, Wallach [l, Ch 11. In this section we will only discuss a variant of Poincare duality for this theory and a few specific results that will be used in this chapter. 6.1.2. Let oobe a fixed element nonzero of A"(g/m), where n = dim(G/M). We define a sesquilinear pairing of (Aj(g/m)*)c with (A"'(g/m)*)c as follows. Let conj(j?)denote the complex conjugation of j3 E (Aj(g/m)*)crelative to the real form A'(g/m)*. If a E (Aj(g/m)*),and if B E (A"'(g/m)*), then we define (a,0)by CAconj(j?) = (a,B)oo. Assume that det(Ad(m)) = det(Ad(rn)l,) for rn E M . If V E C(g, M ) then we define V' to be the space of all conjugate linear functionals, p, on V such that
6.1.
175
Relative Lie Algebra Cohomology
M p spans a finite dimensional space. There is a natural pairing ( , ) of V ” with V given by ( p , u ) = p(u). We look upon C j ( g , M ; V ) as a subspace of (Aj(g/nr)*)c0 V. The restriction of the tensor product of the above pairings induces a sesquilinear pairing of C ”  j ( g ,M; V ” ) and Cj(g,M ; V ) . We will denote this nondegenerate pairing by ( , ). If a E C n  j p l ( g ,M ; V ” ) and then ( d a , 8) = (  l ) j ( a , dfi).
(1)
P E C j ( g ,M ; V )
This is proved by direct calculation (cf. Borel, Wallach [l, p.151). Let Bj(g, M ; V ) = d C j p ‘(9, M ; V ) and let Zj(g,M ; V ) be the kernel of d on Cj(g, M; V ) .(1) implies that
(2) ( Z ”  J ( g ,M ; V”))’ = Bj(g, M ; V ) and (B”j(g,M ; V # ) L = Zj(g,M ; V ) relative to ( , ). (2) clearly implies that
(3)
(
, ) induces a nondegenerate pairing of H”j(g, M ; V ” ) with H j ( g , M ; V ) .
6.1.3. Let W be an (m, M)module. We form a(g, M)module U(g,) @”(,,,) W endowed with the gmodule structure given by left multiplication and the Mmodule structure given by m (0 ~ W)
=
Ad(m)g 0mw.
This result is a special case of Lemma 6.A. 1.5. 6.1.4.
We now recall another result that will be useful in the next few sections. Let U, V E C(g, M ) . Suppose that T E Hom,,,(U, V ) . Then T induces a linear map of C’(g, M ; U ) into C j ( g ,M ; V ) given by T P ( X , , . . . , X j ) = T ( P ( X , , ..., X j ) ) . The formula for d implies that Td = d T . So T induces a linear map of H’(g, M ; U ) into H.’(g, M ; V ) . If O+U+
V+
w+o
is an exact sequence in C(g,M ) then the corresponding maps on the C j also induce exact sequences. The standard method of cohomology theory now yields a long exact sequence + H’(g,
M ; U ) + H’(g, M ; V ) + Hj(g, M; W )+ Hj’
(g, M ; U )
+
176
6.2.
6.
A Construction of the Fundamental Series
A construction of ( f, K)modules
Let K be a compact Lie group. Set H ( K ) equal to the space of left (hence right) Kfinite smooth functions on K . We look upon H ( K ) as a (€, K ) module in two different ways. We set L ( k ) f ( x )= f ( k  ' x ) and R ( k ) f ( x ) = f ( x k ) for f E H ( K ) and x , y E K . If V is a complex vector space then we define Cm(K;V ) to be the space of all functions, f,from K to V such that f ( K ) is contained in a finite dimensional subspace W of V and f is smooth as a function from K to W. On C"(K; V ) we also have two actions L and R of K given by the formulas above. We set H ( K ; V ) equal to the subspace of those functions in Cm(K;V )that are Kfinite under both actions. Let V be a (€,K)modulewith action given by n. If u E V and if f E H ( K ) then we set L,(u f ) ( k ) = f ( k ) n ( k  ' ) u . Then L , maps V O H ( K ) into H ( K ; V ) .An obvious calculation yields 6.2.1.
L,(n 0L)(k)= L(k)L,
(1)
for k E K or U ( € ) .
H ( K ; V ) then f ( K ) spans a finite dimensional subspace 5 of V. Let u l , . . ., u d be a basis of 5. Then f ( k ) = C J(k)uj. We set Q v ( f ) = C J 0 uj. It is clear that Q,(f) is independent of all choices used in its definition and that it defines a linear map of H ( K ; V ) into H ( K ) @ I/. Set S , = Q,L,.
If f
E
(2) S, (n@ L ) ( k )= (L(k)@ I ) S , and Sv ( I @ R(k)) = ( R @ n)(k) S, for k E K or U(f). 0
0
0
0
This observation is proved by the obvious direct calculation.
(3) S, is bijective. It is obvious that L , and Q, are injective. Thus S, is injective. We prove the surjectivity. Let f E H ( K ) ,let u E I/ and let u l , . . ., u d be a basis for the linear span of Ku. Let pl,. . . , pd be the dual basis set c,,,(k) = pr(n(k')ut). Then S,(ujOf) = C ~ , . ~ f O u Since ,. K is compact, we may assume that Cj c , , ~conj(qj) = ~5,.~. So
6.2.2. Let M be a closed subgroup of K . Let V E C(f, M) with action 7c. Then we look upon V O H ( K ) as a (f, M)module under 7c 0L and also as a (f, K ) module under I @ R . We define
rj(v)= ~ j ( f M; , vo H ( K ) ) .
Here the cohomology is relative to the first action above. We look upon rj(V ) as a (f, K)module under the action induced by the (f, K)module structure
6.2. A Construction of (t, K)Modules
177
I 0R . Then rj is a functor from the category C(f, M ) to the category C(f, K ) . These functors are special cases of Zuckerman’s derived functors. We will show, in the next section, that one can construct the general ones from these. Let F E C ( f, K ) . We define, for each j , two functors from C(t, M ) to C( f, K ) . The first is V + Ij(V 0 F ) = AF(V ) and the second is V + ri(V)0F = BF(V). If C and D are categories and if A and B are a functors from C to D then a natural transformation of A to B is an assignment X H T ( X )for each object X E C of a morphism T ( X )E Hom,(A(X), B ( X ) )such that if S E HomJX, Y ) then the following diagram is commutative
4x1
T ( X )+ B ( X )
B(Sq
.IS11
A ( Y ) T ( Y ’ B( Y )
If T ( X ) is an isomorphism for every X equivalence.
E
C then we say that T is a natural
Lemma. Let F E C(f, K ) then there is a natural equivalence TF of A , with B,. Furthermore, if W is a (f, K)module and if S E Horn,,,( W,F ) then, if we set Us = TJ(S0I ) , the fdlowing diugrum is commutative A w ( V ) T w ( v ) , BW(V) U W I
Furthermore,
if
p
s
A F ( V ) T F ( v ) >BF( V ) X , Y E C(f, K ) then
T*@YV)= (G(V)@ I)T,(V@ XI. We note that if X is a vector space over C, which we look upon as a (€,K)module with the trivial action, and if
V E C(f, M ) then HJ(f, M ; V 0X ) = H J ( €M, ; V ) 0X .
This is immediate from our definition of relative Lie algebra cohomology. Let S, be as in the previous paragraph. We put TF(V)= Hj(S,). Then 6.2.1(2),(3)imply the all but thelast assertion of the Lemma. We now prove the last assertion. A direct calculation shows that
s,
@
y
= (S,
0” 1 0S y ) .
To complete the proof apply the cohomology functor, H j , to both sides of this equation and use the fact that H j takes products to products.
I78
6.
A Construction of the Fundamental Series
6.2.3. We now come to a critical result in this theory. We look upon V (f ) as a (f, K)module under the adjoint action. If V is a ( f , M)module and if F is a ( f , K)submodule of V (f ) then we have a (t, M)module homomorphism m: V @ F , V given by v 0y H yu. Lemma. Assume that M acts trivially on A"(f/m) (n = dim t/m). Let V E C(f, M) then the following diagram is commutative
rJ(v)
rqv0
I
Identity
.I;(V,I
m
rJ(V )
rqv)@f
We first prove the result for j = n. The formula for d and combined with our hypothesis implies that H"(f, M; V ) = (V/fV)M for V E C(f, M). Thus T"(m):( V 0f 0H(K))/f(V 0f 0H ( K ) )+ ( V 0H(K))/f(V 0H ( K ) )
is given by It is easy to see that u 0sk(x 0f )maps under 1 0 m to  u m S k ( x uj 0xk
0f ; & )
=
c
uj
0L ( X ) f .Hence
0L ( x k ) & , k .
Since, Xjvk 0f;,k + vk 0 E €(V @ H ( K ) ) the result follows for j = n. We now prove the result by downward induction on j . Assume that the result is true for j + 1. Let Z be the kernel of the natural mapping of V(f,) @Ll(m) V onto V. Since ( U (fc) @u(m) V )0H ( K ) is isomorphic with
W,) @ o c r n , ( V OHW)) (A.6.I), Lemma 6.1.3 implies that
rj((U (fc)
@U(m)
V )0H ( K ) ) )= 0 for
j < n. The long exact sequence of cohomology now implies that we have the
exact sequence
. . . + rj(U (f,) @ V ) ,rj(V ) ,rj+(z), U(m)
This yields the following prism
6.3. The Zuckerman Functors
179
where the edges starting with a 0 are exact. The inductive hypothesis implies that the square and the righthand triangle is commutative. Thus the lefthand triangle is also commutative. This proves the Lemma.
6.3. The Zuckerman functors 6.3.1. Let G be a real reductive group and let K be a maximal compact subgroup of G . Let M be a closed subgroup of K such that M acts trivially on AtoP€/m. We look upon U ( g c ) as a ( f , K)module under the adjoint action. If V E C(g,M) then we have the (g, K)module homomorphism U(g,) @ V + V given by g @ v H g v , which we denote by m. We will also look upon V as a (€,M)module. We can therefore apply the functors of the previous section to V. Lemma. Let V E C ( g ,M ) then there is a unique structure of a ( g ,K)module on r j ( V )such that the action of (€, K ) is as in the last section and the following diagram is commutative (V = U ( g ) ) .
rqv 0 u )X rqv) 0 u rwl
rq v)
 Im Identity
rqv).
Let rn be the linear map such that if m is replaced by m" on the right arrow in the above diagram then the diagram is commutative. We must therefore show that m" is a U(g)module structure. T o do this, we analyze the following cube
All of the faces are commutative except possibly the top and front faces. The content of the Lemma is that the front face is commutative. Since all of the ''T" mappings are isomorphisms, it is, enough to show that the top face is
180
6. A Construction of the Fundamental Series
Now apply Lemma 6.2.3. 6.3.2. The above result implies that the T J define functors from C(g, M ) to C(g,K ) . They are usually called Zuckermun's functors. We now give some of their basic properties.
Lemma. Let V E C(g, M ) and
if' V, E y E K then
Horn,,,( V,, rj(V ) ) = H j ( € , M ; V 0(I(,)*). The PeterWeyl theorem implies that
as a (f, K)bimodule. To complete the proof we will use the following result.
Scholium. Let X be a (f, M)module and let L be a compact Lie group such that X also has the structure of a (1, L)module with the two structures commuting. I f y E LA then set X [ y ] equal to the yisotypic component of X . Then H ' ( f , M ; X )=
@ H'(f,M;X[y]).
ycL"
It is clear that the spaces X [ y ] are (f,M)submodules of X . Also, each space C'( f, M ; X ) is an (1, L)module under the action (up)(x,, . . .,x i ) = u(p(x,, . . . , x i ) for u E L. Clearly, d(ub) = udp. Thus each (I, L)isotypic component of C*(f, M ; X ) is a subcomplex. Let E, be (as usual) the projection onto the yisotypic component. The by the above, dE, = E,d. It therefore follows that H * ( f , M ; X ) is the direct sum of the cohomology spaces of the complexes C * ( €M , ; X ) [ y ] . Since it is also clear that C'(f, M ; X ) [ y ] = C'(f, M ; X [ y ] ) ,the result follows. We now complete the proof of the Lemma. As we have observed before, the (t, K)structure on V 0H ( K ) given by I 0R commutes with the (f, M ) structure that we are using to calculate cohomology. Thus the Lemma follows from the above Scholium and the observation preceding it.
181
6.3. The Zuckerman Functors
6.3.3. Lemma. Let V be (9, M)module. I f I/ is admissible then r j ( V ) is an admissible (g,K)module. Let I = { y E U ( g ) l g acts by 0 on V } then I . rj(v)= 0. We note that C j ( € M , ; V 0(V,)*) is finite dimensional if V is admissible. Thus the first assertion follows from Lemma 6.3.2. We note that I is a (g, K ) submodule of U ( g ) .Hence the second assertion follows from 6.2.2 and the definition of the (g, K)module structure on r i ( V ) . 6.3.4 Proposition. If F E C(g, K ) and if I/ E C(g, M ) then TF(V ) is a (g, K)isomorphism from rj(I/ F ) onto rj(V ) F.
We must show that
rqvOF)Ou%rJ(vOF) lTF(v)
(rJ(v) o F ) u 3 rqv)o F is a commutative diagram. To prove this we examine the following prism
rqv@F O u ) r l ( m ? r J ( v OF ) (**I
rJ(v)o F O u Lr q v ) O F The triangles in the diagram are both commutative with invertible maps. The rear face is commutative by the definition of m. The bottom face is (*). Thus, if we can show that the top face is commutative then the result will follow. Let A: U ( g )+ U ( g )0 U ( g )be as in 6 . A . l . l .Let for, Y, a vector space over C, T : Y @ U 0U + U O Y 0U be defined by T ( y O u 1 0 u,) = u1 0y O u , . We consider the following diagram
If this is commutative and if y , p i ,8 are all invertible then it is easy to see
182
6. A Construction of the Fundamental Series
ALB (***)
71
C
‘I
4
+D
is commutative. We apply this observation to the case when A = I”(V @ F @ U ) , B , = r i ( V @ F @ U @ U ) , B, = r j ( V @ U @ F @ U ) , B3 = l?(V@ U @ F ) , B = r j ( v @ F ) , c = r j ( v ) @ F @ U ,D , = r j ( V ) @ F @ U @ U , o 2 = r J ( v ) @ U @ F @ U, D 3 = r j ( V ) @ U @ F, D = T’(V)@ F, c(, = r j ( I @ I @ A ) ,u2 = TJ(I @ T ) , a3 = Tj(I 0 m), a4 = r j ( m @ I ) , y = TF@u, p, = TFsosu(V), P 2 = ‘ G @ F @ U ( V )P3 > = TUB,(V ) , 8 = TF(V), = I@ I@ A, p 2 = I@ T, /13 = I @ m, P4 = m @ I . (The reader should write out this diagram sideways on a piece of paper.) All of the squares except for the last one are obviously commutative. Since the diagram (***) is the top face of (**), we will have (finally!) proved the result if we show that the last square is commutative. Let us write it out. rj(V@ U @ F )
I rj(v) u
(****) T i . , , ( V )

0 0F
r q m 0I ) mO1
rJ(V @ F ) 17,(VI
rj(v)0 F,
To prove that this is commutative we examine the following prism
6.3.5. The next result is basic to the later developments of this theory. The idea is due to Zuckerman the result was first proved in Enright, Wallach [2]. Let dim(f/nr) = p . Theorem. Let V be a (9,M)module. Then there is a nondegenerate sesquilinear pairing between r j ( V ) and r p  j ( V # ) Furthermore, . if p = 2n with n a
183
6.3. The Zuckerman Functors
natural number and if V admits a nondegenerate (g, M)invariant Hermitian form then I”( V ) admits a nondegenerate (9,K)invariant Hermitian form.
We should warn the reader that the proof of this result (involving the material in 6.1.2) will be as important to us as the statement. Let fi denote the sesquilinear perfect pairing between H j ( f , M ; 1/ O H ( K ) ) and H P  j ( f ,M ; ( V O H ( K ) ) # ) (6.1.2). In light of the Scholium above, fi induces a perfect pairing between H J ( f M , ; V O H ( K ) ) [ y ] and H P  J ( fM , ; ( V 0H ( K ) ) # ) C y ] .Now, as in the last number, H P  j ( f ,M ; ( V O H ( K ) ) # ) [ y ]= H P  ’ ( f ,M ; ( V O H ( K ) ) # ) [ y ] ) .
Let 6 be the (t, M)module homomorphism of V # 0H ( K ) into ( V O H ( K ) ) # corresponding to the tensor product of the canonical pairing of I/ with V’ and the L2inner product on H ( K ) . Then 6 is an isomorphism of V # 0H ( K ) [ y ] onto ( V O H ( K ) ) # [ y ] .This, in light of the definition of rj (6.2.2) implies the result in the special case when g = f. We will abuse notation and denote by the ( f, K)invariant, nondegenerate, sesquilinear pairing of r j ( V ) with r p  j ( V ’ ) . We now prove that fi is ginvariant. We have the following commutative diagrams
rj(v@g ) @ rpj(v#)r,cv,o I 9 (rj(v)09) 0 r p  j ( v”) m 0I r J ( m )o 1 1 I rqv)o r p  j ( v # ) r j ( v ) or p  j ( v # )
I
I
I
and
rj(v)0rpj(v#09)
’rj(v)0(rpj(v#)0g )
I 0 7b(v”)
I 4
1 0r P  J ( m )
rj(v)8 r p  j ( v”) C
The definition of
p
I
’rj(v )0 r p  j ( v#)
I
’c.
ID
now easily implies that
( r j ( m )8 I )
=
p
(I
o rpj(m))
(I
o T,) (50
~)l.
This is the content of the first part of this result. If V admits a nondegenerate
I84
6. A Construction of the Fundamental Series
(g, M)invariant Hermitian form and if p = 2n then we can look upon sesquilinear pairing of T n (V ) with itself. One checks that
fl as a
P(u, w ) = (  1)" conj(P(w, 0)).
Thus, if n is even P is Hermitian. If n is odd multiply j by i . This completes the proof of the theorem. 6.4. Some vanishing theorems In this section we will prove some vanishing theorems for the Zuckerman functors. Let G be a real reductive Lie group of inner type and let 0 be a Cartan involution of G. Fix, 5, a 0stable Cartan subalgebra of g such that 5 is fundamental. Let € be, as usual, the Lie algebra of the maximal compact subgroup of G corresponding to 0. Let € = € n 5. Let H E if. ad H is semisimple with real eigenvalues. We set 1 = {X'E g I [ H , X ] = 0). Let 11 denote the direct sum of the eigenspaces of ad H corresponding to strictly positive eigenvalues. We will call q = I, + u a 8stable parabolic subalgebra. Notice that 0 restricted to I is a Cartan involution of I and that Ou = u. If q is a 8stable parabolic subalgebra then qk = q n f, is a parabolic subalgebra of 1,. We set m = f n q = f n I and set uk = u n f,. Then q k = m, uk. Let L = {g E GI Ad(g)H = H } . Set M = K n L. We leave it to the reader to prove that M acts trivially on A'oP(f/m).If W is an (m, M)module then we look upon W as a ( q k , M)module by letting uk act by 0. We set M(qk,W ) = U(€c)@u,qr)W. Then M(qk,W ) is a (f,M)module with f acting by left multiplication and M acting by m(k @ w ) = Ad(m)k @ mw for m E M , k E U(f,) and w E W. We note that if dim u, = n then dim f/m = 2n.
6.4.1.
+
Lemma. r'(M(qk, W ) )= 0
for j < n.
As a (€,M)module M(qk, W )0H ( K )is isomorphic with M(qk, W 0 H ( K ) ) by Lemma 6.A.l.l. Since dim fc/qk = n the result now follows from Lemma 6.A. 1.5.
Lemma. Let V be a (f, M)module such that V has a (€,M)module jiltration 0 = V, c V, c V, c . . . with y/q isomorphic with M(qk,Wj) for some (m, M)module Wj and 5 = I/. Then r j ( V ) = 0 for j < n. 6.4.2.
u
185
6.4. Some Vanishing Theorems
We first prove that rJ(V) = 0 for all i and all j < n. If i Assume this for i then the ( f , M)module exact sequence o+i+
y+'
9
= 0 this
is obvious.
y+l/v+o
induces the (f, K)module exact sequence
rj(V;) + rj(q+ + rj(q+ l~v).
+
Thus Lemma 6.4.1 implies the assertion for i 1. Now let fi E Cj(f, M; V 0H ( K ) )with j < n. Then there exists i such that 8, E Cj(f, M ; q 0H ( K ) ) . The preceding results now imply that p = dcr with a E Cj '(€, M ; V 0H ( K ) ) .This completes the proof.
Corollary. Assume that V is as in the previous Lemma and in addition that V is admissible and admits u nondegenerate (€, M)invariant Hermitian form. Then r J ( V )= 0 f o r j # n. 6.4.3.
6.4.2 implies that r j ( V ) = 0 for j < n and r j ( V ) = 0 for j > n by 6.3.5. Let W be an (1, M)module. We extend W to be a (4, M)module by letting u act by 0. We write M ( q , W ) for the (9,M)module, U(g,) @, W with g acting by left multiplication and M acting by m(g 0w) = Ad(m)g 0mw for m E M, g E U(g,) and w E W .
6.4.4.
Lemma. M(q, W ) has a (f, M)module Jiltration as in Lemma 6.4.2. In particular, rj(M(q, W ) )= 0 for j < n. Let X denote the complex conjugate of X in gc relative to g. Then gc = u @ 1, @ U. Thus V = M(q, W )= U(U) 0 W as an (I, M)module. Set U, = { X E U OX =  X } . Then U(U) = U(U,) symm(S(ii,,)).Set Z, = U(f,)(l 0W ) . Put Zjt I
=
U(€,)(symm(Sj+'(ii,,))0W ) + Zj.
Notice that 2, is isomorphic with M(q,, W ) . We also note that u,(symm(Sj+'(U,,)) 0 W ) is contained in symm(Sj+'(ii,)) 0 W Zj. Thus modulo Z j , symm(Sj+'(U,,)) 0 W is the u,module, Sj+'(gC/Uk04)) 0 W. These observations now easily imply that Zj+,/Zj has a filtration of the desired form. The Lemma now follows.
+
6.4.5. We continue our discussion with q a &stable parabolic subalgebra of gc. Let b be a Cartan subalgebra of 1., Let @ be the root system of gc rel
186
6. A Construction of the Fundamental Series
ative to b. Fix @+ a system of positive roots in @ such that if we set nf equal to the sum of the positive root spaces of gc relative to @+ then n+ contains u. Set (@,)+ equal to the set of roots of I, relative to 9 in a+.Put @(b,u) equal to the set of weights of b on u. Let p be half the sum of the elements of @+ (as usual). The following Lemma is a special case of a more general result that allows W (below) to be infinite dimensional.
Lemma. Let W be a an irreducible (I, M)module. Then
w = w, @ . . . @ w, with WJ an irreducible (I, Mo)module. Assume in addition that W is finite dimensional. Let Aj be the highest weight of Wj relative to (@,)+. If (Re Aj + p, a) 5 0 for all CI E @( b, u ) and all j then M(q, W ) is irreducible. Let W, be an irreducible, nonzero, (I, Mo)submodule of W (4.2.1). Let M I = { m e M l m W , = W,}. Then M , contains M o . Hence M / M , is finite. Let {a,,.. ., a,} be a set of representatives for M / M , . We assume that a, = 1. Then ajW , is an irreducible (1, Mo)submodule of W. Let j be the smallest index such that aj W, intersects W, in 0. If j doesn't exist then W = W, and we are done. Otherwise, set W, = ajW,. Then the sum W 2 = W, @ W, is direct. Let i be the smallest index such that oiW, intersects W zin 0. If i doesn't exist then W 2 = W. Otherwise, set W, = a, W,. The sum W 3 = W, @ W 2 @ W, is direct. It is now obvious how one completes the proof of the first assertion. For the proof of the second assertion we use Scholium. Let F , and F, be irreducible finite dimensional (I, M ')modules. Let n(F,) denote the set of weights of F , relative to 9. Let A be the highest weight of F, relative to (@'I)+, Then F , 0F, splits into a direct sum of irreducible jinite dimensional (I, M ' ) modules with highest weights of the form A + p with p E n(F,).
If p, 6 E n(F,) then we write p 2 6 if ,u  6 is a sum of elements of ((I+)+ Let f,,.. ., fd be a basis of F , with 4 an element of the p j weight space of F, and such that if i 2 j then p i 2 p j . Set n, equal to the intersection of 1, with n + . Then nIJ is contained in Cf.Let v be a nonzero element in the weight space of F,. Set 5 = U(lc)(Xi2 Cfk 0 u). We leave it to the reader to check that V, = F, 0F,. Now nl(J 0 u) is contained in Q + l . Hence is either zero or is irreducible with highest weight A + p j . This proves the Scholium.
Y/Y+,
187
6.4. Some Vanishing Theorems
We now prove the second assertion of the Lemma. We will use the notation of the first part of this proof. Since MI contains M o , we can choose each aj such that Ad(aj)b = b and Ad(aj)*(cDl)' = (cDl)+. Thus ajW, is an irreducible (1, Mo)module with highest weight o j A , . This implies that (Re Aj + p, Re Aj p ) is independent of j. Let U be the sum of the root spaces corresponding to the elements of (P(l),u). Then as an (I, M)module M ( W ) = M(q, W ) is isomorphic with U(U) 0 W . This implies that the highest weights of the Misotypic components of M ( W ) are of the form A j  Q where Q is a sum of (not necessarily distinct) elements of cD( 6, u). Let V be a nonzero (g, M)submodule of M ( W ) . Then V" is nonzero. Let p be a highest weight in this space. Then, since the infinitesimal characters of M ( W ) are of the form xA we must have p + p = s(Aj + p ) for some element of the Weyl group of gc relative to b and some j. This implies that (Re p p, Re 11 p ) = (Re A j p, Re Aj p). But then p, Re A j  Q + p ) =(Re Aj p , Re A j + p )  2(Re A j + p, Q) (Re A j  Q (Q, Q) 2 (Re Aj + p, Re Aj p ) (Q, Q) by our hypothesis. Thus Q = 0. But then V contains 1 0W. Hence V = M ( W ) . This completes the proof.
+
+
+
+
+
+ +
+
+
+
If g E U ( g , ) then we write conj(g) for complex conjugation of g relative to U ( g ) .If g E U(g,) then we set y* = (conj(g))'. We note that PBW implies that
6.4.6.
U ( e c )= U(1c) 0(UU(ec) + U(gc)u).
Let p denote the corresponding projection onto U(Ic). Let W be a (I, M)module. We now define a (9, M)invariant, sesquilinear pairing of M ( W ) with M ( W " ) . If x, y E U ( g c ) and if w E W, w # E W # then set ( x 0w, y
0w " )
= ( p ( y * x ) w , w").
It is easily checked that if q E U(q) then ( x q 0w

x
0qw, U ( g c )0 W " ) = 0
and ( U ( g c )0 W , yq
0w #

y 04 w # ) = 0.
Thus, ( , ) "pushes down" to a sesquilinear pairing of M ( W )with M ( W # ) . We will also leave it to the reader to show that this pairing is (g, M)invariant. Set R"(q, W ) equal to the set of all (I, M ) submodules, N , of M ( W ) such that N n (1 0 W ) = 0. Then it is easily seen that if N , and N , E R" then N , + N , is also. Set R(q, W )equal to the sum of the elements of R". Then
188
R
E
6. A Construction of the Fundamental Series
R“ and it is easily seen that R(q, W ) = { m E M ( W )I (m, M ( W ” ) ) = 0}
and R(q, W ” ) = { m E M ( W ” ) ( M ( W )m) , = O}.
Proposition. Assume that W and M ( W )are irreducible. Then the form ( , ) is nondegenerate. In particular, Tj(M(W ) )= 0 for j # n. The first assertion is an immediate consequence of the above observations. ( = 0 for j < n. Thus, Lemma 6.4.4 implies that Ij(M( W ) )= 0 and T J ( M W”)) the second assertion follows from Theorem 6.3.5.
6.5.
Blattner type formulas
We retain the notation of the previous section. We also assume unless otherwise specified that G is connected.
6.5.1.
(1) M is connected.
Let H E it be as in the definition of q. Then M = { k E K I Ad(k)H = H } . Let T be the maximal torus of M with Lie algebra it. Then T is also a maximal torus of K . If m E M then there exists m, E M o such that Ad(m,m)t = t. Thus Ad(m0m)induces an element s E W ( K ,T ) . Fix, @,: a system of positive roots for @(fc, tc) such that m(H) 2 0 for c1 E @: . We set @: = 0: n @ ( M ,T ) . Then ~(0): @ ); = (@:  0;).There exists s1 E W ( M o ,T ) such that s,s@; = @.: Thus s,s@: = 0;.We may thus assume that Ad(m,m) acts as the identity on t. This implies that mom E T, since K is connected. Hence m E M o . Thus m E M o so M = M o . Let @: be a system of positive roots for @(€,, tc) that is compatible with q,. If p E it* is (D: dominant integral and T integral then we denote by V, an irreducible (f, K)module with highest weight p. If F is a finite dimensional (f, K)module then we write ch(F) for the character of F restricted to T. We will also write e’ for the character t H t’. Let (D: denote @: n @ ( M ,T ) and set pm equal to the half sum of the elements of @.: If y E MA fix E, E y. If V E C(m, M ) is admissible then we set ch,(V)
=
yoM”
dim Hom,(E,, V)y.
This expression has meaning as a formal sum.
6.5.
Blattner Type Formulas
189
Let A,,, = ePmIIaEam (1  e  = ) . Let iy be the highest weight of y relative to @.: Then the Weyl character formula says that
A,,, ch y
=
c
det(s)es(ay+Pm!
SEW(M.T)
Notice that there is exactly one term for each Weyl chamber. Thus A M chM(V ) makes formal sense on T. Furthermore, we can read dim HomM(Ey,V )as the coefficient of e’v+pm. Lemma.
With the above notation and conventions
As an Mmodule M(qk, E,) is isomorphic with s(iik)@ E, = 0 s’(iik) 0E,. chM(s’(fik) 0E,) = Chy(sj(iik))ChM(y). NOW, Ch,(S’(&)). = c eQ, the Sum over Q that are sums of j (not necessarily distinct) elements of @(&,tc). Thus
c Ch,(SJ(Uk))
=
Since AMepkpmlI a r E a ( U k , t C ) (1
n
11
(1

e‘)
(6.A.2.2).
‘E@(Uk,tC‘)
~
C a )=
A,, the Lemma now follows.
Let Pk denote the partition function of @(u,,t,) with multiplicities equal to 1 (see 6.A.2.1).
6.5.2.
Lemma. If there exists s E W ( K ,T ) such that s(2, dominant and Tintegral then
1( 
l)’ ch, l  j M ( q k >
Ey)
=
det(s) ch,
+ Pk)

Pk is 0;
I/s(l,+pk)pk.
otherwise C (  1)’ ch, ljM(qk,E , ) = 0. We note that M(qk, E,) has infinitesimal character x ~ ~Hence + ~the~ same . is true for rj(M(qk, E,)) (6.3.3).Thus Hj(f, M ;M(qk,E,) 0(V,)*) = 0 if A, Pk is not in W ( K ,T)(E,,+ pk). This implies the last assertion of the Lemma. Fix p such that P k E W ( K ,T)(A,+ Pk). We must compute
+
+
1( 
l ) j dim Hj(€,M ; M(qk,E Y )0(V,)*).
Since the cohomology we are studying is the cohomology of a finite dimensional complex, we may apply the EulerPoincark principle, which says that the alternating sum of the dimensions of the graded components of the cohomology is equal to the alternating sum of the dimensions of the graded
190
6.
A Construction of the Fundamental Series
components of the complex. We are thus left with the calculation of
1(
1)' dim(Aj(f/m)* 0M(qk,E Y )0(V,)*))".
As an Mmodule M(qk,E ) is isomorphic with S(iik)0E . Thus we are computing
c 1(  1)' dim(Aj(f/m)*0S r ( i i k )
@ E,
0(%)*)".
r
Let w be the order of W ( M ,T )then the Weyl integral formula says that w times the number that we are computing is ( 1)'
1T IAM(t)I2ch(Aj(€/m)*)ch(S'(ii,)) ch(E) conj(ch V,)dt.
Now C (  1)' ch(Aj(t/m)*) = IIaEa(u,t, (1  ea)(l  e'). Also, conj(ch V,) = C det(s)e"("M+Pk'/conj(A,)by the Weyl character formula. After the obvious algebra is done w times the number we are computing is
c det(s4 s
r.s,u
T
(1  t  a )
~ae@D(u.t)
ch(S'(iik))tS(ayCPk)t~u(a,+Pk)
dt.
This in turn is equal to
If we apply A.6.2.2 then we have
The individual integrals in the above expression are nonzero if and only if s ( l ? + P k ) = u(& + pk). Since we are assuming that A, pk = t ( l , + Pk). The nonzero terms are those with to st = u. Thus we have w terms each equal to det(t). This completes the proof of the Lemma.
+
6.5.3. Let p n be the partition function of 0 for a E @.: Hence (pit + pk,a) < 0 for CI E 0:. This implies that ssK(plt P k ) = plt Pk Z n,a the sum over CI E with n, 2 0. Hence X n,a =  Q. Let H E it be such that I,( = f)) is the centralizer in 9, of H and a ( H ) > 0 for a E @(u,b) = @+. Then we have 0 IC n,a(H) =  Q ( H ) I 0. Thus Q = 0. Since det(s,) = (  l)",(2) follows. We now prove (3). If dim Hom,(V,,T"(M(q,E))) > 0 then (1) implies that there exists w E W ( K ,T )such that W ( [ T P k ) = (PIt P k )  Q with p , ( Q ) > 0. Hence S K W ( ~ P k ) = SK(p1t Pk)  sKQ. Now, S K W ( G Pk) = Pk  R with R a sum of elements of m i . Thus [T pk = sK(p P k )  s K Q + R . If we write R =  s K ( s,R) then (3) follows.
+ +
+
+
+
+
+
+
+
193
6.6. Irreducibility
6.6. Irreducibility 6.6.1.
We retain the notation and assumptions of the previous section. Let b = b + u be a @stable Bore1 subalgebra and let @+ be the corresponding system of positive roots. The purpose of this section is to prove
Theorem. Let A conditions
E
s,(A
(1)
b* be pure imaginary on b n g and satisfy the following two
1, + Pk)

Re(A
(2)
Pk is @:dominant and Tintegral.
+ p,
ci)
0 for all a E P then 7cp.A = rr is square integrable.
Let 6 be the half sum of the elements of P. We assume by going to a finite covering of G (if necessary) that there exists an irreducible finite dimensional (g,K)module with highest weight 6 relative to P. Let d be the action of G on F. We define a representation of G on End F by P(g)T= o(g)To(dg)'.Let k > 0 be an integer such that if [ is a weight of a on
6.9. The Case of Disconnected G
203
c,
End F then (kA + a) > 0 for all a E P. Thus D p , k A 0 (End F ) = 0 DP,k,,+i the sum over the weights of End F taken with multiplicity. Let f (9)= tr j ( g ) Z . Then f ( k , a k , ) = tr(a(a)') for k , , k , E K , a E A . Hence, if a E Cl(A+) then f ( k , a k , ) 2 a Z p Thus, . since our hypothesis and the previous theorem imply that each D p , k A + [ is tempered, we see that if c is a K finite matrix coefficient of np,k,, then Ic(k,ak,)(a2PI C < co for a E Cl(A+). Hence 2.4.2 implies that n p , k A is square integrable. Theorem 5.5.4 implies that if c,(kA)is nonzero then ksA'), E +a*. There exists p , a positive integer, such that p is a highest weight of a finite dimensional irreducible ( g , K ) module. Hence cs(A kpA) = c,(A) for all s E W. Hence, if k is sufficiently large, then (1 + kp)sA'),E +a* if c,(A) is nonzero. If we divide both sides by (1 + k p ) we find that sA'I,, E +a* if cs(A)is nonzero. Now Theorem 5.5.4 implies that nP,,,is square integrable.
+
6.8.3. The reader should be warned that the above statement is false if the condition t, = b is removed. It is a good exercise to see how this assumption was used in the above proof. Our use of tensor products with finite dimensional representations to obtain estimates on matrix entries is based on ideas taken from HechtSchmid [I]. An immediate consequence of the preceding result is the following fundamental theorem of HarishChandra [ 131. (We will prove the converse in Chapter 7.)
Theorem. I f G contains a compact Cartan subgroup then G has irreducible square integrable representations. 6.9. The case of disconnected G 6.9.1. In this section we drop the assumption that G is connected. We assume that G is of inner type and that G = OG. We also assume, throughout this section, that there exists, T, a Cartan subgroup of G contained in K and that for some (hence every) choice of positive roots relative to t, = l~, is T o integral (this can be achieved by going to a finite covering of G). Let Z = { g E G I Ad(g) = Z >.
Lemma.
T=ZTo
B y d e f i n i t i o n T = { g ~ G I A d ( g ) l , = Z } . L et ~ T . 1 af E O = @ ( g c , t ) ) t h e n Ad(t)(g,), = (g,),. Hence each a E @ extends to a one dimensional character
204
6. A Construction of the Fundamental Series
of T. Let P be a system of positive roots of @ and let A be the corresponding system of simple roots. If x , y E T and X" = y" for all c1 E A then X" = y" for all c1 E @. It is clear that there exists to E T o such that (to)" = t" for all u E A. Thus t ( t o )  ' E Z . This implies the result. 6.9.2. Set G , = ZGO, K,, = ZKO. Let T j denote the Zuckerman functors from C(g, T )to C(g, K ) , r{ those from C(g, T )to C(g, K , ) and T i those from C(g, TO)to C(g, KO). If V E C(g, K,) we define IndEl(V) to be the space of all functions, f , from K to V such that f ( k , k ) = k,f ( k ) for k, E K , and k E K. If k E K, X E g and f E IndE,(V) we set kf(x) = f ( x k ) and X f ( x ) = (Ad(x)X)f(x) for x E K . We leave it to the reader to show that with these actions IndEl(V) is a (9,K)module.
Lemma. I f V E C(g, T ) then r j V
= Indil
r{V.
We leave this as an exercise (which is an easy consequence of the definitions). 6.9.3. Let y E T" and fix E, E y. Since T o is central in T there exists p, E (To)" such that t E T o acts on E, by p , l . Fix P, a system of positive roots for 0,such that A = A, = p, + pn  pk is Pdominant. Let b = b u be the 0stable Bore1 subalgebra of gc corresponding to s,P. Let k E KO be a representative for S , and let s,y be the element of T" with representative E, with action y ( k  ' t k ) . We set E ; equal to this Tmodule tensored with C s K p .
+
Lemma. As a (9,K)module T"M(b, E ; ) = DP,, is isomorphic with the modTnM(b,E,)) @ & , A , haoing ( g , K o ) acting on the second ule Homg,K(Dp,A, factor and Z acting on t h e j r s t factor. Furthermore, Dp,yis irreducible if DP,* is irreducible. As a (9, To)module M(b,E,^) is just dim E,^ copies of iVf(b,csK(A+pf). Thus M(b, E ; ) = Homg.,(M(b,C,~,,+,,),M(b,E,^))
@ M(b>cs~(A+p))
with Z acting on the first factor, Let X , denote this Z module. It is a simple matter to see that DP,, is isomorphic with X @ Dp,Awith Z acting on the first factor and ( g , K o ) acting on the second. The lemma easily follows from this.
205
6.9. The Case of Disconnected C
6.9.4. We let OH"") be the unitary representation of Go associated with Dp,A in 6.7.6. We form a unitary representation ( l n p , y , l H P , yof) G, as follows: l H p * y= Xy@O H p , with Go acting on the second factor and 2 acting on the first (the compatibility is guaranteed by the construction of Xy).Let l ~ p , denote y this action.
Lemma. Set (nP,?,HPgy)= Ind~o(lnp,,)(unitary induction, here G/G, = K/K, which is j n i t e ) . Then ( H P . Y ) Kis isomorphic with T " M ( b , E , ^ )as a (9, K)module. Indeed, ( H P * , ) ,= (Ind$ ' H p S),y
=
Ind,(('HP*Y)K) = Ind, T:M(b, E,^ ).
6.9.5. Theorem. If (A?,a) > 0 for all CI E P then (nP,?, H P 9 ? )is a nonzero, irreducible, square integrable representation of G with injnitesimal character xA. Since G / G , = K/K,, for each element of G I G , we may choose a representative in K. If k E K then Ad(k)t is a maximal abelian subalgebra of €. Hence there exists an element k , E KO such that Ad(ko)t = Ad(k)t. Thus the representatives of the cosets can be chosen so that they normalize t. For such k, Ad(€)l,E W(g,,l)) (we are assuming that G is of inner type). We may thus choose a set of representatives 1 = y , , . . .,y , for G I G , such that (1)
yj E K and Ad(yj)t = t.
(2)
If sj is the element of W(g,, l)) corresponding to yj then sjP contains @.:
(3)
If sj
=
s, then j
= r.
with the elements of DP.? = T " M ( b , E , ^ )supported on We identify K,. Then it is easily seen that yfD,,, is isomorphic with lDsp,sy.Thus, if y f DP,, is isomorphic with y,' Dp.y then we must have
(*I
Ds,p,s,y is isomorphic with Ds,p,s,y.
+
In the left hand side of (*) the Ktype with highest weight sj(A p)  2pk occurs. The highest weights of the Ktypes that occur on the right hand side are of the form s,(A + p )  2p, + s,Q with Q a sum of elements of P. Thus if (*) holds then there must exist Q as above such that sj(A + p )  2p, = s,(A + p + Q)  2p,. Now this implies that (112 p1I2 = IIA + p + QII'. Since (A + p , Q ) 2 0 this implies that Q = 0 and hence sj(A + p ) = s,(A + p). But then s, = s, since A + p is regular.
+
206
6.
A Construction of the Fundamental Series
Thus the yf DP,?are mutually inequivalent. This easily implies the irreducibility assertion. The square integrability is clear since it can be tested on Go. As a (g,Ko)module ( I T Y ) , = 0 DsJp,sJA and each of the summands has infinitesimal character xh. The proof of the theorem is now complete. 6.9.6. The above discussion is a modification of the arguments in HarishChandra [14, pp.176 1771. 6.10. Notes and further results Let M be a closed subgroup of K . Put K , = MK'. If V E C(g, M ) then set V,, equal to the space of all u E V such that span(U(f)Mu) is the underlying (f, M)module of a finite dimensional representation of K,. Set V, = Indz1(VK,)(6.9.2). Then V + V, is a left exact functor from C ( g , M ) to C(g, K ) . The Zuckerman functors are usually defined to be the right derived functors of V + V, (c.f. Cartan, Eilenberg [11). Let us recall what this means. I E C(g, M ) is said to be injective if whenever we have 6.10.1.
0  A L E
I with a, p morphisms in C(g, M ) and a injective then there exists o,a morphism of B into I in C(g, M ) such that oa = p. In C ( g , M ) one has injectives given as follows. Let W E C(m,M). Put I ( W ) = (Hom,(,,(U(g,), W ) ) MHere . g acts by right translation and M acts by (mf)(g) = mf(Ad(m)'g). We leave it to the reader to show that Z(W) is injective. If V E C(g,M ) then V injects into Z(V) (we forget the gmodule structure) under the map i(u)(g) = gu. In the jargon of homological algebra this implies that C(g, M ) has enough injectives. If V E C(g, M ) then an injective resolution of I/ is an exact sequence with each Ij injective. One can find such a resolution by taking I , I , = I(Z(V)/i(V)), etc. We note that the cohomology of the complex ('O)K3(Il)K%'.'
= [(V),
6.A.l. Some Homological Algebra
207
is, up to a natural isomorphism, independent of the choice of the resolution. The jth cohomology space of this complex is the jth right derived functor. One of the key results in Enright, Wallach [2] implies that our functors rj are naturally equivalent with the right derived functors of V,. It is this formulation of the Zuckerman functors that is studied in Vogan [2]. Zuckerman introduced these functors to give an algebraic analogue of the sheaf theoretic constructions in Schmid’s thesis (Schmid [11) which proved a substantial part of Langlands’ conjecture on the discrete series.
6.10.2. Our calculations of Kmultiplicities are based on Lemma 6.5.2. This result can be sharpened as follows. If .YE W ( K , T ) then denote by I(s) the number of a E CP: such that sa is negative. Then det(s) = (  l)’(‘). In the notation of 6.5.2 one has Theorem. If there exists s E W ( K ,T ) such that s(A + p k )  pk is (Dldominant and Tintegral then l  ’ M ( g k , E ) = 0 if j is not equal to I(s) and T“”M(g,,E) is isomorphic with K(,,+pk,pk.Otherwise l’M(g,,E) = 0 ,for all j .
This theorem is substantially, the Borel, Weil, Bott theorem (see Enright, Wallach [2] for a proof using the formalism in this chapter, there are also related results in Chapter 9).
6.10.3. We now move to the situation in Section 6.7. Let g be a &stable parabolic in gc. Let A E i ( I / [ l , l ] ) * be Tintegral. Let C, be the corresponding one dimensional (I, M)module. Suppose that p E i(I/[l, I])* vanishes on I n p and is such that M(q, C,,,,,) is irreducible for t 2 0. Then we set P M ( g , C,,) = Bq(A).Theorem 6.7.4 implies that B,(A) is either 0 or it is a (9, K)module that admits a positive definite (9, K)invariant Hermitan form. This result implies a conjecture of Parthasarathy [2] and of Zuckerman which was first proved by Vogan [3]. Our discussion follows the proof in Wallach [4]. We will study the modules B&A) in more detail in Chapter 9. 6.A. Appendices to Chapter 6
6.A.1. Some homological algebra 6.A.l.l. In this appendix we will compile several results on algebraicly induced modules that are used in this chapter. The first theorem is taken from Garland, Lepowsky [l]. Let g be a Lie algebra over (say) C and let m be a subalgebra of g. Let W be an nrmodule and let I/ be a gmodule.
208
6. A Construction of the Fundsmental Series
OU(,,,) (W 0 V ) are
Lemma. The gmodules ( U ( g )@u(m) W )0 V and U ( g ) isomorphic.
Let A: U ( g )+ U ( g ) 0 U ( g ) be defined by A ( l ) = 1 0 1 and A(X) = X 0 1 1 0X for X E g. Let S(x) = x T (see 0.4.2). Let m: U ( g )0 U ( g )+ U ( g ) be given by multiplication. We leave it to the reader to check the following identities. (Hint: Test them on elements of the form X".)
+
(1)
(1 0 m)(A 0 I ) ( I 0 S)(A(g))= g 0 1
for g E U ( g ) .
(2)
( I 0 m ) ( l 0S)(A 0 I)(A(g)) = g 0 1
for g E U ( g ) .
Recall that if X and Y are gmodules then X 0 Y is a gmodule with action
g(v 0w) = A(g)(v 0w). We define a mapping, a from U ( g )&,cm,(WO V ) to ( U ( g )@Li(m) W )0 V by a(g 0 (w 0v)) = A(g)((l 0 w ) 0 v). Then it is easily
seen that CI is well defined and is a gmodule homomorphism. We define a map fl in the opposite direction by B((g 0w ) 0 v ) = p((Z 0S)A(g)((l 0 w ) 0v)). Here p is the projection of U ( g )0W 0 V onto U ( g )&,cm,(W 0 V ) . (1) and (2) imply that a and fl are mutual inverses. The Lemma now follows. 6.A.1.2. The next results have to do with the Koszul complex. Let V be a finite dimensional vector space over (say) C. Let S ( V ) denote the symmetric algebra over V. Let S j ( V ) denote the elements of S( V ) that are homogeneous of degree j . We define
a: s ~ ( 0 v )A ~ ( v+) s j + l ( V )0A k  ' ( V ) by a ( u @ v , A v , A . ~ . A v , = ) C (  l ) p u v p @ V , A ~ ~ ~ A ~ ?Here ~ Athe ~ carat ~ ~ A ~ , . means delete. Lemma.
The following sequences are exact. O  + S j ( V ) @A"b'+Sj+'(V)@ A "  ' V + . . . +
~ j + n 1
( V )@ A ' V'
+
Sj+"(
V)
+
0.
We look upon S ' ( V ) as the space of all polynomial functions on I/* that are homogeneous of degree j. If p E V * and if u E S j ( V ) then we set ~ , u ( c I=) d/dt,=,u(a tp). Let vj be a basis of V and let p j be the dual basis. We define
+
d: S j ( V )0 A kV + S' ( V ) @ A k + l V
d(u 0u) =
1a,,u
0 v,Av.
209
6.A.1. Some Homological Algebra
+
We leave it to the reader to check that d a + ad is j n times the identity on Sj(V ) 8 AkV. The Lemma follows from this observation. 6.A.1.3. Let g be a Lie algebra over C and let b be a subalgebra. Let W be a gmodule. We define a gmodule homomorphism, a, from U(g)@,(,, (A'(g/b) 8 W )into U ( g )@u(b)(Aj '(S/b) 8 W )by
a(g 8 XIA . . . A X j 8W ) = C (
1)k~ Xk 8 X,A .. . AXkA . . . AXj Q w A

+~ (  l ) k ' ' u ~ X , i \ . . . / \ ~ k / \ . . . ~ x j ~ x k W r<s
(  I)*+ ' u 8 [X,,X,] A XIA . . . A
grA . . .A z s A.. .AXj 8w
x
here denotes the projection of X E g into g/b. It is an exercise (which we leave to the reader) to show that a'
=
Here the last map is the obvious natural 9module homomorphism g
8 w H gw.
0.
Set q ( g ) = Uj(g)U(b). We set Ej,k
=
q ( g )@((Ak((S/b) U(b)
8W ,
then a maps Ej,k to Ej+l,k Let I/ be a subspace of g such that g = b 0 V. Then q(g) = symm(Sj(V))U(b).Here
a induces
a:
a
a map Ej,k/Ejl,k+E ~ + ~, / ,E~~ , ~ It] .is easily seen that is given by the map a of 6.A.1.2.Thus if u E Ej,kand if u = 0 then there exists u l € Ej_,,,+,suchthatu av, E E j  , , , , t h e r e i s t h u s v , ~ Ej_,,,+,suchthat u  a u ,  a u , E E j  Z . k etc. , This implies the lemma.
6.A.1.4. Now let g XI b XI m with b and m subalgebras of 9. Let W be a bmodule, then U(g)&,) W is easily seen to be isomorphic with U(g) @U(b)(U(b)BU(,,,) W ) .Thus Lemma 6.A.1.3 implies (r = dim(b/m))
210
Lemma.
6. A Construction of the Fundamental Series
Let W be a bmodule. There exists a gmodule exact sequence
6.A.1.5. Let G be a real reductive group. Let K be a maximal compact subgroup of G and let M be a closed subgroup of K . Let g, 6, m be as in 6.A.1.4 with Ad(M)b = b. Here g and m denote the complexified Lie algebras of G and M respectively. If W is a (b, M)module then the exact sequence in 6.A.1.4 is a (9, M)module exact sequence.
Lemma. Let W be a (6, M)module. Then H'(g, M; U ( g )@, j < dim(g/b).
W ) = 0 for
We first prove the result in the special case when b = m. Let and Ej,k be as above. Set Ej = and C: = Hom,(Ak(g/m), Ej). Then dC: is a subspace of C,",':. Let V be an Minvariant complement to m in g. Then the corresponding graded complex is given by d : Hom,(AkV, S j ( V ) 0 W )+ Hom,(Ak+' V, Si"(V) 0 W ) with da(U0,.. . , u k )
=
(  l)r(U,.
0I ) a ( U o , .. ., 6,,.. . ,u k ) .
If we choose a nonzero element of A " V (n = dim V ) then we can identify Ak((V*)with A"kV. The corresponding map is thus given as in the Koszul complex with indices n  k rather than k. The special case of the Lemma now follows from Lemma 6.A.1.2. We now prove the general case of the Lemma. Consider the exact sequence in Lemma 6.A.1.4. Set
X j = a ( U ( g ) @ (Aj( b/m) 0 W ) ) . Wm)
Then the exact sequence yields the short exact sequences 0 + X j + U ( g ) @ ((Aj '(b/m)) 0 W )+ X j U(m)
+
0.
The above special case of the Lemma combined with the cohomology long exact sequence implies that if p + 1 < n then H P ( g ,M;Xj,) is linearly isomorphic with H p f ' ( g ,M ; Xj). Thus if p + r < n then HP(g,M ; X o ) is linearly
21 1
6.A.2 Partition Functions
isomorphic with H P + ' ( g ,M ; X,). But X , = U ( g )@u(,,,)((Ar(b/m)) 0 W ) and X , = U ( g )@, W . The result now follows.
6.A.2.
Partition functions
6.A.2.1. Let V be a finite dimensional vector space over C. Let S be a finite subset of V. To each element, s, of S we assign a positive integer, m(s),which we call the multiplicity of s. We will think of S as containing m(s) copies of s. We also assume that there exists p E V * such that p ( s ) is real and greater than 1 for all s E S. If v E V then we define p,(v) to be the number of ways that v can be written in the form s1
(1)
+ " . ' + s,
with sj E S (allowing multiplicity). Clearly ps(v) = 0 if v is not in CsssNs. For example if S = {s}and m(s) = 2 then ps(ns) = n 1 if n E N.
+
Lemma.
If v
E
V then 0 I ps(v)
rj for all j . Since this is ridiculous the lemma follows.
6.A.2.2. We retain the notation of the previous number. We look upon I/ as the space of linear functionals on V * . Lemma.
Here the expression can either by interpreted formally or a convergent series on V + = {v* E I/* I v*(s) > 0 for s E S } .
This is proven by doing the obvious (formal expansion) using (1  e?) X n o Sens.
=
6.A.2.3. We now assume that S is the union of two sets S , and S , and that each s E Sj is assigried multiplicity mj(s).We define a multiplicity on S by setting m(s) = m , ( s ) + m2(s).(Here mj(t) = 0 if t is not an element of S j . )
212
Lkmma.
6. A Construction of the Fundamental Series PS(4
=
L" Ps,(V

W)P,,(W).
We note that the previous Lemma implies
If we expand the right hand side and collect terms the Lemma follows. 6.A.3. Tensor products with finite dimensional representations 6.A.3.1. In this appendix we prove several results that involve tensor products of finite dimensional and infinite dimensional modules. The basic ideas are due to Zuckerman [l] and Jantzen [l]. Once we have developed the theory of characters we will prove somewhat better results. For the purpose of this chapter the relationship with the Jacquet modules will be of great technical importance. Let g be a reductive Lie algebra over R with Cartan involution 0. Let be a Cartan subalgebra of gc. Fix, @+, a system of positive roots for @(gc,b) and let b denote the corresponding Borel subalgebra. If x: Z ( g , ) + C is a nonzero homomorphism then there exists p E I)* such that x = xb,, (Theorem 3.2.4, there xb,, was denoted x,). Furthermore, xb,, = xb,,, if and only if p' = sp for some s E W = W(gc,t)).If 6, is another Borel subalgebra of gc then there exists y E Int(g,) such that g b = b,. Set bl = gb. If p E f)* then set r ~ p= p 0 g'. Then x b , l r = &,l,oa. Also if g' E Int(g,), if g'b = g b and if 0' depends on g' as above then x b l , a a = ,ybl,a'p. We may thus parameterize all infinitesimal characters by fixing one Borel subalgebra, b, and one Cartan subalgebra, b, in b. In this appendix it will be convenient to use Borel subalgebras contained in complexifications of minimal parabolic subalgebras of g. For applications we will use other Borel subalgebras (say 6stable, 6.4.1). Let p be a minimal parabolic subalgebra of g. Set n equal to the nilradical of p and let m = 6(p) n p. Let h0 be a Cartan subalgebra of ni and set b equal to the complexification of 6,. Let b denote a Borel subalgebra of gc contained in pc. We will use b and b for our parameterization and we will write x p for X b , p . 6.A.3.2. Let G be a real reductive group of inner type with Lie algebra g and Cartan involution 6. Fix ( P ,A ) , a minimal ppair and let b and b be as in the end of the last number. Let V be the category of (g, OM)modules of 4.1.1. If E is a finite dimensional (p, OM)module then we set M ( E ) = U ( g , ) @, E
6.A.3.
Tensor Products with Finite Dimensional Representations
213
with g acting by left multiplication and OM acting by m ( g 0e) = Ad(m)g 0 me for g E U(g,), e E E and m E OM. Clearly M ( E ) is an object in f Assume that E is irreducible. Let p be the highest weight of E (we use @+, as above). We denote by, 6, the half sum of the elements of @+. (1) M ( E ) has infinitesimal character Indeed, PBW implies that
U(g,)
=
xp +
U(mc) 0(Q(n)U(&)+ U(gc)n).
Let p denote the corresponding projection onto U(m,). If z E Z(g,) and if g 0e E M ( E ) then z ( g 0 e) = g 0p(z)e. Now p ( z ) is central in U(m,) hence p ( z ) e = x(z)e. Let q and y be as in 3.2.2 and let e be a highest weight vector for E. Then p ( z ) e = q ( p ( z ) ) e = q(z)e = p(q(z))e = ( p 6)(y(z))e.Thus x = x p + s .
+
( 2 ) M ( E ) has a unique nonzero irreducible quotient, L(E). If V E lr is irreducible then there exists an irreducible, finite dimensional ( p , OM)module such that V is isomorphic with L(E). If E and E' are irreducible finite dimensional ( p , OM)modules then L ( E ) is isomorphic with L(E') if and only if E is isomorphic with E'. Let H E a be such that a ( H ) = 1 for all simple roots in @(P,A). M ( E ) = U(O(n),) 0E as an amodule. Thus H acts semisimply on M ( E ) with eigenvalues of the form p(H)  n with n a nonnegative integer. The p ( H ) eigenspace for H is 1 0E . Set R  = { N I N a submodule of M ( E ) such that N n (1 0E ) = O}. Clearly, a submodule, N , of M ( E ) is in R" if and only if the p(H)eigenspace for H on N is zero. This implies that R" is closed under addition. Set R ( E ) equal to the sum of all elements of R  . It is obvious that R ( E ) E R". It is also clear that R ( E ) is proper and that if N is a proper submodule of M ( E ) then N E R". This proves the first assertion. If I/ is a gmodule then set V" = { u E V ;nu = O } . If V E "fis nonzero then V" is a nonzero, finite dimensional ( p , OM)module. Let E be an irreducible, nonzero submodule of V". Then we have a nonzero (g,OM)module homomorphism of M ( E ) into V. Thus if V is irreducible then V is isomorphic with L(E). Finally, it is easy to see that L(E)" is isomorphic with E as a ( p , OM)module.
6.A.3.3. Let V E % If p E f,~* then we set V, = { u E V ( h  ~ ( h ) ) = ~0 u for some k and all h E 6). Then dim V, < 00 (4.1.3). We denote by ch V the formal expression
1(dim Vp)e". Set A,
=
A
=
esn(l C a )the product over a E CD'.
214
6. A Construction of the Fundamental Series
Lemma. Let E, p, M(E) be as in the last number. Then ch M(E) =
1det(s)es(”+@/A
the sum ouer s E W(m, 6). As an 6module, M ( E ) = U(O(n,)) 0 E . Let p denote the partition function of @(nc,6)= C (the weights of 6 on nc) with multiplicities equal to 1 (see 6.A.2). Let @(nC,b)= {al,..., ad}. Let be a nonzero element of the  a i weight space. Then PBW implies that the monomials Y;l... Y p form a basis of U(On,). Thus ch U(On,) = Cp(,u)e”. So ch M(E) = ch(E) rIuEx(le’)’, The Weyl character formula implies that
1
ch(E) = SE
W(m.h)
det(s)es(p+6)/A,
with 6, the half sum of the elements of cD(m,,
1
ch(E) = earn’
(
6) n a’.
Hence
)
det(s)es(”+’’/A, .
S E w(m,h)
Since (earn’rIuez(1  ea))AM= A,, the Lemma follows. 6.A.3.4. Lemma. (1) Let E be an irreducible finite dimensional (p,’M)module with highest weight p. Then there exist integers c,(p) such that (W = W(g, 6)) 7
ch L(E) =
1w cs(p)es(”+6)/AG.
SE
(2) If El,. . ., Ed are mutually inequivalent then ch L(E,), . . . , ch L(E,) are
linearly independent. (3) If V E V has generalized injinitesimal character gers cs(V)for s E W such that
xc then there exist inte
If pl, p 2 E b* then we write p1 2 p 2 if pl  p 2 is a sum of (not necessarily distinct) elements of @+. We observe that if L(F) occurs as a subquotient of M(E) and if E and F are irreducible with highest weights p and (T respectively then p 2 (T with equalify if and only if F is isomorphic with E. Also (T 6 E W ( p 6) under the above condition. Fix p E b*. If (T 6 E W ( p + 6) is a minimal element relative to the above partial order and if E is an irreducible finite dimensional (p, OM)module with highest weight (T then L(E) = M(E). Indeed, any highest weight of R(E)
+
+
+
215
6.A.3. Tensor Products with Finite Dimensional Representations
is strictly less than o. Thus assertion (1) is true in this case. Assume that (1) has been proved for all y + 6 E W ( p + 6) with y < 6. Then ch(L(E)) = ch(M(E)) ch(R(E)). Since the elements of V have finite length (4.1.3). ch(R(E)) = C ch(L(Ej) with Ej a finite dimensional irreducible (p,'M)module with highest weight y j such that y j < o. Thus (1) is true for CJ. We now prove (2). It is clear from the preceding lemma that ch M ( E , ) , . . . , ch M ( E d ) are linearly independent. It is also clear from (1) that it is enough to prove (2) in the case when E , , . . . , Ed are a set of mutually inequivalent irreducible (p, OM)modules such that if E is an irreducible finite dimensional (p,OM)module with highest weight of the form s ( p + 6)  6 then E is isomorphic with Ej for some j . If the Ej are numbered compatibly with the partial order on their highest weights then the matrix relating the ch M ( E , ) and the ch L(E,) is triangular with ones on the main diagonal. This implies (2). If V E Y then V has finite length. Let V,, . . . , Vd be the irreducible constituents of V. Then ch V = C ch V,. (3) now follows from (1) and 6.A.3.2(2). 6.A.3.5. Corollary. I f V,, V2 E Y  are irreducible and V, is equivalent with Vz.
if ch
V,
= ch
V, then
6.A.3.6. Let Vj denote the subcategory of all objects in V that have generalized infinitesimal character x,,. Clearly, Vp= Vsjfor s E W.
Lemma. Let F be a jinite dimensional (g, 'M)module and let V E V j .Then V 0 F = @ 2, with 2, E V,, and the sum is over a subset, S, of the weights of F such that {xp+,; CJ E S} = {xu+, 6 a weight of F ) and such that if a, B E S and x , + ~= xp+p then a = B. We note that ch(V 0 F ) = ch( V) ch(F). If o is a weight of F let m(o)denote the dimension of the CJ weight space. Then m(so) = m(o)for all s E w. Now ch(V) = E c,(V)esp/A. Thus ch(V 0 F ) = C m(o) C cs(V)es(p+u)/A. In light of 6.A.3.4 and 6.A.3.5 this implies that the irreducible constituents of V 0 F have precisely the infinitesimal characters described in the statement. This implies the Lemma. 6.A.3.7. Let V E V ; and let F be an irreducible finite dimensional object in V. Let o be a weight of F. We set QF,, equal to the projection of V O F onto the summand with generalized infinitesimal character xu+,.
Lemma. Let @+
Let p E b* be such that Re(p, a) is nonzero for all a E O(gc, l)). { a E Q(g,,b)l Re(p,a) > O}. Let o be the highest weight of F
=
216
6. A Construction of the Fundamental Series
relatiue to CD+ and set 0"= QF,,. Then CD' is an equivalence of the categories Vwand V,,,. Let X,Y E V, and let A
E
Hom,,,(X, Y). Then
A 0 I E Hom,,oM(X0 F, Y 0 F ) .
Set CDF,,(A)= A 0 I restricted to CDF,,(X) for CJ a weight of F. Then this construction shows that each CDF,, is a functor that is easily seen to be exact. Set, for V E *y;, CD, = CDF.,,. We put T ( X )= CDQDU(X))
for V E V,. We now come to the two main observations. (1) If V E V, and if A ch V =
c,espthen A ch(CD"(V)) =
1c,es(p+u'.
Let bR be the real span of the Ha for c1 a root. If E b* then we write Re p for the real part of the restriction of p to b R .We have
the sum over the weights of F (see the proof of 6.A.3.6). If p is a weight of F and if there exists s E W such that s ( p + p) = p + CJ then s Re p Re p = CJ  sp. This implies that s Re p  Re p is a sum of positive roots. Our assumption on implies that s Re p = Re p  Z '?,a with nu 2 0 (the sum is over 0'. Hence s Re p = Re p. But then s = 1 (again by our assumption). This implies (1).
1c,es("+")then A ch @  , ( V ) = c,esp. Indeed assume that p is a weight of F* and that p = s ( p + p + for some s E W.We note that p = + Q with Q a sum of positive roots. Thus Re p = s(Re p + Q). But then (Re p, Re p) = (Re p + Q, Re p + Q) = (Re p, Re p) + 2(Re p, Q) + ( Q , Q )2 (Re p, Re p) + (Q, Q). Hence Q = 0. Thus s = 1, j? =
(2) If V E Vp+, and if A ch V
=
CJ)
 CJ
 CJ.(2) now
follows. The upshot is that if V E Vpthen ch T ( V )= ch V. We assert that T is an equivalence of V, with itself. If X, V E V, let y: Homg,oM(X,V 0 F * ) + Hom,,o,(X 0 F, V ) be defined by y ( A ) ( x 0 f ) = ( I 0 f ) ( A ( x ) ) . Here ( I 0 f ) ( u 0 f*) = f * ( f ) u , f E F, f* E F*. Then y is a bijection. Now, V, (D,(CD"( V ) ) )= V , CD"( V ) 0 F*). Set Hom,,o,( V, T(V ) )= S, = y  ' ( W ( I ) ) . Then S,: V T ( V ) is a (g,OM)module homomorphism which is 0 if and only if V = 0. If V is irreducible then T ( V ) is irreducible (ch V = ch T(V)).Thus S,: V ,T ( V ) is an isomorphism. We note that +
217
6.A.3. Tensor Products with Finite Dimensional Representations
V H S , is a natural transformation (6.2.2) from the identity functor to T. Assume that we have shown that if V has length r then S , is an isomorphism and assume that V has length r 1. Let L be an irreducible and nonzero submodule of V. Then we have the following commutative diagram with exact rows:
+
0 0
    L
V
V/L
0
T(L)
T (V )
T (V / L )
0
Since our hypothesis implies that S, and S,,, are isomorphisms this implies that S , is also. The Lemma now follows.
6.A.3.8. We now transfer the above material to the category X (4.1.4). Let j:%'., V be the Jacquet module functor (4.1.5). If V E %' then V = 0 V x (see the proof of Lemma 4.1.4) here the sum is over homomorphisms of Z(g,) to C and I/, = { u E V ;( z  ~ ( z ) )=~0u for
some d and all z E Z ( g c ) > .Let P, be the corresponding projection of V onto V x .Let X xdenote the full subcategory of %' consisting of objects with generalized infinitesimal character x. Then P,:X.,Xx is an exact functor. If x = x, then we set P, = P, and X x= A?,. If F E %' is finite dimensional, if V E X p and if 0 is a weight of F then we set (DF,,JV)= P , + J V @ F). Then is an exact functor from the category 2''to the category X p + "If. x is a homomorphism from Z(g,) to C then set x*(z) = ~ ( z ' ) .Note that j : X Z +V,.. Also (x,)* = x,.
Lemma. I f V E X pand if ( V 0F ) , is nonzero then x weight of F. Also, j((DF,g(V ) )= OF*,Jj ( V ) ) .
=
x,+,, for some, 0, a
j ( V 0 F ) = j ( V )@ F* under the identification of ( V 0F)* with V* 0F*. Indeed, there exists r such that (n)'F* = 0. Thus, if p E j ( V ) and if (n)kp= 0 then(n)k+r(p@F*)= 0. Soj(V)@F*iscontainedinj(V@F).Letf,, . . . , f d be a basis of F* such that nfi is contained in F , + , = Z j z i + , C J . If p E j ( V 0 F ) then it can be written uniquely in the form C p i0fi. Let j be the smallest index such that p j is nonzero then modulo V * 0F j + l , (n)kp= (n)kpj@ A. Thus, p j E j ( V ) . The assertion now follows. We now prove the lemma. If (V @ F ) , is nonzero then j ( ( V 0F),) is also nonzero and it is equal to ( j ( V )0F*),*. The Lemma follows from this and the observations preceding it.
218
6.
A Construction of the Fundamental Series
6.A.3.9. We now assume that F is irreducible. Let p E b* satisfy the condition of Lemma 6.A.3.7 and let @+ be as in the statement of that lemma.
Lemma. Let c be the highest weight of F. Then categories between Xfl and Xfl+a.
@F,a
is an equivalence of
Let for V E X " , T (V ) and S , be defined as in the proof of 6.A.3.7. We note and that j ( V )E "K,,so the pertinent functors on j ( V ) are @F*,a = @F,a = Qu. In the proof of Lemma 6.A.3.7 we showed that S,(,,:j(V) + j ( T ( V ) )is an isomorphism. Lemma 6.A.3.8 implies that j ( T ( V ) )= T ( j ( V ) ) . We now show that S, is an isomorphism. We first look at the case when V is irreducible. It is clear from the definition of S, that it is injective. We therefore have the exact sequence in Hfl S
0 + V 4 T (V )9 T (V ) / S " (V )+ 0

induces the exact sequence in V(Theorem 4.1.5) 0+,mV)/uV))
+mu j ( & )
.
J(V)
+
0.
As we have observed above, j ( T(V ) )is isomorphic with j ( V ) .Hence, j ( S , ) is bijective. The exactness of the above sequence implies that j ( T ( V ) / S , ( V ) ) = 0. Hence T ( V )= S,(V) (4.1.5(1)). This proves that S, is an isomorphism if Vis irreducible. The rest of the proof is now identical to the last part of the proof of 6.A.3.7. 6.A.3.10. Corollary. W e retain the assumptions and notation of the previous number. I f V E X p f a is irreducible then @F*,a(V) is irreducible. The proof of the previous result shows that @F,a is an equivalence of categories between Hfl and H p + uwith inverse functor 6.A.3.11 The rest of this section is devoted to a proof of a theorem of Kostant [3]. Although this result is not seriously used in the text, the proof that we give is an application of the theory that we have just developed. Let g be a reductive Lie algebra over C. We put the direct product Lie algebra structure on g x g. Set H(X, Y ) = ( Y , X ) . Then there is a real form of g x g for which 6' is a Cartan involution. Indeed, let g, be a compact form of g. We look upon g as the complexification of g,. If we consider g as a Lie algebra over R then gc is isomorphic with g x g. f3 is the Cartan involution associated with (g,g,). Set f = { ( X ,X ) IX E g}.
219
6.A.3. Tensor Products with Finite Dimensional Representations
Lemma. Under the identification, U ( g x 9)with U ( g )0 U ( g ) ,Z(g x 9) is isomorphic with Z ( g ) 0 Z(g). This is an easy exercise and is Ierr 10 rne reauer. 6.A.3.12. If V is a gmodule then we set Ann(V) = {g E U(g)lgV=O}. Then Ann(V) is a two side ideal in U ( g ) . We define a g x gmodule structure on U ( g ) by ( X , Y)g = X g  gY. Then U = U ( g ) is a (g x g,f)module and Ann(V ) is a submodule under this action. Hence, U/Ann(V) is also a (g x g, €)module. If xl, xz are homomorphisms of Z ( g ) into C then set (x1,x2)(z10 z 2 ) = Xl(ZI)XZ(ZZ).
Lemma. I f V has generalized infinitesimal character generalized infinitesimal character (x,x*).
x then U/Ann(V) has
This is also left to the reader. 6.A.3.13. We now prove the advertised result of Kostant.
Proposition. I f V is a gmodule with generalized injnitesimal character xp and if F is a finite dimensional g module then V 0 F splits into a direct sum of g modules with generalized infinitesimal characters. Furthermore, the generalized injnitesimal characters that can appear are of the form x ~ with + o~ a weight of F. We may assume that V = Uu with u E V. Fix a nonzero element f E F. Let T(g 0A ) = gu 0A f for g E U and A E End(F) then T defines a gmodule homomorphism of (U/Ann(V))0 End(F) onto V @ F. Here, g acts by left multiplication on both factors. We make End(F) into a g x gmodule by setting ( X , Y ) A = X A  AY. Then the action of g on (U/Ann(V)) 0End(F) is the same as that which is gotten under the identification of g with g x 0. U/Ann( V) is a finitely generated (g x g, €)module with generalized infinitesimal character (xp, (x,)*). Hence, it is admissible (3.4.7. Corollary). Thus, Lemma 6.A.3.8 implies that (U/ann(V)) 0 End(F) splits into a direct sum of invariant subspaces, q,with generalized infinitesimal character of the form ( X ~ + ~ , X  ,  ~with ) 0, 6 weights of F . Hence, V @ F = C T(U;.).Since, each T(Uj)has a generalized infinitesimal character of the correct form, the result follows.
220
6. A Construction of the Fundamental Series
6.A.3.14 Let A denote the category of all gmodules. Let A ' denote the full subcategory of modules with infinitesimal character x'. The above result implies that if F is a finite dimensional gmodule and if (T is a weight of F then we can define the functor Q,F,a from the category A' to the category A'+a as in 6.A.3.1.
Infinitesimally unitary modules
6.A.4.
6.A.4.1. Let G be a connected real reductive group with maximal compact subgroup K and associated Cartan involution 8. Let g = f p be the corresponding Cartan decomposition of g. Let B be a (9,K)invariant symmetric bilinear form on g such that if we set ( X , Y ) =  B ( X , OY) for X , Y E g then ( , ) is positive definite. We set llXll = ( X , X ) " , for X E g. Let V be a (g,K)module endowed with a preHilbert space structure, ( , ) that is ( g , K ) invariant. Let C denote the Casimir operator of g corresponding to B. Let H denote the Hilbert space completion of (V,( >).
+
?
Lemma.
There exists a positive constant, E, such that if C acts on V by pI then Cllx"ull/n! < cc
for X
Eg
with llXll < E and u E V.
Let C, be the Casimir operator of f corresponding to B. If X E g then we write X = X , + X , with X , E p and X , E f. Then llX112 = JIX,I12 11X21J2. Let X E g and let u E V(y)for some y E K" . Then
+
l l X " ~ 1 1=~IIX1Xn1u112+ 2 R e ( X , X " ~ ' u , X , X "  ' u )+ IJX2Xn1u112 I 211x1X"'u112 =
+ 211X2x"'u112
 2 ( ( X 1 ) 2 x "  l u, xn1 u )

2 ( ( X 2 ) 2 x " 1 v, x n ' u )
Let t be a maximal abelian subalgebra of f and let Q, be the set of nonzero weights o f t on gc. Fix Q,:, a system of positive roots for Q,(f,, tc). The weights of t , on U n  ' ( g c )are of the form 8, ...Pn,with Pj E Q,. Hence the highest weights of Unl(gc) 0 V(y)are of the form A? P1 ... Pn, with E Q, (see Scholium 6.4.5). Let Z , , . . .,Z , be an orthonormal basis of p such that X , = llXlllZl and let Wl, ..., W, be an orthonormal basis of f such that X , = ~ ~ X , [ ~ W Then l.
+
+ + +
22 1
6.A.4. Infinitesimally Unitary Modules
C,
= C
(Wj)' and C  C,
=C
(Zj)'. We have
1
((X,)2X"1u,X"1u) I  ~ ~ X l ~( (~Z j2) 2 X n  1 ~ , X "  1 ~ ) and ((X,)'X"1U,X"1U)
(3)
I IIX,II'
IIX"UII' I ((n  ~ ) ! ) ' c ( Y )
n
c ((q.)'X"'
( ( c ( y ) / ( j 1)'
2<j 0 be as in the previous Lemma. Set U , = {X E g ; llXll < E } . If X E U , and if u E V then we set
eye'
(3
A(X)U=C Then X
H
7 X"UEH.
A(X)u is real analytic on U,.
(1) Let F be a finite subset of K " , u E,n,(exp X ) u .
E
V and X E U , then P,A(X)U =
Let 0 < 6 I E be such that C I(Xnu,w)l/n! < co for llXll < 6 and w E V, (3.4.9). Then (n,(exp X)u, w ) = C (X"u, w)/n! for llXll < 6 and w E V'. Thus, if llXll < 6 then (n,(exp X ) u , w ) = C (Xnu,w)/n! = Z (E,X"u, w ) / n != C (E,X"u, AFw)/n! = C (XnU, AJ'w)/n! = (A(X)u, AFw) = (PFA(X)o, w ) Thus (1) is true for llXll < 6. Since both sides of (1) are real analytic for X E U,, the assertion follows.
(2)
If u, w E V and if X
E
U, then (A(X)u, A ( X ) w ) = ( u , w ) .
We note that x , y H (u, n l ( x ) n l ( y ) w is ) real analytic on G x G . Thus if F is a fixed finite subset of K" such that u, w E V, then there exists 0 < 6 < E such that if u, w E V, and if llXll < 6 then
C (u, X"+"'W)( l)"/n!m! = (u,w).
+ + ... + X m / m !E U(g). If
Set for X E g, expm(X)= 1 X llXll < 6 then ( A ( X ) u , A ( X ) w )=
lim (exp,(X)u, exp,(X)u)
=
mm
lim ( u , expm( X ) exp,(X)w)
mao
=
lim ( A i l u , expm( X ) exp,(X)u)
mm
=
u, w E V,
and if
6.A.4.
223
Infinitesimally Unitary Modules
(AT'u, w)= ( u , w). This proves (2) for llXll < 6, so (2) follows from analyticity. In particular (2) implies
(3) If u E V and if llXll < E then IIA(X)ull associated with ( , ).
= Ilull.
Here
11. . 1 ( is the norm
This implies that A ( X ) has a continuous extension to H which we also denote A ( X ) .
(4)
A ( X ) * = A(  X )
A ( X ) * A ( X )= I
and
for X
E
17,.
This follows from (2) and (3). We assume by choosing a (possibly) smaller E that exp is a diffeomorphism on U,. Set U = exp(U,,) and let log denote the inverse function on U to exp restricted to U,. If g E U then we set n(g) = A(log(g))on H. Then if k E K n U, n(k)u = ku for u E V. Also if g E U then n(g)* = n(g)l = n(9l) on H.
( 5 ) If g l , . . ., gn are either in U or K and if F is a finite subset of K" then PFn(g1)'"n(gfl)u = EF7tl(gl)"'nl(gn)ufor all u E V. If n = 1 this follows from (1). Assume the result for II  1 2 1. Let Fl c F2 c ... be finite subsets of K A whose union is K " . Set Ej = EF and 4 = PF with F = 4 . Let w E VF then (x(g1). .. n(sfl)u,w>
=
lim (47c(92).'.n(g,)u, n ( q l )  w .
j m
lim (Ejnl(g2) '. . nl(gn)u,X ( Y ~ )  'W > = lim (n(gl)Ejnl(g2).. .nl(gn)U, W >
j m
=
lim
j m
(PFn(gl)Ejn1(g2)"'711(gn)u,
w>
j m
= lim jm
(EFnl(gl)Ejnl(g2)."nl(gn)u,AFw)
= (EFnl(gl)"'nl(gn)u,
=
(EFnl(gl)"'nl(gn)u,w > 
This proves (5). ( 5 ) implies that if g1,..., gn, x 1,..., x, are elements of U or K and if g l " . g n = xl"'x,,, then n(g,)...n(q,,)= n(x1)...n(x,,,) . Thus if g E G and g = g1 " . g n then n(g)= n ( g l ) . . . n ( g f ldepends ) only on g. We clearly have: (6) If x, y E G then n(x)n(y)= n(xy),n(x)* = n(x') and n(k)u = kufor k and u E V.
E
K
(7) (n,H ) is a unitary representation of G In light of (6), we need only show that if gj + g then lim n(gj)u = n(g)u for
224
6. A Construction of the Fundamental Series
u E V. Now there exists N such that if j > N then gj = g x j with x j E U and lim xj = 1. Since n(xj) = A(log(xj)) (7) follows. We finally note that if X E g then n(exp(tX))u = A(tX)u for tX E U , and u E V. Thus d/dt,=,n(exp(tX))u = Xu for u E V. The proof is now complete.
6.A.4.3. We note that one can use the CambellHausdorff formula to prove (slightly more directly, but using some topology) that the conclusion of the above theorem is valid under the hypothesis of Lemma 6.A.4.1.
7
Cusp Forms on G
Introduction In this chapter we study a variety of integral transforms that were first introduced by HarishChandra. The goal of this chapter is to lay the groundwork for the proof that the representations that we called the discrete series in the last chapter exhaust the irreducible square integrable representations of a real reductive group. The first step is to introduce the space of cusp forms on G. We show that matrix coefficients of irreducible square integrable representations are cusp forms (eventually we will show that the space of cusp forms is the span of these matrix coefficients).Thus the analysis of cusp forms gives information about irreducible square integrable representations. The key theorems in this direction are 7.6.1 and 7.5.2. These results are based on HarishChandra’s theory of orbital integrals which is also critical to his other monumental achievement, the “local L’theorem” for characters. Our main contribution, in this chapter, to HarishChandra’s original method is Lemma 7.4.3, which, in particular, allows us to defer the character theory to the next chapter. The key Lemma that allows this simplification is the result in 7.A.l.l which is a special case of Kostant’s convexity theorem. Another simplification in our exposition is the observation that Theorem 7.6.1 is a consequence of the material in Appendix 5. In HarishChandra’s original
225
226
7. Cusp Forms on C
development, an analogue of Theorem 7.6.1 is proved for all f E W(C). The material in Appendix 5 is then used to calculate the constant C., The more general theorem will be proved in Volume 11. In the first section, we introduce a general method of constructing Frtchet convolution algebras of functions on a real reductive group. We are mainly interested in two examples, the space Y ( G )(whose importance will be more apparent in the next chapter and in Volume 11)and the HarishChandra space %(G) which is critical to the theory of cusp forms on G (and plays the leading role in this chapter). The exposition in Sections 7.3 and 7.4 is strongly influenced by the notes of Varadarajan [l] and (of course) by the original papers of HarishChandra. The key results on cusp forms are contained in Sections 5 through 7. As usual, in this book, there are several important, but (even more technical) results that are deferred to the appendices. The deepest are in 7.A.2 and 7.A.3. 7.A.5 contains an exposition of a technique of GelfandShilov [11 for finding fundamental solutions of certain constant coefficient differential operators. The main result in that appendix is due originally to de Rham [l]. The Theorems in this chapter are all due to HarishChandra. His motivation can only be surmised. However, the earlier work of the Russian school must have had an important influence on this work. But it was HarishChandra (and only he) who realized that the key to the representation theory and harmonic analysis on real reductive groups is the discrete series and hence the harmonic analysis on the space of cusp forms. 7.1. Some Frechet spaces of functions on G 7.1.1. Let G be a real reductive group with maximal compact subgroup K and corresponding Cartan involution 8. We denote by L and R respectively the left and right regular representations of G and U ( g )on Cm(G).Let a and b be smooth positive Kbiinvariant functions on G such that (1)
If r > 0 then the set {g E G f a(g) I r } is compact. b(x)= b(x')
(2)
(3)
There exists a constant do > 0 such that a(xy) I a ( x ) a ( y )
(4) (5)
(6)
for x
E
G.
j l ~ ( x ) ~ a ( x )  24. Set D = llpk1I2 + C K .Then D'EJ = E,D'f: So Eyf = IIP + PkllzrE,w.
(*I
Let Sj(y)denote the set elements of K" that occur in V, 0 U j ( g ) .If CJ E Sj(y) then pu = p, + 6 with 6 a weight of the action of T on Uj(g,). This we have (iii) Let f
If E
+
$0') then lipu p k l l 5 CjIIPy + Pkll + Dj with Cj and Dj positive constants depending only on j .
CJ E
9 Then a ( g ) * b ( g )  ' I L ( . ~ ) R ( y ) E r ~= (g)l
Now, Ix,(k)l I d(y) and luj(k)l I Cx for k
E
K . We therefore conclude that
In the integral above we may replace L ( x j ) R ( y ) f ( k  ' g )by
If we apply (iv) and (*) above we have
230
7. Cusp Forms on C
The Weyl dimension formula implies that d(y) 5 CIIp, 1 0 ' 1 . Thus if ( ( p Y pkll > Dj/2Cj then
+
+ pJm with rn =
This combined with (ii) easily implies that ZyEyfconverges in 9 The argument in 1.4.7 easily implies that the above series converges pointwise to f. This completes the proof of the theorem.
7.1.2. We now give two examples which will be most important to later developments. We take (a,F ) to be a finite dimensional representation of G with compact kernel. We put an inner product on F, ( , ), that is K invariant. Set llgll = tr o(g)a(g)* + tr a(g')a(g')*. Then llgll 2 1. We take a(g) = llgll and b(g) = 1. Then the material in 4.A.1 implies that a, b satisfy (1)(6) in the previous number. With this choice, we denote the space %.,(G) by Y ( G ) . We call Y ( G )the space of rapidly decreasing functions on G. Notice that we may use any norm (4.A.l) on G to define Y ( G ) . The next example is due to HarishChandra. Following his usage we will call it the Schwartz space of G and denote it by %'(G). Let Z be as in 4.5.3. We set b(g) = E(g) and a(g) = 1 + log 11g11. Then (l), (2), (4) of the previous number are clear. We leave (5) to the reader. (3) follows from Theorem 4.5.3 and 5.A.3.1. To prove (6) we note that if x E G is fixed and if
(here we are using the notation of 3.6) then h, is Kbiinvariant and if x E U ( g ) K then xh, = p(yo(x))hp.Thus the material in 3.6 implies that h, = c S,. Clearly c
=
h,(l)
=
J E,(xk)dk
K
=
S,(X).
If we recall that 8 = Eo then (6) now follows with equality. The observation about arbitrary norms on G applies in this case also. The rest of this chapter will be devoted to analysis on this space.
7.2. The HarishChandra transform 7.2.1. Let G be a real reductive group such that G = OG. Fix a maximal compact subgroup K of G and let 8 be the corresponding Cartan involution. Let Z and II...)) be as in 7.1.2. Let for a = 1 + log 1111 and b = E,pa,b,x,y,ro ~ ,(we ~ ,use ~ the notation of the previous section). Let V(G) be the corresponding Frtchet algebra of functions on G (7.1.2).
23 1
7.2. The HarishChandra Transform
Fix (Po,A , ) a minimal parabolic pair for G. Let (P, A ) be a corresponding standard ppair with P = OMAN a standard Langlands decomposition (see 2.2). Let p = p p . If f E %(G) and if m E M , m = aom, a E A and m E OM then we set f'(m)
= ap
{ f(nrn)dn. N
Theorem. (1) If f E %(G) then the integral dejining f converges absolutely. (2) If f E V ( C )then f P E % ( M ) and the map f + f' is continuous from %(G) to %'(M).
'
By the definition of %(G), for all r 2 0. 5.3.4(2) implies that
E(xy)= J aP(kx')ap(ky)dk. K
If R is compact in G then a ( k y ) I C,, for y E SZ and k E K . Thus E ( x y ) I C,E(x) for y E SZ, x E G. Also (1 log Ilxyll) I Ch(1 log Ilxll) under the in mind, we see that Theorem 4.5.6 implies ( 1 ) . same circumstances. With this We also note that if we use the above argument with the seminorms c , , ~ , ~ , x E U(m) then it is easy to see that f' E C " ( M ) . We note that
+
= (R(x)f)'
+
6)
R(x)f'
(ii)
L(x)fP= (~(x)f)'
(iii)
If h E U J ( a )and if h , , . . . , hd is a basis of Uj(a)
for x E U(m). for x
E
U("m).
then with ak linear in h. Thus to prove (2) it is enough to prove that there exists k > 0 and for each d > 0 there exists c d such that if f E C(G) then ( *)
If'(m)l
5
cdaF(m)(l
+ log l \ m l l )  d C l ,
l,d+k(f).
Here P = PF and EF is as in 4.5.5. Let 6 be a Cartan subalgebra of m. Let @+ be a system of positive roots for @(gc,bc) such that n is contained in the sum of the root spaces for @+. Let 6 be the half sum of the elements of 0'. Let ( p , F ) be the irreducible finite dimensional gmodule with highest weight 6. Let G, be a finite covering group
232
7.
Cusp Forms on G
of G such that p is the differential of a representation of GI. Fix an inner product on F such that the compact form of GI acts unitarily on F. If T is an endomorphism of F then denote by 11 TI1 the HilbertSchmidt norm of T. Let llgll = llp(g)11. Then 11...11 is defined on G. 4.5.3 implies, (iv) There exist positive constants C,, C,, d,, d, such that Let MI be the subgroup of GI corresponding to M . Then as a representation of M I , F = F , ... + F,, a direct sum of irreducible MI modules and F , has highest weight 6. Relative to this decom'position of F we may also assume that p ( n ) for n E N has the block form
+
= I + y(n) with y(n) in the above upper triangular block form with zeros on the main diagonal. Thus p(nm) = ( I y(n))p(m).So
+
llflml12 = IIAm)l12 + Ilr(n)cl(m)l12.
This implies that Ilnml12 2 llml12. On the other hand, llnll = Ilnmm'!I < ~ ~ n m ~ ~ ~ /Since m  l \ llmll ~ . = IIO(rn)ll = ~ ~ r n this ~ ' ~ implies ~ , that llnmll 2 [ ~ n ~ ~ ~We ~ mhave ~ ~  l .
(4
llnm1I2 2 llnll
and
llnmll 2 Ilmll.
Let n E N , a E A and m E OM. Then IlnamlI = llO(n)ulO(m)ll. Let 'A' be the for OM relative to P,noM, A , n OM. Then Q(m)= k , a , k , with ki, k , E K n OM and a, E CI(OA'). Let uo be a unit highest weight vector for (p, F ) . Then (Inam112= I10(n)a'k,a,k,l12 = 11O(n)k1a~'alll2.Set u = O(k;lnk,). Then Ilnam((22 l l ~ a  ~ a ,2l l Ilp(u)p(a~'al)uol12 ~ = ~  ~ ~ u ~ ~ l l p ( u Now )u,ll~. u' = n(u~')a(u~')k(u~').Thus IIp(u)u,l12 = a ( ~  ' )  ~If ~we . put all of these inequalities together with (iu)we have ( P = PF) "A"'
=(narn) IaPEF(m)(l+ log ~ ~ m ~ ~ ) d l ~ ( u  l ) ~ .
Hence
+ log Ilnrnll)2d' I ZF(m)(l + log ~ ~ m \ ~ ) ~ d ~ + d + ~ ulog ( uJ~l u) (( (~)  ~ ~ .
aPS(nam)(l
233
7.2. The HarishChandraTransform
Let d, be so large that (Theorem 4.5.4, Lemma 4.A.2.3)
s
+ log [ [ x [ [ )  d ' d x< 00.
a(x)"(l
OV"
We also note that there exists r > 0, C > 0 such that lImall* 2 Cllall.
Since m is fixed, we have N
+ log I[man[[)2d+'Pdn I S F ( m ) ( l+ log Ilmll)d(l + log E(nam)(l
[[ul[)p.
This implies (*) above. The theorem now follows. 7.2.2. We say that f E V ( G )is a cusp form if (L(x)R(y)f)' = 0 for all proper parabolic subgroups P (2.2.7) of G and all x, y E G. One of the key points in the theory of HarishChandra is that the space of cusp forms on G is the closure in %(G) of the Kfinite matrix coefficients of the discrete series. The following result is a key step in this direction.
Theorem. Let f form on G .
E
V(C) and assume that dim Z&)f
< co. Then f is a cusp
It is enough to prove that f'( 1) = 0 for all proper parabolic subgroups of G if dim zG(g)f < 00. If P is a parabolic subgroup of G and N is the nilradical of P then N is contained in Go. Thus, since Z(g)is a finitely generated ZG(g)module, we may assume that G is connected. If y E K" then define
for g E G. Then E y f E V(G) is left Kfinite and C E y f converges to f in V(G). Thus the previous result implies that we may assume that f is left Kfinite. Thus L(U(g,))f = V is a finitely generated admissible (g, K)module (3.4.7). Let P be a proper parabolic subgroup of G. Let P = OMAN be a standard Langlands decomposition (2.2.7) of P. Set T ( h )= hP for h E V. Then it is easy to see that T(nV) = 0 and that T is an (m, K n M)module homomorphism from (V/nV) 0 C, into %'(M). Thus, since V/nV is admissible and finitely generated as a (m, K n M)module, dim U(a)T(h)< co for all h E V. In particular, this implies that T ( f ) ( a )= C a"p,(log a) a finite sum with p E a2 and pp a polynomial on a (8.A.2.10).The previous theorem implies that if we
234
7. Cusp Forms on G
set T(f)(exp H ) = p ( H ) for H E a then p is rapidly decreasing on a. Now Lemma 4.A.1.2 applied to p ( t H ) , t E R (limt+mB(tH) = 0) implies that T ( f ) ( a )= 0 for a E A. Thus, in particular, f'(1) = 0. 7.3. Orbital integrals on a reductive Lie algebra 7.3.1. We retain the notation and assumptions of the previous section. Let B denote an Ad(G)invariant, symmetric, nondegenerate bilinear form on g such that if ( X , Y) = B(OX, Y) then ( , ) is symmetric and positive definite and B restricted to [g, g] is the Killing form of [g, g]. If X E g then
set llXll = (X,X)'/'.If x E G then denote by llxll the HilbertSchmidt norm of Ad(x). Then 11...11 is a norm in our sense. We assume that f contains a maximal abelian subalgebra t that is a Cartan subalgebra of g. The results in this section are due to HarishChandra (some of the proofs differ from the originals). There exist nonnegative integers p , q and a positive constant C such
Lemma. that
llxll I(det(ad YI,)Ip for all x
E
s CIIAd(x)Y + B(Ad(x)YI14
G, Y E f. Here p is (as usual) the
 1 eigenspace
for 6'.
Let Po be a minimal parabolic subgroup of G with 6' standard Langlands decomposition OMAN. If a E @(Po,A ) then let n, denote the corresponding arootspace. Let a+ = ( H E a I a ( H ) > 0 for a E @(Po,A ) } (as usual). Then G = K Cl(exp a+)K.If g = k , a k , with a = exp H , H E Cl(a+),k , , k, E K then )1g(('= dim
(1)
rn + 2 1 dim n, cosh 2a(H).
Let a',. . .,ad be the simple roots in @(Po,A). Let H,, . .. ,Hd E a be defined by aj(Hk)= dj,k. If tl E @(Po,A ) then a = Z rnj(a)tljwith mj(a)a nonnegative . (1) implies integer. Set r = max{mj(a) I1 I j s d, a E O ( P o , A ) } Then (2) There exists a positive constant C such that
for g
=
k, exp H k , , k , , k ,
(3) If X
Ef
E
K , H E Cl(a+).
and if p is an eigenvalue of ad X on p then llXll 2 JpJ.
This is clear.
235
7.3. Orbital Integrals on a Reductive Lie Algebra
(4) If X E f, x E G then IIAd(x)X
+ 8(Ad(x)XI(’ = 211Ad(x)X[[’+ 211x11’.
Indeed,
+
IIAd(x)X 8 Ad(x)XI[’ = (Ad(x)X, Ad(x)X) 2(Ad(x)X,O Ad(x)X) (0 Ad(x)X,O Ad(x)X) = 211Ad(~)X11~  2B(Ad(x)X, Ad(x)X) = 211Ad(~)X11~211x11’.
+
+
+
As asserted.
(5) If a
= exp
H with H E Cl(af) and if X E f then
IIAd(a)X
+ 8 Ad(a)XII’ 2 r’IIIHj,X](l’(cosh
for all j = 1,. . .,d. Indeed, X = Z C, (X,
+
IIAd(a)X
’
Clearly, IIX,, 1) 2 r’
+ OX,) with Z E m,X, E n,.
E~(H))’
Thus (4) implies that
+ 8 Ad(a)X11’ 2 4(cosh ctj(H))’llX,ll’.
11 [H,, X] 11 ’.
(5) implies (6)
d
IIAd(a)X
+ 8 Ad(u)XIIZd2 C n (II[Hj,X]I(’(cosh aj(H))’) j= 1
for H , X as above. Let y(X) be the smallest absolute value of an eigenvalue of ad X ,1 for X E f. Then Ilad(X)Hjll 2 y(X)llHjll.Let s = dim p . (6) implies that d
2
n IIIHi,X]I1’ cosh mj(H) CIIUII”~n IICHi,xIII’
2
C,y(X)2dllal11’r
IIAd(a)X + 8 Ad(a)X112d2 C
i= 1
d
i= 1
with C, > 0 (here we have used (2)). Let p l , . . ., ps be the eigenvalues of ad X ,I counting multiplicity, with lpll = y(X). ( 3 ) and (4) imply that Ipil I IIAd(a)X 0 Ad(a)XII. I d e W X1,)I = lpl.~.psl,so
+
(IAd(a)X+ 6’ Ad(a)XI12d+2d(s1) 2 ~llldet(ad(X)~,)I12dll~l11~r. This implies that if p = 2 dr and q
=
2 rds then
llall Ildet(ad(X)I,)IIP2 CIIAd(a)X + 6 Ad(4X1Iq.
236
7. Cusp Forms on G
If X E G then x = k,akz with k,, k,EK and a E C l ( A + ) . llxll = Ilall, Idet(ad(Ad(k,)X)I,)I = Idet(ad(X)I,)I and IIAd(k,)YII = 11 YII for Y Eg. The Lemma now follows.
7.3.2. We set f" = {X E f det(ad(X)I,) # O}. Then our assumption on t implies that f" is nonempty. Let Yl,. . . , & be a basis for f. Let y,,. . ., yk be the corresponding coordinates on f. Let XI,. . . , X, be a basis of g relative to ( , ). Let xi,. . .,x, be the corresponding coordinates on g. We will use standard multiindex notation for higher derivatives in they and x coordinates (see 5.A.2.1). Set for r E R, m E N and f E Cm(g) qr.rn(f) =
1
111 srn
SUPXE~
IIX + QXIIrtalrl/ax'f(X)I.
Proposition. Let f E C" (9) be such that qr,rn(f)< co for all r, m then the integral
1f(Ad(9)Y)dg
G
converges absolutely for Y E f " and defines a smooth function, g( Y), of Y E f". Furthermore, there exist constants u and such that ( p , q are as above)
II YItrI(a"'/aY')g(Y)I with u = pill
cj,rIdet(ad Y [ p ) I  u q u , w ( f )
+ a, v = r + qIZ( + u and w = 111.
with
Thus
The previous result says that
237
7.3. Orbital Integrals on a Reductive Lie Algebra
If we use (*) again and (4) from the previous number we have
7.3.3. We set t" = t n f". We fix a system of positive roots @+ in @(gc, tc). Set n = Ilacm+ CI.Let On= {aE @(gc, tc) l(gc), is contained in pc}. Set n, equal to the product of the a E 0 ' n @,,. If H E t then Idet(ad(H)I,)( = (n,,(H)I2.Let T be the Cartan subgroup of G corresponding to t. Let Y ( g ) denote the (usual) Schwurtz spuce of 9. That is, the space of all f E Cm(g)such that pr,s(.f)
=~
~
~
IIXII' x c
C
g 111 s.7
Ial'l/ax'f(x)I
< 00,
endowed with the topology given by the seminorms P,,~. Notice that the seminorms qr,sare continuous on Y ( g ) . I f f E Y ( g ) and if H E t " then we set
@T(W= 4 H ) Gj f(Ad(g)H)dg. Then @; E Cm(t"). Let H , ,..., H, be an orthonormal basis of t and let t , ,..., t, be the corresponding coordinates on t.
Lemma. There exists a constant, u, such that if I is an rmultiindex then there is a continuous seminorm pI on Y (g) such that lal'l/at'@,,T(H)l 5 I.n,,(H)I"p'(f) .for f '
E
.V(a), H
E
t ".
We use the notation of 7.A.2.6 except that the b in the appendix is now t. If p E S(g,) let j? be as in 7.A.2.9. Let I = {PI p E S(g,)'}. Then S(t,) is finitely generated as an Imodule (see the proof of 3.2.4). Let pj E S(t,), j = 1,. . . , d, be such that S(t,) = C I p j . Let t' = { H E t I n ( H ) # O}. Then t' is contained in t". Let H E t' and let W be an open neighborhood of H in t'. Set U = Ad(G)W. I f f E Y ( g ) then we define g ( X ) = j .f(Ad(g)X)dg c
238
7.
Cusp Forms on G
for X E U. Then the preceding results imply that g E Cm(U). Clearly, g(Ad(x)Y) = g ( Y ) for Y E U, x E G . Hence Theorem 7.A.2.9 implies that pglw = n'pngl, for p E S(g,)G. This implies that p@'/T(H)= @ l f ( H )
(1)
Let HY ... H> ( = al'l/at' uj E S(g,)'. Then
for f
E
Y ( g ) and H
E
t'.
as a differential operator) equal C Ujpj with
I pj@fT(H)I
CInn(H)I r p j ( f )
with p j a continuous seminorm on Y ( g ) and rj depends only on the degree of p j . If we make a suitable choice of p j we can replace rj with r, the maximum of the rj. Thus Ipj@,T(H)I5 CInn(H)I'C for H
E
pj(f)
t ". Hence lal'l/&'@T(H)lI Cln,(H)I'
1p j ( u j f )
for H E t'. Since both sides of the above inequality are continuous on t", the result follows.
7.3.4. If U is an open subset of t then we define Y ( U )to be the space of all f E Cm(U)such that p u , r , s ( f ) = S U P X E L ~ IIXttr
C
IIlSS
Ial'l/at'f(x)t
< a.
It is easy to see that Y ( U )endowed with the topology given by the seminorms pu,r,sis a Frechet space. We are now ready to state (and prove) the following basic result of HarishChandra. Theorem. If f E Y ( g ) then 0;E Y(t").Furthermore the map f Y ( g ) to Y(t") is continuous.
from
Let C be a connected component of t". We note that if H it then a(H) E R for LY E @ = O(gc,t,). Thus if L Y EQ,, then ia is either strictly positive or strictly negative on C. We define IaIc to be ia in the first case and otherwise to be ia. Thus Inn(H)I= I'Ilal,(H) (the product over aE0 ' non for H E C . Fix x,, E C. Let x E Cl(C). Then lalC(x fx,) = lalc(x) tJaIC(xO) 2 flal(:(xo) if t 2 0. Let f E Y ( g ) . The preceding result
+
+
239
7.3. Orbital Integrals on a Reductive Lie Algebra
now implies that if F
=
Qfand if q
I@+ n QnI then
=
I P W + tx0)l 5 t””Jf) for t > 0 and p E Y(t).Here p p is a continuous seminorm on Y ( g ) . Now d k / d t k p F ( x+ t x , ) = ( ~ , ) ~ p F+( xtx,). (Don’t forget that we are identifying S(t) with the constant coefficient differential operators on t.) This implies that if we set u(t) = p F ( x + t x , ) then u‘‘)(t) = ( ~ , ) ~ p F + ( xt x , ) . Hence d k ) ( t )I tU4p,,( f
(1)
)l~n(xo)lu.
Scholium. Let u E Cm((O,11) and suppose that lu‘k’(t)l
f o r 0 < t i 1 and k
= 0,
i tPak
1, 2 , . . . . Then lu(t)l 5 C ( ~ + O . . . am + 1 )
f o r 0 < t i 1. Here C depends only on m.
We may assume that m 2 1. I f m > 1 then since,
we see that l ~ ‘ ~ ’ ( i t ) la k + l t  m + l / ( m 1 ) we have (2)
+ a  k + l + ak for 0 < t s 1. Hence
+
l ~ ( ~ ) (It )2l t ~ ~ + ’ ( a a~k )+ ~ for 0 < t 5 1.
If we argue as above using (2) we find that (3)
+
Idk)(t)t I2 m  1 t ~ ’ ( a k“ ‘ a k + m )
If we apply (3) to the case k
=
for o < t I 1.
1 we find that
for 0 < t i 1. If we integrate this inequality we get the estimate asserted in the Lemma for do)= u. We now complete the proof of the theorem. The Scholium combined
240
7.
with (1) above implies that if X
E
Cusp Forms on G
C then
with E a constant independent of f.We also note that if p ( X ) =  B ( X , X ) then p ( X ) = llX112 for X E t. This implies that @’p’.,(X) = JIX((2k@;(X). The Theorem now follows from 7.3.2.
7.3.5. We now study the analogous integrals for other Cartan subalgebras.
We will be constantly referring to material in Chapter 2. Let ( P o , A o ) be a minimal ppair for G . Let b be a Cartan subalgebra of 9. Then Proposition 2.3.6 implies that there exists a standard, cuspidal ppair, (P,, A,) and x E G o such that lJ = Ad(x)f), (see 2.3.6 for the terminology). Let H (resp. HF) be the Cartan subgroup of G corresponding to (resp. .)&I By definition, 6, = t, + a, where t, is maximal abelian in Om n f. Let T, be the Cartan subgroup of OMF corresponding to t,. Then it is easy to see that HF =
(1)
TFAF =
XHX‘.
7.3.6. On HF we take the invariant measure dt, da, where dt, is normalized invariant measure on TF and da, is Lebesgue measure corresponding to an orthonormal basis of a,. On H we take the pullback measure corresponding to h H xhx’. It is easily seen that this measure is independent of the choices made in its definition. We fix an invariant measure on G and take the quotient measure, dgH, on G f H. Let @+ be a system of positive roots for @(gc, bC). Let n denote the product of the elements of @+. Let @; be the set of all real roots in @+ (2.3.5). Set b’ = { h E 6 ; a(h) # 0 for a E @}. If h E 6’ then set ~ ( h=) sgn(ll,a(h)) the product taken over @)t. If .f E Y ( g ) we will be using the following notation:
with the domain of 0: equal to the set of all h E 6’ for which the integral converges absolutely. We note that 07 depends on the choice of @+ but only up to a sign. If in @(gC,(bF)C) we choose the positive roots to be { a 0 Ad(x)’ I o! E a+}. Then
(2)
@T(h)= @,H‘(Ad(X)h).
Thus we loose no real generality in studying these integrals if we assume
24 1
7.3. Orbital Integrals on a Reductive Lie Algebra
(as we do) that H = H,.On G / A F we take the quotient measure corresponding to our choice of invariant measure on A,. Then it is clear that
Now Lemma 2.4.1 implies that the invariant measures on K , OM,, NF can be normalized so that
@y(h) = &(h)n(h)
(4)
K
l
f(Ad(kmn)h)dk dmdn.
X u M X N
We now begin our analysis of this formula.
7.3.7. Let h E 6. If n E NF then we set T,(n) = Ad(n)h  h. If n E N, then n = exp X with X E n,. If we expand the exponential series for Ad(n)= ead it is easy to see that T,(n) E nF. The obvious calculation gives (dT,),(X) = Ad(n)CX,h l for n (1)
E
N F ,X
E
n,. This easily implies
If det(ad(h)l,) is nonzero then T, is everywhere regular.
Lemma. I f det(ad(h)l,) is nunzero then T, is a diffeomorphism of NF onto nF. Furthermore there is a choice of Lebesgue measure on 11, such that det(ad(h)l,)
J f'(Ad(n)h h)dn = J f ( X ) d X NF
nF
for (say) f a rapidly decreasing function on nF. If we show that T, is a diffeomorphism of NF onto n, then the integration formula will follow from the above formula for the differential of T,. . a, = exp(th,). Let h, E a, be such that cc(h,) > 0 for c( E @ ( P F , A F ) Set The obvious calculation shows that T,(a,na_,) = Ad(a,)T,(n). Since G is, in particular, regular at 1, there is an open neighborhood of 1 in N F and a neighborhood U , of 0 in nF such that T, is a diffeomorphism of U , onto U,. Now Ut,, Ad(a,)U, = ?tF. Thus the above equivariance implies that T, is surjective. Suppose that T,(n,) = T,(n2).Let t be such that afnja, E U , for j = 1, 2. Then T,(a,nla,) = Ad(a,)T,(n,) = Ad(a,)T,(n,) = q(arn2a,). Thus a,n,a, = a,n,a,. Hence, n, = n2. This completes the proof of the Lemma.
7. Cusp Forms on C
242
7.3.8. We now choose @+ such that if a E @+ and if a1, is nonzero then al, E @(PF,AF).Let C be the set of all a E 0’ whose restriction to aF is nonzero. If h E t, then Idet(ad hln)l = lllaEz a(h)l. It is an easy matter to see that
n
Idet(ad h ( J = &(h)
a@).
U€Z
This combined with the previous Lemma implies that (2) @y(h)= C,&(h)
n
a(h)
aEO+Z
J
KxMxn
f(Ad(k) Ad(m)(h + X ) ) d k d m d n .
If f is a smooth function on g then we set for X E 9.
f ( X )= J f(Ad(k)X)dk K
Since Ad(OM) preserves d X on nF we have (in the above notation)
(3)
@y(h)= CF
If f E Y ( g ) and Q (4)
n
UEUJZ
=
a(h) J f(Ad(m)h OMxn
PF then we set for Z f‘Q’(Z)= J f ( Z
E
+ X ) dm d X .
mF
+X)dX.
nF
If f E Y ( g ) and if h E 6 then write h = h + h,, h E aF and h , set u ( Z ) = u ( f , h _ ) ( Z )= ?(”(h t Z ) for Z E OmF. We have proved: (5)
E tF
then
@/H(h) = C F e  ( h + )
Let OF,“= { a E @ ( ( t ~ ~hC)! ) ~ ((mF)c)a , E p c } . The above calculations imply the following result of HarishChandra Theorem. (1) The integral dejining @/H for f E Y ( g ) converges absolutely for h E f)‘. (2) Set 6” = { h E b a(h) # 0, a E I f f E Y ( g ) then 0: extends to a smooth function on b“. Furthermore, i f f E .Y(g)then @y E Y(6“)and the map is continuous. of Y ( g ) to Y (6”) given by f’ H
This follows from the above material and Theorem 7.3.4.
7.3.9. If X E g then set det(ad X  G I )= X t’Dj(X). Here n = dim 9. Let D = D,(X) with r = dim 6. The preceding theorem has the following corollary which will be important in the next chapter. Corollary. 1D11’2is locally integrable on g.
7.4.
243
Orbital integrals on a Reductive L i e Group
We use the notation of 2.4.3. Then 2.4.3 says that
jf(x)dX
=Ccj
9
j Irj(h)12 j f(Ad(g)h)d(gHj)dhj
hi
GilIJ
in the sense that the right side converges if the left side does. Now ID(h)l = Inj(h)12for h E bj. Let f E Cp(g) be nonnegative. Then the preceding theorem implies that @, E S(( bj)) for j = 1,. . . ,r. Thus cj
a>
=
j
I@f/(h)Idh=
hJ
C cj h’j Inj(h)I
j f(Ad(g)h)dgHjdhj G/Hj
1I D ( X ) 1  ” 2 f ( X ) d X . 9
Since f is an arbitrary smooth, compactly supported, nonnegative function on g the corollary now follows.
7.4.
Orbital integrals on a reductive Lie group
7.4.1. We retain the notation of the previous section. We also assume that G is of inner type. Let K “ = { k E K I det((1  Ad(k)I,) # O}. If b is a Cartan subalgebra of g let H denote thc corresponding Cartan subgroup. Let H ‘ be (as usual) the set of all h E H such that det((1  Ad(h))I,l) is nonzero. Put
G[H’]
= (yhg’
I h E H‘, g E G}.
Then G[H’] is an open subset of G (see the proof of 2.4.4).The proof of 2.4.4 yields
Here dg is a fixed choice of invariant measure on G, we fix an invariant measure on H and we take dgH to be the quotient (Ginvariant) measure on G / H . Also w is the order of the finite group N ( H ) / H where N ( H ) = { g ~ G ; g H g  l =H } . 7.4.2. We now assume the f contains a Cartan subalgebra t of g (under this assumption K ” is not empty). The displayed formula in the previous number easily implies (apply it to both K and G )
7. Cusp Forms on G
244
Lemma. There exists a positive constant, c, such that
J
G[K"]
7.4.3. If
E
f(9)dg = c
J I d W  Ad(k))I,)l jc f(gkg')dydk.
K
> 0 then we set Ge,c
= (9tgl Ildet((Ad(t) 
I)l,l)l
>
&I,
K,
=
{ k E K I (det((Ad(k) I ) l , ) l > E ) and
(G"),,,
=
GCKE1.
Fix a norm (2.A.2.1),11...11 on G, which we assume is given as the operator norm corresponding to a representation (x,F ) of G on a finite dimensional Hilbert space F. We also assume that x(g)* = z(Og)land that det n(g)= 1 for g E G.
2.4.2 says that, up to constants of normalization, i f f is integrable on G then
= K,
If X
Idet((Z Ad(k))1),
J
(I
+ log [ [ a k a ' Il)dZ(aku' ) da dk.
A+
E
p then n ( X ) is selfadjoint. If a E A then a
Ilaka'II
=
llexp H exp(Ad(k)H)II  Ad(k)H)Il/(dimF 1 ) ( 7 . 4 1.1).
IIaka'k'II
>  elldH
= exp
=
=
H with H
E
a.
So
Ilen(H)enAd(k)H II

Thus we see that log Ilaka' Let for k
E
11
2 Ilx(H  Ad(k)H)ll/(dim(F  1).
K , p ( k ) denote the minimum of the absolute values of the
7.4.
Orbital Integrals on a Reductive Lie Group
245
eigenvalues of (Ad(k)  I ). ,1 Then we have shown
(2) There exists a positive constant C such that log Ilaka~'II2 Cp(k)log Ilall. Let pl,. . . , p Z sbe the eigenvalues of ( I  Ad(k))I, counting multiplicity. If we assume that k E K" then we may relabel so that Ipjl = p ( k ) for j = 1,2. It is clear that Ipil i 2 for all i. Hence if k E K , then E
< lpl...p2qlI p(k)'2'4'.
Set C = 2?+'. Then we have proved
If k E K , then p ( k ) 2 CE"'.
(3)
This combined with (2) and the calculations already done implies that
I CdCdi' j Idet((Ad(k)  I)l,)l K
< CdC'Ed'2j+y(a)(l A
+ log
j y(a)(l + log Ila(()dZ(akal)dadk
A
Ilall)d
j a(aka')dkda.
K
Now
j E(aka')dk
K
=
Z(U)E(C') = E(a)2
(see the discussion in 7.1.2). If we put this together with the preceding inequalities the lemma follows.
with domain the set of k
E
K for which the integral converges absolutely.
Lemma. If f E C:(G) then the domain of Qr contains K". Furthermore, Qr E Cm(K"). Set h(g) = If(g)l. Then
s h(gtgl)dg
G
=
s
A+ x K
y(a)h(autu'a')dadu.
7. Cusp Forms on C
246
The argument used to prove 7.4.3(2) proves (3)
If k
E
K , and if a E A then log laka'l 2 &'I2Clog Ilall.
This implies that if u is a compactly supported function on G and if supp u is contained in { g E GI log llgll I r } then there is a constant C > 0 independent of u such that u(auku'a')= 0 for k E K,, U E K and log IJa(J> C&1/2r.Thus the integral converges for k E K " . Let X , , . .. , X,, be a basis of g. If Y E f then Ad(g)Y = C c j ( g ) X jwith each cj a matrix coefficient of (Ad, g ) hence dk/dtkIt = J ( g k exp t Y g  ' )
= (Ad(g)Y ) " f g k g  ' ) =
(1cj(g)Xj)"fgkg')*
There exist constants D > 0 and u 2 0 such that Icj(g)l I D11g11''. Hence Idk/dtkIt=Of(gk~ X tYg')I P I CIIgIIk"
1I u j f ( g k  ' ) I ,
where uj is a basis of Uk(gc). This and the argument we used to prove the first assertion of the present Lemma implies the second. 7.4.5. Fix, O', a system of positive roots for O(gc, tc). We assume that p is Tintegral (this is always possible by going to a covering group of G ) .Set W = Wg,, tc) then
A(t)= tP
fl
( 1  t") =
USID+
for t E T. Set T" = T n K". If f
SE
E
w
det(s)tsP
C,"(G) then we write
FF(t) = A ( t ) f ( g t g  ' ) d g G
for t
E
T".
The previous Lemma implies that F/T E C"(T"). Set T, = Ge,cn T
=
{ t E TllA(t)12> E } . Clearly, T, is a subset of T".
Lemma. Let d be such that
S
(1
+ log Ilall)dZ(a)2y(a)da< 00
A+
(see 4.5.3). Then there exist positive constants C and u such that iff then
(see 7.2.1 f o r r
~ ~ , ~ , ~ ) .
E
CP(G)
7.4.
247
Orbital Integrals on a Reductive Lie Group
= C ~ ~ " ~ a ( f( I)
J
+ log Ilgll)dz(g)dg
Gex
5 Cdo(f)&'d+')'2 y(U)(l k log IlUll)Cdz(a)2da A+
by Lemma 7.4.3. If we take u 7.4.6.
(1)
= (d
+ 1)/2 the Lemma now follows.
If one argues as we did to prove 7.3.3(1) (using 7.A.3.7) one proves F.$(t)
= y(z)FF(t)
for f
E
C;x(G),t E T' and z E Z(g).
We will now use the notation of 7.A.4.2. We label the elements of cD+ as q,. . . , a,. Then the set T' in 7.A.4.2 is our T'. Let B (T' )be as in that number. We can now prove the following basic theorem of HarishChandra. Theorem. (1) I f f E C:(G) then F: E B(T'). (2) The map f H FT of C,"(G) into B ( T ' ) extends to a continuous map of V(G) into B( T). Let I/ = V ( G )and W = C,"(G),S(w) = F,, A = Z(g,), y = y and o = Co. Lemma 7.A.4.2 implies that S ( f ) E B(T') for all f E C,"(G) and S extends to a continuous map of %(G) into B(T').The result now follows.
Note: The original proof of HarishChandra used the theory of the characters of the discrete series (see Varadarajan [l, Part 111 for a nice treatment of HarishChandra's original proof). The key new ingredient in our proof is Lemma 7.4.3. We note that the above theorem is stated in G. Warner [II,8.5.6] but the proof therein is based on a result on discrete series characters that is deferred to "Volume 3".
7.4.7. The next result will be used in Section 7.7.
248
7. Cusp Forms on G
Theorem. Let p be a continuous seminorm on B(T'). Then there exists d 2 0 and a continuous seminorm CJ on W(G) such that
J P ( G ( k g ) f ) d k5 (1 + log Ilgll)d%dCJ(f)
K
for f E W(G) and y
E
G.
If we argue as in the proof of the previous theorem it is enough to show that there exists a continuous seminorm 0 on W(G) and q, d > 0 such that if f E V ( G )then
Thus log llxyll
+ log llyll 2 log IIxII. Hence
We therefore conclude that the expression in (*) is less than or equal to
7.4.8. As in the last section we now study the analogous results for general Cartan subgroups. We assume (for simplicity) that for each Cartan subgroup of G the corresponding p is integral. Let b be a Cartan subalgebra of CJ and let H be the corresponding Cartan subgroup. Set E(h)= for CI E @(g,, $,) h E b (here is the complex conjugate of X E gc relative to 9). Let @+ be a system of positive roots for @(gc,bc) such that if u E @+ and if ii #  u then
x
a(h)
249
7.4. Orbital integrals on a Reductive Lie Croup 2E
@+. Set
X
#  a } . We set
= { a E @+ I a
n
AH(h)= hP
a€@+
(1  h")
z
Clearly, AH(h) = k A ( h ) for h E H (see 7.A.3.6 for A). If f E W(C) then set
The measure on GIH is chosen as in 7.3 and the domain is the set of all h E H for which the integral converges absolutely. As in 7.3.6 we assume that H = HF = TFAFwith ( P F , A F )as standard cuspidal ppair. We also assume that (D(PF,A F )= {aIoI c( E C}.AS in 7.3.7 we have F/H(h)= AH(h)
1
KxoMxN
.f(kmnhn'm'k')dkdmdn.
Define for h E H , Th(n)= K'nhn' for n E NF 7.4.9. Lemma. (1) I  h ( i v F ) is contained in N F . (2) If det((Ad(h')  I ) l n F ) is nonzero then r, is a diffeornorphism of NF onto N F . Furthermore, i f f is integrable on NF then
Idet((Ad(h') 
J .f(h'nhn')dn
N.=
=
J f(n)dn.
NF
The proof of this result is essentially the same as that of Lemma 7.3.7. We leave the details to the reader. 7.4.10.
If f
E
C(G) let
7be as in 7.4.4. We note that if h
E
H then
Thus, as in 7.3.8 (3) we find that (1)
F/H(h) = CFhPAM(h) J
OM x N F
=
CFh'AM(h)
J
f(rnhm'n) dm dn
OM x NF
f(nmhm')drndn
where AM is the "A" for n @(mC,bC). This implies that if we set for a E A,, rn E OMF,u,(m) = f'(ma) (7.2.1) then
(2)
F/H(ta)= FTa(t).
250
7.
Cusp Forms on C
If U is an open subset of H and if g E C W ( U then ) we set for each p r 2 0 qp.r,dg) =S
U P ~ ~ (1 L ~
E
U(bc),
+ log Ilhlll’lPg(h)l.
Let % ( U ) be the space of all g E Cm(U) such that qp,r,a(g)< co for all p , r endowed with the topology given by these seminorms. Set H” = { h E HI ha # 1 for all c1 E @(ntF, bc)}. As in 7.3.8 we now have Theorem. (i) The integral dejining FY, f o r f E CT(G) conoerges absolutely for h E H“ and dejines an element of %‘(HI‘). (ii) Furthermore, f H FY extends to a continuous mapping from W(G) to W (H ’I).
+
7.4.11. If x E G then det(Ad(x)  (1 t)!) = C trdr(x).Let 1 be the rank of gc and set d ( x ) = d,(x). The following result is a basic ingredient in HarishChandra’s proof of the “local ,!,‘theorem” for characters.
Corollary. Idl’/’ is locally integrable on G.
which is finite by the preceding theorem. 7.5.
The orbital integrals of cusp forms
7.5.1. We begin this section with some calculations on SL(2,R). As in most of this chapter, the results are due to HarishChandra. Let L be a connected Lie group locally isomorphic with SL(2,R). We identify the Lie algebra of L with sl(2,R). Set
‘1,
h = [  1O 0
.[I
0
101,
x=[;
;I.
251
7.5. The Orbital Integrals of Cusp Forms
Let T (resp. A', resp. N ) be the connected subgroup of G with Lie algebra Rh (resp. RH, resp. RX). Let A be the Cartan subgroup of G corresponding to a. Set for 0 E R, t(0) = exp(n0h). If f E %(G) then we set F;(t(O)) = FJ(0). Notice that T" = T' = ( t ( 9 ) 0 E R  Z}. A direct calculation using the integral formulas in 7.4.3 (see also 7.4.4) yields a
Ff(8)= 2i sin n0 j sinh(2t)f
(1)
0
Set u
=
InOlcosh 2t. Then we have for nonzero O
with z(0,u)= u + (u2  (n0)2)1'2. The two values (mod 2 2 ) for which we can have (jump) singularities are 9 = 0 or 1. The above formula shows that there is no jump singularity for 0 = 1. We concentrate on the case 0 = 0. (2) implies that 3
lim F,(O)
= 2i
00+
(3)
j f(exp 2 u X ) d u 0
r
lim F,(H)
=
2i
j f(exp  2 u X ) d u 0
60
This implies (4)
lim F,(e)
eo+

lim F,(O)
eo
3c.
=i
j f(exp u X ) d u .
3)
If we differentiate formula (2) then we have = (0(nOcos
(')Ff(%)
710  sin n o ) / n e 2 sin no)FJ(e)
 2ni(sin nO/nO)f(exp
n0h) + E ( 0 )
with lime+0E ( 8 ) = 0. We therefore have
We now interpret (4) and (5) in terms of orbital integrals on A . We set
H,(t) = F;(exp t H ) . Then 7.4.10(1)says (in this case) that
(6)
HJ(t) = e'
x,
1 f(exp t H exp x X ) d x . x,
252
Cusp Forms on G
7.
We therefore conclude that lirn F,(B)
(7)

0o+
lirn F’(0) = i lirn H,(t). 80
The definition of F’ implies that F,(a)
t+O
= F’(a’).
Thus
lirn (d/dt)H,(t) = 0. IrO
This implies (8)
eo+ lim (&)F’(B)

/ly (&)Ff(B)
= 0 = lirn 10 (%>.(t).
Now let C E Z(1,) be such that yT(C)=  d 2 / d B 2 and yA(C)= d 2 / d t Z . Here yJ is the HarishChandra isomorphism associated with the Cartan subgroup J. C is (up to scalar multiple and subtraction of a scalar) the Casimir operator of 1.7.4.6( 1) and its analogue for F A combined with (7) and (8) imply that eo+
oo
($)iFI(B)
= (i)k+’
lirn t0
It is this formula that we will use in the rest of this section. 7.5.2. U p to now we have been assuming that G = OG. We have made this assumption in order to simplify the statements of the main results. We now assume (only) that G is of inner type. Let A be a split component of G. If H is a Cartan subgroup of G then A is contained in H . The formula for FfH(h) is meaningful for h E H. If f E U ( G ) and if a E A then R(a)f = u E %?(‘G). Furthermore, FfH(ha) = F f n o G ( h )for h E H n OG. This device allows us to transfer our results in the case of G = OG to the more general situation. We can now state the main result of this section. The rest of the section will be devoted to its proof. In the course of the proof several results will be proved that are theorems in their own right (for example the formula generalizing (9) above).
loG
Theorem. Let f E U ( G ) and assume that FfH = 0 for every Cartan subgroup of G that is not fundamental. I f H is fundamental then FfH extends t o a smooth function on H.
We will in the course of this proof use orbital integrals for several different real reductive groups. If L is a reductive group and if J is a Cartan subgroup of L then we set LF: for the corresponding “FY.This will keep track of the group over which the integration has taken place.
7.5.
The Orbital Integrals of Cusp Forms
253
We prove this result by induction on the dimension of G. If dim G = 0 or 1 then G = H is the only Cartan subgroup and FfH = f . So this case is trivial. We now assume that the result has been proved for all reductive groups, L, of inner type with 0 I dim L < dim G. If G is not equal to OG then dim OG < dim G. Let A be a split component of G. If J is a Cartan subalgebra of OG then J A is a Cartan subalgebra of G and every Cartan subalgebra of G is of this form. Thus the discussion at the beginning of this number combined with the inductive hypothesis implies the theorem in this case. We may therefore assume that G = 'G.
7.5.3. Now suppose that H is a noncompact fundamental Cartan subgroup of G. Then we may assume that H = HF and PF is proper. Set Q = PF, L = OMF and T = TF.If J is a Cartan subgroup of L then J A is a Cartan subgroup of G and (7.4.10(2))
Thus the inductive hypothesis prevails. We are thus left with the case when G contains a compact Cartan subgroup, T. We return to the notation of the parts of 7.4 preceding 7.4.8. Let Qn = { a E 0 ' I (gc), c pc}. Let OX denote the complex conjugate of X E gc with respect to 9.Then o(gc), = (gc),. Let a E On,let Z E (gc), and W = 02. If Z is nonzero then Z W is a nonzero element of p (not just pc). We may normalize 2 so that a ( [ Z ,W ] )= 2. Set H = Z W, h =  i [ Z , W ] and X = ($)([ZW , ] + i(Z  W ) ) .Then one checks that H , h, X have the same commutation relations as the elements with the same designation in 7.5.1. Let I" = R H R X t. Then [I",1"] is isomorphic with s@,R). We can thus use the calculations of the previous number. Let T, = { t E Tit" = l}. Set T:, = ( t E T, I t S # 1 for p E Q+  { w } } . Then T:, exp(Rh) is open in T. Let L" be the connected subgroup of G with Lie algebra I". Then T, is in the center of L". Set k,(8) = exp 8nh. If t = uk,(8) E T" and if f E C,(G) then
+
+
+
We set A,(t) = tP'/'
(3)
+
II
(1

L  ~ ) Then .
A(uk"(0)) = 2iA,(uk,(B)) sin(n8).
Set R f ( g ,u, 0) = A,(uk,(O)) sin(n8) ff(guxk,(B)x'g') dx. L
7.
254
Cusp Forms on G
Then
Let f E CF(G).Let u E Th. Fix p E U(t&). We note that if 161 is sufficiently small and positive then uk,(8) E T'.We calculate ( phk)F/(uka(o)) =
7(;)(
(i $YJpA,(uk,(O))( i $Isin rc8 G / L R,(g, u, 8)dgL".
Let J be the centralizer in G of T, exp RH. Then 7.5.1(9) implies that (up to a multiplicative constant) (4)
lim phkFfT(uka(0)) lim phkFfT(uka(0))
e+o+
e+o
Since both sides of (3) are continuous on %?(G),(3) holds for f E %?(G). This is the jump condition we mentioned at the beginning of the proof of the theorem. The above formula implies that if FfH = 0 for all nonfundamental Cartan subgroups, H, of G then F, is smooth in a neighborhood of each t E Ta, a E 0;. Suppose that a E 0:.Let I" = g n (tc (gc), (gc),). Let La be the connected subgroup of G corresponding to I". Then La is compact. We may now argue as above and see that Lemma 7.4.4 implies that there are no "jumps" in this case. We have therefore shown that if FfH = 0 for all nonfundamental Cartan subgroups of G then FfT is smooth in a neighborhood of each t E T such that ta = 1 for at most one a E 0+. The theorem now follows from 7.A.4.3.
+
+
7.5.4. Corollary. Let f E %(G) be a cusp form. I f H is a Cartan subgroup of G that is not compact modulo the center of G then FfH = 0. I f H is compact modulo the center of G then FfH extends to a smooth function on H .
The first assertion follows from the definition of cusp form (7.2.2) and 7.4.10(2). The second is a consequence of the preceding theorem.
In the next section we will derive some consequences of this result. 7.6. Harmonic analysis on the space of cusp forms Let G be a real reductive group of inner type such that G = OG. We will use the notation of 7.A.2. Thus, we look upon S(g,) as the algebra of differential operators with constant coefficients on 9.Fix 8, K , etc. as in the previous sections. Let q = dim f and p = dim p .
7.6.1.
7.6.
255
Harmonic Analysis on the Space o f Cusp Forms
Let '%(G) denote the space of all cusp forms on G (7.2.2). If G has a compact Cartan subgroup, T and if a+is a system of positive roots for @(gc,t,) then we set
n
w =
H,
E
V(tc).
a€@+
Theorem. If G has no compact Cartan subgroups then 'W(G) = (0). If T is a compact Cartan subgroup of G then there is a nonzero constant C, such that if f E '%'(G) then mF,T(l) = C,f(l). (Notice that Ff E C"(T) by 7.5.2).
This result is a special case of a much more general theorem of HarishChandra which asserts a similar limit formula for any f E%?(G)with T replaced by a fundamental Cartan subgroup (c.f. Varadarajan [l; 11, p.2201 for an exposition of HarishChandra's original proof). We will only need the above statement which is much easier to prove. As usual, the proof takes some preparation. There is however, one case where the result has already been proved. Assume that all of the Cartan subgroups of G are one dimensional. Then it is easily checked that either G is one dimensional, g = sl(2,R) or g = su(2). In the first case the result is obvious. In the second case it is a restatement of 7.5.1(5). In the last case G is compact and the result is a consequence of the PeterWeyl theorem and the Schur orthogonality relations (we leave this as an exercise to the reader). We will thus assume that the Cartan subgroups of G are at least two dimensional. We also note that we can replace G by T' x G and extend f E % ( G )to T1 x G by f ( t , g ) = f(g). This will not change the statement of the theorem but the Cartan subgroups will all have dimension at least 2.
7.6.2. We use a pseudoorthonormal basis of g relative to B to identify g with R". We set P ( X ) = B ( X , X ) . Then the L of 7.A.5.1 is the o of 7.A.2.8. We set F = F , = FPTq(7.A.5.8).Then F(Ad(g)X)= F(X) for g E G and X E g. We will also use the notation in 2.4.3. For each j let rcj be as in 7.A.2.9 for bj. Then ID(h)l = Ixj(h)12for h E bj. Thus if we apply 7.6.1 and 2.4.3 we have for
f E %I)
256
7.
Cusp Forms on G
Here we are using the notation of 7.3.6. Let Gj E S ( g ) be as in 7.A.2.9. Then Theorem 7.A.2.9 implies that f(0)
=
C cj f j
17tj(h)l&j(h)F(h)G~’21@~(h)dh.
hi
7.6.3. Let Dj be as in 7.3.9. Let r be the rank of gc (recall that we are assuming that r 2 2) and let n = dim g. For each t > 0 we set R, = {X E g lDj(X)(< t, r I j < n}. x in this number will denote 3.14.. . .
Lemma. Suppose that G is semisimple. I f 0 < t < x  1 then exp restricted to d,is a difeomorphism. Lemma 7.A.1.4 implies that X H e a d Xdefines a diffeomorphism of R, onto an open neighborhood of I in Int(g). Now, Ad(exp X ) = e a d X Since, . Ad is a covering homomorphism, the Lemma follows. 7.6.4. Let W be an open neighborhood of 0 in j(g) such that exp restricted to W is a diffeomorphism. Let R, be as above in [g, g]. We set W, = W 0 R,. Then
(1) If 0 < t < x  1 then exp is a diffeomorphism from W, onto an open neighborhood, V, of 1 in G. Let u E C?(R), 0 I u(s) 5 1 be such that u(s) = 1 for s I (x  1)/2 and u(s) = 0 for s > 2(x  1)/3. Let 0 E Y c CI(Y) c W with Y open and CI(Y) compact. Let h E Cp( W )with h ( X ) = 1 for X E C1(Y). We define a function, p, on gas follows: If X E W, I and if X = Z + T with Z E W and T ER n _ then P ( X ) = h ( Z ) n , , j , n  u(Dj(T))otherwise P is 0. Then
(2) (3) (4) If
P E Cm(g),SUPP B P(Ad(g)X)= P ( X )
c
for X
Wn 1. E
g and g E G.
b is a Cartan subalgebra of g then supp b n I) is compact.
The last assertion follows from 7.A.1.3. We now introduce a function a on G that will be used later. If X E W n p then set a(exp X ) = b ( X )otherwise a = 0. Then a is a smooth function on G and ct(gxg’) = a(x) for x, g E G. 7.6.5. If f E Cm(G) then set f ” ( X ) = b(X)f(exp X) for X f (O) = f(1). Let H be a Cartan subgroup of G. Then
E g.
Clearly,
7.6.
257
Harmonic Analysis on the Space o f Cusp Forms
We note that A,(exp h ) / z ( h ) is nonzero for h E W, n b. Thus n(h)/A,,(exp h) defines a smooth function on W, I n b. Since the map f H FY extends to a continuous map of V ( G ) into %(H") (7.4.10)we have (2) The map f
H
@)s" extends
to a continuous map of W ( G ) into CF(6").
7.6.6. We now begin the proof of Theorem 7.6.1. The material in the previous number combined with the results in 7.6.2 imply that if f E V(G) then
We note that the above integrals are over compact sets. Now (1) implies that if f E '%( G ) and if G contains no compact Cartan subgroup (recall that we are assuming that G = OG) then f(1) = 0 (7.5.4).Now, if f E O%(G) then R ( g ) f E O%(G). Thus, if G contains no compact Cartan subgroups then O%(G) = ( 0 ) .This proves the first part of the theorem. We now begin the proof of the second part of the theorem. Recall that r > 1. 7.6.7. that
We assume that H ,
=
T is compact. I f f E OV(C)then 7.6.6(1)implies
Recall that FfT E P ( T )(Theorem 7.5.2).The following result is one of the keys to our proof. Lemma.
There exists a nonzero constant M , such that
j F(h).(h)6["'2'g(h)
if
g
E
Y(t) then
dh = M,g(O).
t
( M g will be, essentially, computed in the course of the proof.)
If p E P(tc)we look upon p as a differential operator of order 0 on t. We will use the following commutation identities are easily proved by induction. (2) Let X , Y be endomorphisms of a vector space then (  l)kj"XJ((ad X)kiY) = j=O
7.
258
Cusp Forms on G
We now prove the lemma.
S F(k)n(h)G["'21g(k)dh = S F(h)G["/']n(h)g(k)dh t
t
J t
F(k)[d"'21, n]g(h)dh = I

11.
Now our assumption that T is a Cartan subgroup of G implies that p (= dim p) is even and q  dim t is even ( q = dim €).Thus Theorem 7.A.5.8 implies that there is a nonzero constant B, such that I = ~ , ( ~ [ n / 2 1  [ r / 2 1)(ns)(O)*
(3)
We now compute 11. We first note that since n  r is even ( n  r ) / 2 = [n/2]  [ r / 2 ] . We apply (2).I1 =
Now degn = [n/2]  [ r / 2 ] .Thus (ad is)t"'21jn = 0 if j < [ r / 2 ] . Hence we may again apply 7.A.5.8 and find that I1 =
We now apply the second formula in (2)to the "ad" terms. We observe that the coefficients of adt"'21jn vanish at 0 for j > [ r / 2 ] (see Scholium 7.A.2.9).Thus (as the reader should check) if j 2 [ r / 2 ] then wj
[r/21(ad~
" 2 1 j n )
.g(o) = (ad
,jjtn/21[r/21
n) * do).
We combine this with the above formulas for I and I1 and we have
S F(h)n(k)i$"21g(k)dh = C((ad G["121[r/21 n)s)(O) t
with
Now Scholium 7.A.2.9 implies that
Since
the lemma follows.
7.7. Square Integrable RepresentationsRevisited
259
7.6.8. We will also use Lemma. Let W = W(g,, t,). Let u be a Winvariant smooth function dejined on a Winvariant neighborhood of 0. Then
Let a be a simple root in @+. If F c @+ then we set F" = s,F if a is not in F and F" = (s,(F  { a } ) )u { u } otherwise. Then F H F  is a bijection of the set of subsets of 0'. Let p denote the left hand side of the formula that we are proving. Then sap =
(Here we have used xu(0) = (sx)u(O)for x E S(t,).) Thus sp = det(s)p for s E W. This implies that p = q naccD+ Hawith q E S(t,). A comparison of degrees shows that q is constant. If we compare homogeneous terms we see that q = u(0).
7.6.9. If we apply 7.6.7(1) and Lemma 7.6.7 we have (1)
n
f ( 1 ) = c 1 4 ( a t @ + Ha),((x/A(exp .))PF/T(exp' ) l h = O .
Now, /3 is identically equal to 1 in a neighborhood of 0. Set u(h) = x(h)/A(exp h). Then u E Cm(W, l ) w and u(0) = 1 (see the proof of the Weyl dimension theorem). In light of the preceding Lemma we have completed the proof of the theorem.
7.7. Square integrable representations revisited 7.7.1. We continue to assume that G = 'G. Let cF2(G) denote the set of equivalence classes of irreducible square integrable representations of G (1.3). If CT E E 2 ( G ) then fix ( x u , H u )E 0. If u, w E then the matrix coefficient c,,,(g) = (xu(g)u,w) is an element of U ( G )(Theorem 5.5.4) which is also Z(g)finite. Thus c",, E ' C ( C ) (7.2.2). Theorem 7.6.1 combined with Theorem 6.8.3 implies the following deep theorem of HarishChandra [13].
Theorem. & ( G ) is nonempty if and only if G has a compact Cartan subgroup.
260
7. Cusp Forms on C
7.7.2. In light of the above result we assume that T is a compact Cartan subgroup of G. As in 6.9.1, we write T = ZTO. If p E T" let denote the character of p and d ( p )the dimension of p. p restricted to T o is d ( p )times a character, A(p) of TO. If f E O%(G) then F, E Cm(T)(7.6.3). Thus the PeterWeyl theorem implies (1.4.5, 1.4.7)that
c,,
Here if h E C"(T) is Tcentral then
The second part of Theorem 7.6.1 implies that there is a nonzero constant CGsuch that
If z
E Z ( g c ) then
we have seen that Fzs
(3)
= y(z)F'
on T', hence on T.
so (K,)" (PI = NP)(Y(Z)j(F,j" (PI.
(4)
Now if o E G?~(G)and if f = ,c, u, w E (H,,), then zf = x,(z) f with xu the infinitesimal character of o.Thus if we put all of this material together we have proved another Theorem of HarishChandra. Theorem. Let o E C Y ~ ( G then ) there exists p E T" such that (A(p),a) is nonzero for all a E CD(g,, tc) and such that the injinitesimal character of o is X,,(,,).
7.7.3. The above theorem has an important corollary (as usual, due to HarishChandra). Corollary. Let y nonzero is ,finite.
E
K" then the number of
CT
E
&(G) such that (Ha)&) is
Let C be the Casimir operator corresponding to B. Let C, be the Casimir operator for K corresponding to B restricted to f. Let XI, ..., X , be an orthonormal basis of p relative to B. Set C, = X ( X j ) 2 .Then C = C, C , . Fix y E K " and let p7 be the eigenvalue of C, on any representative of y. We note that
+
(1) If (n,H ) is a unitary representation of G with C acting by c l and if H,(y) is nonzero then c Ipy.
261
7.7. Square Integrable Representations Revisited
Indeed, if u E HK(y)is a unit vector then c
= c(u, u ) =
(cu,0 )
=
(CKU, 0 )
+ (C,u,
0)
= py 
(xju, xju) 5
pLr.
If ~ E & ~ ( Gthen ) let A, denote an element of (TO)" that gives the infinitesimal character as in the preceding theorem. Let p be the half sum of a choice of positive roots. Then x,(C) = ~ ~ A , ~IIp1(2. ~ z Hence
ll~,IlZ 5 llPllZ + PLy. We have fixed y. The A, "wander over" the lattice (TO)",thus the above inequality implies that there are only a finite number of possibilities for infinitesimal characters of square integrable representations whose yisotypic component is not zero. Since there are only a finite number of isomorphism classes of irreducible (g, K)modules with a fixed infinitesimal character (5.5.6), the result follows from 3.4.1 1. 7.7.4. We also record the following implication of the main theorem of this chapter.
Proposition. Let G = OG. If (n,H ) is an irreducible tempered representation with infinitesimal character x,, with A E (tc.)* such that (A,@)E R  {O} f o r 2 E @( gc, tc) then n is square integrable. 5.2.5 implies that there exists a standard ppair, ( P F , A F )with , PF = P =
OMAN a standard Langlands decomposition such that (n,H ) is equivalent with 0 E &(OM) and p E a*. We may assume that TF = T which is with lp.o.l,l contained in OM is a Cartan subgroup of OM. Set Ij = t F + a. Then relative to bc, IP,o,iahas infinitesimal character given by A, ip with A. an element of (TF)"that gives the infinitesimal character of 0 . Our hypothesis implies that A, ip is real valued on a. Hence p = 0. If P is proper G then b must have a real root, c1 (2.3.5). If a E @(gc, tc) corresponds to c1 then (A,a) = 0. This contradicts our hypothesis on A. Hence P = G and the result follows.
+
+
7.7.5.
We now introduce a construction that will be useful in later chapters. Let (as usual), R and L denote respectively the right and left regular representation of G and U ( g ) on C x ( G ) .Let A ( G ) denote the space of all smooth right and left Kfinite functions, .1; on G such that dim Z ( g c ) i < m.
Lemma. Let f E A ( G ) then there exists an admissible Hilbert representation (n,H ) of G and u, w E HK such that f = c , , . ~(recall , that c,,.,(y)= ( n ( g ) u , w)).
262
7. Cusp Forms on G
Let V = U(g,) span{R(K)f). Then V is an admissible finitely generated (9,K)module under the obvious actions (3.4.7). Let
w = L(U(9,)) span{L(K)f}. If
fl E
V and if f 2 E W then
fi = 1am,nR(xrn)R(kn)f and fz = X b,,,L(u,)L(y,)fwith x,, y,
E
U ( g )and k,, us E K . We assert that
Carn,nbr,s(R(xm)R(ks)L(u,)Lof(') is independent of the expressions for fl and f 2 .Indeed, the formula in question is
(1 br,sL(ur)L(ys)fl)(l) which clearly only depends on fi. Also it is
(1arn,nR(xrn)R(kn)f,)(l) which only depends on f 2 . We set the value equal to sesquilinear pairing of V with W. If k E K then (R(k)f,, f 2 ) =
(fl,f2).
This defines a
1 urn,n(R(k)R(xrn)L(k")fi)(l)= 1arn.n(L(k ')R(xrn)R(kn)LN1) =
(fl,L ( k  ' ) f z ) .
Similarly, if X E g then (R(X)fl,fz) = (fi,L(X)fz). Thus, ( , ) is a (9,K)invariant pairing of V and W. Suppose that h E V and that (h, W ) = 0. Then, (L(k)R(x)h)(l)= 0 for all k E K and x E U(g).Now, h is real analytic on G (see the material in 3.4.9) and G = KG'. Hence, h = 0. Similarly, if g E W and if ( V , g ) = 0 then g = 0. We have thus proved that the pairing of V and W is nondegenerate. Let (n,H ) be a realization of I/ as a Hilbert representation of G. Let (n*,H ) be the conjugate dual representation of G (1.1.4). Let V be the space of K finite vectors of (n*,H).Then the above results imply that there is a (9,K ) module isomorphism, T, of V onto W such that if u E V and if u E V then ( u , u ) = (u, Tu). Let w E V be such that Tw = f. Then we assert that (n(g)u,w) = f(g) for g E G . Indeed, the left hand side is a real analytic function, u, on G with R(x)R(k)u(l)= (n(x)n(k)u, w) = (R(x)R(k)u,w ) = R(x)R(k)f(l)for x E U ( g ) and k E K . Thus u = f.This completes the proof.
263
7.7. Square Integrable Representations Revisited
7.7.6. We are now ready to prove a result of HarishChandra that is one of the essential ingredients of his proof of the Plancherel theorem. Theorem. Let f E '%(G). I f f is right Kfinite then dim Z(g)f < co.
If h E '%(G) is right Kfinite and if p Then (1)
(2) (3) (4)
E
T" then set T ( p ) h ( g )= (F&,,/)"(p).
T ( p ) ( z h )= &4(Y(Z))T(Cl)h
T(P)(R(S)h)= R ( g ) T ( p ) h
for z E Z(g,).
for 9 E G.
T ( p ) h E C"(G). There exists a continuous seminorm, q, on %(G) and d such that
I T(p)h(g)l 5 q ( M 1 + log I1911)d=(s)
for 9 E G.
(2)is obvious. (1) and (3) have already been observed in 7.7.2. (4)follows from Theorem 7.4.7. For the moment, fix y E K " . Set u = L(a,)T(p)f (1.4.6). Then u is left and right Kfinite. We set V = U ( g c ) span{R(K)u}.Then (l), (4) and the previous Lemma imply that V is an admissible, finitely generated, tempered (9,K)module. We now assume that ( A ( p ) ,a) is nonzero for all cc E @(gc,tc). Then Proposition 7.7.4 implies that every irreducible constituent of V is square integrable. 5.1.3 implies that V splits into a direct sum of irreducible square integrable (g,K)modules. Let S ( y ) be the set of equivalence classes of the constituents of V. Let F be the (finite) set of Ktypes of span{R(K)f}. If w E S(y) then there exists o E F such that Horn,( V,, H,) is nonzero. Set R = {oE g2(G)IHom,(V,,H,) is nonzero for some c E F ) . Then R is a finite set by Theorem 7.7.3. Clearly, S ( y ) is a subset of R. Let Z = { p E tz xp = x for some p E R}. Put r = { p E T" I A ( p ) E C}. Then r is a finite set. We have shown that if T ( p ) f is nonzero then p E r. 7.7.2(2) implies that
c n (NPLXa)T(P)f(s)
f(s)= C G p e r
UEO+
for g E G. The theorem now follows from (1). 7.7.7. It will be shown, in the next chapter, that the span of the functions u in the course of the above proof is the span of the matrix coefficients of the discrete series representation of G corresponding to R (see 6.9.5).
264
7. Cusp Forms on G
7.8. Notes and further results 7.8.1. We first expand a bit on the material in Section 7.1. Let a and b be as in that 7.1. If we drop conditions 7.1.1(3) and (6) then the space Y&(G) is still a Frechet space (the seminorms and the topology defined in exactly the same way as in 7.1.1). Furthermore, it can be shown (without difficulty) that Theorem. Ya,b(G)is a smooth representation of G x G under the left and right regular representation. In addition to the two examples of 7.1 one now has the spaces %?”(G)which are given by a ( g ) = 1 + log llgll and b(g) = Z(g)2’pfor 0 < p < a.Clearly, V 2 ( G )= W(G). The spaces V”(G) are usually called the LPSchwartz spaces. 7.8.2. We now look at the material in Section 7.2. The transform ‘f was originally introduced by HarishChandra in his work on spherical functions (HarishChandra [9, p.5951). In this chapter, we have used this transform basically to reduce calculations of orbital integrals to the case of a compact Cartan subgroup. In the next chapter, we will see that we can calculate the character of a representation induced from P in terms of .f ‘. This will also give a better understanding to our (unmotivated) definition of cusp form.
7.8.3. As we have seen, Theorem 7.6.1 is a powerful tool in the analysis of cusp forms. We have also pointed out that this result is a special case of a more general theorem of HarishChandra, which we now state. Theorem (HarishChandra [13, Lemma 38, p.471). There exists a nonzero constant C, such that if H is a fundamental Cartan subgroup of G and if f E %?(G)then lim mF’(h)
h+ I
=
CC,f(l)
with C a nonzero constant depending only on the choice of invariant measure in H .
This theorem plays a basic role in the proof of HarishChandra’s Plancherel theorem. A full discussion will appear in Volume I1 of this opus. Although Lemma 7.7.4 is not difficult,the result will play an important role in our discussion of HarishChandra’s “philosophy of the constant term”
7.8.4.
7.A.l.
Some Linear Algebra
265
(also to appear in Volume IT), since it allows us to transfer the results of Chapter 4 from matrix entries to elements of A ( G ) . 7.A.
Appendices to Chapter 7
7.A.1. Some linear algebra 7.A.l.l. We put the usual inner product ( , ) on C" with corresponding norm 11. . 1 1. On End(C") we put the operator norm. The following result is based on an ingenious trick of Thompson [11.The use of the spectral radius in the proof was suggested by Roger Nussbaum. Lemma. Let X , Y E End(C") be such that X*
=X
and Y*
=
(1)
lleXeYll2 IleX+YII.
(2)
If furthermore tr X = tr Y = 0 and n 2 2 then Yll/(n  1). log lleXeYll2 IIX
Y. Then
+
If X E End(C") then set r ( X ) = lim sup IIXkll"k (Here the limit is as k 4 +m. Also it is well known that we may replace lim sup by lim.). Then r ( X ) I llXll and if g E GL(n,C)then r(gXg') = r(X). We will also use the fact that Il(XX*)kll = IIXI12k. In particular if p E GL(n,C) is selfadjoint and positive definite then r ( p ) = llpll. Let a, b be selfadjoint and positive definite matrices. Ilab((2k Ill(a2b2)kll
(i)
for k
=
1, 2,. . . .
Indeed, (lab)12k= Il((ab)(ba))kll = r(((ab)(ba))k) = r(ab2a2. * .b2a) = r((a2b2)k) Il
l(~~b~)~ll.
Since I(Xkl(I IIXllk,(i) implies (ii)
Il(ab)2kll 5 Il(a b 2
2 k
1 11.
This in turn implies that (iii)
Il(ab)2klI 5 lla
2kb2k
II.
If we apply (iii) to a = e ~ p ( X / 2 ~ )b, = e ~ p ( Y / 2 ~then ) we have If we now take the limit as k + a,(1) Il((e~p(X/2~) e ~ p ( Y / 2 ~ ) ) ~I *IleXeYII. ll follows.
266
7. Cusp Forms on G
To prove (2),it is enough to show that if X is selfadjoint, tr X then
= 0 and
n>1
Let p l , . . .,p,,be the eigenvalues of X, counting multiplicity, labeled such that Ipll 2 (pjl for all j . Let p be the largest eigenvalue of X . llXll = (pl( and lleXll = e'. Thus we must show that p 2 Ipll/(n  1). If pl 2 0 then p1 = p and the assertion is clear. If pl < 0 then lpll = pl = p 2 + ... p,,I (n  l)p, as asserted.
+
7.A.1.2. The other results from linear algebra that we will need in this chapter are of a different nature. Let U = { X E M,,(C)I if p is an eigenvalue of X then IIm pI < n}. It is clear that U is an open subset of M,,(C). Lemma.
The exp is a diffeomorphism of U onto an open subset of GL(n,C ) .
As is well known d exp, Y = ex((I  e  a d X)/ad X)Y. The eigenvalues of ( I  eadx)/adX are the numbers (1  e " ) / p with ,n of the form a  y and a, y are eigenvalues of X (here ( I  e Z ) / z=  1 if z = 0). Since, (1  e z ) / z = 0 if and only if z = 2nik with k a nonzero integer, it follows that exp is everywhere regular on U. Thus to prove the lemma, we need only show that exp is injective on U. If X E End(C") then X can be written uniquely in the form X = X, + X,, with [X,, X,,] = 0 and X , diagonalizable, X,, nilpotent. If g E GL(n,C ) then g can be written uniquely in the form gsguwhere g, commutes with gu and gsis diagonalizable and g.  I is nilpotent. Suppose that X, Y E U and that e x = e y . Then we must have exp(X,) = exp(Y,) and exp(X,,) = exp(Y,). But then X,, = We may thus assume that X and Y are diagonalizable. If (T, y are eigenvalues for X and if ea = ey then a  y = 2nik with k an integer. Thus since X E U this implies that a = y. This implies that the e" eigenspace for e x is the c eigenspace for X. If we apply this observation to Y we see that since e x = e y , X = Y.
x.
7.A.1.3. Let oj denote the jth symmetric function on C".We set a, k > n or k < 0. Recall that (Xj
=0
if
+ t ) = 1 t"'aj(x,, . .. , x,,).
Lemma. If C > 0 and if loj(x,,..., x,)l C + 1 f o r j = 1, ..., n.
sC
for j
=
1,..., n then lxjl I
7.A.1.
267
Some Linear Algebra
The definition of the aj implies that = Oj(X1,.
Oj(X)
. . , x,,)
+ xnajl(Xl,.
. . , x,
1).
After relabeling (if necessary) we may assume that Ix,I 2 Ixjl. We may also (clearly) assume that x, is not equal to 0. (1)
(c+ I a k ( x l , . * . ,
\ a k  l ( x l > . . . ? xnl)l
Indeed, C 2
IGk(x)I
=
Iak(xl,...,
2
xnl)~)/~xn~~
xn1)
+
xn@kl(xl,...,
IXnlla~l(X1....,XnI)I
a,,(x)= X l " ' X , ,
 lak(xl,..., x n  l ) l .
so
I an  1(x 1 > . . ., xn
(2)
xnl)l

1)
II C / IxnI.
An easy argument using induction (1) and (2) shows that
In particular, (3) implies that la,(x,
+ ... + x n  J
+ + C ( l + . . . + ~ x , , ~ ~ ~ ~ )Hence /~x,,~~~.
I C(1
" '
IXln2)/IXnlnl.
+ ... + x,I 2 Ix,I (x,(~ I C(l + . . . + Ixnln') I C ( ~ X , ~ ~ ~ ~ ) / 1) ( \ X , ,if~Ix,I
Thus C 2 Ixl


> 1.
This implies that if Ix,I > 1 then Ix,J"+l  Jx,~" C(x,l". SO Ix,l"+l S (C + l)Ix,,ln. Hence (x,( I C + 1 if I x , ~ > 1. If Ix,I I 1 then it is clear that Ix,I I C + 1. This completes the proof. 7.A.1.4.
If X
E
End(C") then define the polynomials D j ( X )by det(tI  X ) =
tj(
I)njDnj(X).
If X has eigenvalues pl, ..., p,, counting multiplicity then it is easy to see that D j ( X ) = aj(pl,.. .,pn). The preceding lemma now implies that if lDj(X)l < n:  1 for j = 1,. , . , n then X E U (7.A.1.2). Thus Lemma 7.A.1.2 implies Lemma. Set V, = { X E End(C") I lDj(X)l < r ) . If r difeomorphism of V, onto an open subset of GL(n,C ) .
n:  1 then exp is a
268
7.A.2.
7.
Cusp Forms on G
Radial components on the Lie algebra
7.A.2.1. The discussion of radial components in this appendix is based upon the results in HarishChandra [ 6 ] . Let G be a Lie group with Lie algebra g. Set L = G x g which we look upon as a Lie group with multiplication given as foll0ws (1)
(x,X)(y, Y)
= (xy, Ad(y’)X
+Y)
for x, y E G, X, Y E g.
The Lie algebra, 1, of L is g x g with bracket given by
(2)
[ ( X , Y), ( X ’ , Y’)l
= (CX, X’I,
[Y, X’l
+ [ X , Y’I).

L acts on g by (9, X ) Y = Ad(y)(Y + X ) . This makes g into an Lspace. Let DO(g) be the algebra of all differential operators on g with smooth coefficients. If X E I then set T ( X ) f (Y ) = d/dtf(exp( t X ) Y) It=,, for f E Cm(g).Then T is a Lie algebra homomorphism of 1 into D O ( g ) . Hence T extends to an algebra homorphism of U(1,) into DO(g). If ( X , Y) E I then T ( X , Y ) is a smooth vector field on g which we can look upon as a smooth function from g to g. We leave it to the reader to check that T ( X ,Y ) v = [v,X ]  Y.
(3)
In I, g x 0 is a Lie subalgebra isomorphic with g and 0 x g is a Lie subalgebra with 0 bracket operation. Thus (4)
U(1,)
=
U ( g c )@ S(g,) with a complicated multiplication.
l.A.2.2. Let XI,. . . ,X,, be a basis of g and let xl,. . .,x, be the corresponding coordinates on g. If D E D O ( g ) then Here we use the standard multiindex notation. If I = ( i l , . . . , in) with ij a nonnegative integer then 111 = C ijand 8’ = al’l/xy . . . xk. If X E g then we set Then D, is a constant coefficient differential operator on g. Clearly, T (1 @ S(g,)) is the algebra of all constant coefficient differential operators on g. We will thus identify S(g,) with the algebra of constant coefficient differential operators on g. It is convenient to introduce a slight twist on T. We define R(x 0 y ) = T (1 @ y ) T ( x 0 1)for x E U ( g , ) and y E S( gc). If Y E g and if u E U(1,) then we define R y ( u )= R ( U )E~S(gc).
7.A.2.
269
Radial Components on the Lie Algebra
7.A.2.3. Now let E) be a Lie subalgebra of g such that there is an ad(4)invariant complementary subspace, V, of E) in g. We also assume that f)’ = { H E 6 det(ad H 1“) # 0) is nonempty. Set V = symm(S(V,))in U(g,). We filter U(1,) as usual. This filtration induces a filtration of V@S(E),) with (V 0S(6,))j =
C
symm(SP(l/,)) 0Sq(bc).
p+qsj
We filter S(g,) using the filtration associated to the gradation by homogeneous degree. We denote this filtration by Sj(gc).
Lemma. If H E E) then R,((”1@ S(E),))j) is contained in Sj(gc). The map H H R, restricted to (V 0S(t),.))’ is a polynomial mapping from t, to L ( ( V @ S(1lC))j,Sj(g,). I f H E 1)’ then R, is a bijection from (V 0S ( 6 , ) ) j to Sj(gc).
We prove this by induction on j. If j = 0 then the result is obvious. Assume the result for j  1 2 0. Let H l , . .., H, be a basis for b and assume that X , , . . . , X , is a basis for I/. If p + q = j then R,(symm(X,, . . . X i r )0Hi, . . . Hi,)
= (
l)pfq[H,Xi,]. .. [ H , Xir]Hjl ... Hjr mod S j  ,(gC).
The proof of the inductive step is now clear. 7.A.2.4. Set r, = R, restricted to Y ‘ @ S(6,) and rH,j equal to r, restricted to the jth homogeneous component. Then H H (rH,j)l is a rational map with singularities contained in 1)  6’. Let E be (as usual) the homomorphism of U(g,) to C given by ~ ( 1 = ) 1 and c(g) = 0. We identify 1 0 S(g,) with S(g,). If p E S(g,) and if H E E)’ then we set 6,(p) = ( F 0I)((r,)l(p)). Then if H E E)’ then 6, defines a linear map of S(g,) into S( I),). Let 6,. be the restriction of 6, to Sj( gc). Then H H 6,. is a rational map from E) into L(Sj(g,), Sj(E)c)).If D E DO(g) then let 6,(D) = S,(DH).
Lemma. I f D E DO(g) then there exists a diferential operator 6 ( D ) on E)‘ such that 6 ( D ) , = dH(D) for H E 6’. This is clear from the above discussion. 7.A.2.5. Let U and U , be open subsets of g and let W be a neighborhood of I in G. We assume that Ad(W)U, is contained in U.
270
7. Cusp Forms on C
Lemma. Let f E Cm(U) be such that f(Ad(x)Y) = f ( Y ) for x Y E U , . If D E D O ( U ) and if R = 6' n U , then
E
W and
( D f ) ( n= W)(fIn).
7.A.2.6. We now assume that G is a real reductive group such that Ad(G) acts trivially on the center of g. Let be a Cartan subalgebra of g. Let B be as in the definition. Set V = ' 6 relative to B. Let H = { g E G Ad(g)h = h for all h E [I} be the corresponding Cartan subgroup. Lemma. Let h, E 6' then there exist neighborhoods U and U , of h, and W a neighborhood of 1 in C such that Ad(W)U, is a subset of U and such that if B ( U , U , , W) = { f E Cm(U)lf(Ad(x)Y)= f(Y) for x E W and Y E U , } then BW, U , , W ) l " q = Crn(t)'n Ul). Let p be the natural projection of G onto G / H . Let (D(gH,h)= Ad(g)h for g E G and h E 6. Then it is easy to see that (D is everywhere regular on G / H x 6'. Hence there exist an open neighborhood W, of 1H in G / H and U , an open neighborhood of h, in 6'such that 0 restricted to W, x U , is a diffeomorphism onto an open neighborhood, U , of h, in 9.Let W, be an open neighborhood of 1 in G such that p(W,) = W,. Let W be an open neighborhood of 1 in G contained in W, and such that W W is contained in W,. Set U' = @(P(W)x U2). We assert that if we choose a possibly smaller W then U , n 6'= U,.Indeed, if Ad(x)h, = h,, h , E U , and h, E 6'. Then Ad(x)t) = 6.If N = {g E G I Ad(g)$ = 6) then N / H is a finite group. Thus we may choose W such that N n W = H n W. This implies the assertion. If u E Cm(U2)then define f on Cm(U) by f((D(x, h)) = u(h) for x E W and h E U,. Then f is clearly in B(U, U , , W) and f = u on U , .

7.A.2.7. If f E Cm(g) and if y E G then set y ( g ) f ( X )= f(Ad(g')X) for X E 9. Let T ( g )denote the algebra of differential operators, D , on g such that y ( g ) D = D y ( g ) for all g E G. Lemma. 6: T ( g )+ DO( 6') is an algebra homomorphism. Let h, E h' and let U, U , , W etc. be as in the preceding Lemma. Let u E C m ( U ,n 6') and let f E B(U, U , , W) be such that f = u on U , n 6' = R. Let
7.A.2.
27 1
Radial Components on the Lie Algebra
D,, D 2 E T(g). Then (D1D2f )In = W I W 2 . f )since l Q )D 2 f E WJ, ul,V. This in turn equals 6(D,)6(D2)fl,. Now (Dl D , ) f ( h ) = G(D,D,)u(h) for h E R. Thus we have shown that 6 ( D , D 2 ) u = 6(D,)6(D2)u for u E Cm(R).The Lemma now follows. 7.A.2.8. Our next task is to derive a formula for 6 ( D ) for D E S(g)' = S(g) n T(g).We first look at the element w E S(g)' with w = C XjyJ where {Xk} is a basis of g and B(Xj, yk) = 6j.k. Let @ = @(g,, 5,) and let @+ be a system of positive roots for @. If c( E @we choose E, E (g,), such that B(Ea,E,) = 1. Then [E,, E,] = H,(B(h, H,) = ~ ( hfor ) h E b,). Let { H j } be a basis of 5, such that B ( H j , H k )= Sj,k. Then w = C(Hj)2 2 CaeQ+ EaE,. Now
+
R((E,E,
+
L
E
U
)
0 1)
=
CE,, 1[E,,
1 + [EL, .ICE,, *I.
.
If X,Y are vector fields on g then (X Y),, = X,,Y+ X(h)Y(h). Thus rh((EmEa
+ E,E,) 0 I ) = a(h)([E,,E  a l 
=
 CEa,
a(h)2(EaE,
Eul)
+LEE,)
2(a(h)Ha c((h)2EaE,).
Hence r,,(symm(EaE,) 0 1 )
(1)
= +)H,

u(~)~E,E,.
Thus ( 2 )I,(

2
1cc(h)2symm(E,E,) 0 1 + 2
a(h)' 1 0Ha +
10 H f ) = w.
We therefore see that 6(w) = 2
(3)
c x(h)'H, + c Hf.
7.A.2.9. We define an isomorphism of S(g,) onto P(g,), p ~ p by # X # ( Y)= B(X,Y )for X,Y Eg. Let p ++ppbe the inverse map. If ,f E S(g,) then we define E S(6,) by f" = f # lh. We set n(h)= naeQ+ a(h) for h E 6. We can now state HarishChandra's formula for 6 (HarishChandra [6, Thm 1, p1001).
7
Theorem. If D
E S(gC)' then S ( D ) = a diferential operator of order 0.
n'Dn. Here a function is looked upon as
We first check the formula for o.We note that (1)
1 H ~ T=C0.
272
7. Cusp Forms on G
Indeed, if s E W(g,, 6,) then sn = det(s)n. If f E P (6) and if s,f = f then f vanishes on the hyperplane c1 = 0. Thus if sf = det(s)f for s E W(g,, bc) then f = ng with g a polynomial on 6. Now C H 3 n is also skew symmetric relative to W(g,, 6,). Since it has strictly lower degree than n we are forced to conclude that it is 0.
This proves the formula for w. To prove the full formula we use an ingenious trick of HarishChandra which is based on the following Scholium which will be used in another context. Scholium. If f E Pj(g) then 2’j!fb = (ad w)v. Here we look upon P(g)as multiplication operators contained in DO(g) and if x, y E DO(g) then ad(x)y = xy  yx. Let X E g. We must show that (ad w)j(X”)j = 2j(j!)Xj. We compute
(ad w)’(X“)j
Now ad OX’
= (ad
=2
w)J’(ad w(X”)j)
C B(Xj, X)
= 2X.
(ad o)J(X#)j= 2(ad 0 ) j  I
So
1
(X#)kX(X#)jk’.
Osjsk1
It is clear that if f is a polynomial of degree strictly less than j  1 then (ad w)j tf = 0. Thus if we put our calculations together we have (1)
(ad w)’(X’)’
= 2jX(ad w)j’(X’)j’.
If we use the obvious argument by induction the scholium follows. ~. We now complete the proof of the Theorem. Let D ~ S j ( g ) Then D = (1/2jj!)(ad o)jD#. Thus NOWh ( D # ) = D#
Ih.
S ( D ) = (1/2jj!)(ad S(w))’h(D’).
Thus
Ib)
6(D) = (1/2jj!)(ad S(w))j(D”
= (1/2jj!)(ad 6(w))j(n1(D#Ih)n).
7.A.3. Radial Components on the Lie Group
273
Since the Theorem has been proven for o we have (ad S(w))jD” Ih
= n’((ad(G))j(D#lh))n= (2jj!)C1Dn
by the Scholium applied to b. This completes the proof. 7.A.3.
Radial components on the Lie group
7.A.3.1. Let G be a Lie group with Lie algebra 9. We put a Lie group structure on G x G by (g, h) (u, u) = (gu, ulhuo). We leave it to the reader to show that with this multiplication G x G is a Lie group that is Lie isomorphic with the usual product group. Let L denote this Lie group. Let 1 be the Lie algebra of L. We look upon G as an Lspace with action ( x , y ) . g = x ( y g ) x  ’ . Let T :1 + DO(G)(differential operators with smooth coefficients) be defined by

Then T extends to an algebra homomorphism of U(1,) into DO(G). A direct calculation shows that if X , Y E g then
Here I is looked upon as 9 x $1with a twisted bracket operation. Thus U(1,) = U(g,) 0 U(g,) with the corresponding multiplication. The first factor is U ( g , x 0) and the second is U ( 0 x g,). In this appendix we use this formalism to prove analogues (also due to HarishChandra) of the results of the last appendix. The only results that are essentially different are the last two. Thus for the most part we will leave it to the reader to fill in the analogous arguments. 7.A.3.2. Let H be a closed subgroup of G with Lie algebra b. We assume that g = 60 V as in 7.A.2.3 and in addition that Ad(H)V = V. Set Y’ = Symm S(V,) in U(g,). We filter Y ‘ 0 U ( b c )using the standard filtration of U(1,). As before we set R ( x 0 y) = T (1 0y ) T ( x 0 1). We look upon V( gc) as the algebra of all right inuariant differential operators on G. That is, we identify it with T(l 0 U(g,)). If D E DO(G) and if g E G then there exists a unique 0, E U(g,) such that Df(g) = Q f ( g ) . Define for u E U(lc), g E G, R,(u) = R(u),. Let H‘ = { h E HI det((Z  ad(h))I, # O}. Clearly, if 6’ is nonempty then so is H‘. We assume this. If h E H then we write r, for R, restricted to*” 0 U ( f ) , ) .
7. Cusp Forms on G
214
Lemma. I f h E H' then r,,is a linear bijection o f V 0 U(5,) onto U(g,) which respects the jiltrations (we use the standard Jiltration on U(g,)). 7.A.3.3. We assume that there is a complex Lie group H, contained in GL(g), such that Ad(H) = H, n GL(9). Lemma. The map h Hr h lu8s(b)) E L((V 0 ~ ( f ~ , ) ) jUj(g,)) , is real analytic in h, factors through the homomorphism h H Ad(h) and extends meromorphically to H,. This is clear from the definitions and 7.A.3.1(1).
7.A.3.4. If h E H' and if x E Uj(g,) set 6h(x) = ( E 0 I)(rh(x)). As in the preceding appendix, if x E U(g,) then h H 6h(x) is meromorphic from H, into uj(b,). If D
E
D O ( G ) then Set 6h(D) = 6h(Dh).
Lemma. I f D E DO(G) then there exists a differential operator 6 ( D )E D O ( H ' ) such that 6,(D) = 6(D),. 7.A.3.5. Let U and U , be open in G and let Q be a neighborhood of 1 in G such that x U , x' is contained in U for x E Q. As in the previous appendix, we set B(U, U , , Q ) equal to the space of all C" functions on U such that f ( x y x  ' ) = f ( y )for x E Q, y E U1. The following result is proved in exactly the same way as Lemma 7.A.2.5. Lemma. Let f
E
B(U, U , , Q ) and let D DflUnH'
E
DO(G). Then
= 6(D)(fl"nH').
7.A.3.6. We now assume that G is a real reductive group of inner type. Let Ij be a Cartan subalgebra of g and let H be the corresponding Cartan subgroup. We take V = 'jI relative to B. Let U be open in G such that X U x  l = U for all x E G. Let D ( U ) denote the algebra of differential operators on U, D , such that y(g)D = D y ( g ) for g E G. Here y(g)f'(x)= f ( x g x  ' ) . The following result is proved in exactly the same way as Lemma 7.A.2.7. Lemma. 6 is an algebra homomorphism from D ( U ) into DO(H'). Our next task is to find a formula for 6 ( z )for z E Z(gc). Fix @+ a system of positive roots for @(gc,bc). We assume that the corresponding p is the
7.A.3.
275
Radial Components on the Lie Croup
differential of a homomorphism of H into C". This can always be guaranteed by going to a covering of G. Set
n
A(h) = hP
(1

h")
UE@+
1
= SE
W(s,h)
det(s)hsP.
Let y be the HarishChandra homomorphism from Z ( g c ) to U ( 9 , ) . Here is the formula of HarishChandra in this case. 7.A.3.7.
Theorem. I f z
E
Z(gc) then
6 ( z ) = A  y(z)A. Let p E (bc)* be 0 ' dominant integral. Again, by going to a finite covering of G we may assume that p defines a character of H. Let q,be the character of the corresponding finite dimensional representation of G. If z E Z(gc)then (1)
ZG, =
(2)
zaplH'
(P + p)(y(z))a,
=
and
6(z)(G, I H ' ) .
We note that A(h) is nonzero for 11 E H'. Set
(A

6(Z) *
A  l ) h= P h
for h E
We set q h = ph  y ( z ) E U(I],). ( 1 ) and (2) combined with the Weyl character formula imply that det(s)s(p+ p ) ( q h ) h s ( ' L + = P)0
1
(3) SE
for h E H ' .
W(!%b)
We note that the coefficients (in h) of q h extend to meromorphic functions on Ad(H,). Let 5' denote the set of all h E such that a(h)E R and a(h) > 0 for all a E 0 ' . Then s ( k p + p ) ( h ) ( k p p)(h)+  co as k + co. Write q,,, for the homogeneous component of qh of degree j . If /?E b* then r @ ( p h ) = Cj d / ? ( p h , j ) . Let q be the maximum of the j such that q h , , is nonzero. If h E exp(b') then
+
0 = lim km
k4hW'P)
+
1det(s)s(kp + p ) ( q h ) h s ( k a + P=) p(qh,q).
The p E b* that are highest weights of irreducible finite dimensional representations of G are Zariski dense in f)*. We have shown that if h E exp($') then q h = 0. Since H , is connected and q h is meromorphic in h this proves that qh = 0 for all 11 E H,. 7.A.3.8. In the next chapter we will need a generalization (also due to HarishChandra) of the above theorem. Let 8 be a Cartan involution for G.
276
7.
Cusp Forms on G
Let h, E g be such that Oh, = Ao. Then ad(h,) has real eigenvalues. Let m be the centralizer in g of h, and let n be the direct sum of the eigen spaces for ad h, corresponding to strictly positive eigenvalues. Let V = n 0Qn.Let M = {g E G IAd(g)h, = A,). Let M’ = { m E M Idet((1  Ad(m))I.) # 0). Let b be a Cartan subalgebra of ni and let H be the corresponding Cartan subgroup of M (also of G). We denote by 6G,Mthe “6” from DO(G) to DO(M‘), aM,” the one corresponding to D O ( M ) to DO(H’) and by 6 the one going from DO(G) to DO(H’). Set for m E M ,
AG,&)
=
Idet(Ad(m)I,)/
det(l

(Ad(m)I,).
We define a homomorphism ye.,, from Z(g,) to Z(m,) as follows. PBW implies that U(g,) = U(m,) 0(Qn,U(g,) + U(g,)n,). Let q denote the corresponding projection of U ( g c ) onto U(m,). Let q be the homomorphism of U(rn,) to U(nr,) given by q ( X ) = X  (9) tr(ad XI,,) for X E m. Then Ys.m is given by q q restricted to Z(g,). CJ
Let U be an open subset of G such that x U x  ’
l.A.3.9. x E G.
=
U for
Proposition. Let z E Z(g,) and f E C“(M n U ) be such that f ( x y x  ’ ) f ( y )f o r x E M a n d y E M n U . T h e n o n M ’ n U w e h a v e h G, h & )f
=
= AG.IMYG.M(Z)A.C.M.f.
Let x E M n U n G‘ and choose H such that x E H . Let V, be an open neighborhood of 0 in V such that if @ ( X )= exp X M , X E V, then @ is a diffeomorphism of Vo onto an open neighborhood of 1M in G / M . Let W be an open neighborhood of x in M ‘ n U such that if u ( X , y ) = exp X y exp(  X ) for X E Vo and y E W then u is a diffeomorphism onto an neighborhood of x in G. Let W, be an open neighborhood of x in W and let P be a neighborhood of 1 in M such that yWy’ is a subset of W for y E P. Let U , = u(V, x W,). Set Q = exp V,P. Then if we argue as in the proof of 7.A.2.6, it is possible to choose V,, W , W,, P so small that B(U, U i , Q ) l w
(1)
Let f
=
B(W, Wi,P).
B( W , W,,P ) and let h E B(U, U , , P ) be such that h = f on W. If then dM.H(6G,M(z)) = 6 ( z ) . HarishChandra’s formula implies that ~,,,(A,,,YC.M(Z)(AG,’M)) = A Y ( Z ) A  l . Thus E
z E Z(g,)
6M,~,(AG.MYG,M(~)(A.C,M)
S,,,(z))g
=0
277
7.A.4. Some Harmonic Analysis on Tori
for
(T
E
C x ( W ,n H ) . Thus
'f
AG.MYG.M(d&.M)
=
&Ldz)f
for f E B( W, W,, P ) . Thus the result has been proved on G' n U n M . Since this set is dense in M ' n U and the desired formulas are real analytic on the larger set, the result follows.
Some harmonic analysis on Tori
7.A.4.
7.A.4.1. The purpose of this appendix is to collect some technical results that will be used in Section 7.4. Let T be a compact torus with Lie algebra t. Then exp is a covering homomorphism (if we look upon t as an abelian Lie group under addition). Let l = Ker(exp). Then is a lattice in t that contains a basis. We identify T" with { p E t * I p(r)c 27cZ). That is, if p is such a functional then t" = e i p ' Hif) t = exp H . Let ( , ) be an inner product on t. We also use the notation ( , ) for the dual inner product on t*. Let X I , .. . , X , be an orthonormal basis of t. We set A = C (Xj)'. Clearly, depends only on ( , ). It is clear that Atp= (p,p)tp
(1)
If f
E
C m ( T )and if p
E
for p
E
T".
T" then we set f^(p) = j f ( t ) t C " d t . T
Let k , be the distribution on T with Fourier series
That is, if f
E
C " ( T ) then kr(f)
=
C (1 + (n + k ) / 2 then k ,
E
Ck(T).
278
7. Cusp Forms on C
If r > n / 2 then
Thus the Fourier series defining k, converges absolutely. Hence k, E C o ( T )if r > n / 2 . Xjk,has Fourier series
Since I(1
+ (p,p))rip(Xj)I
(1 + < P , P > )  ~ + " ~
it follows that if r > ( n + 1)/2 we can differentiate the Fourier series defining k, term by term. So k, E C ' ( T ) if r > ( n + 1)/2. The lemma follows from the obvious iteration of this procedure. 7.A.4.2. If x, y E T then set d(x, y) equal to the Riemannian distance between x, y corresponding to ( , ). That is, d ( x , y ) = inf{IIX  Y I( I exp X = x, exp Y = y}. Here llXll = (X,X)'l2. We now come to the first main result of this appendix. Let V be a Frechet space. Let a',. . . , ad E T "  { 0 } be distinct. Set T' = { t E T ituz# 1 , i = 1,. . . ,r } , if E > 0 then set TL = { t E T I 11  tall > E for i = 1,. .. ,r } . Let A be an algebra of continuous linear operators on I/ containing the identity. Let C be a subalgebra of U(t,) such that D = I  A E C and such that there exist pl,. . ., p q E U(t,) such that U(t,) = C Cpj. Let y be a surjective algebra homomorphism of A to C. Let W be a dense subspace of V. Suppose that we have a linear map, S, of W into C"(T') such that S(Tu) = y(T)S(u)for u E W, T E A . Finally, we assume that there is a continuous seminorm u on I/ and u 2 0 such that
for V E Wand all0 < E < 1. If p E U(t,) and i f f E C m ( T ' )then we set
Let B ( T ' ) be the space of all f E C"(T') such that ap(f)< GO for all p , endowed with the topology given by these seminorms.
Lemma. S extends t o a continuous linear map of V into B(T'). Let i3T' = T  T' (as usual). If x E T set u(x) =
(4)inf{d(x,y) ly E T'}.If
279
7.A.4. Some Harmonic Analysis on Tori
X
E
t then we denote by B r ( X )the r > 0 ball in t centered at X relative to
( , ). Let ro be such that exp is a diffeomorphism on Br(X)for all X E t. If x E T' then set u(x) = min{u(x), 1,r0/2}. Let h E C"(R) be such that 0 I h(x) I 1 and h(x)= 1 for 1x1 I 2  " ' and h ( x ) = 0 for x 2 1. Fix x E T' and set u = u(x). Let x = exp X . Define g E Cm(T)as follows: g(exp Z ) = 0 if 2 is not an element of B r ( X ) and g(exp Z ) = h(ll2  Xl12/v2) if 2 E B r ( X ) . Let f E Cm(T).If p E V(t,) then
PfW
= P9f(X).
Fix p E Uj(t,). Let d be the maximum of the orders of the p i . Set s = n + j + d + 1 . Then ( ( P , , ) ~is the formal adjoint of p looked upon as a differential operator in y see 8.A.2.7)
PfW
=
jT W
Y  ) w M f ( Y )dY
=
jT ( P , ) ' ~ s ( x Y p ' ) s ( Y ) ~ s f ( & Y ) + j (P,)'ks(xYl"s? T
9 l f ( Y ) dY.
Now [Ds,g] is a differential operator on T of the form C pj(y)al'l/y' (here we are using coordinates on T corresponding to our basis of t) with p j ( y ) = 0 if d ( x , y ) I u/2l'' or if d ( x , y ) 2 u. Also from the choice of g it is clear that lalJl/ayJp,(y) I I C,uP with C,, q, constants independent of y and x (see 5.A.2). We write al'l/ay' = C ai,'pi with a,,' E C . We therefore have
Pf(4
=
jT (P,)Tks(XY1)9(Y)DSf(Y)dY
+1 j ((pi),)T(p,(y)(p,)Tks(xy'))a,,if(y) 1.i T
dy.
+
Now, a l J l / y J k , ( x y p ' ) defines an element of C o ( T ) for IJI I j d. We therefore find that there exist cj E C, q = q,, and a constant C,, such that Ipf(x)I 5
Cp~l'
1
j
d(x.y) 5 u
Icjf(Y)I dy.
If 0 < E < 1 and if x E T: then u(x) 2 CE,with C a fixed positive constant. We therefore have Ipf(x)I 5 Cp&'
C T'j Icjf(t)Idt.
We apply this to f' = S(w) for some w E W. We have shown: (1) If p E U(t,) then there exists a constants q ( p ) and a continuous seminorm a,, on I/ such that if 0 < E < 1 and if t E T, then IpS(w)(t)lI e'(p)~,,(w).
280
7.
If p E U(t,) then p then M j ) = q ( P j ) )
=
E y(uj)pj with uj E A. Thus if w E W and if t E TL
IpS(w)(t)I5 C Ipjr(uj)s(w)(t)I5
by (1). If we set q
Cusp Forms on G
= max{q(j)} and
CpE4"'ap(ujw)
p,,(u) = E op(ujw)then we have shown:
E U(t,) then there exists a continuous seminorm p,, on V with the property that
(2) There exists a constant q such that if p
IPS(W)(t)l 5 &4Pp(w)
forallO<E< 1 andalltET,. Let c E T and let x = exp X . Let y E t be such that exp(X + t Y ) E T, for 0 < t < 1. In light of (2) we can apply Scholium 7.3.4 to pS(w)(exp(X t Y ) ) to find that if p E U(t,) then there exists a continuous seminorm p p on V such that IpS(w)(t)l I pp(w)for all w E W. This completes the proof.
+
7.A.4.3. We now assume that if clj is a nonzero multiple of clk then j = k. Let Tl be the set of all t E T such that taJ= 1 for exactly one j. Then it is clear that T' u Tl = T" is open in T.
Lemma. Let f E B(T') be such that for each p E U(t), pf extends to a continuous function on TI'. Then f extends to a smooth function on T. We may clearly assume that n = dim T 2 2. If a:[O, 13 + T is a smooth curve then we set 1
L ( 4 = 1Ila'(s)ll ds. 0
(1) If x , y E T" and if E > 0 then there exists a smooth curve joining x, y with values in T" such that L(o) I d(x, y ) E.
+
To prove this we will use the following simple result. Scholium. Let PI,...,pd be real valued linear functionals on R" that are pairwise linearly independent. Set for i < j ,
Then U = { ( x , y ) € R " x R " I I I c j p i j # O } is open and dense in R" x R".
7.A.4.
Some Harmonic Analysis on Tori
28 1
We first show that y = ni 0). The obvious calculation yields LPZ+' = 2(2 + 1)(22 + n)PZon C
(1) If f
E
for all z E C.
Y(R") and if Re z > 0 then we set
P W ) = j P'(x)f(x)dx. C
Lemma. P","(Lf) = 2(2 + 1)(22 + n ) P ; ( f ) for all f Re z > 0.
E
Y(R") and all
Set S = { x E R" IP(x) = 0). Then S  (0) is a smooth hypersurface of R". Let for each E > 0, S, = { x E S I llxll 2 E } . Put R, = S , U { x E R"I P ( x ) 2 0 and llxll = E } . Set C, = { x E R" I P ( x ) > 0 and llxll > E } . Then R, is the boundary of C,. R, is piecewise smooth so Stokes' theorem is applicable on C,. Let u E P ( C ) be such that u and ( a / a x i ) u have continuous extensions to Cl(C) and (a)
lu(x)l +
1I(a/ax,)u(x)l s Cllxlld
d> 0
(b)
4s) = (0).
and
allxER".
for some C > 0
and
7.A.5.
283
Fundamental Solutions of Certain DiRerential Operators
If we apply Stokes' theorem, we find that i f f
J u ( x ) ( a / a x i ) f ( xdx )
= lim
C
&0
=

E
Y ( R " ) then
J u(x)(a/ax,)f(x)dx
c
j (a/axi)u(x)j(x)dx.
C
(We leave the details to the reader.) If Re z > 1 then u ( x ) = PZ,"(x) and u ( x ) = ( a / a x i ) P 2 + ' ( x )satisfy (a) and (b) above. Hence
J P z +' ( x ) L f ' ( . ~=) J L P " ' ( x ) f ( x ) d x . C
C
This combined with (1) above implies the Lemma for Re z > 1. Since both sides of the equation that we are proving are holomorphic in z for Re z > 0, the Lemma follows. 7.A.5.2. The above result implements a meromorphic continuation of P ; for z E C. More precisely we have Lemma. I f f E 9'(R") then z H P Z + ( f ) has a meromorphic continuation to C. The poles are contained in the union of the sets { 1,  2 , , . . } and {  n/2,  n / 2  1,. . .}. I f n is odd then the poles are all simple. Furthermore, P ; and Res,=,P: dejne tempered distributions on R".
The first part of the Lemma follows from
(1) P:+j(L'f) = 2j(z + 1)...(z + j ) ( 2 z + n).,.(2z + 2 ( j  1)
+ n)P;(f).
Since this tells how to define P Z ( f )for Re z > j. The last assertion follows from
IK(f)l
5
J
R"
ll~llZReZlf(X)IdX
for Re z > 0 and (1). 7.A.5.3. We now do a different analysis of P:. Let S, (resp S,) be the unit sphere of R P (resp. R4). Let do, (resp. do,) be respectively the rotationally invariant measures on S , and S,. Then (up to a constant depending only on P? 4 ) (1)
P Z ( f )=
m r
J J rP1s41(r2 s2)' j f(rq,,sa,)do,do,dsdr 0 0
s1 x s 2
Let 1, denote the characteristic function of C . (1) implies
284
7.
If Re z >
Lemma.

Cusp Forms on G
1 then (1,P)' is locally integrable on R".
Let f be a nonnegative, smooth, compactly supported function on R". Assume that supp f is contained in {(x,y) I llxll 5 N , llyll I N ) . Then
J
R"
N r
llcP(x)(ReZf(x)dx I ~ ( f 1) J rPlsqO1( r Z  s Z ) R e z d s d r . 0 0
If we use the coordinates r and t with s = tr for 0 5 t 5 1 then the second integral becomes N
1
s r2
dr t4'(1
which is finite for Re z >

t2)Rezdt,
0
0

1.
7.A.5.4. We continue with the analysis of the previous number. Set for r, s E R, u(r, s)
=
f f ( r o l ,so,) do, do,.
s , XS,
Then u E 9'(R2) and u is even in both variables (1) If g E Y(Rz)is even in both variables than h(x,y) = g(x"z,y"z) defines an element of Y ( U )where U = {(x,y ) x, y > 0). (See 7.3.4 for the definition of Y ( U )for U open in a Euclidean space.) Taylor's theorem implies that if x > 0 and if y E R then there exists 0 < 0 < x such that I(am/aYm)f(x> Y)= ( x N +l
1 (ak+"laxkaY")f(O,y)(xk/k!)l
ksN
+ i)!)l(aN+i/axN+i)f(e,y ) l .
/ ( ~
Thus if 0 < x I 1 then
with pr,m,Na continuous seminorm on Y ( R 2 ) , Now (ak+yaxkaym)j(o,
y)=o
if k is odd. Thus, if we substitute xl/' for x in the above inequality we find that if 0 < x I 1 and if y E R then
7.A.5.
Fundamental Solutions of Certain Differential Operators
285
Thus, u(x, y ) = f(x’”, y ) defines a Schwartz function on {(x, y ) 1 x > 0, y E R} which is even in y. We may thus repeat the above argument in the y variable to finish the proof of (1). (1) implies that u(r,s) = u ( r 2 , s 2 )with u E Y ( C ) . Hence
We make the change of variables (x, y) = ( r 2 , s 2 ) and then (x, y ) = (r,tr) with r > 0 and 0 < t < 1 and obtain
p;(f)=
n
j
1
yz+n/Zl
0
j

‘(I

t)’u(r, tr) dt dr.
0
Set 1
@(z,r ) = j P’’ ’( 1

t)”u(r,tr) dt.
0
Taylor’s Theorem implies that
with p m a continuous seminorm on 9’(C). Thus if Re z > 0 we have
Furthermore, E(r, z ) is holomorphic for Re z >  m and I E(r, z)l I C,(z)q,(u) with qm a continuous seminorm on .‘Y’(R2) and Cm a continuous function of z for Re z > m. We can argue in the same way to get similar estimate on (1
+ r 2 ) k ( a i / a r i ) @ r). (z,
We theretore see that @(z,r) has a meromorphic continuation to C with at worst simple poles at  1,  2 , . . . . Furthermore, @(z,) E Y(R) where it is holomorphic and the residues at the poles are Schwartz functions. We observe that x
p:(f)= j
(*I
0
We note that
7
1
rZ+n/2l
,.z/2+nl
@(z, r ) dr.
@(z, r ) dr
286
7. Cusp Forms on G
is holomorphic wherever @(z,r) is. Thus to implement the analytic continuation of (*) away from  1, 2,. . . we may look at 1
j rzi2
(**I
+ "
~
@(z,r ) dr.
0
7.A.5.5. We now look at the case when n is odd. Then (**) above implies that the poles other than  1,  2,. . . are at most simple poles at the points n/2,  n / 2 1,  n / 2  2,. ... Furthermore
+
Res,=.,,P:(J')
=
@(n/2,0).
The calculations in 7.A.5.4 imply that
Let B(z, w) denote the classical beta function ( B ( z ,w) = r ( z ) r ( w ) / r ( z+ w) and T ( z ) is the classical gamma function). Then the usual integral formula for B(x, y ) yields 1
1
tqi2l(i

tydt =
~ ( q / 2z ,
+ 1 ) = r ( q / 2 ) r ( z+ i ) / r ( z + 412 + 1).
0
If q is even the value of this function at shown
Lemma.

n / 2 is nonzero. We have therefore
Assume that n is odd. Set F(x) = lc(( l)qx)~P(x)l1/2.
There exists a nonzero constant Cp3q such that i f f
E
SP(R2) then
j F(x)L""~!~(x) dx = C,J(O). If q is even then this follows directly from (1) and 7.A.5.2( 1). If q is odd then replace P by  P and L by  L.
7.A.5.6. We now look at the case when n is even. We first assume that p and q are odd. As before we begin with the material in 7.A.5.4. In this case it is clear that @(z,r) has a pole at z = n/2. Thus P : ( , f ) has a double pole at
7.A.5.
z
=
Fundamental Solutions of Certain Differential Operators
287
n/2. If we argue as in the previous number, we find that
(1)
(Z
+ n/2)'PZ(f)lz=n/,= C p , q f ( O ) .
We have
Lemma. I f p and q are odd then there exists a nonzero constant CP,,such that
J 1 C(X)L"'Y(X)dx = c p , , m Jiir all f
E
Y(R").
l.A.5.1. We now analyse the case p and q even. In this case one checks that @(z,0) is holomorphic at  n/2. Thus we see that P'( f )has a simple pole at z =  n / 2 whose residue is a nonzero multiple of f (0). On the other hand 7.A.5.2( 1 ) implies that ) C(d/dzl,,,)P'(L"~'f) Res,=  n i 2 P Z ( f=
with C a nonzero constant. This implies
Lemma. that
If p and q are even then there exists a nonzero constant CP,,such
J 1c(x) log I &)I
L"'2ff(X)dx
=
C,,,f (0)
for .f E .Y(R"). 7.A.5.8. We now put all of this material together, and we drop the assumption that p , q 2 1. Theorem. Let n 2 2. Let p , q be nonnegative integers such that p Define FP,,as follows
+ q = n.
Fp,,(x) = lC(x)lP(x)l"' if n and p are odd, Fp,,(x)= l C (  x ) ~ P ( x ) ~ ~if"n' and q are odd, F,,,(x)
=
l,(x) if p and q are odd,
F,,,(x)
=
log IP(x)l if p and q are even.
Then F,,, is locally integrable and there exists a nonzero constant CP,, such
288
7. Cusp Forms on G
that
for f
E
Y(R").
If p , q 2 1 then the result follows from the above discussion (note the change in the case p , q even). The only case we have not checked is n even and q = n. We leave this to the reader (Hint: Argue as in the previous number using C = { x IP(x) < O}. Only the "rintegral" plays a role.)
8
Character Theory
Introduction
The purpose of this chapter is to develop HarishChandra’s theory of characters of real reductive groups. In his early papers, HarishChandra, realized that the correct infinite dimensional generalization of the usual character of a finite dimensional representation was as a distribution given as the trace of an operator on the representation space (see Section 8.1). Although the definition of the character of a (g, K)module is quite natural, it is not at all clear how to apply it as a computational tool. The power of the character theory of real reductive groups rests on HarishChandra’s regularity theorem (8.4.1). As a consequence of this theorem it can be shown that the character of an irreducible (9,K)module is given by a formula that is (formally) quite similar to the character of a finite dimensional representation. HarishChandra gave two important (intimately related) consequences of his regularity theorem. The first was a characterization of tempered representations in terms of the growth of their characters. The second was his determination of the irreducible square integrable representations of a real reductive group. We conclude this chapter with these applications. Our exposition of these results does not stray very far from HarishChandra’s original papers.
289
290
8. Character Theory
We have benefited from Varadarajan's treatment of the regularity theorem (Varadarajan [I]). Our exposition is a bit simpler than that of the original since we have avoided the use of the notoriously difficult Theorem of HarishChandra on analytic Ginvariant differential operators that annihilate the Ginvariant functions (c.f. Varadarajan [l, Thm23, p.143, part 13). In order to achieve this simplification, we prove a stronger theorem on the Lie algebra (8.3.3) than the original of HarishChandra. The key to our approach is Lemma 8.A.3.7, which was suggested to us by Duistermaat. As is usual in this book, we have included several appendices to this chapter that either contain standard results that will be applied in the body of the work (e.g., traceclass operators, elementary Fourier theory and basic distribution theory). There are also several technical results (that could very well have been included in the pertinent proofs) that we have opted to include as appendices in order to help clarify the flow of the arguments.
8.1. The Character of an admissible representation 8.1.1. Let G be a real reductive group and let K be a maximal compact subgroup of G. Let Y ( G ) be as in 7.1.2. Fix a norm, 11. . 1 1 (2.A.2.3),on G. Let p,,,,,denotetheseminormp,,,,,,,,,of7.1.2.witha = Il...llandb = 1.Fixdsuch that
j Ilgllrddg < 0.
G
Let ( n , H ) be a Hilbert representation of G. Lemma 2.A.2.2 implies that there exist positive constants r and C such that
This implies that we can argue as in 1.1.3 to define, for each f operator n ( f )with
E Y ( G ) , an
with C, depending only on IT.
Lemma. Assume that (TC,H ) is admissible and j n i t e l y generated. Let {uj} be an orthonormal basis of H such that each uj is contained in a K Oisotypic component of H, Then there exists a continuous seminorm, p , on Y ( G )such that
C IIx(f)ujII P ( S )
jbr f E Y ( G ) .
29 1
8.1. The Character of an Admissible Representative
Since G = K G o and G/Go is finite, there exist k , = 1, k , , . . ., k , E K such that G = kiCo and each subset k i C o is a connected component of G. If f ’ E Y ( G ) then we define for each i, J ( g ) = , f ( k i g )for y E Go. We extend f, to G by 0. Then f i E Y ( G ) ,f = C L ( k , ) f ) .It is also clear that the maps f H f i are continuous on Y’(G).We note that if g E G then n(L(y)f) = n ( g ) ? z ( f )We . assume (as we may) that 7c restricted to K is unitary. Suppose that we have found { u j ) and p such that the assertion of the lemma is true for f E Y ( G ) such that supp f c Go. If f E Y ( G ) then C lln(f’)ujll= C IIC n(k,)7c(fi)ujllI Iln(f;)ujllI XI 5 i i m p(fi). Thus if we set q ( f )= C p ( f i ) then the result follows from the special case. Now 7c restricted to Go is admissible and finitely generated (4.2.7). The result will therefore follow if we prove it in the special case when G = Go. So assume that G is connected. Let C, be the Casimir operator of K corresponding to B , ,x t . If y E K” then let I,, denote the eigenvalue of C, on any representative of y. Let T be a maximal torus of KO and let P be a system of positive roots for K with respect to T. Let p be (as usual) the half sum of the elements of P . If Ay is the highest weight of y then
u,
Also
This implies that there is a constant, C > 0, such that 4 7 ) 5 CO~,+ IlPIl2)P
with p = IP1/2. There is a positive integer N such that the number of y E K” with highest weight A, is at most N . As in 7.A.4.1
for r > dim T/2. This implies that
6)
1d ( ~ ) ~ (+2 llPl12 , + l)r
2p
+ dim T/2.
Proposition 4.2.3 says that there exists a finite dimensional representation, 0,of Po, a minimal parabolic subgroup of G, such that H , is (9, K)isomorphic with a submodule of X“ (see 4.2 for the pertinent notation). Frobenius reciprocity implies that
(ii)
dim H ( y ) 2 rl(a)d(y)’
for y E K”.
292
8. Character Theory
8.1.2. Let ( n , H ) be as above. The preceding Lemma implies that if f~ Y ( G )then n ( f ) is of trace class (8.A.1.5). We set 0,(f)= tr n ( f ) . Let { u j } , p be as above. Then Itr n ( f ) l I C Iln(f)ujllI p ( f ) , f E Y ( G ) .Thus 0,defines a continuous linear functional on Y ( G )which we call the distribution character
of
71.
We may also assume that each uj is contained in some isotypic component of H relative to K . If y E K" then set F ( y ) = {jI uj E H(y)}. Then if E , is the orthogonal projection of H onto H ( y ) then
Lemma. I f (n,H ) and (a,V ) are admissible finitely generated Hilbert representations of G such that HK and V, are (9,K)isomorphic then 0, = 0,. In light of (1) above, in order to prove this result it is enough to show that 4; = 4: for all y E K " . Since H , and VK are isomorphic it is clear that x $ i = x4: for all x E U ( g ) ,k E K and y E K ". Since 4; and 4: are real analytic and G = K G o this implies that they are equal. 8.1.3. The above Lemma implies that if V is an admissible finitely generated (9, K)module and if (n,H ) is a realization of V then 0, depends only on I/. We
may therefore write 0, for 0,. We will also call 0, the distribution character of V.
8.1.
293
The Character of an Admissible Rcpresentative
Lemma.
Let O+V~W+Z+O
be an exact sequence in H (4.1.4). Then 0,
=
0,
+ 0,
Let (n,H ) be a realization of W. We assume (as we may) that V is a submodule of W. Let H , = Cl(V). Then H , gives a realization of V and H / H , gives a realization of Z . As a Hilbert space H / H , = (H,)'. The lemma follows if we split the sum giving the trace into the part corresponding to H , and the part corresponding to (Ifl)'. 8.1.4. Lemma. I f V,, . . . , V, ure nonzero mutually nonisomorphic objects in H then O,,, . . . , O,, are linearly independent.
Let for each j , nj be a realization of V,. Set for y E K " , 4:, = 4;. In light of the material in 8.1.2, it is enough to show that for each y E K " , the nonzero 4J are linearly independent. Fix 7 E K A .After relabeling we may assume that V,(y) is nonzero for j I r and is zero otherwise. 3.5.4 (3.9.79)implies that each of the U(gc)KmodulesHom,(K,, 5) is irreducible. If x E U ( g , ) K then set p j ( x ) equal to the trace of the action of .Y on Horn,( V,, 5). A direct calculation (which we leave to the reader) yields d(y)pj(x)= xqhY(1)
for x E U ( g J K .
Thus Corollary 3.A. 1.3 implies that 4 ; , . . . , 4; are linearly independent, 8.1.5. If V E X then V is of finite length as a (g, K)module (4.2.1). Let V = V, 2 V, 3 . . . 3 V, 1V,+ = {O) be a JordanHolder series for V. If W is an irreducible object in H then we say that the multiplicity of Win V is the number of indices, j , such that 5/V,, is isomorphic with W. Notice that the previous Lemma implies that the multiplicity is independent of the choice of JordanHolder series. If W has positive multiplicity in V then we say that W is a constituent of V.
,
Theorem. I f V, W E H and if 0, = 0, then V and W have the same multiplicities for their irreducible constituents. This is an immediate consequence of Lemmas 8.1.3 and 8.1.4. 8.1.6.
If f
E
Let V be an irreducible ( g , K)module with distribution character 0,. , V ( G ) then set .r(g)f'(x) = .f'(s 'xg) for x, g E G.
8. Character Theory
294
Lemma. (1)
(2)
0, ~ ( g=) 0,
for g
If V has infinitesimal character
x then z0,
9
E
G. = x(z)O, for
all z
E Z(g,).
Let (n,H ) be a realization of V. Iff E Y ( C )then n ( z ( g ) f )= n ( g ) n ( f ) n ( g )  l . Thus @,(z(g)f) = tr n ( g ) n ( f ) n ( g )  l= tr n ( f )(Corollary 8.A.1.10). If f E Y ( G ) and if z E Z(gc) then n ( z 7 )= n ( z ) n ( f )= ~ ( z ) n ( f )Thus . 0 , ( z T f ) = x(z)O,(f). Hence (2) follows from the definition of the action of U ( g )on D'(G) (8.A.2.7). 8.1.7.
A continuous functional 0 on Y ( G )is said to be central if
0 z(g) = 0
for all g
0
E
G.
It is said to be an eigendistribution with infinitesimal character x if z 0 = x ( z ) 0 for all z E Z(g,). Thus if V is an irreducible (9, K)module then 0, is a central eigendistribution. In the next section we give the relation between the Kcharacter and the Gcharacter. In Sections 3 and 4 we will prove several theorems of HarishChandra that give the local structure of invariant eigendistributions. We will then apply these results to distribution characters to (in particular) complete the theory of the discrete series. 8.2.
The Kcharacter of a (9, K)module
We retain the notation of the previous section. If y E KA then we use the notation yly for the character of y and d(y)for the dimension of any element of y. We endow C " ( K ) with the topology defined by the seminorms
8.2.1.
VD,K(f)
I
= sup{lDf(k)l
K},
U(f)
Lemma. Let V be an admissible Jinitely generated (g, K)module. I f y E K" then set m,(y) = dim HomK(Vy,V ) . I f f E C m ( K )then the series
converges absolutely and dqfines u continuous linear functional Q K , , on C"(K). We will use the notation and results in the proof of 8.1.1. As in 8.1.1 we may assume that G = Go. Set
w1 J yl,(k)f(k)dk =
K
for 7 E K A .
295
8.2. The KCharacter of a (g, K)Module
+
Then T,((I + CK)'f) = (1 ,$,)rTy(f) for f Iqy(k)l I d ( y ) for k E K . Thus
E
Iq(f)l5 d ( y ) v l , K ( f We therefore find that if we set D(r) = (I
Cm(K). It is obvious that ).
+ CK)lthen
+ 'l*),)r d(?)VD(r),K(f ).
I q(f)l5
We have also seen that mv(y) I Cd(y) for all y E K" (here of course C is independent of 7). Thus the series that we are estimating is dominated by
8.1.1(1) implies that the above series converges if Y is sufficiently large. If r is that large then 5
l@K,V(f)l
crvD(r),K(f
1.
8.2.2. We will call eK,, the Kcharacter of V. We now relate the Kcharacter to the distribution character. We will assume that G has a compact Cartan subgroup. Let K" be as in 7.4.1. If .f E Y ( G )then we set
$,@I
=
IdW
for k E K" and equal to 0 if k
Theorem. Let
E
E
> 0 and let f

G
K  K". .Y(G) be such that supp f c Ge,e(7.4.3). Then
E
(1)
(2)
Ad(k))I,)l J f ( &  ' ) d g
Ijr E @V(f)
P(K). = @K,V($f).
Here we are using the same normalization of d g t o dejne 0, and $f.
We have seen in 7.4.4 that Q,
E
C " ( K " ) .Our assumption on the support of
f implies that Q, has compact support in K ' . This implies the first assertion.
We now prove (2). Recall (7.4.4) (i)
If k E K" n GL,e, and if g E G then log Ilgkg'II 2 C E " ~log llgll with C a positive constant independent of k and g.
If f E Y ( G )then set F,(k) = Idet((I  Ad(k))I,)lf'(gkg'). The argument at the end of 7.4.4 proves (ii) If u E U(t), E > 0 then for each r > 0 there exists a continuous seminorm
296
8.
Character Theory
v , , ~ on , ~ Y ( G ) such that
IuF,(k)I 5 119II  “ V u , r , z ( f ) for f
EY
( G )with supp(f) c GC,@ and all k E K .
Let (n,H) be a realization of I/. Let { q , } be an orthonormal basis of H(y) for y E K ” . (2) combined with the argument in the previous number implies that if f E Y ( G )has support in GE,pthen (iii) for all r > 0. Here vr,E is a continuous seminorm on Y ( G ) . This implies that i f f is as above and if Tg = (nlK)(Fg)then T,is summable on H(8.A.1.4) and llTglll I ve,,(f)llgllr for all r > 0. A direct application of the integral formula in 7.4.2 implies that if u, w E H then ( n ( f ) u ,w> =
j (n(g)T,n(g’)u,w >dg.
G
The above inequalities allow us to interchange summation and integration to find that O,(f) = j tr(MT,n(g)F 1 dg = j tr T,dg. G
G
This in turn implies that
In light of (iii), we may interchange the G and K integration and the summation. The theorem now follows.
8.3. HarishChandra’s regularity theorem on the Lie algebra 8.3.1. We retain the notation of the previous sections. If X E g and g E G then we write gX for Ad(g)X. If R is an open Ginvariant subset of g and if f E P ( Qthen ) we write ~ ( g ) f ( X=) f ( g  ’ X ) for g E G and X E Q. Set, as usual, D’(R)‘ = { T E D‘(R) 1 Tz(y) = T for g E G}. Let D be as in 7.3.9 and set g‘ = {X E g 1 D(X) # O}. Put R’ = R n g’. Let b l , . . . , b, be a complete set of nonconjugate Cartan subalgebras of g. We set R; = G(R’ n 4).Then R’ = RJ. If g E G and if H E R’ n bj then set y.(g, H ) = gH. 8.A.3.3 implies
u
(1)
y.is a submersion of G
x (Q’ n bj) onto Q;.
297
8.3. HarishChandra’s Regularity Theorem on the Lie Algebra
Fix d X , a Lebesgue measure, on 9. We will also write dH for a Lebesgue measure on each bj. As in 7.A.2, we look upon S(g) as the algebra of constant coefficient differential operators o n g. Put I ( g ) = S ( g ) G (7.A.2.8).
8.3.2. Fix a Cartan subalgebra, E), of g. Let @ = @(gc,bc). If a E @ and if a(b) c R (resp. a(b) c iR) then we say that a is real (resp. imaginary). Let @, and @, denote respectively the sets of real and imaginary roots. Set rR= OR, r, = i @ , and r = rR u r,. Put 6’’ = { H E b 1 a ( H ) # 0 for all a E r}.Clearly, E)” 3 6’. Lemma. Let C be a connected component of b“. Then there exist yl,. . .,yq E r such that y1 ,. . . , yq are linearly independent,
(1)
C = { H E E) I y j ( H ) > 0, j
(2)
= 1,.
.., 4 ) .
Furthermore, C n 9’ is connected.
If LY E then a(j(g)) = 0. We may thus assume, without loss of generality, that g is semisimple. Set [), = { H E bc a ( H ) E R for a E @)>. Then E) = (b, n E)) @ (ib, n b). If a E r R (resp. a E r,) then @(it), n b) = 0 (resp. a(bRn 5) = 0). Set ‘(bRn b) = { H E t), n b I a ( H ) # O for a E r R } and ‘((iE),) n b) = { H E ((ib,) n b) I a ( H ) # 0 for o! E TI}. Then a connected component of b” is of the form C, x C, with C,(resp. C,) a connected component of ‘(bRn 6) (resp. ‘((if),) n b)). Since r, and r, are both root systems the first assertion follows from 0.2.4. Set E = @  (0, LJ @,). If LY E C then it is clear that the real and imaginary parts of a are linearly independent. Thus ( b j ) a = { H E bj I o!(H) = O} is of codimension 2 in bj. Thus
I
C n (bj)’ = C 
u
Cbj)u
U E Z
which is connected.
8.3.3. For the rest of this section we will assume that Ad(G) acts trivially on the center of g. If . . , $d are homogeneous Ad(G)invariant polynomials on [g,g] then we set for r > 0 Q($l>...> $d>
I
= { X E [S, 91 Id)i(X)I < r, i = 1,.
Let U be an open connected subset of j(g) = j. Put Q Q($l > . . . 6 d > r ) } . 9
. ., d } .
=
{X
+ Y IX E U,
298
8. Character Theory
Lemma. R is connected. Furthermore, if t, is a Cartan subalgebra of g and if C is a connected component of b’ then C n R is connected. If X E U and Y E R ( ~ ,...,& , r) then X + ~ Y E Rfor O s t l l . This clearly implies that R is connected. We now prove the second assertion. It is enough to prove it in the case when g = [g,g]. Let B be a convex neighborhood of 0 contained in R n lj. Let C be a connected component of 5“. Lemma 8.3.2 implies that C is convex. Hence C n B is convex. If X E R n C then there exists t > 0 such that t X E B n C. Thus C n R is connected. The second part of Lemma 8.3.2 now implies the result.
Theorem. Let R be as in the previous section. Let T E D’(R‘) be such that dim l(gc)T< co on R‘. Then there exists an analytic function FT = F on R‘ such that
8.3.4.
(1)
T
=
TF on R‘ (see 8.A.2.2 for TF),
(2) l f f~ is a Cartan subalgebra of g then there exists an analytic function p on 6‘‘ which is an exponential polynomial on each connected component of If (8.A.2.10) such that Flnnbs = 1D1”28.
Furthermore, if we extend F to R by setting F = 0 on R  R‘ then F is locally integrable on R.
We may assume that Ij = bj. 8.A.3.5 implies that Y Y ( p T )= IDI”2plD1”2YY(T) for p E I ( g ) . Thus dim Z(9)(ID1’’2YY(T)) < co. We have seen that S(bC)is finitely generated as an l(g)module. Thus dim S(b,)YY(T) < 00. Lemma 8.A.2.10 implies that there exists a function pj on R‘ n Ijj whose restriction to every connected component is an exponential polynomial and is such that Y:(T) = ID1”2T@I.If X E R; with X = g H , H E bj, then set B ( X ) = 13,(H). If F = IDI”2p thenT = ‘ 7 on R’. We note that if we extend 8 to R by 0 then fi is locally bounded. We have seen (7.3.9) that 1D)”2 is locally integrable. The last assertion now follows. Lemma 8.3.3 implies the asserted extension properties of each pi. We now come to the main result of this section which is an extension of a fundamental theorem of HarishChandra. Let X , , . . ., X , be a basis of g and define X j by B ( X i , X j ) = hlj. Put 0 = 1 X i x i . Then 0 E l ( g ) .
8.3.5.
299
8.3. HarishChandra's Regularity Theorem on the Lie Algebra
Theorem. Let R be us in 8.3.3. Let T E D'(C2)' be such that dim I(g)T < 00 on R' und dim C [ l J ] T < a3 on Q. Let F = Er (8.3.4). Then T = TF.
The proof of this result will take up the rest of this section. Before we enter the details of the proof, we first develop some results on distributions on R that are supported in U @ Jlr(8.A.4.2). We note that if f is a Ginvariant polynomial on g then f ( X ) = f ( X , J for X E g (see 8.A.4.1 for Xs). Thus R n ( 3 @ J V ) = U @ A:
8.3.6. Until we specify otherwise we assume that g = [g,g]. Let .A'" = 0, u 0, u".u 0, with Oj = GX, and 0, open in N, 0, open in JV  O,, etc. (Corollary 8.A.4.7(2)).Set A; = ui2,0i.Then SS, is closed in 8. We may assume that X = X j and that X is nonzero. Let H , X , Y be a standard basis for a TDS, u, in g (Lemma 8.A.4.1). As a umodule under ad, g is a direct sum of irreducible submodules, V", with dim V" = p m + 1 and p,,, is a natural number. The eigenvalues of ad k on I/" are simple and are given by p,  2k, for k = 0 to pm. The  p m eigenspace is g y n V" and X V " is the sum of the eigenspaces for ad k with eigenvalues strictly greater than pm. This implies that 9 = !Iy 0 IX591.
(1)
Set V = gy. If g E G and if Z E I/ then set @ ( g , Z )= g ( X + Z ) . Then dmg,& V ) = g(V + [X,g]) = 9. This implies that there exists an open neighborhood, V '  , of 0 in V such that X + V" c R and @ restricted to G x I/" is a submersion onto its image. We note that @(G x V " )n 4; is open in .A;;. Let W be an open Ginvariant subset of g such that W n lJ: = Oj. Let V;. = { Z E V" ( @ ( g , ZE) W for g E G}. Then V, is an open neighborhood of 0 in V" and @(G x 5)n .4= Oj. If X = 0 then we take V;. = R. The main result of this number is (2) Let Oj c R and X j E Oj. Let 5 be as above for X = X j . There exists a neighborhood, q, of 0 in V;. such that if we put Q j ( g , Z ) = g ( X + Z ) for g E G and X E Uj then
(i)
is a submersion onto an open neighborhood,
(ii)
= 0. RJn 4; I J'
(iii)
( X j + uj) n Oj = { X j } .
Qj,
of X in R.
It is clear that any open neighborhood of 0 in V;. satisfies (i), (ii). We must therefore show that we shrink V, to satisfy (iii). If X j = 0 take V, = uj. We
300
8. Character Theory
therefore assume that X = X j is nonzero. Let { X , Y, H } be as above for X . Let W" denote the sum of the eigenspaces for ad H on V" with eigenvalues strictly less than pm. Set W equal to the sum of the W". Then ad X is a linear isomorphism of W onto [X,g]. Ths implies that there exists a neighborhood, W,, of 0 in W and a neighborhood U' of 0 in vj such that x, Z
+ Qj(exp x , Z
)
is a diffeomorphism of W, x U' onto an open neighborhood of X in g. Let W, be an open neighborhood of 0 in W, such that ead('"l)X is a neighborhood of X in If we shrink W, and U ' we may assume that Qj(exp W,, U ' ) n c exp(ad W,)X. Suppose that Z E U' and X + Z E Oj. Then X Z E Oj n Qj(exp W,, U'). Thus X + Z = ead" X with u E W,. Hence, Qj(1,Z) = Qj(exp v, 0). This implies that v = Z = 0. Thus we may take U, = U' in order to satisfy (iii). We now assume that g = j 0[g, g ] . Let L$be as above. We will now use the notation vj for U 0 vj. We will also write Qj for the map by the formula in (2) above.
4.
+
8.3.7. Let E be the vector field on g defined by
If x , , . . ., x, are linear coordinates on g such that (xi}i5qare linear coordinates on [g,g] and are coordinates on j then
Lemma. Let F be the space of all distributions supported on (j0 A'") n R. If T E F then dim C [ E ] T < cc and the eigenvalues of E on F are all real and strictly less than q/2. Let j be fixed and let Oj c R. Let X E j 0Oj and let Q j , V,, V, Rj be as in 8.3.6(2). Assume that Oj # {O). Let y,, . . ., y, be linear coordinates on V n [g, g ] such that yk(V n V " ) = 0 if rn # k. If 2 E V, write Z = C Z , with Z , E V n V". We note that ad H Z , =  p m Z m . It therefore follows that
=
Qj(g,Z)
for g E G and Z
E
vj.
30 1
8.3. HarishChandra’s Regularity Theorem on the L i e Algebra
Since Qj is a submersion, we may define 0: as in 8.A.3.2(2). (1) implies
(2)
Q:(ET)
=
(1
(+pm
+ 1)Yrna/aytn)m,9(T).
The choices in 8.3.5(2) imply that if supp(T) c ( 3 0 4 ) n R then supp Qg(T) c U x (0). Let F j denote the subspace of those elements of F with support contained in ( 3 0 . 1 n R. We prove by downward induction that if T E FJ then dim C [ E ] T < cc and the eigenvalues of E on F;. are strictly less than q/2. We assume (as we may) that 0, = (0). Then 8.A.5.4 implies that E acts semisimply on F, with eigenvalues strictly less that q < q/2. Assume the result for F;.+l we prove it for FJ. Let T E 4. Then Q:(T) has support in U x ( 0 ) . (2) combined with 8.A.5.4 implies that there exist a,, . . . , a, E R such that  a , > d C p, such that @y(n( E  a,)T)= 0. Now C (p, + 1) = q. So C p, = q  d. Thus a, 2 +(d + 4). By the above supp((Il, ( E  a , ) ) T )c F,, The result now follows.
+4
,.
8.3.8 Set ta(Z + X) = B ( X  , X ) for Z E 3, X E [g,g]. Let Xibe a basis of g such that xi E 3 for i > 4 and B ( x i , x j )= cidij with E~ = k 1. Set 0, = C,,,c,d2/Sx~ and 0,= 0  0,. We look upon o as a differential operator under multiplication. Set h = E (4/2)l, x = *o and y = 0,. Then a direct calculation yields
+
(1)
[h,X]
=2
~ [ .h , 4’1 =

2y, [x, y ]
=
h.
In other words, x, y , h is a standard basis for a TDS, LI. F is a umodule that satisfies the hypothesis of Corollary 8.A.5.1 (Lemma 8.3.7). Hence Corollary 8.A.5.1 implies Lemma. If’ T E F and if p is p ( 0 , ) T = 0 implies that T = 0.
(1
nonzero polynomial in one variable then
8.3.9. We now record a result that will be used at the end of the proof of the regularity theorem. Lemma. If S E F and if p is u nonzero polynomiul in one variable such that p ( 0 ) S = 0 then S = 0. We first show that if S E F, ( E C and if (0  [ ) S = 0 then S = 0. Assume not. Let S = X s, with ( h  / ~ ) ~ S , = 0for some d. Then O=(O[)S= C (0,  [)S, + C O,S,. Let E. be minimal among the p such that S, is nonzero. Since ( h  ( p  2 ) ) d ’ 0S,, = 0. We find that 0, S, = 0. Thus
302
8. Character Theory
8.A.5.1 implies that S, = 0. This contradicts our definition of 1.This proves our assertion. We now prove the lemma by induction on the degree of p . If deg p = 0 then the result is clear. Assume the result for all nonzero polynomials of degree d  1 2 0. If deg p = d then p(t) = (t  [)q(t) with E C and q is a polynomial of degree d  1. Thus 0 = p ( 0 ) S = (0 [)(q(O)S). Our assertion above implies that q ( 0 ) S = 0. The inductive hypothesis now implies that S = 0.
8.3.10. We now prove Theorem 8.3.5 in two special cases. g = su(2) 0 3 and g = d ( 2 ,R ) 03. We do this for two reasons. First of all the proof we give in these two cases contains most of the ideas in the proof of the full theorem. Secondly, these two cases are needed to initialize the induction that will be used to prove 8.3.5. In both of these cases we prove 8.3.5 under the assumption that Ad(G) = Int(g). So we take G = SU(2) if [g,g] = su(2) and G = SL(2,R)if [g, 91 = 542, R). The rest of this number is devoted to the proof of the result in the case g = 5 4 2 ) 03. We note that g  g’ = 3 @ {O). Thus T  TF is supported on U 0(0). Let 6 be the element of D‘(su(2))given by 6 ( f ) = f(0). Then Theorem 8.A.5.2 implies that if xl, x2, x3 are linear coordinates on su(2) and if we use multiindex notation then T  TF=Ca’6@T,
(1)
a finite sum with
r, E D’(U).
Since T and TF are invariant under the adjoint action of SU(2), we see easily that there exist To,.. . , T, E D ’ ( U ) such that
We now compute OT,  TGIF. We shall see in the general case that this is one of the key steps in the proof.
OTF
(3) Let 4
E

TLIF= 6 @ S
with S E D’(U).
C?(Q). Put p ( 4 ) = OTF(4) TLJF(4). Then
P ( 4 ) = j F ( X ) W ( X ) W X ) 4 ( X ) d X . 9
Set
“_[i 0
0i 1 .
Then b = Rh 03 is a Cartan subalgebra of g and all Cartan subalgebras of g are conjugate under Ad(SU(2))to b. We can apply the Weyl integral formula
303
8.3. HarishChandra’s Regularity Theorem on the Lie Algebra
(2.4.3) to find that (up to a scalar multiple) co
p($)
=
j J t2F(th+ Z )J
G
3 oc
U$(t Ad(g)h
+ Z)d g d t d Z
+
Set O+(t,Z)= t JG $ ( fAd(g)h Z ) d g . Then O+E C:(R), (a/at)O+(O,Z ) = 4(O,Z).We also note (7.3.3(l), 7.A.2.9)
@+(O,Z)= 0,
+ U,O+(t,Z) and t O F ( t h + Z ) =  ( a 2 / a t 2 ) t F ( r h + Z ) + tO,F(th + Z ) . OL+(t,2)= (a2/i3t2)O+(t,Z)
Set Q(t,Z )= t F ( t h + Z ) . Then ?I
p(4) = J j { Q ( t h + Z ) ( ~ 2 / ~ r Z ) ~ ~ ( t h + Z )  ( $ 2 / $ t 2 ) Q ( t h + Z ) O + (dtdZ th+Z)} 3
 1
30
+J J 3
{Q(th
30
+ Z)O,O+(th + z ) OoQ(rh+ Z)O@ + Z ) } d t d Z .
The properties of F in 8.3.4 imply that the second integral above is 0. We calculate the first by integrating by parts twice. If f is a function on t, such that J” restricted to (0,m) x j extends continuously to [O,m) x j and f restricted to (  m, 0) x j extends continuously to (  m,O] x 3. Then set l ( Z )= lim f (k t h Z ) . The obvious integration yields
+
1o+
~ ( 4=) J (Q+( Z ) Q  ( Z ) ) 3
The above calculations of @+ now imply that if we set for f E Cp(U), S ( f ) =  j ( Q + ( Z ) Q  ( z ) ) J ( Z ) d z 3
then p = 6 0S. This proves (3). We now prove that T = TFon 51. The hypothesis of Theorem 8.3.5 implies that there exists a polynomial p ( X ) = X r + lower order such that p ( O)T = 0. Thus p(O)F = 0 on 51’. We note that (2) implies that there exist distributions So,.. ., Sr on U such that
P(O)TF= Tp,, ) F
+ i s1 r
1
( 0 , ) i 6 O Si
304
8. Character Theory
Hence
= 
c
isr 1
(0,)'6@Si.
On the other hand (2) says that
If we compare the two formulas (for the same distribution) we find that the coefficient of (!31)d'r6 in the second formula is Td whereas the highest derivative of 6 that occurs in the first formula is ( 0l ) r  6. This implies that & = 0. The argument can now be iterated to show that T, = 0 for all j . This completes the proof in this special case. 8.3.11. We now look at the case g = d(2, R)0j. This case will be done in essentially the same way as the previous one. However, there is the additional complication that ,,Y' is not just 10). We now begin the analysis in this case. We set G = SL(2,R). We note that if X E d(2, R) then the characteristic polynomial of X is t Z + det X . Thus if det X is nonzero then X is regular. This implies
QQ'=U@.N:
(1)
We therefore find
(2)
SUPP(T  TF) c U @
Let F denote (as above) the space of S E D ' ( Q ) such that suppS is contained in U 0A: Let E be as in 8.3.7. In that number we proved that if V E F then dim C [ E ] V < m and the eigen values of E on F are all real and <  3/2. This implies (3) There exist I*,, i = l , . . . , q such that 3/2 > i, > A2 > ... > A,, T, E F such that ( E  i,)"T, = 0 for some m with T  TF= q.
As in the previous case the key to the argument is the calculation of
OT',  TLIF= p. Set
8.3. HarishChandra's Regularity Theorem on the Lie Algebra
If f
E
C:(51(2, R)) then set ( K
=
305
SO(2)) a
6,( f )
=
JJ
f (is Ad(k)X) ds dk.
K O
We also denote by S the evaluation at 0. We prove
(4)
There exist distributions S,, S  and So on U such that
p=6+0S+ +60S
+soso.
Set h = ( l0
H=('
 I o).
0
"). 1
Then R h O 3 and RH 0 3 is a complete set of nonconjugate Cartan subalgebras of g. If we apply the Weyl integration formula then we find that p = c l p l c 2 p 2 with e l , c 2 constants and
+
m
tZF(th
J
pl(+) = j 3
+ Z) J O+(t Ad(g)h + Z ) d g d t d Z G
OD

j
is
3 m
t20F(rh
+ 2 )J $J(tAd(g)h + Z)dgdtdZ, G
co
1 t 2 F ( t H + Z ) J Q&t
pZ(4) = J 3

17
3 m
We put for
+
E
G
In
t20F(tH
Ad(g)H
+ Z)dgdtdZ
+ Z) J + ( t Ad(g)H + Z)dgdtdZ. G
Cp(Q) l@b(t,Z= ) t J 4(fAd(g)h G
2@4(t,Z)= ItI
We note that K
=
J +(r
G
+ 2)dg
and
Ad(g)H + Z)dg.
SO(2) = exp Rh. We set for 4 a continuous function on g $'( Y)
=
J +(Ad(k) Y)dk. K
Then as in 7.5.1 we find that
306
8. Character Theory
If we make the change of variables u
It( sinh 2s we have (as in 7.5.1)
0
1 u
This easily implies that if (**)
=
u
+ (UZ  t
+
(u2 
0
y
t
y
)+
Z)du.
4 = u 0t’ then
,@(O+,Z)= G+(U)U(Z),1 @ ( 0  , Z )
=
6_(U)U(Z)
@,(O,Z)= U(O)U(Z).
and up to a scalar multiple
We apply these results to the calculation of pl.As in the previous case if we set Q(t,Z)= t F ( t h + Z)then a,
pi($) = J
J
3 m
+j
{Q(t,z)(a2/at2)i(D,,(t,Z) ((a2/at2)Q(t,z),@(t,z)} dtdZ
7 (Q(t,z)no(l@,,(tJ))

3 m
(o,Q(t,Z)),@,(t,z))dtdZ.
The second term is 0 since there are no jumps in the Zvariable. If we integrate by parts in the first integral (twice) then (**) implies that p, has the desired form. We now analyse p 2 . 7.3.8(2) implies that 2 @ ( t , Z )=
m
J
+ sx + Z ) d s .
$O(tH
x
Hence 2(D(t,Z)is smooth and (by the original formula) even in t. This implies that if 4 = u 0u then (***)
2 @ b ( 0 , Z )= (6,
+ &)(U)U(Z)
(a/at),@,,(O,Z)= 0.
and
Repeating the argument above in this case shows that p 2 has the desired form. (5)
E6,
=
26,
and
E6
=
36.
The last equation follows from 8.A.5.3. We leave the first as an exercise to the reader (hint: use linear coordinates x l , x 2 , x j such that x l ( X ) = 1). As in the previous case, (5) implies
(6)
okT, 7; k p =
jsk I
(n,)jh+ @ A j t
for appropriate A j , Bj, Cj E D’(U).
8.3. HarishChandra’s Regularity Theorem on the Lie Algebra
307
+
Let p ( x ) = xr lower order be a polynomial in one variable, x such that p ( O)T = 0. Then (7) implies
+
c
jsr 1
(n,)Jmq
with S j , V,, V, E D’(U). Since p(O)T = 0, p ( 0 ) F = 0 (recall that we are extending F by 0). Hence
So(7)impliesthatp(O)(T &.) = C  2 2 j 2  2 r + lW j w i t h E y = jWj.On the other hand (4) implies that
If we expand this in terms of the generalized eigenvalues of E we find that the term that corresponds to the lowest eigenvalue is (0 The corresponding eigenvalue is  2r + iq. Since 2, <  3/2 the above equations are consistent only if (0l)rq= 0. Lemma 8.3.8 now implies = 0. Continuing in this way we find that TJ = 0 for all j . Thus T = q.as asserted. This completes our discussion of the two special cases.
0 such that 14i(X,)I < r  E for all i. We may therefore choose an open neighborhood, U , , of X , such that Cl(U,) is compact and Idi(Y)I< r  E for all Y E Cl(U,) and all i. Set C = SUP%,”l(&k,/l ~ , , ~ , ~ u ~ ( Set Z ) l ) t. = ~ / 2 C b . Put ,Q(u, ,..., u b )= { Y E [ ~ ~ , ~ ] I I u ~ 0. Let U , be an open connected neighborhood of X, with compact closure in U , such that Iq( Y)l > A / 2 for Y E U , . If we argue as above we may choose 0 < t’ 2 t such that if Y, E U , and if Y E, R ( U , , . . , , u b , t ’ ) then Iyl(Y, + Y)l > A / 4 > 0. Set ,R = U 0 U , 0,R(ul,. . ., U b , t ’ ) . Then ,R is a neighborhood of X in m” n R. If g E G and if Y E ,R then set Y ( g , Y) = g Y. Then Y is a submersion onto an open neighborhood W of X in g. Fix a choice of Lebesgue measure on m. Let ,O be the “0” for m, B. Then 8.A.3.5 implies that
+
+
dim I(mc)lyl11i2Yo(T)I,n, < 00 and dim C[,0]lql”2Yo(T) < co. The inductive hypothesis now implies that T = TF on W. This contradiction implies (1). We note that if p E I ( g , ) then FpT = pF,. Thus supp(p(T TF))= supp(pT  p T , ) and supp(pT  TpF)are contained in A‘“ @ U by (1). This
8.3. HarishChandra’s Regularity Theorem on the Lie Algebra
309
imp1ies If p
(2)
E I(g,)
then SUPP(PTF T,F) c JV 0U.
8.3.13. As in the two special cases of the theorem that were proved above we must calculate OT,  TLF.With the inductive hypothesis in hand and since dim[g,g] > 3 (because of the two cases above) we can actually prove OTF =
(1)
TUF.
Let H , , . . ., H, be the Cartan subgroups of G corresponding to b,, . . ., 6,. We will be using the notation in 7.3.6 and the formula in the proof of 7.3.9. Let 4 E Cp(s1).Then
=
1cj J E j ( H ) ( n j ( H ) F ( H ) o @ , , H J ( H )K j ( H ) O F ( H ) q J ( H ) ) d H . 
h;
Here c j is a locally constant function on h” that takes on the values k 1. Let O j be the element of S(hj) defined as in 7.A.2.9 for 0. 7.A.2.9 implies that the expression that we are calculating is equal to cj
S E ~ ( H ) ( ~ ~ ( H ) F ( H ) O @ O~ j((Hn j)F ) ( H ) @ t ~ ( H ) ) d H . 
6;
Set Q j ( H ) = & , ( H ) n j ( H ) F ( H )Then . Theorem 8.3.4 implies that Q j extends to an exponential polynomial on each component of 6;. Let Cj,k be a labeling of the connected components of 6;. We must therefore calculate Ij,k =
S (Qj(H)Oj@,,HJ(H) njQj(H)@,,HJ(H))DH. h;’
Let y I , . . .,y, E r be such that Cj,k = { H E bj I y,(H) > 0, 0 I i I p } . (see p } . Set U j , k = vjtkin bj Lemma 8.3.2).Set 6 . k = { H E bj I y , ( H ) = 0 for 0 5 i I relative to B. Let h , , . . ., h, E q , k be defined by yi(h,) = hi,,. We also note that if H y zis defined by B(Hy,,H ) = y , ( H ) for H E [ j j then Hyi E Uj,,:Let X , , . . . , X , be a basis of 6 , k such that B ( X a ,X,) = q,fi,,b with qa = 5 1. Set 0; = C q , X i . Then 0 = C h,Hy 0;. Since neither Qj nor @p has jumps in the directions in 6,k (7.3.8) it follows that
+
S
Q j ( H ) O J @ p ( H ) d H=
,J,k
We therefore find that
o>Qj(H)@P(H)dH. cj.k
310
8. Character Theory
If we use the y j as coordinates on q , k then C j . k = q,kx ( x p(O, 03)). If 1 I s I p then set C j . k , s = { H E bjl y,(H) = 0 and y i ( H ) > 0 otherwise}. If u is a continuous function on C l ( C j , k ) then define u:(H) = lim,,,, u(H + th,) for H E C j , k , s . If we integrate by parts twice in the above expression for Ij,k then we find that
+ Ca e a . j . k C Jf, k , a ( Q j l l f
(H)(Hy,@,,HJ)J ( H I dH
with da,b,j,kand ea,j,kconstants and dH some choice of Lebesgue measure on the pertinent hyperplanes. We are now ready to prove (1). Let u E Cm(R) be such that u ( x ) = 0 for x I 1, u ( x ) = 1 for x 2 2. Set for E > 0, [ , ( X ) = u ( o ( X ) ~ / E Since ~ ) . [, is 0 in a neighborhood of U @ 8.3.12(1)implies that if f E C?(Q) then p ( [ , f ) = 0 for E > 0. We note that Ely,[&= ( 4 y i o / 2 ) u ' ( W 2 / & 2 )
which vanishes on the set Ij.k([cf)
=
Cj,k,i.
Thus
C
d a , b ,j , k
i
Ca e a ,j , k J
a,b
J
cj.k.b
i,(H)(haQj)b+(H)(@F) db H+(H)
CJ.k.o
Ce(WQj)lf
(ff)(Hya@F)J (H)
If X E g and if E > 0 is sufficiently small then l e ( X )= 1 if w ( X ) is nonzero. Let x be the characteristic function of the set { X E g I o ( X ) # 0). Then limc+oi,= x.Our hypothesis on g (dim[g, g] > 3) implies that xis one almost everywhere on each C j , k , . . Thus the dominated convergence theorem implies that lime+,oI j , k ( [ , f ) = I j , k ( f ) . We therefore find that
This completes the proof of (1). If p E Z(g) then FpT = pF,. Thus (1) implies
(2) If p is a polynomial in one variable then p ( 0 ) T F= Tp(0)F. 8.3.14.
We are now ready to complete the proof of the inductive step and hence of the theorem. Let p be a nonzero polynomial in one variable such that p ( O ) T = 0. Then p(n)(T 
TF)
= p(OlT  p(n)TF
=
p(n)TF
=
 T p ( , j)F
=0
since p ( 0 ) F = 0 and R' and we are extending by 0. Now supp(T  TF)c U @ M Hence Lemma 8.3.9 implies that T  TF = 0 as asserted.
8.4. HarishChandra’s Regularity Theorem on the Lie Group
8.4.
31 1
HarishChandra’s regularity theorem on the Lie group
The next theorem is one of the most profound results of HarishChandra. After its statement the remainder of this section will be devoted to its derivation from the main result of the last section. Let d ( g ) be as in 7.4.1 1 and set G’ = (g E G I d ( g ) 0).
8.4.1.
+
Theorem. Let T be an invariant Z(g)finite distribution on G. Then there exists a locally integrable function F = F , that is real analytic on G’ such that T = TFon G. If H is a Cartan subgroup of G, if h E H and if C is a connected component of { X E 1) I h exp X E H n G’} then X + F(h exp X ) l d ( h exp X)1’l2 is the restriction of an exponential polynomial to C .
Set H‘ = G‘ n H. Let $(g, h) = ghg’ for g E G and h E H’. Then $ is a submersion of G x H’ onto an open subset U of G. Let AG,H = A and 6G,H= 6 be as in 7.A.3.6. We use the notation of 8.A.3.2. If z E Z ( g ) then $O(ZT)= S(z)$’(T). Since 6 ( z ) = A I y(z)Aand since U (1)) is finitely generated as a y(Z((1))module, we see that dim U(b)A$O(T)
0 (in the indicated range of indices. If U is a connected neighborhood of 0 in j(g) such that exp is a diffeomorphism of U onto its image then exp is a diffeomorphism of V@R(D,, . . .,On n  1) onto its image (7.6.3, notation as in 8.3.3). If 0 < t < 1 then
,,
tR(D,, .. . ,D, l , n  1) =) R(D,,. . .,Dn ,,t n  ' ( n  1)).
This implies that if Y E R(D,, . . . ,On t" '(n  1)) and if A is an eigenvalue of ad Y then 111 < tn (7.A.1.3,7.A.1.4). We may now argue as in 8.3.12 to find a neighborhood, R, c I/ 0R(D,, . . . ,On t" '(n  1)) of the desired type of 8.3.3 for m. If we take t > 0 sufficiently small then it is clear that U , = R, has the desired properties. Set R = $(G x (XU,)). Let j be the "j function" for M (see 8.A.3.6). Set ((9,u ) = $(g, xu) for g E G and u E U , . Let ( ' ( T ) E D'(U,)Mbe as in 8.A.3.2. We note that M" n G' = M" n M ' where M ' is the set of all regular elements in M relative to its action on m. 8.A.3.2 implies that
,,
dim ~,.,(Z(g))Y0(T) < 00. Now, h f ( 4 Y 0 ( T )= (AG.M)
lYG.M(Z)AG,MSo(T).
This implies that dim Z(rn)AG,,io(T) < a, since z(m) is finitely generated as a yG,M(Z(g)module. We can now apply Lemma 8.A.3.6 to see that dim I(m)jl/' exp*(A,,,('(T))
on U , n m'.
< 00
+
If C, is the Casimir operator of m then Y ~ , ~ (=CC), 11 with A E C. Let 0,be the corresponding "0" for m. 8.A.3.7 implies that dim C[0,]j112 exp*(AG,,C0(T))
0 such that C
(1
+ a(g))dE(g)dg
1. If g E G, x , y E '(OH)A: and i f 9x9 = y then there exist k E N,,,(OH), h E H such that g = kh. (Here M is a standard Levi factor for P.)
The preceding Lemma implies that g normalizes H. Since H is &stable, this implies that g E K H . We may thus assume that g E K . Let x = ua, y = u l a l with u, u1 E '(OH) and a, a , E A : . Then gxg' = y implies that O(g)0(x)O(g)' = O(y). Hence g0(x)g' = 8(y). This implies that gx0(x)'g
1
= y0(y) 1.
So ga2gfollows.
' = a:. This implies that g E P. Now P n K
8.5.10.
Fix y E O H . Let Co c
=
M n K , so the Lemma
be open, convex and such that
exp:C,
+
exp C ,
=
C,
is a diffeomorphism. We also assume that Cl(C,) is compact and that y exp (Cl(Co))c '(OH).Let for t > 1 Y : G / H o x yCl x A:
+G
be given by Y ( g H o , y c , a ) = gycag'. Lemma. (1)

There exists an open neighborhood V of 1 H o in G / H o such that Y(V,yC,,A:) = R, is open in G.
319
8.5. Tempered Invariant Z(g)Finite Distributions on G
Y :V x yC, x A:
(2)
+ Q,
is u difleomorphism.
Y is everywhere regular. Thus Q, is open for all choices of I/. Thus we need only show that we can choose V so that Y is injective. If
V ( g H 0 ,yc, a) = Y(xHO,yc',a')
then gycag' = xyc'a'x'. If we set u = x'g then 8.5.9 implies that u = kh with k E M n K and h E H o . But then a = a' and kak' = a. Now this implies that mod O H o the possible "k's" vary in a discrete set. Since g H o = x k H o it is clearly possible to choose V so small that k E H o . 8.5.11.
Fix V, C, y, t > 1, Q, as above. Let c1 E C,oO(V)be such that
j
G/H"
a ( 2 ) d x = 1.
Here we have chosen a Ginvariant measure, d x , on G / H o . Let Y ( y C , A : ) denote the space of all ,f E C"(yC,A:) such that with w c yC,A: compact and s > 1.
supp f c w A i
(1) (2)
Vd,D(f)
=
sup,(l
+ a(h))dlDf(h)l< co
for all d 2 0 and all D
E
U(b).
We say that a net f, + f in this space if
( 3 ) There exists w c yC,A: compact and s > 1 such that supp f, c o A ; for all large q and vdTD( f,  f )+ 0 for all d 2 0 and D E U ( 6). We define S : Y ( y C , A : )
+
CXJ(Q,) by
S ( f ) ( Y ' ( x yc, , 4)= 4 w y c a ) .
Lemma. I f f
E Y(yC,A : )
and if we extend S(f ) by 0 to G then S ( f ) E C"(G).
We use the notation xyx' = y" and { y " l x E S) = ys. Let E c G be compact and such that E H o = supp c1. Then supp S ( f ) = (supp f )". (Here S ( f ) is looked upon as an element of C"(Q,).) Now supp f c w (CI(A:)) for some t' > t. Thus supp S ( f ) is closed in G and the result follows.

Lemma. Let gR(G)and the map
8.5.12.
P(f)
=
l D l  ' " S ( f ) for f
P:LY()JC~A:) is continuous.
%?R(G)
E
,Y(yC,A:). Then
p(f) E
320
8. Character Theory
Let R
I
= C[(1  h") c1 E @] c
4 ( h )=
C"(yC,A:). Put
Iurnm+
(1

h")
JL
.
It is an easy exercise to show that if D E U ( b ) then D 4 = 4'f with r a positive integer and f E R[h", conj(h") I a E @+I= R". We also note that ID(h)l' = aO4(h) for h = yca, c E C,, a E A'. We will now be using notation and results in 7.A.3. If g E U ( g ) and if h E @ , A : then ghX = rh(cLjai @ bj)
Thus if x
E
with fij E R", a, E
u(g)and bj E u(b).
E (see the previous number) then
' ( x h x  ' ) ( l + o(xhx'))"lg
G
*
B(f)(xhx')l
+ o(h))" 1 Ifli(h)llaic1(x)llbj(hP4f)(h)l I Const. (1 + rr(h))" 1 Iuir(h)l14'Dif(h)l I Const. Z(h)'(l
with Di E U ( b ) ,uir E R". Here the "consts" depend on E but not on f . This proves the result since the elements of R " [ 4 ] are easily seen to be bounded on yC,A: fort > 1.
8.6. HarishChandra's basic inequality The following theorem of HarishChandra is the key to the study tempered, invariant eigendistributions.
8.6.1.
Theorem. If TE D'(G) is central and Z(g)$nite then T is tempered if and only if there exist constants C and d such that if H is a 8stable Cartan subgroup of G then C(l ld(h)11'21FT(h)lI
+ a(h))d
for h E H .
The sufficiency of the condition is an easy consequence of Theorem 7.4.10 and will be left to the reader. We now begin the proof of the necessity. If z E Z ( g ) and if h E H' then (7.A.3.7)
z * F,(h)
=
A(h)'Y(z)(AFT)(h).
Here we have used A corresponding to a choice of positive roots in @(gc,bC).
32 1
8.6. HarishChandra's Basic Inequality
This implies that dim U ( [ i ) A(FT,,,) . < co.
(1)
Let f = A(FTIH,). Then (1) combined with 8.A.2.10 (more precisely proof of that result) imply that there exist AI, . . . ,A, E I),*and an integer d 2 0 such that if 'J E H' and of 0 E U c I)is open, connected and such that y exp U c H ' then (2)
1
f ( y exp h) = pi,y(h)eA1'h' with pi,?a polynomial of degree at most d depending only on 'J and T.
Let Let H = ' H A , as usual. Set A" = { u E A I a" # 1 for ct E 0  0,}. ( c . , A ) , i = l , . . ., p, be the ppairs with split component A . Set i A + = { U E A I U ' > l , A ~ @ ( ? , A ) } . T h e nA " = U i A + . P u t ' ( o H ) = { h E o H I h a # l for all c( E @,}. We have seen that H"
(3)
=
'(OH).A" c H'.
Clearly, H" is open and dense in H. Let C be a connected component of ( H). Then C iA' is connected. Thus if p,, . . ., p q are the distinct restrictions

$ 0
of the Aj to a then (4)
f ( c a ) = C +ijk(c)pj(logu)aWk,
c
E
C and u E i A + with
+ijk
i,k
a function that extends continuously to Cl(C) and pj a polynomial of degree at most d. Thus to prove the theorem we must show (5)
If
1 &ijk(c)pj(loga) f 0 then Re pk is nonpositive on i a + . j
8.6.2. We now prove (5) above. Fix a ppair (P,A ) ( A as above). Let y E OH, C,, C , , A + and as in 8.5.12. We write (A' = iA+)4ijk(c)= & j k ( y ~ )(notation as in (4) above).
a
Let 4 E C,Z(yCIA+).Then it is clear that
4 E .Y(yC,A:) for some t > 1. Now
322
8. Character Theory
Set v(h) = ld(h)ll”/A(h) for h E H ’ . Then Iv(h)l
=
1 for h E H ’ . We have
8.6.3. Corollary. Let T E D‘(G) be central and Z(g)finite. Then T is tempered if and only if there exist constants C 2 0 and d 2 0 such that
+
~d(x)~l’*~I F ~C(( x1 j ~ a ( x ) ) d for x
E
G’.
The preceding theorem implies the sufficiency of this condition. We must therefore prove the necessity. Since there are only a finite number of conjugacy classes of Cartan subgroups in G, it is enough to prove the inequality on G [ H ‘ ] = { ghg’ I g E G, h E H ’ } for a fixed &stable Cartan subgroup of G. We note that d(ghg’) = d(h). Thus the result will follow from the previous theorem if we can show that there exists a norm I(...)) on G such that if g E G and if h E H then
Ilshg’ll 2 llhll. To prove this it is enough to observe that if h E GL(n,C ) is diagonal and if
g E GL(n,C) then
tr(ghg’(ghg’)*) 2 tr(hh*).
We leave this exercise in linear algebra to the reader.
8.6.4. Corollary. Let T be a compact Cartan subgroup of G. L e t S E D’(G) be a tempered, central Z(g)finite distribution. If f E O%‘(G) (7.6.3)then S(f) = w
j A(t)Fs(t)F,T(t)dt.
T’
Let J E Cp(C) be such that limj+m = f in %(G). Let T, H , , .. .,H, be a complete set of representatives for the conjugacy classes of Cartan subgroups of G . Then S(J) = w
J A(t)FAt)FTJ(t)dt+ 1 ci J Ai(h)FT(h)F?(h) dh
T’
H;
by the Weyl integration formula and HarishChandra’s regularity theorem. For simplicity of notation we set HI = T.
323
8.7. The Completeness of the n,
Theorem 8.6.1 implies that lim
j
jco
H;
Ai(h)FT(h)F;l(h)dh=
j Ai(h)FT(h)FyL(h) dh. H;
This limit is 0 for i 2 2 since f the result.
E
' % ( G ) (7.5.4).This formula with i
=
1 implies
8.6.5. In the next section we will show how the above Corollary can be used to complete our discussion of the irreducible square integrable representations. 8.7. The completeness of the x , We continue with the assumption that G is of inner type and that G = 'G. We assume in addition that there exists T c K a Cartan subgroup (which we fix). We will use the notation of 6.9. If z E T" set A(z) = Ar be as in 6.9.3. Let = t, and set 0 = O(gc, 6). z is said to be regular if (A(z),a) # 0 for all a E 0.If z is regular then P = P ( z ) = { a E 0 I (A(z), a) > 0) is a system of positive roots. Set (n,,H') = (7cp,,, I f p , ' )in the notation of 6.9.4. Then (n', H ' ) is an irreducible square integrable representation of G. The purpose of this section is to prove 8.7.1.
Theorem. If w E G" is the class qf a square integrable representation then there exists a regular element T E T" such that n, E o.Furthermore, n, is equivalent with n,,if and only if there exists s E W, such that z' = zs. The proof will take some preparation which we now begin.
8.7.2. Fix a regular T E T " . Put P = P(z). Set V = ( H T ) KWe . first calculate @K,r (see 8.2). For this we must calculate mv(y)for y E K " . In the notation of 6.9.3 and 6.9.4 V
=
Indi,(T; M ( 6, E: ),
and
r; M(b, E,^ ) = xr
@ &,A(r).
Let yo E (KO)".Then Theorem 6.7.6 says that (A
=
A(z))
324
8. Character Theory
If y1 E (K')'' is given by ( , @ y o where z(z) = (,(z)I for z E Z then dim H o m K f ( V y l T;M(b, , E : ) ) is given by the right hand side of (1). If y E K" then y = Ind$(y,) with y1 E ( K ' ) ^ . Thus we see that if y,y1 and yo are related as above then
Suppose that ,f E C " ( K ) and that y calculate
=
Indtl(yl) with y1 E ( K ' ) " . We
j f(k)rl,(k)dk = tr W).
K
Since we have realized
T>, as
an induced representation it is clear that
Now dim V, < cc so it is easily seen that
K ' = Z K o and q,,(zk) hand side of ( 3 ) is
= [,(z)qyc1(k)for
z
E
Z and k
E
KO. Thus the right
We now apply the Weyl integration formula which yields (4)
j f(k)rly(k)dk = ( l z l I W ( K ,
r)l)'
K
1
z t Z To
IAK(t)12
j f(kztk')rl,,(t)i,(t)ddkdt.
K
Here we have chosen a system of positive roots @; c @( f,, b) and A K is the corresponding Weyl denominator. We will assume (for the sake of simplicity) that we have gone to a covering of G so that ( t + t p k )E ( T o ) ^ . The Weyl character formula combined with the fact that conj(A,) = (  l)"Akwith n = I@;[ implies (5)
(  1)"IZlI W K , U =
c
1
I 1f ( k ) u l , ( k ) d k
z t Z s€W(K.T)
K
det(s)
j
Tox K
A,(t)f(kztk~J)i,(z)ts~a~'+Pk)dtdk.
Here i y= A,". Let py.*be the character of the representation of T given by Z acting by z,,and T o acting by t". Then we have proved (6)
jK f ( k ) r l y ( k ) d k = (  ')" T jx K Ak(t)py.r(t)f(ktk
dt dk.
325
8.7. The Completeness of the n,
This in turn implies that
In 6.5 we have seen that the expression on the right hand side of (7) is given by &(t) (  1)"
1T
C
SEW(K.7)
det(s) tr(Ts(t))
4Sf)
K
f'(ktk')dkdt.
This implies (finally!)
8.7.3. Let f E '%'(G). Choose for each c > 0, 4e E Y ( G ) with supp dZE GZ,e and 4e = f in W(Gk). Then (see Theorem 8.2.2) @V(o&)
=
@K,V(y&)
x = (
s I1

W ( K , T)I
= (
d e W d ~ l J I ( A k ( ~ ) / A A Ntl r ( m
1)"'l W ( K ,T)I
j 4AYtSl)dSdt
G
1tr(s(f))Fir(t)dt. 7
As in the proof of 8.6.4 we have lim&+o@"(4&) = @,(f). Thus (1)
@,(f) = (  l)lpll W ( K ,T)I(F/T)"(T*) the dual representation of T.
with T*
Theorem 7.7.2 now implies that
(2) If .f E O%'(G) and if @ ( f )= 0
for all regular
T
E T" then j'(1) = 0.
8.7.4. We can now complete the proof of the main theorem in this section. Let o E G" be a class of an irreducible square integrable representation of G. If x, $ o for all regular T then if f is a Kfinite matrix coefficient of o then (2) above implies that f'(1) = 0 (we already know that j' E '%?(G)). This is ridiculous. Thus there exists T E T Awith t regular such that x, E o. If n, 'v x,,then the equality of Kcharacters implies that T ' = ST' for some s E W ( K ,T ) = W(C,T ) .This completes our determination of the irreducible square integrable representations of G.
8. Character Theory
326
8.A. 8.A.1.
Appendices to Chapter 8 Trace class operators
8.A.l.l. The purpose of this appendix is to develop the elementary aspects of the theory of trace class operators. Let H be a separable Hilbert space with inner product ( , ). An endomorphism, T, is said to be compact if T maps bounded subsets of H onto subsets of H with compact closure. In the literature, the term completely continuous is also used for compact. It is obvious that a compact operator is bounded. If T is a bounded operator on H with finite dimensional image then T is said to be of j n i t e rank. Obviously, an operator of finite rank is compact. We set L ( H )equal to the algebra of bounded operators on H. Then L ( H )is a Banach space relative to the operator norm IITII
= S~PII"II=lIIT~II
Let K ( H ) denote the space of compact operators in L(H). Lemma. K ( H ) is a closed ideal in L ( H ) .
It is clear that K ( H ) is an ideal in L(H). Let {q}be a sequence of operators in K ( H ) that converge to T in L(H). We show that T E K ( H ) . Let { f,} be a sequence in H with Ilf,ll I C . The diagonal process yields a subsequence {u,} such that T u n converges for each j . Let E > 0 and let r be such that IIT  T,ll < E for j 2 r. Let N be such that if rn, n 2 N then IIT,(u,  u,)ll < E. If m, n 2 N then IIT(un  um)ll =
ll(T  T,)(u,  u,)
+ T(u,

~,)ll
I IlT T112C
+
E
I (2C + 1 ) ~
The result follows from this.
8.A.1.2. If T E L ( H ) then we define T* by ( T v , w ) = ( u , T * w ) for all v , w E H. T is said to be selfadjoint if T = T * . The following result is standard. Lemma. Let T be bounded and selfadjoint. Then T is compact if and only i f there is an orthonormal basis { u j } of Ker T I such that (1)
Tvj = Ajvj with l j E R,
8.A.I.
327
Trace Class Operators
lim A j = 0.
isl
We may assume that Ker T = 0. We first prove the sufficiency. Set PN(u)= C j S N(u,uj)uj. Then TP, = PNT for all N . Let E > 0 be given and let N be such that [ A j \ < E for j 2 N . Then ll(T  P,T)uII I E I I u I I for all M 2 N . Hence IIT  P,TII I E for M 2 N . Since P,T is of finite rank, the sufficiency follows from the previous Lemma. We now assume that T is compact and selfadjoint. Let H , be the span of the eigenvectors of T. We show that H i = (0). Assume the contrary. Let m be the norm of T as a bounded operator on H i . There is a sequence {u,} of unit vectors in H i such that lim (Tuj, Tuj) = m 2 . Since T is compact, we may assume that lim Tuj exists and is a vector u E H i . Let Q denote the restriction of T to H i . Then llull = m = IIQII. If m = 0 our assertion follows. We therefore assume that m > 0. Set x = u/llull. Then we have
IlQll 2 IlQxll = lim IIQ2~jllIllQII2 lim (Q’Uj,~j>/llQII = IlQll Thus IlQxll = IIQII. Hence, IIQ112 = (Q2x,x) I llQ2xllI IIQII’. Schwarz inequality implies that Q’x and x are linearly dependent. Thus Q2x = IIQ112x. We conclude that if H i # 0 then T has an eigenvector in H i . This contradiction implies our assertion. Since T is compact, the eigenspaces for T are finite dimensional (we are assuming ker T = 0). We can therefore find an orthonormal basis for H that satisfies (1) in the statement. If (2) were not satisfied then there would be an infinite sequence { u j } of unit vectors in H such that Tu, = pjuj and IpjI 2 C > 0. Hence T would not be compact.
If T E K ( H )then T*T E K ( H ) . Let { u j } and { A j ) be as in the previous Lemma for T*T. We define an operator I TI as follows
8.A.1.3.
ITI(Ker T * T ) = 0,
lTluj = ( A j ) l i 2 u j .
The preceding Lemma implies that IT1 E K ( H ) . Let V = ITI(H), W = CI(TH). We define a mapping U of V to W by UlTlu = Tu for u E H . Then U is linear and IIUull = llull for u E V. We extend U to H by setting U(ker IT[)= 0. U is a so called partial isometry. We have proved the following standard decomposition (1)
8.A.1.4.
T
=
UlTl
Let T E L ( H ) . Then T is said to be summable if there exists
328
8. Character Theory
an orthonormal basis {w,,} of H such that
Lemma.
Let T E L ( H ) . I f { u j } is an orthonormal basis of H then
1 j
llTujll
j.k
ll.
Let wj be a unit vector for each index j . Then
Choose wj = uj if Tuj = 0 and wj follows.
=
Tvj/llTujll if Tuj # 0. The inequality now
8.A.1.5. Let T E L ( H ) . Then T is said to be of truce class if there exists an orthonormal basis { u , } of H such that
C IITvnII < n
The above Lemma implies that if T is summable then T is trace class.
Lemma.
I f T is trace class then T is compact.
Let {u,,) be an orthonormal basis of H such that X IITu,)I < co. Define ( u , uj)vj. Let 0 < E < 1 be given. Then there exists N such that
PkV = x j s k
If k > N then
If j 2 N then llTujll < 1, hence IITujl12< IITujll. We therefore conclude that
ll(T
 Tpk)uII
2
EIIuII*
Hence lim TPk = T. Lemma 8.A.l.l now implies that T is compact.
8.A.1.6. If TE L ( H ) then we say that T is of HilberrSchmidt class if there exists an orthonormal basis { uj} of H such that
329
8.A.l. Trace Class Operators
(1) If A is of HilbertSchmidt class then so is A*. Furthermore, if { u j } and {wj} are orthonormal bases of H then
1 IIAUjII’ 1 IIAwjI12. 1
Indeed, [lAWj11’
= C,
= c k I(Wj,A*U,)l2.
I(Awj,Uk)[’
1 llAwjllZ =
j,k
=
I(wj,A*vk)l’
Thus
1
IIA*ujll’.
(1) now follows.
(2) If B
E L(H) and if A is of HilbertSchmidt class then BA and AB are of HilbertSchmidt class.
Indeed, let {ej} be an orthonormal basis of H. Then C IIBAejlI2 I ((BJI’C IIAejl12. Thus BA is of HilbertSchmidt class. Now A B = (B*A*)*. So the second assertion follows from (1).
Lemma. Let T E L(H). Then T is of trace class if and only if T is compact and if ,Ij are as in 8.A.1.2 for I TI then C A j < 00. Let { u j } be an orthonormal basis of (ker [TI)’ such that Tuj = L j u j . Define = S by S(ker IT\)= 0 and Suj = ( A j ) l i 2 u j . Clearly
C IISujI12 = 1
Aj.
This implies that S is of HilbertSchmidt class if and only if
LAj < a. i
Suppose that T is of trace class. Let {ej} be an orthonormal basis of H such that X 11 Tejll < 00. Then
1 IITejII = 1 IIuITIejII = 1 IIITIejII 2 1 (ITIej>ej>= 1 IISejII’. This implies that if T is of trace class then S is of HilbertSchmidt class. The lemma now follows since it is clear that ker T = ker 1 TI and
1IITVjII = 1 IIITIujII = 1
ibj.
8.A.1.7. Lemma (1) Let T be truce class. If {en}is an orthonormal basis of H then X (Te,,en) converges absolutely and is independent of the choice of {en}.We set (Ten,en)= tr T.
(2) Let TE K ( H ) . Then T is of truce class if and only
if for
each choice of a
330
8. Character Theory
pair of orthonormal bases {e,}, { f , } of H
c I(Te,,ffl>l < Furthermore, tr IT1 < co.
if T is of trace class then the supremum of such sums is
Let {x,} be an orthonormal basis of H such that C 11 Tx,ll < co. Let { e n } be another orthonormal basis of H . Then
This implies that
Now C ( T x , , ~ , ) and C (Te,,em) are both rearrangements of the absolutely convergent series
1 (Txn,em)(em,xn).
m.n
This proves (1). We now prove (2). C I( Ten,f,)l = C I( UIT le,, f,)l. Let {x,} and {A,} be as in 8.A.1.2 for IT].Then the above formula implies that
by Schwarz's inequality. S.A.1.8. Lemma. Let T E L ( H ) . Then T is of trace class if and only can be written in the form A B with A , B of HilbertSchmidt class.
if
T
Assume that T is of trace class. Let T = UITI, as usual. Then [TI = S2 with S of HilbertSchmidt class (see the proof of Lemma 8.A.1.6). Clearly, e S is also of HilbertSchmidt class. Set A = US, B = S . Suppose that T = A B with A , B of HilbertSchmidt class. Let { e n }and { f,} be orthonormal bases of H . Then
C I(Ten7 f , ) = 1 I(ABen7 f, > l = c I(Be,,A*fn)l (91(IIBe.1l2 + llA*f,1I2) < 00. The result now follows from the previous Lemma.
8.A.2.
33 1
Some Operations on Distributions
Set L , ( H ) equal to the space of all trace class operators on H. If T E L,(H) then set llTlll = tr IT[.Lemma 8.A.1.7 implies that 11...11, defines a norm on L , ( H ) and that
8.A.1.9.
IlTll 5 IITIII.
(1)
We leave it to the reader to prove
Lemma.
L , ( H ) is a Banach space relatioe to 11...11,.
Lemma. I f T E L , ( H )und if'A Furthermore. tr AT = tr TA.
8.A.1.10.
E
L ( H )then AT, TA, T* E L , ( H ) .
Write T = XY with X and Y HilbertSchmidt. Then AX and YA are also HilbertSchmidt (8.A.1.6(2)). Now, AT = (AX)Y, TA = X(YA) and T* = Y*X*. The first assertion follows from 8.A.1.8. Let { e n }be an orthonormal basis of H such that C 11 TeJI < co.Then
C I 0. If X E U and if Y E g then we set T(X)Y equal to the element of g such that d exp,( Y ) = T(X)Y,,,,. Set gij(X)= B(T(X)X,,T(X)Xj).We assume that U is connected. Then y(X) = det(gij(X))> 0 for X E U. Set u(X) = s ( X ) " ~for X E U. On U we take the linear coordinates xi defined by the equation X = 1 x,(X)X,. Then if [ g ' j ] = [gij]' we have (exp*)C(exp*)'
= ul
C a
ax,
a
. ax,
grsu
r.s
34 1
8.A.3. T h e Radial Component Revisited
This formula is a direct consequence of the standard formula for the LaplaceBeltrami operator on a pseudoRiemannian manifold (see any book on Riemannian geometry). The standard formula for the differential of exp implies that
We note that
(3)
Y ( X )=
.w2.
Indeed, j ( X ) = det T ( X ) .Clearly, g ( X ) = (det T ( X ) ) ' . We now assume (as we may) that B ( X i , X j ) = cihij with D = (exp*)C(exp*)'  j''2A(C)j''2.Then D
=Jl
a
a
ax,
ax,
CE,E,B((T(X)T(X))'X,,X,)~r,s
E~
=
f 1. We set
a ax;
j1/2C~i2j1/2
D1 = O .
(4)
The previous result implies that D annihilates the Ginvariant smooth functions on g.
Thus D
=j
'cE E a

r,s

=j'
j1
c r,s
ax,
Es(
a
B((T ( X ) T ( X ) )  ' X r , X s ) j ax,
& & i)
a 2 
Es
a
E,E, ( B ( ( ( T ( X ) T(X ) )  '
ax,

a
z)xr,x,)j).ax, a
Now if F is a smooth function from U to g then C, &,B(F,X,)  is the ax,
vector field on U corresponding to F, Thus Zs&,B(((T(X)T( X))'

I ) X r ,X,) l ? is the vector field correax,
342
8.
Character Theory
sponding to the function ( ( T ( X ) T (  X ) )  '  I ) X r .We note that this function can be written in the form [ X , G,(X)] with G, an analytic function on U. We have therefore shown that (6)
D
= j'
1 ax,a j V , E, 
with V, the vector field corresponding to
r
[ X , Gr(X)I. The Lemma now follows from Scholium. Let G be a unimodular Lie group, let U be an open Ad(G)invariant subset of g and let Y be a vector jield on U of the form X H [ X , C ( X ) ]on U. Then Y T = 0 for all T E D'(U)'.
Let X I , . . . , X,, be a basis of g. Then G ( X )= C g i ( X ) X i .Thus Y = C gi y with y the vector field corresponding to X H [ X , X i ] . Thus Y T = C g i y T . Thus we may assume that G ( X ) = Z E g. If f E C""(U)then Yxf = d/dt,=, f(Ad(exp(  t Z ) X ) .Thus Y =  Y. Hence d 0 T ( f 0 Ad(exp t Z ) ) = Y T (f ). dt, = 0
This completes the proof. 8.A.4.
The orbit structure on a real reductive Lie algebia
8.A.4.1. Let G be a real reductive group with Lie algebra g. We continue to write g X for Ad(g)X for g E G and X E g. If X E g then we say that X is semisimple if ad X is semisimple on gc. If X E [g, g] and if ad X is nilpotent then we say that X is nilpotent. If X E g then ad X can be written uniquely in the form ad X = S N with [S, N ] = 0 and S semisimple, N nilpotent (Jordan canonical form). It is easily seen that S and N are derivations of g. Thus there exists X,, E [g, g ] such that ad X , agrees with N on [g, g ] . Set X , = X  X,,. Then ad X , is semisimple. We have proved
+
(1) If X E g then X can be written uniquely in the form X , [X,,X,,] = 0 and X , is semisimple, X , is nilpotent.
+X,
with
The key to the orbit structure of the action of G on g is the following Lemma of Jacobson, Morosov (c.f. Jacobson [l, Lemma 8, p.991. Lemma. If X is a nonzero nilpotent element of g then there exist H, Y E g such that [ H , X ] = 2 X , [ H , Y ] =  2 Y and [ X , Y ] = H.
8.A.4.
343
The Orbit Structure on a Real Reductive Lie Algebra
We first look at the case when g = d(n,R). If X E g is nilpotent then X is nilpotent as an endomorphism of R". The Jordan decomposition implies that there exists a basis if R" such that X is the direct sum of Jordan blocks 
0 1 0 ... 0 0 0 1 ... 0
i
I
0 0 0 ... 1 00 0 0. ... . 0
It is now an easy exercise to prove the existence of H and Y in this case. In the general case, we may clearly assume that [g, g] = g. If we choose a basis of g then ad defines an isomorphism of g onto a subalgebra of el(n,R) = g1 such that the form B(X, Y) = tr X Y is nondegenerate when restricted to it. We identify g with this subalgebra of g,. Let V = {X E g1 I B(X, g) = O}. Then g, = g 0 V and [g, V ] c V. Let X be a nilpotent element of g. Then there exist Y ' , H' E g1 such that X, Y ' , H' have the desired commutation relations. We write Y' = Z Yl and H' = H + H, with Z, H E g, Y,, H, E V. It is easily seen that [H,X] = 2X and that [Z, X] = H. Set gx = { y E g I [ X ,y] = O}. We assert that
+
(2)
ad H
+ 21 is invertible on gx.
Let us show how one completes the proof of the Lemma using (2). A direct calculation shows that [X, (ad H + 21)Z] = 0. Thus there exists W E g x such that (ad H + 21)W = (ad H + 21)Z. Set Y = Z  W. Then (ad H 21)Y = 0 and [X, Y] = H . We are thus left with the proof of (2). We note that (ad X)mg= 0 for some m > 0. We set (ad X)' = I.
+
(3)
(ad H

j l ) ( g x n (ad X)jg) c g x n (ad X)j' '9.
Indeed, if j = 0 then ad H(gx) = ad X ad Zgx c gx n ad Xg. If j > 0 and if y E g x n (ad X)jg then y = (ad X)'T with T E g. Hence ad Hy
=
ad X ad Z(ad X)'T (ad X)'ad H(ad X)'"T
=1 =j
2(i
(j
+ l)(ad X)'T
+ l)(ad X)jT
~
j ad H(ad X)'T
+ (ad X)'+'ad
ZT
+ (ad X ) j + ' a d Z T
j ad H(ad X ) j T + (ad X)j+'ad ZT.
344
8. Character Theory
This implies that ( j + l)(ad H  j Z ) y = (ad X)j"[Z, TI. Which implies (3). ( 3 ) implies that the eigenvalues of ad H on g x are contained in the set {0,1,. . ., m  I}. ( 2 ) follows from this. 8.A.4.2. Let j be the center of g. For the remainder of this appendix we will assume that Ad(g)X = X for g E G and X E j. Let I ( g ) denote the algebra of all Ginvariant complex valued polynomials on g. Set P ( g ) be the subalgebra of elements that vanish at 0. Lemma. Let A'" denote the set of .Ar = { X E g I I+(g)(X)= 0).
all nilpotent elements of g. Then
g = 3 0 [g,g]. If ;1E j* then extend ;1 to g by setting A([g,g]) = 0. Thus If(g). It is now clear that if X E g and if I+(g)(X)= 0 then X E [g, g]. We may thus assume that g = [ g, g]. j* c
(1) If X
E
sl(n,R) and if tr X j = 0 for j
=
1,. . . ,n then X is nilpotent.
This is well known and left to the reader. Let n = dim g . If we choose a basis of g then ad g c 41(n,R). The polynomials J ( X ) = tr(ad X)j are in Zf(g) for j > 0. Thus (1) implies that if I+(g)(X)= 0 then X is nilpotent. If X is nilpotent then there exists H E g such that ad HX = 2 X . Thus if f E Z+(g) then j ( X ) = f(Ad(exp(tH))X) = f ( e  " X ) for all t > 0. If we take the limit as t ,+ cc then we see that f(X) = 0. 8.A.4.3. Our next goal is to prove a basic result of Kostant. The proof will use the following Theorem of Whitney [ 11. Theorem. Let f i , . . .,f , be polynomials on R". Then ( x E R" I A(x) = 0 for i = 1 , . . .,m } has a finite number of connected components.
This theorem has an immediate corollary. Corollary. Let f be a nonzero polynomial on R". Then U has a finite number of connected components.
={xE
R" I f ( x ) # 0}
If x E U then set F ( x ) = ( x , f ( x )  I ) .Then F defines a homeomorphism of U onto { ( x , t )E R"+l I t f ( x ) = l}. This reduces the corollary to the Theorem.
8.A.4.
The Orbit Structure on a Real Reductive Lie Algebra
8.A.4.4.
34s
The following theorem is the result of Kostant [l] alluded to above.
Theorem. The set of nilpotent elements of 9 consists of u$nite number of orbits relative t o the adjoint action of G. The idea of the proof is to show that up to the action of Ad(G) there are only a finite number of choices for the “Hpart” of a TDS in g. We will then show that for each choice of an H , the stabilizer of H has only a finite number of orbits in the “Xparts”. Let H be an “Hpart” of a TDS. Then H is a semisimple element of H with integral eigenvalues. GH = {g E G I Ad(g)H = H } is then a real reductive subgroup of G that contains a Cartan subgroup of G (2.3.1).Thus H is contained in a Cartan subalgebra of g, which we may assume (up to conjugacy by G) is 0stable. But then it is easily seen that up to conjugacy we may assume that H E a a maximal abelian subalgebra of p . We can thus choose a minimal ppair (Po,A ) ( A = exp a) such that if a E @ ( P o , A ) then a ( H ) 2 0. Let {a,, . . . ,a,} be the simple roots in @(Po,A ) . We assert that
0I a,(H)I 2
(1)
for i
=
1,. . .,r.
We note that this will prove that up to the action of Ad(G) there are only a finite number of such H . Let X, Y E g be such that X, Y, H is a TDS. Then X is contained in the nilradical of p o . We can thus write X = C a e @ +Xu with X, in the a rootspace. If XuL# 0 then a , ( H ) = 2. Otherwise, q ( H ) 2 0 and if a , ( H ) > 0 then [ Y , g u L # ] 0 by the representation theory of a TDS. If we interchange the roles of X and Y we find that there exists a E @(Po,A ) with a ( H ) = 2 and a  a,E @(Po,A ) . Now a = C mjaj and m i> 0. Thus m,a,(H) I 2. So a i ( H ) 2 since m i2 1. This proves (1). Fix H an “Hpart” of a TDS. Let g’ = ( X E g 1 [H,X] = j X } . If X is an “Xpart” of a TDS with H the “Hpart” then X E g 2 and [X,go] = g2. We set V = { z E g 2 I [z,go] = g 2 ) .Then V is nonempty. We now show that I/ is the union of a finite number of orbits under the action of G H .This will complete the proof of the theorem. Choose bases of go and g2. If z E g 2 then set f ( z ) equal to the sum of the squares of the p x p minors of ad z as a linear map of go into g 2 (dim g 2 = p ) . Then V = ( z E g 2 I f ( z ) # 0 ) .Corollary 8.A.4.3 implies that V has a finite number of connected components. Let U be the identity component of G,. Then U has Lie algebra go and thus if z E V the U z is open in V. Since two orbits are either disjoint or equal this implies that the connected components of V are orbits of U.

8.A.4.5.
We now study more general orbits under the action of G on g. We
346
8. Character Theory
will use the following Lemma. Let X be a complete metric space. Let A be a topological group acting continuously on X . We assume that A is acompact. That is, A = Uwj with w j c w j + and each wj is a compact neighborhood of 1 in A.
,
Lemma. If X is a countable union of orbits under the action of A on X and if x E X then A x is open in its closure in X . If X is the union of a finite number of Aorbits then we can label the orbits O,, . . .,O, such that ,,J. Oj is closed in X for m = 1,. .. ,k. In particular, 0, is closed in X .

u

Let p E X . Let Y be the closure of A p in X . Then Y is a countable union of orbits of A . Let {qi} be a sequence in Y such that { A qi} is the set of all orbits of A contained in Y. A q i = w, qi.If A q ihas interior then A q i is open in Y. Thus, if none of the A q iare open in Y then we may apply the Baire category theorem (c.f. Reed, Simon [l, p.801) and find that u i A q i is nowhere dense in Y. Since, Y = A q ithis is a contradiction. Hence there exists q E Y such that A q is open in Y. If A * q is not equal to A p then Y  A q is closed and contains A p . This is a contradiction. Hence A p is open in Y. We now prove the second assertion. Let X = Oj.Then X = C1(Oj).The Baire category theorem implies that there is an index, j , such that Cl(Oj)has interior in X . We have just seen that Oj is open in Cl(0,). Thus Oj is open in X . If we relabel the orbits, we may assume that 0, is open in X . We can now argue as above for X  0,, etc.



Ujsn
u. ui 





u
If
8.A.4.6. If X E g then we set V' = { Y E g ( Y ) = f ( X )for all f note that V, is the set of nilpotent elements of g. Theorem. If X
E

E
I ( g ) } . We
g then V, is a finite union of Gorbits.
As usual, we may assume that G is semisimple. Let b,,. . ., be a set of representatives for the Gconjugacy classes of Cartan subalgebras of g.
(1) For each j , bj n V' is a finite set. Indeed, we may choose a basis of gc such that ad H is diagonal for each H E bj. As usual, we write det(ad X  tZ) = C t"D,(X). Then each 0,E Z(g) and if H E bj then D,(H) is, up to sign, the n  mth elementary symmetric function in the diagonal entries of ad H . This clearly implies (1). (1) implies that up to the action of G on g, there are only a finite number of semisimple elements in V,, H , , . . . , H N .If 2 E V, then we can write Z = Z , 2, (8.A.4.1(1)).
+
347
8.A.4. The Orbit Structure on a Real Reductive Lie Algebra
Z,, E C d ' , 9"Sl
(2)
= $31.
Indeed, gzs = j1 09,. If W E ,jl then W is semisimple in 9. Write Z, = X I X , with X , E 31 and X , E gl. Then X , is nilpotent and Z = ( Z , + X I ) + X , with 2,+ X I semisimple and [ Z , + XI, X,] = 0. The uniqueness in 8.A.4.1 implies that X , = 0. Up to conjugacy relative to G we may assume that 2, = H, for some j. Then (1) implies that 2,is nilpotent in gHJ. The number of nilpotent orbits in gHJ, relative to the action of GI,,, is finite, say, GH, Zj.mfor m = 1,. .., M j . Thus V, is the union of the orbits G ( H j ~7,,~).
+

 +
8.A.4.7. We now can apply Lemma 8.A.4.6 to V' since V, is clearly closed in g.
Corollary.
(1) If X E g then G . X is open in its closure. (2) If X E g then V, = G XI u"'u G X , with

uJtm G
X j closed in g.
With this material in place we can now prove the following basic theorem of Borel, HarishChandra [ l ] .
.
8.A.4.8. Theorem. Let X E 9. Then X is semisimple if and only if G X is closed in g.
+
If X E g, X = X , X,, there exists H E g such that [H,X,] [H,X,] = 2X, (8.A.4.6(2), Lemma 8.A.4.1). Thus,
=0
and
Iim e r a d H = X X,. t+ar
.
This implies that if X is not semisimple then G X is not closed. If X and Y are semisimple elements of g with Y E V, then ad X and ad Y have the same characteristic polynomials. Hence, in particular, dim G, = dim G,. Fix X a semisimple element. If Y E CI(G X ) then Y E V, and hence G Y is open in CI(G X ) . Since CI(G X ) c V,, there exist X , = X , . . . ,X , E CI(G X ) such that CI(G X ) is the disjoint union of the orbits, G Xiand each is open in CI(G X ) . Thus each is closed in CI(G X ) . In particular, G X is closed in CI(G X ) . So G X = CI(G X ) as asserted.






.





8.A.4.9. We conclude this appendix with several results about semisimple elements. Let G be a real reductive group of inner type. If g E G then g is said to be semisimple if Ad(g) is diagonalizable on g c .
348
8. Character Theory
Lemma. I f g E G then g can he written uniquely in the form g with gs semisimple and X E g nilpotent and Ad(g,)X = X .
= gs exp
X
Let for p E C", (ac)@be the generalized eigenspace for Ad(g) on g, with eigenvalue p. Then (1) c(gc)lc,(9c)vl = ( S C ) , , . Let S be the linear automorphism of gc defined by Sl(gc)r= p l . (1) implies that S is an automorphism of gc. Thus N = S'Ad(g) is also an automorphism. Clearly, N  I is nilpotent. Thus log N = D is given by a finite series and D is a derivation of g, that is zero on the center. Hence D = ad X with X E [g, gc] and ad X is nilpotent. Since ad X is a polynomial in N , SX = X. Let o be conjugation in g, with respect to g. Then Ad(g) = o Ad(g)a = OSOONO. Thus the uniqueness in the Jordan decomposition implies that a N a = N . This in turn implies that OX = X . Hence X E 9. Clearly,
Ad(g exp(  X ) ) = S. So set gs = g exp(  X). The Lemma now follows. 8.A.4.10. Lemma: If g E G is semisimple then nr is reductive and rk(rn) = rk(g).
=
( X E g I Ad(g)X = X }
We prove this by induction on dim g. If dim g = 0 or 1 there is nothing to prove since then G is abelian. Assume for all g of strictly lower dimension. Since G is of inner type $(g) c rn. Hence, if 3(9) # (0) then the result follows from the inductive hypothesis. We therefore assume that G is semisimple. We use the notation of 8.A.4.9. Define for s E R, T', a linear isomorphism of g, by q(gclu = lplsl. Then 8.A.4.9(1) implies that for each s E R , T is an automorphism of gc. Since ~(g,),, = (gc)ii, it follows that aTS= T'o. Hence T s is a one parameter group of automorphisms of 9.Thus T' = ead with YE g such that ad Y is semisimple with real eigenvalues. Now m c g" and g E G". Thus if Y # 0 then 2.3.1 and the inductive hypothesis complete the induction. We may thus assume that the eigenvalues of Ad(g) all have absolute value 1. With this assumption H = C / ( { g kI k E 2 ) )is a compact subgroup of G. This implies that there is a compact form of Int(g,), U, such that H c U. Since U is connected there is a maximal torus T of U such that g E T. This clearly implies that rk(m) = r k ( g ) . Let u be the Lie algebra of U. Then m, is isomorphic with the complexification of
"
(X
E 11
I Ad(g)X Ad(g)'
=
X}
whicn is the Lie algebra of the compact Lie group {U E
Thus rn is reductive.
U I Ad(g)u Ad(g)'
= u}.
8.A.5.
Some Technical Results for HarishChandra's Regularity Theorem
349
8.A.4.11. Lemma: If T E D ' ( G ) is central and if for each semisimple element 9 E G there exists an open neighborhood U of' g in G such that q, = 0 then T = 0. If Ti, = 0 then 7ixLixl = 0 for all x E G. Let y E G. Write g = gs exp X as in Lemma 8.A.4.9. Lemma 8.A.4.1 combined with Lemma 8.A.4.10 imply that there exists a TDS { X , Y, H } with Ad(g,)Y = Y, Ad(q,)H = H. Now exp(tH)g exp(  tH) = gTexp(e2'X).This implies that if U is an open neighborhood of gs in G then there exists t > 0 such that exp(tH)U exp(tH) is an open neighborhood of g. Thus our hypothesis implies that T vanishes in a neighborhood of g. Since g is arbitrary, T = 0.
8.A.5. Some technical results for HarishChandra's regularity theorem 8.A.5.1. In this appendix we collect several results that will be used in Section 8.3. Let H, X , Y be a standard basis for a TDS, u, over C. Set b = CH + CX.
Lemma.
Let M be a umodule such that
dimU(b)m < CE for all m E M . (2) N o eigenvalue of H on M is a nonnegative integer. (1)
Then the action of C[Y] on M is torsion free. Let m E M be nonzero. If there exists p E C[ Y] such that p # 0 and prn = 0 then dim C[Y]m I deg p < co. U(u) = U(b)C[Y]. Hence (1) implies that dim U(u)rn < 00. Now 0.5.5 implies that H must have a nonnegative integral eigenvalue on U(u)m and this contradicts (2). Corollary. Let M be a umodule such that if' m E M then dim C[H]m < co and such that the eigenvaluesof H on M are real and strictly less then 0. Then the action of C[ Y] is torsion free on M . M is a direct sum of the generalized eigenspaces for H acting on M . Let m be an element of the Ageneralized eigenspace. Then X"m is an element of the eigenspace for A 2n. Thus our hypothesis implies that X"m = 0 for some positive n. Hence, if rn E M then dim U(b)m < 00. The corollary now follows from the preceeding lemma.
+
8.A.5.2. We now collect a few results about distributions on R". Let p , q E N with q > 0 and p + q = n. We write R" = R P x R4.Let U , be an open subset of R P and let U , be an open neighborhood of 0 in R4.Set U = U , x U , .
350
8. Character Theory
Theorem. If T E D ' ( U ) and if supp T c U , x 0 then there exists TI E D'(U,) such that T = C T, 03'6. Here 6 is the Dirac delta function on R 4 supported at 0 ( & f )= f ( 0 ) ) . For a proof see Schwartz [ 1, p. 1021. Corollary. Let w E P(R4)be such that w(0) = 0. W e extend o to R" by setting w(x, y) = o(y). If T E D'( CJ) and if supp T c U , x 0 then there exists k such that o k T = 0. As above, T = T, 0D6. Let m = deg D. Let 0 E V c CI(V) c U , with V open and Cl(V) compact. Let Ic, E CF(U,) be such that $ is identically 1 on V. Extend 11/ to U by setting $(x,y) = $(y). Iff E C F ( U )then T(f )= T(f )= T($f).If r = ( r , , ..., r,)EN"thensety'= y ; ' . . . y ~ . I f f ~ C F ( U ) a n d i f
f ( x , Y) =
1
Irlsm
ar(x)yr + Rm(x, Y)
is the Taylor series around 0 of f ( x , .) to order m in y at 0 then
Thus, if f vanishes in y to order rn at 0. Then T (f )= 0. It is clear that there exists k such that okvanishes to order m in y at 0. Thus okf vanishes to order m in y at 0 for all f E Cm(U).Thus o k T (f ) = T ( w k f )= 0 for all f E C F ( V ) .
8.A.5.3. We retain the notation of 8.A.5.2. Write D;,(U) for the space of distributions on U supported on U , x 0. Lemma. Let Ej = yj a/ayj for j = 1,. . .,n. Then each Ej acts semisimply on D ; , ( U ) with eigenvalues of the form  k with k > 0, k E Z. Iff
E
Cm(U,) then E j T ( f 1 = (a/ayj(yjf )NO)
=
f(o)
Ej6 = 6. If I is a multiindex then [Ej,al/ayl] Ej al/ayl 6 =  (1
=
=
6(.f).
ijal/ayl. Thus
+ ij)al/ayl 6.
The Lemma now follows from Theorem 8.A.5.2. 8.A.5.4.
Lemma. Let aj be nonnegative real numbers for j
=
1,. . . ,n. Set
8.A.5. Some Technical Results for HarishChandra's Regularity Theorem
+ l)Ej. Then D acts semisimply A such that 1 2 n + C a.i.
D
= C(aj
Since [ E , , Ej]
= 0 for
351
on D;,(U) with real eigenvalues
all i, j this follows directly from the preceding Lemma.
8.A.5.5. For lack of a better place to put the following material, we will conclude this "hodgepodge" of an appendix with it. As usual, let Y(R") denote the space of all f E C"'(R")such that P,,,(f)
=
SUP (1
XER"
+ Ilxll)rla'f(x)/ax'l
2 (in the case that dim V = 2 the result that we are proving is an easy exercise). We set @+ =
{ z(p, '  p,) I 1 I r < s I k } u { i ( p r + ,us)I 1 I r < s I k } .
Then @+ is a system of positive roots for @(g,,t,). The only dominant weights in (1) are  i(pl . . . & ) / 2 and  i ( p l . . . pk  pk)/2. The assertion for n even now follows from the theorem of the highest weight.
+ +
+ +
~
9.2.3. The module ( p , S(V ) )for so(V) is called the spin module. Lemma.
There exists a preHilbert space structure ( , ) on S( V ) such that (y(u)u,w> =
(u,
y(o)w>,
fJ
E
V, u, w E S ( V .
If n = 1 this is easy and is left to the reader. So assume that n > 2. If n is even then we may adjoin a unit vector, u o , orthogonal to V. We may thus assume that n is odd and n 2 3. Let G be as in 9.1.2. Then G is compact. A direct calculation yields (*)
(
p(exp(tX(u,u)))=y sin
(3 (3 ) ( (3 
u+cos

u y cos

u+sin
.
u
(:) ) 
(*) implies that p ( G ) is the set of all products y(ul) .. . y(u,,), u l , . . . , u2p
unit vectors in V. Set G" equal to the set of all products y(ul)...y(u,),with u l , . . ., up unit vectors in V. Then G contains G as a normal subgroup. Also, if u E V is a unit vector then it is easily seen that G" = y(u)p(G)u p(G).
363
9.2. Spinors
Thus G  is compact. This implies (the unitarian trick) that there exists a G"invariant inner product ( , ) on S(V). If u E V and if (u, u) = 1 then y(u)' = I. Thus y(u)' =  y ( u ) . The Lemma now follows. 9.2.4.
Lemma.
Let R be the nufurul representation of 5o( V ) on A V,.
(1) I f n is even then R is equiuulent with p @ p. (2) I f n is odd then R is equiuulenr with the direct sum of two copies of p @ p.
This follows from the calculation of the weights of p in 9.2.2. 9.2.5.
The next two results expand a bit on the previous Lemma.
Lemma. Let F be a ,finite dimensional vector space over C. Let 6 be a linear ~ (v,u)I ,for u E V. I f n is even then map of V into End(F) such that 6 ( 1 1 )= there exists a vector space U and a linear isomorphism, T, of S ( V )0 U onto F such that T(y(u)@ I ) = 6(v)T jiir v E V. If n is odd then there exist spaces U + and U  and a linear isomorphism T of S ( V ) @ U + @ S ( V ) 0 U  onto F such that
T(y+(u)@ I +
+ y ( U ) @ I  ) = 6(v)T
for u E V (here I + is the identity map on U + ) . We use the notation in the proof of 9.2.1. Let F, = { f E F I G(oW)f = 0). As in 9.2.1 it is easily seen that Fo is nonzero and if n is odd then
6(v,)F,=F,. If n is even then set U = F, if n is odd then set U + = { f ~ F , 1 6 ( v , ) f = i f } , U  = { f ~ F ~ I 6 ( u , ) f =  i f } If. f e F o then let Tf be the linear map of W into F given by T f ( u , A . . ~ A u ,= ) 6(ul)...6(u,)f. If n is even then the argument in 9.1.2 implies that Tfy(v) = 6(u)Tf.If n is odd and if f' E U' then Tfoy'(u)= d(r1)T'. Set T ( s X f )= T'(s). The result now follows. 9.2.6. On if v E V and if u E V then set E ( U ) U = vAu. If u l , . . ., u, E V then set
i(u)(ulA... Aur) =
Set 6 + ( v )= E ( U ) + i(u)and & ( u ) It is easily checked that (1)
(2)
6,(v)'
(  l ) i + ' ( t i , u , ) u , A . . ACI,A.. . Au,. =
i(r:(u) i(u)).
=
(u,u)l
6+(u)6(w)+ 6 ( w ) h + ( v )= 0
and for v, w
E
V.
364
9. Unitary Representations and (g, K)Cohomology
Define Q,(Xij) =  ( t ) 6 k ( e i ) 6 k ( e j ) for i < j. Then as above a representation of so(V) on AVc.
Lemma. Q ( X ) = Q + ( X ) + Q  ( X ) and
a,
defines
for X , Y E so(V).
[Q+(X),Q_(Y)]=O
The last assertion follows from (2) above. The first follows from Q ( X i j )= & ( e i ) i ( e j )
+ i(ei)e(ej)
and the obvious calculation. 9.2.7. The following lemma (although easy) is useful outside of representation theory. Fix e , , . . . , en an orthonormal basis of V.
Set y j
c
=
y(ej). In the expressions below all indices are summed. (3) says
RijklYiYjYkYl

1
c
R k i j l ~ i ~ j ~k ~ l R j k i l Y i ? l j Y k Y l
=
1
RijklYiYjYjYk.
We calculate
1
=
RkijlYiYjYkYl
1
RkijlYiYkYjYI

1
RjijlYiYl*
+
Now (1) implies that Rjijl= R j l j i .So  2 C Rjijlyiy1= C Rjijl(yiyl y l y i ) = 2(C Rjiji)l.Also (2) implies that, RkijiYiYkYjYI
Set R
=
1
=
=
Rikjl?iYkYjYI
1
RijklYiYjYkYI,
C R i j j i .Then we have
1
RijklYiYjYkl)l
= 2R1

1
RjkilYiYjYkYl.
Also, as above RjkilYiYjYkYl
1 =1
RjkilYjYiYkYl
RijklYiYjYkYl

 2R1 =
4R1.
c
RjkilYjYkYiYl

4Rr
365
9.3. The Dirac Operator
Hence
3 C Rijk,yi>jjlljyk= 6RI as was to be proved.
9.3. The Dirac operator 9.3.1. Let g be a semisimple Lie algebra over R with Cartan involution 8 and corresponding Cartan decomposition 9 = f @ p . We will use the material of Section 9.2 with I/ = p and ( , ) the restriction of B to p . Let ck be the Casimir operator of f relative to B restricted to f. Set p,(Y) = ad YIP for Y E f. Then p, is a homomorphism of t into so(p). Set s ( Y ) = p ( p o ( Y ) )for Y E f. Then (s,S(p)) is a representation of f with a €invariant inner product ( , ) (9.2.3). Let t be a maximal abelian subalgebra of f. Fix Pk, a system of positive roots for CD(&., tc). Let b = { X E g I [ X , t] = O}. Then b is a fundamental Cartan subalgebra of g. Let 0 = CD(gc, bC). Let 0 act on (be)* by Ho(h)= o(Bh), h E 6. Then 190= 0. We say that a system of positive roots, P, for CD is 0stublr if BP = P. We say that it is compurible wirh fk if it is 0stable and if a E fk then there exists [j E P such that /?It = a. Fix, P, a system of positive roots for 0 compatible with Pk.Set a = { h E b IBh =  h ) . We identify (tc)* with {o E (be)*! 00 = o} and (ac)* with (0E (be)* I 00 = 0;. If cr E (b,)* then write 0 = ' 0 + 0  with ' 0 E (tc)*, 0 E
(ac)*.
(4)
Set p ( P ) = C a e Pa, C)k = (+) CatPk a, p,(P) = p ( P )  p k . We note that since BP = P, p ( P ) , pk, p,(P) E ( t o * . Let c(pk)denote the set of all systems of positive roots that are compatible with Pk. If ,iE (tc)* is Pkdominant integral then set y I equal to the element of K A with highest weight E,. 9.3.2. Lemma.
Let I,
=
dim a. Then
Fix P E C(Pk).If CI E (tc)* then set ( p c ) , = ( X E pc I [h, X ] = a ( h ) X ,h E (tc)*}. Set = ( x E ( ~ , . ) *( p c ) , is nonzero). Put C* = C n ( k p 1 , ) . Set p'(P) = &r (&la. Then pc
=
a,.
0P+(P)0P(P).
366
9. Unitary Representations and (g, K)Cohomology
The analysis in the previous section implies that the weights of t, under s are of the form p,(P)  u1  ...  a, with xi E C'(P) and there exists a subset Q of P such that u, + ... + a, = (Q)' (see 9.A.1.5 for ( Q ) ) . This implies that p , ( P ) is an extreme weight.
s(ck)= cl.
(1)
Indeed, let x 1,..., x, be an orthonormal basis of p. Let y l , . . ., y , be a basis Of f such that B ( y i , y j ) =  8 i j . Set R i j k l = B ( [ X i , X j ] , [ X k , X , ] ) . If y E f then (9.2.2) ~ s ( Y= ) C(C~,xjI,xi)~(xi)~(xj). ij
Thus 16S(ck) =
=
I
ijkl
RijkIr(Xi)r(Xj)Y(Xk)Y(XI).
It is clear that RijkIsatisfies (l), (2) of Lemma 9.2.7. (3) of 9.2.7 follows from the Jacobi identity. Hence 9.2.7 implies (1). As we have seen above yp,(p) occurs in s. So (1) implies s(ck)
(2)
= (llp112 
Suppose that yn occurs in s. Then (2) says that 110 p k l l = llpll. Hence
+
IIP

(Q>' I I
llpk112)1.
CJ = p,
=

(Q)'
with Q a subset of P.
llpll.
Now, P  (Q)' = ( P  (Q))'. SO, IIP  (Q>' I I IIP  (Q)II. Hence IIp  ( Q ) " 2 llpll. p  (Q) is a weight of the finite dimensional irreducible gmodule with highest weight p (9.A.1.5). We therefore see that p  (Q) = wp, for some w E W(g,, bc). Also the inequalities must be equalities, so, Q w = wQ. This implies that w P E c(pk). We have shown that CJ = p ( w P )  pk. We leave it to the reader to check that the dimension of the p ( P )  p k weight space is 2[10'21.
9.3.3. Let G be a connected semisimple Lie group with finite center. Let K be the connected subgroup corresponding to K . Then a (g, K)module is
367
9.3. The Dirac Operator
said to be unitary if there exists a preHilbert space structure ( , ) on V such that if X E g, k E K and u, w E V then (1)
(Xu, w )
=

( v , XW),
(2)
(kv,w)
=
(v,k’W).
Let V be a (g, K)module, set S = S(p). We now define a Kmodule homomorphism, D , = D from V @ S to V S . Let n be the action of g on V. Then if x , , . . . , x , is an orthonormal basis of p set D
=
1 .(xi)
O ?(xi).
If V is unitary then we put the tensor product preHilbert space structure on V O S . Lemma. D 2 =  n ( C ) 01  (llp1I2  IIpkl12) + ( n @ s)(C,). If I/ is unitary then ( D v , w ) = ( v , D w ) . In the calculations below all indices will be summed (unless otherwise specified). Let y , , . . . , y , be a basis of f such that B ( y i ,y j ) =  6,.
9.3.4. Corollary. Assume thut V is u unitary (g,K)module with injinitesimd churacter zh. l f ( V O S)(y,) is nonzero then 110 P k l l 2 IlAIl.
+
Indeed, n ( C )= (11A1l2  llp1I2)1.Hence
368
9. Unitary Representations and (9, K)Cohomology
Now, D 2 is positive semidefinite and ck acts on any representative of yb by 116 + p k 1 I 2  llpk112. The corollary now follows. We will refer to the conclusion of the above corollary as the Dirac inequality.
9.4.
(9, K)cohomology
We retain the notation of the previous section. For simplicity, we take G to be semisimple, the identity component of GR and we assume that Gc is connected and simply connected. If V is a (g,K)module then let H'(g, K ; V ) be as in 6.1. For the next few sections we will be studying these cohomology spaces. Fix P E C(P,). Let F be a finite dimensional irreducible (g,K)module with highest weight A relative to P. The following result is usually known as Wigner's Lemma.
9.4.1.
Lemma. l f V is a (g,K)module with infinitesimal character H'(g, K , V 0 F * ) is nonzero ,for some i then x = x , , + ~ .
x
and
if
Let g, = f + ip in gc. Let G, be the connected subgroup of Gc corresponding to 9,. Then G, is connected and simply connected. Also G, n G = K . Let r' be the i'h Zuckerman functor (6.2) from C(g,, K ) to C(g,, GJ. Let y E G:and let F, be a representative of y. Then (6.3.2) P ( V ) = OH'(g,,K;V@(F,)*)@F,. 6.3.3 implies that T'(V) has the same infinitesimal character as V. Thus H'(g,, K ; V @ (F,)*) = H'(g, K ; V @(F,)*) is nonzero only if V and F, have the same infinitesimal character. Let x denote the complex conjugate of X E U(g,) relative to the real form U ( g ) . 9.4.2.
Lemma. (*)
If V is a unitary (g, K)module with injinitesimal character
x then
x(zT)= ~ ( z ) for z E Z ( g ) .
I f F is a jinite dimensional, irreducible (g, K)module with highest weight A and if the injinitesimal character of F satisjies (*) then OA = A. If z E Z(g,), u, w E V, then x(z)(u,w ) = ( z u , w ) = ( u , F T w ) = x ( ~ ' ) ( uw, ) . This proves (*). We now prove the second assertion. Let 6 denote complex
369
9.4. (g, K)Cohomology
conjugation in gc relative to g,,. Since F is unitary as a (g,,, G,)module, the first assertion implies that xF(ozT)= x F ( z ) for z E U(gc). Since ox = 82 for x E U(gc), (*) implies that xF(Bz)= x F ( z )for z E Z(g,). Let F, be the ( g , K ) module, F with g acting by O(X)u,u E F, X E g. Then we have just shown that F, and F have the same infinitesimal character. This implies that they are isomorphic. Since the highest weight of F, relative to P is 8A, the second part of the Lemma follows. 9.4.3. Proposition. If V is a unitary, admissible, (g, K)module with inJinitesimal character x,,+,, then H'(g, K ; V @ F * ) = Hom,(A'p, V @ F*). Note.
H ' ( g , K ; V 0 F * ) is the cohomology of the complex
C'(g, K ; V 0F*)
=
Hom,(A'(g/f), V 0F * )
The content of the proposition is that d
=
Hom,(A'p, V 0F*).
= 0.
On F we put a G,invariant inner product. On kp put the inner product corresponding to the restriction of B to p. On (h'p)* use the dual inner product. Now, C'(g, K ; V 0F * ) = C' = ((/Zip)*0 I/ 0F*)'. Set D'
= (/lip)* 0 V
0F*.
We put the tensor product inner product and on D' we restrict that inner product to C'. Since V is admissible, C' is finite dimensional. We will use the following standard result. Let (C',d) be a complex with dim C' < co. Fix ( , ), an inner product on each C'. Define d * : C' + C'' by
9.4.4.
(d*x,y) =(x,dy),
XEC',~EC''.
Scholiurn. The natural map from
to H'(C', d ) is a surjective isomorphism. We assert that C' = dC' ' 0d*C'+' 0 S' orthogonal direct sum. Indeed, if (x, dC'' + d*C'+' ) = 0 then dx = d*x = 0 and conversely. Thus, xi = (dci 1 + d * C ' + ' ) I. If u E d C '  ' , u E d * C i + ' then u = dw, u = d * z so (u, u) = (dw,d*z) = ( d 2 w ,z ) = 0. The assertion follows. If z E C' and if dz = 0
370
9.
then write z
= dx
Unitary Representations and (9, K)Cohomology
+ d * y + h with h E 2''.Then 0 = dz = d d * y . So, 0 = (dd*y,y)= (d*y,d*y).
Hence d * y = 0. The first assertion now follows. To prove the second we note that (d d*)' = dd* d*d. If (d d*)'c = 0 then
+
+ + 0 = (dd*c,c) + (d*dc,c) = ( d * c , d * c ) + (dc,dc).
The second assertion is now also obvious. 9.4.5. We now return to the proof of 9.4.3. If x E p define x # E p* by x " ( y ) = B(x,y). If u E (A'p)* and if x E p then set E ( X ) U = x # A u . If x E p and if u E (A'p)* then set i ( x ) u ( z l , . . , z i  = u ( x , z i , .. .,z i p Relative to ( , ) on the D', E(x)*= i(x).Let n be the action of g on I/ and let a be the
action of g on F * . Then d on C'is the restriction of
d
=
C &(xi)Q .(xi)
0I
+ 1&(xi)0I
0 xi)
on D'. Here xl,. .. , x , is an orthonormal basis of p. ) n(x)* =  n ( x ) for x We note that a(x)* = ~ ( xand restriction of d*
=
C
i ( x j )Q n ( x j )Q I
E p.
Thus d * is the
+ C i(xj) Q I Q a ( x j )
on D' to C'. On D' we have d + d * = C d + ( X j ) Q n ( ~ j ) Q I i C d  ( ~ j ) @ I Q ~ ( x j )
in the notation of 9.2.6. Thus, if we apply 9.2.6(2) we find that on D'
( d + d * ) 2 = ( C d + ( x j ) @ ~ ( x j ) Q I ) ' ( C d  ( ~ j ) O I @ ~ ( x j ) ) ' . If we combine 9.2.6 with Lemma 9.3.3 then on D' (d
+ d*)'
= 1
0n(C)0I
+ I@
+ (IIPII'

IIPkll')
I @ a(C)  (llP112  IlPk11')
(b+ adlf)@ 0I ) ( c k ) O
 ((p
@ @ z)(ck)*
Since n ( C )and a(C) act by the same scalar, we find that on D' (d
+ d*)'
= ((p+ 0
ad\,)0n 0I)(C,J  ((p adl,) 01 0 a>(Ck). 0
Thus to complete the proof of the proposition we must show that this expression is 0 on (D')K= C'. Let y , , . . . , y , be a basis of f such that B ( y , ,y j ) =  d i j . Let ~ ( y=) (p+ adl,)(y) and p ( y ) = (p ad(,)(y) for y E f. Then 0
0
9.4.
37 I
(9, K)Cohomology
+
~ ( y ) B ( y ) = ad(y) on (Asp)* (9.2.6). In what follows all expressions will be
looked upon as evaluated on ( D * ) KWe . are studying (*)
2
1Cc(y,)0
TC(Yi)
2
0I
+ 1 10 TC(yi)'
0I
c P(yJ O 10a ( y J + c 1 0 1 0
CJ(Yi)'.
Now, if y E f then ( a + b ) ( y )01 0 I + 1 0 ~ ( y0) I + I0 I 0 a ( y ) = 0 on the Kinvariants. If we apply this identity to the above expressions and do the obvious algebra (which we leave as an exercise to the reader) we find that on the Kinvariants (*) is equal to
1(M(yi)

B(yi))(a(yi)+ P ( y i ) ) O I 0I = (a(C,c)  P(Ck))01 0 I
since ~ ( y and ) P(y) commute for y E f. This expression is 0 by 9.2.6 combined with 9.2.5 and 9.3.2(1). This completes the proof.
9.4.6. We now state a result that sums up most of the material of this section. Proposition. Let P be a $xed Pkcompatible system of positive roots for
@(gc, bC). Let F be an irreducible jinitc dimensional (9, K)module with highest weight A relative to P. If HA # A and if' V is an irreducible unitary (9,K)module
then H ' ( g , K ; V @ F * ) = 0. I f V is unitury with injinitesimal character x and if x # xA+p then H ' ( g , K ; V 0F * ) = 0. Assume that HA = A and that V is an irreducible unitary (9,K)module with injinitesimal character x,,+,,. Then H'(g, K ; V 0F * ) # 0 if and only if there exists y E K" such that Hom,(V?, V 0S) and Hom,(Vy, F 0S) are nonzero. Furthermore, for any such y there must exist PI E c ( p k ) such that A is P,dominant and Pk =A p(P,).
+
+
The first two assertions follow from 9.4.1 and 9.4.2. We now prove the assertions of the last paragraph of the statement. The previous result implies that H ' ( g , K ; V @ F * ) = ((Asp)* 0 V 0F*),.
On ((Asp)* 0 V 0F * ) set
D,
=
0 =
c c 6
0n(xi) 0I (Xi) 0I 0( T ( X i ) .
S+(.xi)
and
In the course of the proof of 9.4.3 we showed that (0,)'

(0)' is 0 on
372
Unitary Representations and (9, K)Cohomology
9.
Since both ( D + ) 2 and  ( D  ) 2 are positive operators this implies that D+((A'p)*0 I/ @ F*)' = 0. Suppose that uo E ((A'p)* 0 I/ 0F*)'  (0). Let C, be the span of all elements of the form
((Amp)*0 I/@ F*)'.
(6'(u,)0101)'~'(6+(u,)0101)(6~(w,)0I0I)~~~(6~(w,)~I0I)u, ui, wj E p . Set C, equal to the span of { ( I 0n(k,) O 4 k 2 ) ) C 1I k , , k 2 E K}. Then C, is a finite dimensional so(p) x so(p) and K x Kmodule with action given as follows: the first 4o(p) factor acts by p 0I 0I , the second acts by p+ 0I0 I , the first K factor acts by I 0 n 0I and the second acts by I0 I0 CJ.All of these actions commute. If we apply Lemma 9.2.5 we find that
with each C,[cqfi] an so(p) x so(p) module which is a direct sum of tensor products of spin modules. We therefore conclude that
(V@S@F*@S)K#O.
Furthermore, on ( S 0 V 0F* 0S)', D, 0 I and I 0 DF*act by 0. Thus if (( I/@ S)(y,)0( F * 0S ) ( Y ~is) )nonzero ~ then
(*I
\\lLyi
+ PklI
=
lIlby2
+ Pkll
= llA
+
This implies everything but the last assertion. Suppose that HomK(Vy,F 0S )
+
+
is nonzero and that p k l l = IIA pII. The weights of F'O S with respect to t are of the form A + pn  (Q)' with Q a subset of P . Thus A y pk = A pn pk  (Q)' A p  (Q)'. Thus
+ +
Iliy + ~
2
k l= l
IIA
+
+
+ P (Q>'II
IIA

+ P (Q>II 
IIA
+ PII
by 9.A.1.5. Hence all of the inequalities are equalities. This implies that (Q) = (Q)' and that there exists s E W(g,,h,) such that p  (Q) = sp and sA = A. Since sp = (sp)', 8s = SO so sP is &stable. Since s(A + p ) is P'dominantsP E C(P,). Thus Ay = A p,(sP) as asserted.
+
In the next section we will give sharper results due to Kumaresan, Parthasarathy, Vogan and Zuckerman.
9.4.7.
373
9.5. Some Results of Kumaresan, Parthasarathg, Vogan, Zuckerman
9.5.
Some results of Kumaresan, Parthasarathy, Vogan, Zuckerman
9.5.1.
In this section we will be using several Hstable systems of positive roots compatible with different systems of positive roots for K . It is thus worthwhile to recall the relationship between W ( K ,T ) and W(gc,bc).The notation will be as in the previous section. Let s E W ( K ,T ) . Then there exists k E K such that Ad(k)I, = s. Since b = { X E g I [ X , t] = 01, Ad(k)I) = 1). We are assuming that G is connected hence Ad(k) = s' E W(g,, b,). Clearly, s' 1, = s. If t E W(g,, bc) is such that t It = s then tC's' is the identity in t. Now t contains regular elements of g. Thus t = s'. We have proved
Ih
Lemma. If s E W ( K ,T ) then there is a unique element s' E W(g,, bc) such that s'It = s.
In light of this we will identify s E W ( K ,T )with s' E W(g,, bC). 9.5.2.
We now continue the discussion initiated in the previous section. Let F be a finite dimensional irreducible (g, K)module. If P is a system of positive roots for @ = @(g,,$,) then we write A(P) for the highest weight of F with respect to P . We assume that if P is Hstable then BA(P) = A(P). Let V be a unitary (g, K)module with infinitesimal character xA(p)+p(p). Fix Pk, = a system of positive roots for tc). Let y E K" be such that (1)
HOmK(A'P, VOJ)O F * ) # 0.
Unless otherwise specified, F , V, Pk,y will be fixed. Let p denote the highest weight of y relative to Pk.The following result is due to Kumaresan [l] for F = C and to VoganZuckerman [l] in general (all of the essential ideas appear in the case F = C). Proposition. There exist Pl E C(Pk)and Pz a 0stable system of positive roots for @ such that A(Pl) is P,dominant and
P
=N
P , 1 + PAPI) + Pn(P2).
We have seen in 9.4.6 that (1) implies that there exists P (A = W),P = P ( P ) )
(2)
Horn,( 5 + p n ( p ) ' V, O S ) # 0.
E
C(Pk)such that
374
9. Unitary Representations and (9, K)Cohomology
Notice that we are denoting Vywby V,. Now S is a multiple of
We therefore must have HomK(VA+~,,(P)>
Vp
@ ( V ~ n ( Q ) ) * )#
for some Q E c(Pk). Let u E W ( K ,T ) be of minimal length (9.A.l.l) such that u ( p  p,(Q)) is Pkdominant.Then V& occurs as a summand in V @ S (9.1.4).The Dirac inequality (9.3.4) implies that IIu(P  P n ( Q ) )
+ Pkll 2 IIA + PII.
On the other hand, 9.1.6 implies IIA
+ Pn(P) + Pkll
PII = IIA
2 IIu(P  P n ( Q ) )
PkII.
Thus all inequalities are equalities. This implies (9.1.6) U(P 
(3)
=
Pn(Q))
A
+ Pn(P).
+
We rewrite (3) as u p  p,(P) = A upn(Q).Let u, f E W ( K ,T )be such that t is of minimal length such that up  p,(P) is tPkdominant and u is of minimal length such that ,u  pn(P)is uPkdominant. Lemma 9.1.7 implies that IluP  pn(P)
+ fPkII
2
[IA
 Pn(P)
+ rpk/l.
9.1.4 implies that the irreducible finite dimensional Kmodule with highest weight f l ( p  p,(P)) occurs in V @ S . Hence the Dirac inequality implies that
Ilup  P n ( P )
+ fPkll 2 IIA
f
On the other hand, IIuP  Pn(P) + lPkll = IIA
+ +
+
uPn(Q)
tPkII
IIA
+
Pn(Q)
Pkll
by 9.1.5. Now, A pn(Q) P k = A p(Q). Let w E W(g,,b,) be such that Q = wP. Then IIA wp(P)II 5 IIA + p(P)II. Hence all of these inequalities are also equalities. We look at the implications of our new equalities. We first look at IIA
+ WP(P)II
=
IIA
+ P(P)II.
375
9.5. Some Results of Kumaresan, Parthasarathy, Vogan, Zuckerman
9.1.5 implies that there exists r E W(gc,bc)such that rA p ( P ) . Thus rw = 1, so WA = A. We have thus shown
=A
and rwp(P) =
(4) A ( P ) is Qdominant. We now look at iup,(Q)
f tPkll
=
IIA
+ dQ) +
This implies that there exists r E W ( K ,T ) such that rA = A, rup,(Q) = p,(Q) and rtp, = p k . Thus r = t  ' . We have therefore shown that (5)
= At
upn(Q) = tpn(Q).
We use this to prove
UA = A.
(6)
If a E Pkn (tPk) then (A,a) = 0. Thus SJ = A. In light of 9.A.1.3 it is therefore sufficient to prove that Pk n (  Upk) is contained in pk n (  tpk). So assume that a is in pk n (  u P , ) but not in Pkn(ttPk). ( 5 ) implies that (a, up,(Q)) 2 0. Also (a,up) I 0. Hence (2,u ( p  p,(Q))) I 0. Hence
(*I
(a, ~
( /1
pn(Q)))= 0.
Since u was assumed to be of minimal length, 9.A.1.4 implies that f'pk
=
{ b E % I (p, P  P,(Q)) > 0) U { b E 4 (b, P  p,(Q)) = 0).
This says that pk =
{a
E @k
(fl, u(P  pn(Q))) > 0) u { Lj E pk I (b, u ( P  pn(Q)))
= 0).
(*) now implies that a E upk. This contradiction implies (6). (3) implies that p = u  ' A p,(Q) u'pn(P). In light of (6), the Lemma follows if we take Pl = Q, Pz = u  ' P (recall our identification in 9.5.1).
+
+
9.5.3. If q is a &stable parabolic subalgebra of g, (6.4.1), q =I, + u then set 11, = u n p, uk = u n f. If h E t,. then set p,(q)(h) = tr(ad hl,,)/2. We say that q is Pkcornpatibleif q n t, contains bk = t, 0 @ a e P k ( € C ) n = t, + n k . (1) If q is P,compatible then 2pn(q)is P,dominant integral.
Indeed, let n = dim u,. Let V = U(tc)(A"(u,,)). Then V is a submodule of A"p. [uk,ii,] c u, and ad(uk) consists of nilpotent elements, hence uk * A"U, = 0. Also 1 n f stabilizes A"11,. Thus, nkAnu,,= 0. (1) now f O ~ ~ O W S , since t, acts on Anu; by 2pn(q).
376
9. Unitary Representations and (g, K)Cohomology
Theorem. Let F be a finite dimensional, irreducible (9,K)module. Let V be an irreducible unitary (9, K)module with the same infinitesimal character as F. Let E be a finite dimensional irreducible Kmodule such that Hom,(E, A'pOF) #O and Hom,(E, V ) # 0. Then there exists a &stable parabolic subalgebra, q, of gc such that F" = { u E F I u v = 0) is one dimensional. Let A be the weight of t on F". (2) If p k is a system of' positive roots compatible with q then E has Pkhighest weight A + 2pJq).
(1)
The proof of this result (mainly due to Kumaresan [l]) is complicated and will take up most of the rest of this section. We use the notation of 9.5.2. In light of the result therein we may assume that E has highest weight p = A + p,(Pl) + p,,(P,) with Pl, Pz 8stable systems of positive roots, PI E C(Pk)and that A is both PI and P,dominant. Our first task is to find a system of positive roots P3 such that p,(P,) = p,,(P,), A is P3dominant and P3 E C(Q,) with p Qkdominant. If CJ E it* is Pkdominant set
9.5.4.
pk(G) =
{aE
I
p k (0, a)
> o} u { a E  pk I (a,0)= 0).
Then Pk(o)is a system of positive roots for 0,(9.A.1.4(1) with by  p k ) . Set Qk Lemma.
= Pk(P) = pk(A
pk
replaced
+ Pn(pI) + Pn(pZ)).
Both A and p,,(P,) are Q,dominant.
Let R, be the system of positive roots for @k such that Pz E C(&). Suppose that a E Qk is such that ( A , a ) < 0. Since A is Rkdominant this implies that tl E Q, n (  Rk). Hence (p,,(Pz),a) 5 0. Similarly, tl E Qkn ( Pk), so (p,,(Pl),a) I 0. But then ( p , a) < 0 contrary to the definition of Qk.The second assertion is more difficult. Suppose that a E p k and (p,,(Pz),tl)< 0. If we show that this implies that (p,a) = 0 then the second assertion will follow. If (p,c() is nonzero then it must be positive. We look for a contradiction. Write p,(P,) =  U C J with u E W ( K ,T ) and CJ a P,dominant form. Let so E W ( K ,T ) be such that S O P k = pk. Then \lpn(pZ)
p
+ sOpkll = llA
 pn(pl)  Pkll =
+
9.5.
377
Some Results of Kumaresan, Parthasarathy, Vngan, Zuckerman
Let r
E
W ( K ,T ) be such that
CJ
IlPn(P2)  P
 p is rP,dominant. Then
+ W,ll
2
IICJ  c1 + TPkII
by 9.1.7. Let
M E
P, be such that (pn(P2),a ) < 0 and ( p , M ) > 0. Then
2(P,(PZ)  P2 4 / ( % = 2(P,(P2), @ ) / ( a >
co
co

co < 2(P,(P,), M ) / ( R , 4.
2(p,a)l(a,
This implies that s,p,,(P,)  p is on the astring of weights in I/sl,(p,(p)a through p,(P2)  p. Also, the above inequality implies that it is not an element of W ( K ,T)(pn(P2)  p). It is also easily seen that [(s,u) < [(tl), hence 9.1.8 implies that if s,uo  p is r’P,dominant then IlPn(P2)  P
+ .w,ll > ll.wJP + T’PklI ~
2
IICJ

p
+ rpkll.
) in l$@ S we have a contradiction to the Dirac Now V  s o r  l ( a  P occurs inequality. Let R, be as in the proof of the previous Lemma. Let r E W ( K ,T ) be such that Q, = rR,. Then we have just proved that both A and p,(P2) are dominant with respect to R, and rR,. This implies that
9.5.5.
rA
(1)
=
A
Set P3 = rP2. Then p,(P3)
(2)
p =A
=
= p,(Pz).
rp,,(P2)= p,(P2).This gives
+ p,(P1) + pn(P3),A is both Pl and P3 dominant,
PI E C(P,), P3 E C(Q,) and
Lemma.
rp,(P,)
and
p(Pl)
11
is both
+ p(P3)is P,dominunt.
pk
and Q,dominant.
378
9. Unitary Representations and (g, K)Cohornology
Indeed, (A
+ p,(Pl) + p,(P3), a) = ( p ,a) = 0 and P(pi) + P(P3) = P,(pi)
Pn(p3)
Pk
+ P(Qd
Hence (3) implies (4). We now complete the proof of the Lemma. Suppose that p(Pl) + p(P3)is not PIdominant. Then there would be a simple root, a E PI such that
(*I
(P(P1)
+ P ( P 3 ) 7 4 < 0.
Thus a would be an element of P3. We now show that this is impossible by showing that  a would be P3 simple (if so then 2 ( p ( P 1 )+ p(P3),a)/(a,a) = 1  1 = 0). So we are left with showing that for such an a,  a is P3 simple. Assume that it exists.
(4
(g,),
is contained in pc (in particular Oa = a).
Assume that (i) is false and that 8a = a. Then (g,), is contained in f,. Thus a defines an element of pk. If a E Pk n Qk then a E P3 which is contrary to our assumption. Thus a E (  Q k )n pk. But then (hdPi)
pn(p3),a) = ( P k
+ P(Qk),
a) = 0
which is also contrary to our assumption. Thus we may assume that (i) is false and that a # Oa. As usual, write a = a+ + a  . Then
0 > (P(P1) + P(P3),4 = (P(P1) + P ( P 3 ) , E + ) . Let X E ( R , ) ~ . Then X + OX E (f&+  {0) since (g,), is not contained in pc. Hence c(+ E Pk. Now (3) and (4) imply that (p(Pl)+ p(P,),a+) 2 0. This contradiction implies (i). Set P" = supl.Since Oa = a, P" is &stable. a is PIsimple so (i) implies that P" E C(Pk)and p,(P") = p,(Pl)  a. Set o = p,(P3) + a. We assert that cr is an extreme weight of S. Indeed, 6 = A + o + p,(P") = A + p,(Pl) + p,(P,). So V, occurs in V @ S. Now apply the Dirac inequality. This implies that there exists a 8stable system of positive roots, P A ,such that p,(P") = p,(P3) + a, is both P A  and P"dominant and p,(P") + p,(P") = p,(Pl) + p,(P3). We can now apply our results for PI and P2 to P" and P" to find that p ( P " ) = p(P3) + a (we leave this chore to the reader). Thus sup(P3)= p(P3) a. This leads to our desired contradiction.
+
9.5.6.
We now complete the proof of Theorem 9.5.3. Let C = ( a E PI I + p(P3))> O}. Put @[ = { a E Pl I(@, p(P, 1 + p(P3))= 0). Set I, = b, o Then q = ,1 u is a Ostable parabolic subu=
(a,p(Pl)
Gael:
+
379
9.5. Some Results of Kumaresan, Parthasarathy, Vogan, Zuckerman
algebra of gc. We have seen that of a E @, then (A,u) = 0. Thus F" is one dimensional. Also, 2p,(q) = p , ( P l ) p,(P,), so p = A 2p,(q), as asserted.
+
9.5.7. Lemma.
+
Let the notation he as in 9.5.3 then if dim u,
Let H E it be such that cy(H) > 0 for [X,H] = O}. =
n we have
0F)""(A+ 2p,(q)) = (Aj"(l n p))A(A"u,,) 0F".
((A$)
pc
=
p, n I,@ u, 0 U,. A'pc
0F
= p+q+1 I
cy E
C (see 9.5.6) and I
=
{X
E
gI
Thus

0F.
Ap(p,. n lc) A4u, A'U,
Now F = U(U)F". Thus the A(H)eigenspace for H on F is F" (which is one dimensional by the above material) and if p is an eigenvalue for H on F then p I A(H). Let x E Ap(pcn I,), y E Aqu,,, z E A'ii,,, w E F be such that ad Hy = ay, ad Hz =  bz, H w = cw. Then H ( x A y A z 0w) = ( a  b + c)(xAyAz 0w). a = 2p,(H)  m with m 2 0 and m = 0 and only if q = n. Also b 2 0 and b = 0 only if t = 0 and c = A(H)  m' with rn' 2 0 and m' = 0 if and only if w E F". Hence A(H) 2p,(q)(H) = a  h c = A(H) 2p,(q)(H)  m  m'  b. Thus m = m' = b = 0. The result now follows.
+
+
+
Note. We have actually shown that
We conclude this section with a vanishing theorem for (g, K)cohomology (due to Kumaresan [l]) and a proof that it is best possible that are immediate consequences of the previous results and those of Chapter 6. If F is a finite dimensional irreducible ( g , K)module then set Q ( F ) = {q I q = I,@ 11 a proper @stable parabolic subalgebra of g, such that dim F" = I } (i.e., if F = C then Q ( C )is just the set of all proper Hstable parabolic subalgebras of gc). Put c ( F ) = min{dim u, I q = I, @ 11 E Q ( F ) } . 9.5.8.
Theorem. I f F i s afinite dimensional (9, K)module and irreducible unitary (g, K)module then ~ ' ( g K; , V @ F * )= 0
if' V is a nontrivial
for i < c ( F )
Suppose that H'(g, K ; V g F * ) is nonzero. Then Theorem 9.5.3 and Lemma 9.5.7 combined with Propositions 9.4.3 and 9.4.6 there exists a
380
9. Unitary Representations and (g, K)Cohomology
0stable parabolic subalgebra, q, of gc such that dim F" = 1 and i 2 dim u,. Also, Lemma 9.5.7 implies that if i > O then q E Q ( F ) .Now H o ( g , K; V 0 F * ) = { u E V 0 F* I ku = v and X u = 0 for k E K, X E g} by the definition of relative Lie algebra cohomology. Thus H o ( g ,K; V 0 F * ) = Hom,,,(F, V ) . Since the only irreducible finite dimensional unitary (g, K)module is C (the trivial (9,K)module) the result follows. We will now use the modules Bq(p) of 6.10.3 to show that the Kumaresan vanishing theorem is best possible. Let F be a finite dimensional irreducible (g, K)module such that if P is a &stable system of positive roots for @ and if A(P) is the highest weight of F relative to P then OA(P) = A(P). We fix P E c(pk). Let q be a &stable parabolic subalgebra of 9, compatible with P and such that dim F" = 1. Let sK E W ( K ,T ) be the element such that sKPk=  P k . Let sLnK be the element of W ( L n K , T ) such that sLnK(Pkn @((I n €)c,t c ) ) =  pk n @((I n f),, tc). Put s o = s L n K s K . Let k E K be such that Ad(k)I, = so. Set q' = Ad(k)'q. Here q = 1, 011. Set p(q)(h)= (tr(ad hl,,))/2for h E I). Put
9.5.9.
iL = s;'(A We set A,(A)
=
+ 2p(q)).
BJj.) (notation as in 6.10.3).
Proposition. A,(A) is a unitary (9,K)module. Furthermore
dim Horn,(
VA
+ zp,(q), A,(A)) = 1.
 P and let s2 E W(Ic,t),) be such Set sq = szsl. We note that j, = s;'(A 2p(q)) = s i ' A  s , ' s q p s o ' p (use sqp  p = 2p(q)). q' is compatible with s,'P. If c c ~ @ ( t ) ~ , A d ( k )  ' then u ) cx = s,'/3 with B E @( bc, u) (notation as in 6.4.5). So, Let s1 E W ( g c ,bc) be such that s , P that s2(Pn@(lc,bc)) = Pn@(l,,b,).
+
+
(3.  s , ' p , ~ r ) = (sG'A Since, s; (i)
=

s , ' s , ~ , c I ) = (A,P)
+ (p,s,'P).
'@(bc, u) c  P, we have ( 3  s; ' p , c() < 0
for CI E ~ ( u 'bc). ,
(i) Theorem 6.7.5 combined with Lemma 6.4.5 implies that A,(A) is unitary. This proves the first assertion. We note that A,(A) = TmM(q',C,) with m = dim uk. (i) also implies that T'M(q',C,) = 0 if i # m. We can thus apply Theorem 6.5.3 to find that dim Horn,( V,, A,(A))
= (
1)"
det(s)pL(i+k
s€W(K.T)
Pk
 s(2, k
Pk)).
38 I
9.6. uCohornology
Here pk is the partition function of @(ilk, tc). We note that pk(a) = p,( sea) ( p , the partition function of @(u,,,1,). Thus p;(i
+ Pk
s(Ap
+
/)k)) =
p,,(s,$(jL,
+ Pk)

sg(i
+
Pk).
We note that det(so)= (  1)"'. We therefore have (after the obvious algebraic manipulation)
We now assume that 3,, Since, sopk = p(q,)
=
A
+ 2p,(q) = A + 2p(q)
+ p(pk n @(€,. )",I
+Pk

2p(qk).
n I,, t,)) we conclude that
=A
+ 2P(q) +
We must therefore calculate
We now show that the only term in thc above sum that is not 0 is the one corresponding to s = 1. This term yields p,(0) = 1, and the second assertion would now follow. Fix H E it such that a ( H ) > 0 for s( E P. Let s E W ( K ,T ) be such that pn(S(jb,f P k )  (j, + Pk)) > 0. Then S()., f pk) = ,(; + Pk) + Q with p,(Q) > 0. Hence Q ( H ) > 0. On the other hand i., + p k is P,dominant so (s(& + p k ) + p k ) ) ( H ) S o . This implies that Q = 0. Since j,, + p k is Pk regular this implies that s = 1. The proof is now complete.
9.6. ucohomology 9.6.1. In preparation for the proof of the VoganZuckerman theorem on (g, K)cohomology we need some results on ucohomology. For the next three numbers g will denote a reductive Lie algebra over C. Let b be a Cartan subalgebra of g and let P be a system of positive roots for 0 = @(g,b). Let b = b(P) = 1) 0 OIEP g,. Let q be a subalgebra of g containing b. Put @ , = [ " ~ Q , l ( g , + g .)cq}.Set C = P @,, l = I ) @ @ a E Q , g a and i i = 9,. Then q = I0 11 and [ I , i i ] is contained in 11. We note that 1 is reductive and acts semisimply on 11. Set 1 1  = g, and q = 10 u. We note that g = u 010 I I  . Thus PBW implies that
BEEX
U ( g ) = U ( 1 )0(11U(g)0 U(g)U). Let p be the projection of U ( g ) into U(1) corresponding to this direct sum decomposition.
382
9. Unitary Representations and (g, K)Cohomology
Let H E b be such that a ( H ) > 0 for x E X and [H,I] { y E U ( 9 )I ad H(y) = O}. Then as in 3.2.1 (1) we have
= 0.
Set U ( g ) H=
(1) U ( d Hn@U(9)+ U ( 9 W ) = U(!IlHn (uU(9)) = U ( d H (U(g)I1). Thus, as in 3.2.1 we find that (2) p restricted to U(g)" is an algebra homomorphism. Let V be a gmodule with action n. Then Cl(u, V ) = Hom,(A'u, V ) is naturally an Imodule under the action (Xp)(Y) = X(p(Y))  p(ad X(Y)) for X E 1 and Y E 11. Also, d(Xp) = Xdp. Hence we have an action of 1 on H ' ( u , V ) for each i. Also, C'(u, V ) is naturally a Z(g)module under (zp)(Y)= z(p( Y)), z E Z(g), Y E A'u. If p E I* is such that p[l, I] = 0 then we set q,(X) = X  p(X)l for X E 1. Then q,, extends to an isomorphism of U (I) onto U(l). Set p,, = q,,p. The following result is due to Casselman and Osborne [I]. The proof below is due to Vogan [l].
Lemma. !f z E Z(g) and if B E Hi(u, V ) then zb (tr(ad h lu))/2,as usual.
= pZPcq,(z)b. Here
p ( q ) ( h )=
We prove this result by downward induction on i. If i = n = dim u then H"(II,V ) = Anti* 0 V/u I/. Thus z acts by I 0p(z). It is also clear that p(z) acts . ( I 0p(z))fi = pzP(,,(z)p.This is the result for i = n. by (  2p(q)0n ) ( p ( z ) )Thus Assume the result for i = r + 1 I n. We now prove it for i = r . Let F be the gmodule U ( g )0 V with g acting by left multiplication. Set a ( g 0u ) = yu. Then a is a 9module homomorphism of F onto V. Let X = Ker r . Then we have the gmodule exact sequence
O+X+F%
V+O.
Now U ( g ) is a free U(u)module under left translation. Thus Hj(u, F ) = 0 for j < n (6.A.1.5). So the long exact sequence of cohomology yields the 1 and Z ( 9)module exact sequence
O + H ' ( u , V )   t H i + ' ( u , x ) + H i + (14 ' F)
+
This injection implies the result.
9.6.2. We now use the above result to give an especially simple proof of a theorem of Kostant [2] (Bott [l] for the case when q = b). Set = P n @(I,b). Let W ' = (s E W(g,t))IsP contains 5 ) . If p E b* is +dominant integral then let E,, denote an irreducible finite dimensional Imodule with highest weight p.
383
9.6. itCohomology
Theorem. Let F be an irreducible finite dimensional gmodule with highest weight relative to P. Then as an Imodule the sum over s E W' with l(s) = i. Z(g) then z acts on F by x ~ + ~ ( zAlso, ) ) . z E Z(g) acts on Hi(u,F ) by acts on Hi(ll, F ) by xA+ ,(z). Thus, if z E Z(g)then Set W, = W(1,b). We denote by ly the HarishChandra isomorphism of Z(1) onto U ( b ) w f .Then [ y ci pp(,,, = y , the HarishChandra isomorphism of
If z
E
pZp&).
Z ( g )onto U(t)IW. As Imodules both F and Ail are semisimple. Thus H'(u,F) splits into a direct sum of irreducible Imodules, E,. Let pij be the highest weight of E,. Then Z ( I) acts on E, by I ~ r , , (lower left subscript corresponds to objects defined for I in the same way as they are defined for 9). This implies that pij pI p(q) must agree with 1 + p on U ( b ) w . This implies that there must exist sij E W such that pij pI = sij(2 p). Since pij pl is 5dominant and regular sij E W ' . As an 1module Hi(u,F) is a subquotient of Nu* 0F. Thus the weights of H'(u, F ) are of the form cr  (Q) with Q a subset of C and cr a weight of E Hence, pij = cij (Qij) with crij a weight of F and Qij a subset of C. We therefore have, sij(l. p ) = crij p  (Qij). Now p  (Qij) is a weight of a finite dimensional representation with highest weight p (9.A.1.5) hence aij = s,3, and sijp = p  (Qij). 9.A.1.6 therefore implies that Qij = ( sijP) n P. Hence l ( s ) = i. Also the multiplicity of this weight is at most 1. We have therefore shown that as an Imodule
+ +
+
+
+
+
+
the sum overs E W' with l(s) = i and m, is either 0 or 1. The above argument also tells us how to construct the corresponding cohomology classes. Set for s E W ' , Q = (  s P ) n P. Then the (sn  (Q))weight space in Au* @ F is one dimensional and is contained in A'u* @ F. Let be a nonzero element of C'(u, F ) in that weight space. The d p = 0 and /3 cannot be in the image of d . Hence m, = 1 for all s E W' with l(s) = i. This completes the proof of the theorem.
a
9.6.3. We now return to the notation of the previous sections. Let q be a Bstable parabolic subalgebra of gc, q = I, 0u, as usual. Fix t), a fundamental Cartan subalgebra of g contained in 1. Then l) = t + a, as usual. Let H E it be such that I = { X E g I [ H , X ] = 0) and such that ad H has strictly positive eigenvalues on u. Clearly, U(gc)' is a subalgebra of U(g#. Hence p is a
384
9. Unitary Representations and (9, K)Cohomology
homomorphism of U ( R , . )into ~ U(lJKnL.Write 11 = uk 0 u,, as usual. Let R = dim u, and fix an element p E AR(u,)*  ( 0 ) . Let oi denote the map of I\'(&)* into Ai+' u * given by a,(cc) = ccAB. Let V be a gmodule. Then a,O I is a K n L module homomorphism of c'(uk, V )O A'(u,)* into C"'(i1, V )which commutes with the pertinent "8s". It therefore induces a map
n i : H i ( u kV, )@ A'(u,)* + H i + R(u, V ) . Welet U(g,)Kact on c'(uk, V )by(zj)(y)=z(/?(y)),~€U(g,)~and y~A'(u~)*. The following result is due to Vogan [ 11. Lemma.
If z
E
U ( g , ) K and i f
GI E
Hi(&, V ) 0 A'(u,)* then
ni((z O 1 ) ~=) Pzp(q)(z)ni(GI). As in the proof of Lemma 9.6.1 we prove this Lemma by downward induction on i. We first look at i = m = dim uk (the largest index for which there is anything to prove). Then rn + R = dim u so Hm(Uk, V
)
AR(U,)* =
Am+Ru*O V/ukV
and
H m + R ( ~V ,) = am+'^* 8 V/uV. Hence n,,,is given by the natural map Am+'u* @ V/ukV+ Am+'u*
@ V/uV.
Thus the result in this case follows in exactly the same way as in 9.6.1. We now assume the result for i + 1 I m and prove it for i. Let F, X be as in 9.6.1. Then the gmodule exact sequence O+X+F+V+O
induces the following commutative diagram with exact rows H'(uk, F ) @ AK(IIn)* + H i ( l l k , V )O AR(U,)*
5
J.
H'+'(l[,F)H'+yu,
V)
H'(uk, V )O A'(U,)*
Hi+'+'
J.
(U? X
)
As before, H'(uk,F ) = Hi+'(u, F ) = 0 for i < m. The result for i now follows from the result for i 1 applied to X .
+
9.6.4. The next result will play an important role in the calculation of H ' ( g , K ; Aq(A)O F * ) . It is a special case of a more general result that is fairly
385
9.6. uCohornology
easily derived using the derived functor construction of the Zuckerman modules. Rather than interrupt our exposition to give the more “sophisticated” result we have opted to give the following cumbersome proof. Set P, = Pkn @((f n tc). Put K W 1= {s E W ( K ,T ) I P, is contained in SPk}. Let so be the longest element of K W 1 .If p is a P,dominant integral form that is Tintegral then let E , denote an irreducible, finite dimensional K n Lmodule with highest weight p . Lemma.
Let y
E
K” have highest weight ?, and let K,
E
y. Then (m
= dim uk)
H”(b L n K ; M(qk, E p ) 8 (V,)*)
(*I
is zero unless p = so(?.,
+ pk)

pk und in this case it is one dimensional.
Lemma 9.4.1 implies that (*) is nonzero only if there exists t E W ( K ,T ) such that p = [(A, + p k )  p k . M , = M(q,,E , ) @ (V,)* has a (f, L n K)module filtration M , 3 M , 3 ... Md 3 M d + = (0) with M J M , , % M(qk,E N  * , )with hia weight of V,. As above the only terms that can contribute to cohomology are those such that p  hi + pk = spk with s E W ( K ,T). For such a term we write 6, = sp,. Then t(2, + p k ) = s ( p i pk). Since pi is a weight of V, this implies that t = s and p i = 2,. We have therefore shown that
+
(**I
H “ ( f , L n K ; M ( q k , E , ) @ ( V , ) * )= Hm(€,LnK;M(qk,E,,,).
We now show that (**) is nonzero only if t = so. We prove this by setting up a “resolution” as in 6.6.2. Let fi E tz be P,dominant and Tintegral. Set M = M(qk,&).Set
Di= U(€,) @
U((ln1)c)
and let a,:D,
+
0Ed)
Di be defined as follows:
ai(k @ x l A x , A ~ ~ ~ A@xe) i = C (  l ) j k x j 0x,A...A5ijl\... Ax, @ e
+ 1 (  1)“’k
8 [xr, xs]A x , A . . . AQI. . . A2J.. . Axi @ e.
r<s
Then as in 6.A.1.4we have the (f, K n L)module exact sequence
0 + D,
+ D,
+
... + D , + Do + M
+ 0.
Let X i = a,D,(X,,, = D,,,). Then we have the following (€,K n ,!,)module exact sequences
O+X,+D,+M+O 0+Xi+
+
Di
+
X i + 0.
and
386
9.
Unitary Representations and (g, K)Cohomology
These induce cohomology long exact sequences
Hi(€, K n L; D o ) + H i ( € ,K
n L; M ) + H i ( € K , n L ; X , ) + H i + ' ( f ,K n L; D o )
and
H i ( f, K n L; Dj) +H i (f, K n L ; Xi) + H i ( €, K n L ; xj+ 1 ) + H i + ' ( f, K n L; Dj) Now H'(f, K n L;Dj)= 0 for i < 2m (6.A.1.5). We therefore find that
'
H"( f, K n L ; M ) z H m ( €,K n L ; X , ), +
H"+j(f,K n L ; X j ) Y, H " + j + ' ( € , K n L ; X j + , ) for j
+ 1 < m. Hence H"(f, K n L ; M ) Y, H Z r n  '( f , K n L ; X ,  l ) .
There is still one more long exact sequence + H2"l(f,K n L;X,
H2" 1 ( € , K n L;D,
1)
+ H 2 m ( €K , n L ; D,) + H2"(f,K n L ; D,
Since H 2 "  ' ( € , K n L; D m  l = ) 0, we have the exact sequence
O+H'"'(€, K n L ; X m  , ) + H 2 " ( € , K n L ; D , ) + H 2 m ( € , K n L ; D , ~ l ) . Now H2"(f, K n L; D,)
=
((u(fc) @ (A"11,O Ea))/fU(f,) U((fnl)c)
@
(A"11,O Ea)))KnL
U(lfnl)c)
which is a quotient of (A"u, 0E#"'*. We now look at the case when 6 = tpk  pk. Then (Amuk0 E,)Kn', is nonzero only if rp,  pk = 2p(q,). But then t = so. 6.4.5 combined We are left with calculating H " ( f , K n L; M(q,, CsnprPr)). with 6.5.1 imply that this is C as asserted. 9.6.5.
We now turn to the notation in 9.5.9.
Lemma. lf y p,(Q) > 0.
E
K" and
if
Aq(A)(y)is nonzero then A? = A
Let M , = M ( q ' , C , ) 3 M , constructed in 6.4.4. Then
3
M,
3
+ 2p,(q) + Q with
... be the (€, K n L)module filtration
Mi/Mi+,= M(qL,LQ,)
with p X Q i ) > 0.
387
9.6. uCohomology
If q is a homomorphism of Z ( tc.)to C then let ( M , ) q = { m E M , I ( z  q(z))lm= 0 for some r and all z
E
Z(f,))
The above filtration implies that M , is the direct sum of the ( f , K n L)modules ( M o ) q .Let p be the infinitesimal character of y,.Then dim Hom,(Ki,T"M,)
=
dim H"(f, L n K ; M , O ( V i ) * )
= d i m H"(f, L n K ; ( M o ) f ' @ ( P ' J * ) . = (M,)"
inherits a ,finite filtration Vo 2 V, =) ... Vd 2 V,, = 0 with v/q+ 2 M(q;, Ei,J and i, Qi + pk = si(jLy + p k ) for some siE W ( K ,T ) . If we use the spectral sequence (9.A.2) corresponding to the filtration
Now, V,
,
F' C'( f, L n K ; V, 0( K9)* ) = C'( f, L n K ; 0( Vi)* ) then the El term is the direct sum of the spaces
K(f,L n K ; (K/ 0 and i.  Q = sO(iy &)  P k . This combined with the definition of and q' implies the result (we leave the algebra to the reader).
+
9.6.6. We continue with the notation of 9.5.9. So F is a finite dimensional (g, K)module satisfying the hypothesis therein. We fix q E Q ( F ) and i.the highest weight of F relative to a 0stable system of positive roots compatible with q. Theorem.
Let R
=
dim
11,.
Tlzrri
H'+K(g,K;A,(i.)O F * ) = Hi(L K n L ; C ) .
In light of Proposition 9.5.9, Lemma 9.5.6 and Proposition 9.4.3, it is enough to show that if q , E Q ( F )and if q , is compatible with PI then Aq(j)(;si,,~,l+ 2 , 1 , < w l , , ) = 0
unless ).(PI)+ 2p,(ql) = j . + 2p,(q). We note that pc = (p n I), @ II,, 6 f i n . Thus if CJ is a weight of t on Ap, then CJ = A + 2pn(q)  B  C with A a weight of A(p n B and C weights of Ailn. Thus if p is a weight of Apc 0 F then p = 6 + (T as above and 6 a weight of F. Thus 6 = i.  Q with Q a sum of elements of P. We therefore see that
388 y
=
9.
A
Unitary Representations and (9, K)Cohornology
+ 2p,(q) + A  B  C  Q. This implies that A(P1)
+ 2p,(q,)
=
A
+ 2p,(q)

B CQ
+ A.
On the other hand L ( P l ) + 2p,(q,) = 1, + 2p,(q) + S with p , ( S ) > 0 by the previous result. Let H E it be as in the definition of 6stable parabolic subalgebra for q. Then if we evaluate the above two expressions on H we find that 0 I S ( H ) =  ( B + C + Q ) ( H )5 0. Thus S(H) = 0. But then S = 0 and the result follows.
9.7. A theorem of VoganZuckerman 9.7.1. In this section we complete our discussion of (g, K)cohomology. If F is an irreducible, finite dimensional (g,K)module as in 9.5.9 and if q E Q ( F ) (9.5.8) let (F,q) denote the action of I on the one dimensional Imodule F”. The Theorem of VoganZuckerman [ 11 is Theorem. Let V be an irreducible, infinite dimensional unitary (9, K)module such that H ’ ( g , K ; V @ F * ) # 0. Then there exists a 6stable parabolic subalgebra of gc, q E Q ( F ) such that V is (9,K)isomorphic with the irreducible summand of A,(L(F, 9)) containing the Ktype with highest weight R(F,q) 2P,(d.
+
Note. This result, combined with Proposition 9.5.9, Theorem 9.6.6 and Theorem 9.4.6 completely calculates the (9, K)cohomology with coefficients in I/ 0 F* for V irreducible and unitary and F finite dimensional. We note that if we argue as in 6.6.2 using a “resolution” as in 9.6.4, it is not difficult to show that the A,(]+)are irreducible. The proof of this theorem will occupy the remainder of this section. We first give an outline of the proof. Theorem 9.5.3 implies that there exists q E Q ( F ) such that V(A(F,q) + 2p,(q)) is nonzero and that V has the same infinitesimal character as F. Choose q E Q ( F ) such that llA(F, q) + 2p,(q) + 2p,ll is minimal subject to the condition V(l.(F,q)+ 2p,(q)) is nonzero. Let y denote the corresponding Ktype. We prove that the multiplicity of y in V is one. Let ,u denote the homomorphism of U(g,JK into C that corresponds to its action on V(y).We show that y depends only on q and F. Since A,(L(F, 4)) has the properties just used for V (9.6.6) we can apply the above argument to
389
9.7. A Theorem of VoganZuckerman
it as well. Thus U ( g # acts in the same way on V ( y )and on A , ( I ( F , q))(y). The theorem now follows from Theorem 3.5.4. We will now give the detailed proofs of the assertions made in the course of the above sketch.
9.7.2. Fix Pk a system of positive roots for @(b,t,) such that q is compatible with Pk.Let b, = t, 0i l k be the Bore1 subalgebra of f, corresponding to Pk. Set = itk n.1, (q = 1O , u, as usual). Let nibe defined as in 9.6.3 and let R = dim 11,.
Lemma.
nn defines an isomorphism of' AR(u,)* 0 V(.))"" onto
Hn(ii, V)""(I(F,q)).
To prove this we analyze the spectral sequences in 9.A.2.3 and 4. We take, u2 = 11,. Then 11, u 1 and u2 satisfy the conditions of 9.A.2.3. Thus we have a spectral sequence with abutment H'(u, V ) and
ti1 = ilk and
EYsq = H q ( i ~ , , A P@~ ,V ) . Set A
=
A(F, 4). We prove the Lemma by showing that (EP.4)"'.k(A) = 0 unless
p = R and q = 0 and that
(EP~q)"'*k((n) = (An(~i,)*0 V(y)"")"'~"(A). This will clearly suffice to prove the Lemma. To this end we use the spectral sequence in 9.A.2.4. This time we have for an E;"term H'(llk, V )0( A 1 l n )  a r  a s .
Here H (i)
E
it is chosen as usual. Let
(T
E
K A be such that
( H r ( ~ i V(a)) k , 0( A ~ i , )  ~ ~  ~ ~ ) " I#, "0.( j ~ )
Since H'(u,, V ( o ) ) is a multiple of Hr(iikrV,) we can apply Kostant's formula. Let p be the highest weight of (T relative to Pk then the K n Ltypes that occur in Hr(uk,V(a)) have highest weights s(p p k )  pk with l(s) = r and s E K W 1= {s E W ( K ,T ) lsPk 2 P, n @((€ n I),, t,)}. Thus (*) implies that
+
(ii)
A = s(P + P k )  Pk

(Q)'
with Q a subset of C ( = @(u,t),)) and (Q)' is a weight of is, as usual, the projection of (Q) onto (t,)*).
A(ii,)
(here (Q)'
390
9.
Unitary Representationsand (9, K)Cohomology
Fix P E c(pk) such that q is compatible with P. Let pn = p,(P). Then (Q)' = 2p,  (Q')' with Q' c P and (Q')' is a weight oft on A"(pc n n(P)) with u = IQ'I. Put Rk = { M E @ k I (a, S,U  p , ) > 0) U { R E S P k I (a, S,U  p , ) = 0). Then S,U p, is Rkdominant (Rk is a system of positive roots for Qk by 9.A.1.4). Let Rk = tPk, t E W ( K ,T ) .Set c = (I2 E SPk/(a, Sp  p , ) < o}. Then
Since C is a subset of Pkn (Spk) there exists a subset, C', of P disjoint from Q' such that C = (12 1, ; a E C'). Put A = Q' u C'. Then sp
(iii)

pfl
+ tPk = 1. + p n + Pk

(A)'
=
1.
+p

(A)'.
Hence lisp  pn
+ tpk\l
=
llA +p
5

IIA + p

ll
5
[IA +
since p  ( A ) is a weight of a finite dimensional representation of g with highest weight p. Let u E W ( K ,T ) be such that p  p , is upkdominant. Then Lemma 9.1.7 implies that Ilsp  P n
+ rPkII 2 lip  pn + upkll.
Since the Ktype with highest weight u  ' ( p Dirac inequality implies that 
pn

p,) occurs in V g S (9.1.4) the
+
+ uPkll 2
This implies that all inequalities are equalities. So there exists q , that p = l+(F,ql)+ 2p,(q,). Our hypothesis on q implies that lip
(iv)
+ 2PkII
2
+
E Q ( F ) such
[IA + 2pn(q) + 2pkII.
+
+
+
We now show that if 11 # i. 2p,(q) 2Pk or if D , = jL 2p,(q) 2pk and p < R or q > 0 then we have a contradiction. Choose a system of positive roots, 6,for @((,,b,) such that if 2p,,k = (Pk n @((€n I),, t,)) then p[,k is &dominant. Set PI = u Z. Then PI is a 8stable system of positive roots for @(g,,t),) compatible with q. Put p,,, = p ( 6 )  p1.k. We rewrite (i) as P
2pk
with Q" c C and (Q")' P
= S'(i
f
2Pn(q) + P k  (Q")')
+ Pk
is a weight of A R  P ( ~ i nHence, ).
+ 2pk = s'(L + 2Pn('I) + 2pk

(Q")'
 (Pk 
Spk))
391
9.7. A Theorem of VoganZuckerman
+ pk
Now, (Q")' Hence

+ 2pk =
(v) (vi)
pn(q)
with B a subset of C and IBI
spk = (B)'
+ 2pn(q) + 2pk
S'()L
+ Pk is &dominant
+

p
+ q.
(B)').
and if a E C then (pn(q)+ pk,a) > 0.
+ =p + pk, a) 2
Indeed, pn(q) pk = p(q) expression implies that (p,(q)

=R
p1.k

p,,n. If a
(pl,k,
E
& then the second
a) 2 0. If a E
I: is simple then
(pn(q) + pk,a) = ( P ? @ ) (pf,n?a).
Now 2p,,, = C rnpbthe sum over all b E P , such that ((lc)p n pc is nonzero and rnp = 1 or $. Since (a, b) 5 0 for fl E P,, (p,,n,a)I 0. (vi) now follows. If we use (v), we find that
+
lI~~+2pn(q)+2pk112  IIp+2pk112 = 2 ( ~ + 2 p n ( q ) + 2 p k , ( B ) )  ( ( B ) ' , ( B ) + ) = 2(1
+ pn(q) + P k , ( B ) ) + 2(pn(q)+ Pk, ( B ) )  ( ( B ) + ,( B ) ' ) .
(vi) implies that the first term in the last expression is strictly positive if B is nonempty. Thus if we can show that 2(pn(q) + Pk, ( B ) )  ( ( B ) + ,< B ) + )2 O,
(vii)
we would conclude from (iv) that B is empty and the Lemma would follow. We are thus left with (vii). Let C be a subset of
6 such that 2p1., = (C)'. C,
and set C,
=
= {aE
C  C,. Put C,
=
Let
C l ( ( B ) + , a ) > 0)
{aE
pr

C I ( ( B ) + , a )> 0}
Let S E W(Ic,bc) be of minimal length such that ( B ) + is s&dominant. Then
 s S = { a ~ @ ( l C , b C ) ; ( ( B ) + , a>) O } U { ~ E ~ ~ I ( ( B ) + , C L ) = ~ } by 9.A.1.4. Thus (sfi)nP, = C , uC,. We note that since B(B)' ( B ) + , s o = 0s. Thus ( C , ) + (C,) = (C,)' + (C,)' and so ( p I = p(P,)) SP, =
PI

(C,)'
 (Cl>+.
This implies (viii)
( 2 P f  (C,>'

( C , ) ' , (C,)+
+ (CI>+) = 0.
=
392
9. Unitary Representations and (g, K)Cohomology
9.7.4. We can now apply Lemma 9.6.3 to see that the action of U ( g ) Kon V(j. 2 ~ , ( q ) ) "is~ given by pzp(q)(.x)on H R ( u ,V)"'.k(E.). Let 3, = 3(I) n t and I, = (j(1)np ) 0[l,I]. Then I = 31 0 I , . Since, (A,a) = 0 for c( E @(Ic, bc), f n I acts trivially in HR(u,I/)"'B~(~,). Hence, Theorem 3.6.6 implies that U ( acts on HR(u,V)"l,k(jb) by a commutative algebra. Thus U(g,JK acts on V(A + 2p,(q)) by a commutative algebra. Proposition 3.5.4 now implies
+
Lemma.
dim V ( 2 + 2p,(q))"* = 1.
9.7.
393
A Theorem of VoganZuckerman
This is the first assertion of our outline. In particular there exists a homomorphism (T of U(g# into C such that gu = o(g)u for
9.7.5.
uE UA
+ 2ptI(q)).
We now compute (T. Fix t),, a maximally split Ostable Cartan subalgebra of 1. Then b,, = t, + a, t, = bo nf and a, = bo np. Let p, be a corresponding minimal parabolic subalgebra of 1, p , = + a, + 1i0, as usual. 31 acts on HR(u,V)’”*k(i) via A13,, Also, U((ll)c)’lnf acts on this space via v yo for some v E (a,): (3.6.6). We now look upon 3, and p(q) as elements of (I),,):. Notice that ( P , ) ~0u is a parabolic subalgebra of gc. Let Q be a system of positive roots for @(g,, ( bo)c) compatible with this parabolic subalgebra. Let p denote the “p” for this system of positive roots. Since I/ has infinitesimal character x i + p . 9.6.1 implies that if z E Z(gc) then P ~ ~ ( ~ , (acts Z ) on H R ( u ,V‘)nf,k(A) by X ~ + ~ ( Z This ) I . combined with the above implies that there exists s E W ( g c ,( I),)J such that 0
(*I
s(2
+ p(q) + t’ + p,)
=
+
/I p
here pm is the “p” for @((Om, + ao)c,(I),)c)n Q. In particular, (*) implies that v €(a,)*. We may (and do) thus assume that (v, c() 2 0 for a E @(po,a,). We rewrite (*) as 1.
(**)
+ p(q) + v + j),n = S  y A + p).
We note that s  ’ i b = 3,  Q1 and s  ’ p = p  Q 2 with Ql and Q 2 sums of elements of Q. Since, (v + p m , i + p(q)) = 0, if we take the inner product of both sides of (**) with i. p(q) we have
+
0,+ P(qX 2 + P(q)) = ( 2 + P(OLj, + P ( q ) )  (2 + P(qX Q i )  ( 2 + p(q),Q 2 ) .
+
+
Now, (A p(q), Q1) 2 0 and (2 p(q), Q 2 ) 2 0. The above inequality therefore implies that (i(q),Q2) = 0. This says that (  s  ’ Q ) n Q is contained (9.A. 1.3). We have thus in a(lc, (11,)~) n Q = Ql. Hence s E W (I,, )&I( shown if po = p p , ,then (v
+
zz ” ‘ ( P O
+ pm).
sCi(po + p,) = p, + pm  S with S a sum of elements of Q I . Hence,  S. Thus SI, = 0. This implies that S is a sum of elements of @(po,ao). Also, Ilv + Pm1I2 = llvl12 + llpmll2and v = p,
394
So, (v, v)
9. Unitary Representations and (9, K)Cohomology = ( p o ,po). On
(v, v)
the other hand,
= (v, Po

S) 5 (v, Po)
= (Po 
s,P o ) 2 ( P O ? Po).
Thus the inequalities are all equalities. In particular, this implies that ( p o , S ) = 0. Thus ( p o , p o ) = (v, v) = ( p o ,po) (S,S). Hence, S = 0. Thus, v = Po. We have therefore shown that the action of V(g# on V(I1 2p,(q)) depends only on F and q. This completes the proof of the steps in the outline of the proof. Q.E.D.
+
+
9.8. Further results 9.8.1. We continue with the nc ation of he previous section. We note that if we combine the vanishing theorem of Borel, Wallach [l;V, 3.41, Zuckerman [2] and 9.5.8,9.5.9 we have Lemma. I f q is a proper 0stable parabolic subalgebra of gc then dim u, 2 rk,g.
Obviously, this result has a direct proof. In fact, there are tabulations the values of c(G) = c F ( G )for F = C. If G is simple and has the structure of a complex Lie group then the tabulation was first given in Enright [l]. We give the table. The first column is the classical name (if it exists), the second column is the name in the Cartan classification and the third is the value of c(G). We now give the table of VoganZuckerman [ 11 for G simple over R such that G, is simple over C (i.e., G has no structure as complex Lie group). This
Classical group
+ +
S L ( n 1,C) SO(2n 1,C) W n , C) S0(2n,C)
n2 1 n22 n>3 n24
dG)
Cartan Label A" B" C" D" E6
n
2n 2n
2n 16
EL7
21 51
F4 G,
15 5
E,

1

I 2
~
9.8 Further Results
395
time we will only include entries for cases when c(G) > rk,(G). In this table the first column corresponds to the classical label (if it exists) the second column gives the Cartan label (Helgason [l, p.5181) and the last gives c(G).
Classical group
Cartan Label
SU*(2n)n 2 3 SU*(6) SO*(2n), n 2 4 SP( P , 4). 1 I p I 4
A11 All DlIl CI I El EII Ell1 EIV EV EVI EVlI EVIII EIX FI FII G
C(C)
2(n

3 nl 2P 13 8 8 6 15 12 11 29 24 8 4 3
1)
9.8.2. We conclude this section with some results for groups of Rrank one that are direct consequences of the theory in this chapter and of the calculations in Borel, Wallach [ I ; VI, Section 41. Theorem. Let G = O(n, 1)' or SU(n, 1). Let V be an irreducible (g, K)module with injnitesimal character x p . Then there exists a &stable parabolic subalgebra, q, of gc such that V is (g,K)isomorphic with A,(O). In particular, V is the underlying (g, K)module qf an irreducible unitary representation. Let n P ( G ) denote the set of equivalence classes of irreducible ( g , K ) modules with infinitesimal character x p . In Borel, Wallach [1, op. cit] it was shown that there is a bijection between n(G) and S
=
(y
E
K A Hom,(Vy,Apc) # O}.
(This was done using the Langlands classification and by explicitly decomposing pc as a Kmodule.) We leave it to the reader to check that each
396
9. Unitary Representations and (9, K)Cohomology
S has highest weight 2, = 2p,(q) for an appropriate 8stable parabolic subalgebra of gc. Thus the counting argument implies the result.
y
E
9.8.3. The connected semisimple Lie groups of split rank one can be listed (up to local isomorphism) as O(n, l)', SU(n, 1) n 2 2, Sp(n, 1) n 2 2 and FII. Let G correspond to one of the latter two examples. Then the vanishing theorems imply that if I/ is an infinite dimensional irreducible unitary (9, K ) module then H ' ( g , K ; V ) = 0. Since there always exists an infinite dimensional irreducible (g, K)module with H ' ( g , K ; V ) nonzero (Borel, Wallach [l; V, 4.61) this implies that the analogue of Theorem 9.8.2 is false for these groups.
9.A.
Appendices to Chapter 9
9.A. 1.
Weyl groups.
9.A.l.l. The purpose of this appendix is to prove a few results about Weyl groups that will be used in the body of this chapter. Let g be a reductive Lie algebra over C. Let b be a Cartan subalgebra of g and let @+ be a system of positive roots for @(g, tj). Let W = W(g,tj) be the Weyl group of @(g, tj) (0.2.3). Let A be the set of simple roots in 0'. Then W is generated by the reflections s,, a E A. If s E W then we define the length of s relative to @+, Z(s), to be equal to min{r Is = s, s 2 . .. s, with each si a reflection about a simple root hyperplane}. It is clear that I(s) = /(sf'), since reflections are involutive. If s E W then set C (s) = { a E @+ I sa E a+>. 9.A.1.2.
Lemma.
Let s E W. Then
(1)
p (s)l = 44,
(2)
s is a product of reflections s,, a E C (s).
If s = 1 then C (s) = 0and (l), (2) are clear. Suppose that we have proved (1) and (2)for 0 I l(s) 5 r  1 and that l(s) = r. Let A = {al,, . ., a [ } and set s, = si if a = ai. Let s = s i t .. .sirbe a minimal expression. Put a = ai, . Then s,s = s i z . . . siris also a minimal expression. Thus l(s,s) = l(s)  1. Since a is simple, sap E @+ if /? E @+  { a } . This implies that if fi E @+ and if s,sp E @+ then /i'E C (s) unless j3 = s'a. If fl E Z (s) and if sg is not equal to  a then p E C (s,s). This implies that if  s  l a is not in C (s) then C ( s ) would be equal to C (s,s). This would imply that
9.A.1.
397
Weyl Groups
s@+ = s,s@+
and hence s
(a)
= s,s
which is false. Thus
.sls(
C (S,S)
(b)
=
E
C (s)
C (s)

and
{ s'a).
This implies that [ C (s)l = IC (s,s)l + 1. So the inductive hypothesis implies that (1) is true for s. We note that (2) combined with (b) implies that ss, is a product of reflections from C (s). Also, s  ~ , = ss,spl. Thus s,s(s,,) = s,s(s*s,s) = s. Hence s satisfies (2). This completes the proof of the Lemma.
9.A.1.3. Corollary. If s E W then s is u product of rejections about roots in @+ n (s@+). By definition @+ n (so+) = C (sI). Thus the previous Lemma implies that sl is a product of reflections about roots in 0 ' n (so+). Since root reflections are involutive, the Corollary follows.
9.A.1.4. Lemma. Let p E b* hr such thut ( p , a )E R, CI E @(g,b). Let s E W be of minimal length such that (sp,a ) 2 0 for a E @+. Then
(*I
s'@+ = j C I E 0 ) ( ~ 1 , a ) > O ) U ~ s ( E 0 + ~ ( p L , a ) = 0 } .
We note that (1)
If o E
b* is such that (0, a ) E R for all a E @ then P,
= (aE
0 I(0, r ) > 0) u (CI
E
0+I (a,o)= 0)
is a system of positive roots for 0. We leave this as an exercise to the reader. (1) implies that the right hand side of (*) is a system of positive roots for 0. It may thus be written in the form with t E W. Let u E W be such that u p is @+dominant.Then u  '@+ contains {CI E
I
@ I ( p ,a ) > 0) u ( a E @ (p,a) = O}.
Hence u  ' 0 + n(0+) contains t  ' 0 + n(@'). Hence I(u) 2 l(t). If n (0'). So u = t . l(u) = l(t) then up'@' n (0')=
9.A.1.5. We conclude this appendix with some results related to the irreducible finite dimensional representation with highest weight p. If Q is a subset of 0 ' then set (Q) = C,,, a.
398
9. Unitary Representationsand (9, K)Cohomology
Lemma. Let F be an irreducible j n i t e dimensional gmodule with highest weight p. Then the weights of F are the linear forms p  ( Q ) with Q a subset of @+ and the multiplicity of a weight ,u is the number of subsets Q of @+ such that P=P
(Q).
The Weyl character formula says that if ch F is the character of F and if A = eplIaEg( 1  ea) then ch F
= SE
Now, A
det(s)eZsp/A.
= C s s w det(s)esP.Thus
ch F
9.A.1.6.
w
= eZp
n
as@
n
(1  eKza)/eP
as@
(1  e')
Lemma. Let Q be a subset of 0 ' and let s E W. Then
(Q> = P  SP if and only if
Q
=
(so+) n @+.
We note that 2sp = Cas@+nso+ a+ 2~
=
a and that
Cas@+nsa+ a  x a s (  @ + ) n s @ +a.
The obvious subtraction implies the sufficiency. We now prove the necessity. The previous lemma implies that p  (Q) is a weight of F, an irreducible finite dimensional gmodule with highest weight p. Our assumption says that p  (Q) = sp. Since the weight sp occurs in F with multiplicity 1 the necessity now follows from the sufficiency. 9.A.2.
Spectral sequences
9.A.2.1. In this appendix we collect some material on special sequences which will be sufficient for the application in this chapter. A detailed account of spectral sequences can be found in MacLane [11. Let A be a vector space over C and let d E End(A) be such that d 2 = 0. Then, as usual, we write H ( A ) = H ( A , d ) = Z ( A ) / B ( A )with Z ( A ) = ker d and B ( A ) = dA. If A is a graded vector space A = A' and if dA' is contained in A"' then we write H ' ( A ) = Z ' ( A ) / B ' ( A ) with Z ' ( A ) = { a E A' da = 0 ) and
Oi2,
9.A.2.
399
Spectral Sequences
B'(A) = dA'+'. We assume that A (resp. A') has a filtration F'A such that each F f i is d stable (resp. dFfi' is contained in F f i j " ) . We also assume that F ' A 2 F ' + f i , F'A = 0 and F'A = A for i I 0. Put Gr A = @ i z o F ' A / F ' + ' A . Then d induces Gr(d) on Gr A . We analyze H ( G r A , Gr(d)). By definition
n
Bit,{ a E F ' A Ida E F'+l}/F'+'A B(Gr A ) = Oit0 (dF'A + F' ' A ) / F '"A.
Z(Gr A ) =
and
+
Hence H(Gr A ) =
@ { a E F'A Ida E F ' + ' } / ( d F f i
it0
+ F'+'A).
'>
+
Set Zi = { a E F'A Ida E F i t and E'; = Zi/(dFfi F'+'A). Then @ E i = H(Gr A ) . To establish a pattern for higher terms in the spectral sequence (which we are both explaining and constructing) we set Z, = F'A. Then
El Set Zi
= {u E
=
Zl/(dZb
+ Zr'),
F'A I da E F'+'>. It is clear that dZ' c Z'+' and that d(dZh
+ Z r ') c d Z r '.
Thus d induces
d,: E;
+ E;".
Let z E E' be such that d,z = 0. Let a E Z' be an element of z (recall that z is a coset). Then da is an element of d Z r Z; . Hence there exists u E ZF'such that d(a  u) E Z;' = F'+'A. Hence a  u E Z i . It is obvious that Z c 2is contained in Z y . Hence we have a linear map of Ker(d, into Z i / Z i + ' . Set E i = Z i / ( d Z ;  ' + Z;+'). Then the above natural mapping induces (El =
+
'
IEi )
'
BE;)
TI:Z'(E,) + E ; . Suppose that T,(z) = 0. Then if a E z , a E dZi' a = dz,
+ Z"',
.Thus
+ u, u E z;'
and u E Zy' c Z r ' . Thus Ker d, is contained in B'(E,). Thus TI induces an injective linear map, s,,of H ' ( E , , d , ) into E,. Suppose that z E E , then there exists U E z with a E Z,. Thus a E Z 1 and d a E Z i + ' = F"+,A. So a defines an element of Z'(E,, dl). This proves that S, is bijective.
400
9. Unitary Representations and (9, K)Cohomology
This sets the pattern, set 2; = {a E F'A Ida E F"'}
and
+ ZLT;).
Ef = Zf/(dZfI';
Then, as above d induces d,: E: + EL+'. We note that E,P = (2; + FP+'A)/(dZ,PII+' + FP+'A)
(1)
and
(2) HP(E,,d,) is isomorphic with E,P+lunder a natural map S, defined in the same way as S , . We now relate these spaces with H(A,d). We note that since each F'A is dinvariant we have a natural mapping Li of H ( F h , d ) into H(A,d) that assigns cohomology classes to cohomology classes. Similarly, if j > i then we have a natural mapping L, of H(FjA,d) into H(F'A,d). Obviously, LiL, = L,. There is thus a decreasing filtration, F'H(A,d) = H(F'A,d) of H(A,d). We assume, for the sake of simplicity that there exists a nonnegative integer, s, such that F"+'A = 0. E'
(3) Indeed, Z"+'
=
=
Z(A) n F';I/B(A) n F'A
=
F'H(A).
{a E F'A da E F'+"+'A)= Z ( A ) n F'A.
Also
Z i  ( s + l ) += l {a E F'Y I da E F ' A }
=
{a E A ! da E F'A)
= B ( A ) n F'A.
We say that the spectral sequence E,P*qhas abutment Ho(A,d). 9.A.2.2. We now assume that A is graded. So A FPA' c A ' and dFPA' c FpA'+'. Set
Then E P =
ZIP"'
= { Z E FPAP'q
E:4
= Z:.q/(dZ,P:;+l*q+r2
=
Biz,,A' and dA' c A"',
Idz E F p + r ( A p + q + l ) ) , P+lsql +zr1
1.
0, Ep*4and d, maps E:*q into Elp+r,qr+'
Lemma. Let B be an endomorphism of A such that BF'AP c F'AP, and Bd = dB. Then BZFVqc Z f P 4and if Bf is the induced map on E: then
9.A.2.
Spectral Sequences
40 1
BFd, = drBF. Furthermore, i f F"+'A = 0 then B!+, induced by B on F P H ( A ,d).
agrees with the map
This is clear from the naturality of the constructions above. 9.A.2.3. We now give some spectral sequences that will be used in this chapter. These spectral sequences are related to the famous HochschildSerre spectral sequences and to a family of spectral sequences used by Bore1 in his study of L2cohomology. Let n be a Lie algebra over C. Assume that n, is a subalgebra of n and that n, is a subspace such that n = it, 0 n 2 , [n,,n,] c n2 and [n2,n2] c n,. Let M be a nmodule. Set A' = C'(n, M ) = Horn,(A'n, M ) . A = Horn,(An, M ) . Set
Then F'A = A for i I 0 and F"+'A = 0 if dim n2 = s. Suppose that u E F'A n A P . Let X,, . . .,Xi E n, and Y,,. . ., YPpi E n,. Then du( Y,, . . . , Ypir X,, . . ., Xi+,) = I + I1 I11 IV V with (indices involving only Y's run between 1 and dim n,, those involving X's run between 1 and s) I = (  I )j 1 qu( Y, ,. . . , $, . . . , Yp  ,x,,. . ., Xi),
+
'
+
I1 =
+ +
c ( I)'+"([r,, ys], Y,,. . ., t,..., 2,..., Yp',X1,...) c l)r+Pi+su"LXs1,Y1,..., K.. , . . . , X , , . . . ,.X.S , . . . ) r<s
111 =
(
I v = (  l ) p  ' ~ (  l ) J + ~ x j u ( Y)...) l Ypi,x, ) . . . ) gj,...) Xi)
( l)'+"([X,,XS], Y,, . . . , X I , .. ., 2',. . ., gs,. . . ,Xi).
v= r<s
This formula easily implies that our filtration is dinvariant. We now calculate the E , term of the corresponding spectral sequence. In other words we calculate the cohomology of Gr A . Let u' E Gr' A then modulo F'+'A, u' is represented by u E Horn,(An, 0 A'n,, M ) . In the notation above du(Y,, . . . , Ypi, X , , . . ., Xi) = I I1 111, since IV and V are 0. We note that (Ain2)*is a n,module under the action induced by ad. A simple rewriting of I + I1 + 111 yields
+ +
(1)
EP
=
H'(n,,(APn,)* 0 M ) .
9.A.2.4. We continue with the example of the previous number, with an additional assumption on n, and n,. Assume that there is a semisimple
402
9.
Unitary Representations and (g, K)Cohomology
derivation, H , of n that stabilizes n, and n, and has positive eigenvalues. Then H acts on (A%,)* with strictly negative eigenvalues if q > 0. Let 0 = a,, >  a , > ... > ad be the eigenvalues of H on (An,)*. Set G' = G'(A'n,)* = CjSi(An,)$. Then G o = (An,)* and G d + ' = 0. Set
F' Hom,(A'n,,An: 0 M ) = Horn,(An,, G'O M ) . Then F' defines a decreasing dinvariant filtration of Horn,(An,, Ant 0 M ) . We note that n, G' is contained in G"'. We therefore have a spectral sequence with

EP.¶ = HP(n1 9 ( G p + q / G p + q +0 l ) M ) = H P ( n , ,M ) 0 (Aqn2)?apaq.
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Index
Admissible (g,K)module, 81 representation, 81 Affine algebraic group, 42 Analytic vector, 34 A R (ArtinRees) property, 14 Augmentation homomorphism, 9 Bore1 subalgebra, 37 Bruhat decomposition (lemma), 52 Cartan subalgebra, 4 of a real Lie algebra, 56 maximally split, 57 fundamental, 57 Cartan decomposition Lie algebra. 43 group, 46 Cartan involution, 42 Cartan subgroup, 59 fundamental, 59 maximally split, 59 Central distribution, 294 Chevalley restriction theorem, 75 Coefficients (matrix coefficients), 22 C"vector, 31 Compact form, 44 Cusp form, 233 Delta sequence, 25 Dirac operator, 367 inequality, 368
Distribution, 332 character, 292 order, 332 Exponential polynomial, 335 Finitely generated module for an algebra, 14 Formal degree, 24 Frobenius reciprocity, 31 ( $1,K 1module, 80 equivalent, 80 finitely generated, 80 tempered, 138 underlying module, 81 Gelfand, Naimark decomposition, 54 Generalized weight space, 108 HarishChandra isomorphism, 78 homomorphism, 93 Homomorphism 9module, 1 1 Gmodule, 18 Induced representation, 31 Infinitesimal character, 34 (ly) equivalent, 81 (ly) irreducible, 81 Intertwining operator, 9module, 1 I group representation, 18
41 I
412 Invariant symmetric bilinear form, 5 subspace (for group representation), 18 Irreducible group representation, 18 Isotypic component in a group representation, 28 Iwasawa decomposition Lie algebra, 45 group, 45 Jacquet module, 111 Kcharacter, 295 Langlands data, 149 Langlands decomposition, 51 Left invariant measure, 1 normalized, 2 Lie algebra compact form, 8 nilpotent. 14 reductive, 4 Lie group unimodular, 2 Locally integrable, 332 Modular function, 2 Natural equivalence, 177 transformation, 177 Noetherian algebra, 13 Nilpotent element, 342 Norm, 71 Operator compact, 326 HilbertSchmidt class, 323 selfadjoin t , 326 trace class, 328 Ppair (parabolic pair), 51 cuspidal, 58 Parabolic subgroup (standard), 51 minimal, 51 PBW, 9 Rapidly decreasing functions, 230 Real reductive group, 42 inner type, 51 Realization, 13 Regular element, 4 character, 323 Representation, conjugate dual, 20
Index direct sum, 24 (strongly continuous of a) group, 18 Hilbert, 18 Lie algebra, 1 1 (right) regular, 22 smooth, 18 square integrable, 22 unitary, 18 Root, 4 real, 58 simple, 4 space, 4 system, 5 system of positive roots, 6,48 Schur's lemma, Dixmier's, 1 1 for (g, K)modules, 80 for groups, 21 Schur orthogonality relations, 23 Scwartz space, 237 of HarishChandra, 230 Semisimple element, 342 Smooth vector, 31 Spin module, 362 Split component (standard), 48 Submersion, 332 Support, 332 Symmetric subgroup, 42 Symmetrization map, 9 TDS (three dimensional simple Lie algebra), 11 0stable parabolic subalgebra, 184 root system, 365 Universal enveloping algebra, 8 canonical filtration, 9 Unitary (g, K)module, 367 Verma module, 37 Weight, 36 space, 36 dominant integral, 36 Weyl chamber, 6,48 character formula, 67 group, 6 integration formula Lie algebra, 63 Lie group, 63 reflection, 6
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W. D. Curtis and F. R . Miller, Differential Manifolds and Theoretical Physics Vol. I I7 Jean Berstel and Dominique Perrin, Theory of Codes Vol. 118 A. E. Hurd and P. A. Loeb, An Introduction to Nonstandard Real Analysis Vol. 119 Charalambos D. Aliprantis and Owen Burkinshaw, Positive Operators Vol. 120 William M. Boothby, An Introduction to Diferentiable Manfolds and Riemannian Geometry, Second Edition Vol. 121 Douglas C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres Vol. 122 Sergio Albeverio, Jens Erik Fenstad, Raphael HoeghKrohn, and Tom Lindstrom, Nonstandard Methods in Stochastic Analysis and Mathematical Physics Vol. 123 Alberto Torchinsky, Real Variable Methods in Harmonic Analysis Vol. 124 Robert J. Daverman, Decomposition of Manifolds Vol. 125 J. M. G. Fell and R. S. Doran, Representations of *Algebras, Locally Compact Groups, and Bunach *Algebraic Bundles: Volume 1, Basic Representution Theory of Groups and Algebras Vol. 126 J. M. G. Fell and R . S. Doran, Representations of *Algebras, Locally Compact Groups, and Banach *Algebraic Bundles: Volume 2, Induced Representations, the Imprimitivity Theorem, and the Generalized Mackey Anulysis Vol. 127 Louis H. Rowen, Rincj Theory, Volume I Vol. 128 Louis H. Rowen, Rincj Theory, Volume I1 Vol. 129 Colin Bennett and Robert Sharpley, Interpolation of Operators Vol. 130 Jurgen Poschel and Eugene Trubowitz, Inverse Spectral Theory Vol. 131 Jens Carsten Jantzen, Representations of Algebruic Groups Vol. 132 Nolan R. Wallach, R e d Reductive Groups I Vol. 116
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