Representations of Real and P-Adic Groups

REPRESENTATIONS OF REAL AND P-ADIC GROUPS

LECTURE NOTES SERIES Institute for Mathematical Sciences, National University of Singapore Series Editors: Louis H.Y. Chen and Yeneng Sun Institute for Mathematical Sciences National University of Singapore

Published Vol. 1

Coding Theoty and Ctyptology edited by Harald Niederreiter

Vol. 2

Representationsof Real and pAdic Groups edited by Eng-Chye Tan & Chen-Bo Zhu

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

REPRESENTATIONS OF REAl INUHIOOC GROUPS D O

Editors

Eng-Chye Tan Chen-Bo Zhu National University of Singapore

SINGAPORE UNIVERSITY PRESS NATIONAL UNIVERSITY OF SINGAPORE

vp World Scientific

Published by Singapore University Press Yusof Ishak House, National University of Singapore 31 Lower Kent Ridge Road, Singapore 119078 and World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA once: Suite 202,1060Main Street, River Edge, NJ 07661 UK once: 57 Shelton Street, Covent Garden, London WC2H 9HE

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REPRESENTATIONS OF REAL AND p-ADIC GROUPS Copyright 0 2004 by Singapore University Press and World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereoJS may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Contents

Foreword

vii

Preface

ix

Three Uncertainty Principles for an Abelian Locally Compact Group Tomasz Przebinda

1

Lectures on Representations of padic Groups Gordan Savin

19

Lectures on Harmonic Analysis for Reductive padic Groups Stephen DeBacker

47

On Classification of Some Classes of Irreducible Representations of Classical Groups Mark0 TadiC

95

Dirac Operators in Representation Theory Jang-Song Huang and Pavle PandZiC

163

On Multiplicity Free Actions Chal Benson and Gail Ratcliig

221

Multiplicity-Free Spaces and Schur-Weyl-Howe Duality Roe Goodman

305

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Foreword

The Institute for Mathematical Sciences at the National University of Singapore was established on 1 July 2000 with funding from the Ministry of Education and the University. Its mission is to provide an international center of excellence in mathematical research and, in particular, to promote within Singapore and the region active research in the mathematical sciences and their applications. It seeks to serve as a focal point for scientists of diverse backgrounds to interact and collaborate in research through tutorials, workshops, seminars and informal discussions. The Institute organizes thematic programs of duration ranging from one to six months. The theme or themes of each program will be in accordance with the developing trends of the mathematical sciences and the needs and interests of the local scientific community. Generally, for each program there will be tutorial lectures on background material followed by workshops at the research level. As the tutorial lectures form a core component of a program, the lecture notes are usually made available to the participants for their immediate benefit during the period of the tutorial. The main objective of the Institute’s Lecture Notes Series is to bring these lectures to a wider audience. Occasionally, the Series may also include the proceedings of workshops and expository lectures organized by the Institute. The World Scientific Publishing Company and the Singapore University Press have kindly agreed to publish jointly the Lecture Notes Series. This volume on “Representations of Real and pAdic Groups” is the second of this Series. We hope that through regular publication of lecture notes the Institute will achieve, in part, its objective of promoting research in the mathematical sciences and their applications. Louis H. Y. Chen Yeneng Sun Series Editors

February 2004

vii

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Preface

A semester-long program on Representation Theory of Lie Groups was held at the Institute for Mathematical Sciences (IMS) at the National University of Singapore (NUS) from July 2002 to January 2003. This is the third research program of IMS since it started operations in July 2001. The goal of the program was to explore recent advances in representation theory of both real and padic groups. The program had three sub-themes: (i) representation of padic groups; (ii) unitary representations of real reductive groups; (iii) multiplicity-free actions and representations. As part of the program, tutorials related to the sub-themes were conducted by leading experts in the fields. These tutorials covered the fundamentals of representation theory as well as some of its recent developments and were meant for graduate students and researchers who would like to prepare themselves for original research in the fields. The current volume collects the expanded lecture notes of these tutorials. In the following, we give a brief indication of the ranges of topics represented in this volume. The first five articles are on the sub-themes (i) and (ii). The article by Tomasz Przebinda directs us to some recent developments in harmonic analysis on locally compact abelian groups, specifically three Uncertainty Principles. In Gordan Savin’s notes, he gives an elementary and elegant introduction, with exercises, to representations of padic reductive groups. The material represents real classics from the past quarter century. Stephen DeBacker’s article examines some distinguished classes of distributions on reductive padic groups and their Lie algebras and discusses one of the deepest connections between them: the Harish-Chandra-Howe local character expansion (and its ramifications). The lecture notes of Marko Tadib deal with the problem of classification of some important classes of irreducible representations of both padic and real classical groups (with more emphasis on padic groups), in particular the classification of the square-integrable representations modulo cuspidal data. Jing-Song Huang and Pavle Pandiib’s article examines the role Dirac operators play in repix

X

Preface

resentation theory and discusses various topics surrounding their proof of the Vogan’s conjecture on Dirac cohomology, an important development in the algebraic aspects of unitary representations. The last two articles in this volume are on the sub-theme of multiplicityfree actions. Here Chal Benson and Gail Ratcliff present a comprehensive treatment of these actions, covering topics such as the classification of linear multiplicity-free actions and eigenvalues of invariant differential operators. Finally Roe Goodman’s lecture notes explore the fundamental duality between the irreducible representations occuring in a linear group action and irreducible representations of the commuting algebra, which of course, is a recurrent theme in many parts of representation theory. Besides the local organizers (us), the other members of the Organizing Committee are Jian-Shu Li (Hong Kong University of Science and Technology and Co-Chairman), Jeffrey Adams (University of Maryland), Kyo Nishiyama (Kyoto University), Dipendra Prasad (Mehta Research Institute) and Gordan Savin (University of Utah). We are very much grateful to their invaluable services. We would also like to express our deep appreciation to Roger Howe (Yale University and Chairman, Scientific Advisory Board of the IMS) who helped to conceptualize scientific aspects of the program and was equally valuable in many other aspects throughout this program. Thanks also to all the participants of this program for their support and stimulating interactions during those few months! We would like to take this opportunity to thank Louis Chen, Director of IMS, for his leadership in creating an exciting environment for mathematical research in IMS and for his guidance throughout our program. The expertise and dedication of all IMS staff contributed essentially to the success of this program. Financial support to the program was generously provided by the IMS and by a grant from the Faculty of Science, NUS. Last but not least, we would like to record our appreciation to the Department of Mathematics, NUS for freeing some of our obligations during this period, and more importantly for providing all kinds of support to our work ever since we joined as faculty members more than a decade ago. Eng-Chye Tan and Chen-Bo Zhu National University of Singapore Singapore

Three Uncertainty Principles for an Abelian Locally Compact Group

Tomasz Przebinda Department of Mathematics University of Oklahoma Norman, OK 73019, USA E-mail: [email protected]

0. Introduction

The purpose of this article is to direct reader’s attention to some recent developments concerning three Uncertainty Principles: the classical Heisenberg-Weyl Uncertainty Principle, The Hirschman-Beckner Uncertainty Principle based on the notion of the entropy, and the Donoho-Stark Uncertainty Principle. We state all three in the first section with some further explanations and proofs in the following sections.

1. The main results Let S(R) denote the Schwartz space on R,[lo], and let f E S(R)be a real valued function. Then, by the Cauchy inequality,

Moreover,

Thus

7

T. Przebinda

2

If the equality holds in (l),then there is a real number a such that f’(z) = - 2 a z f ( z ) , so that

f(z) = *e--ax2-c,

(2)

where c E R. Since (1 f 112< 00, the number a must be positive. Let us write f as the inverse Fourier transform of the Fourier transform f of f :

f (z) =

s,

e2KixEf(E)dE.

Then

and therefore

Thus for

1) f

(12=

1,

s,

1f. (.)I2

dz.

s, lE!(4I2

dz 2

1

S’

(3)

A few more easy steps lead to the following theorem of Herman Weyl, (see

POI ). Theorem 1.1: (H. Weyl, 1931) Let f E S(R), with

(1 f

112=

1. Set

Then 1 0.524lr’ and the equality occurs if and only if

f(.)

= e-ax2-bz-c

,

where a > 0 , and b , c E @. In other words f is a constant multiple of a translation

Three Uncertainty Princaples

3

and a modulation f(z) + e i y o x f ( z ) of a Gaussian e-axc2

The above theorem states that a function and its Fourier transform cannot both be arbitrarily concentrated. In computer applications one deals often with finite cyclic groups, rather than with the real line. In this context there is no obvious or straightforward generalization of the theorem 1.1, because the quantities involved ( p ,c,...) do not seem to make sense. In order to circumvent this difficulty, Donoho and Stark, [7],have introduced a different, elementary, measure of uncertainty, which we explain below. Let A = {0,1,2,3, ..., N - 1) be a finite cyclic group of order ( A (= N . Let A denote the dual group of all the characters (i.e. group homomorphisms 6 : A 4 C"). It is customary, and sometimes convenient, to identify A with A by the formula

6 ( b ) = e*ab For a function f : A

-+

(a,b E A ) .

C define a Fourier transform of f by

f(i)=

c

(6 E A).

f (a)&(-a)

aEA

Theorem 1.2: (Donoho-Stark, 1989, [7])For any non-zero function f : A --f C, b P P f I . ISUPPfI

2 1-41.

The equality occurs if and only i f f is a constant multiple of a translation

f (a> f ( a + 4 +

and a modulation

f(a)

+

k4.f( a )

of the indicator function of a subgroup of A .

In applications one deals often with multi-dimensional signals, i.e. with functions defined on a finite product of finite cyclic groups. Hence it is natural to ask for a generalization of the theorem 1.2 to this more general context. This has been done by K. Smith. Theorem 1.3: (K. Smith, 1990, [19]) The above theorem 1.2 holds f o r any finite Abelian group A (a finite direct product of finite cyclic groups).

T. Przebinda

4

The theorem 1.3 also follows from theorem 1.10 below. In addition, a proof which uses no more than basic concepts from finite dimensional linear algebra over complex numbers and the structure of finite abelian groups is available in [12]. Clearly, the cardinality of the support is not the most precise measure of the concentration of a function. One possible improvement is based on the notion of entropy.

Definition 1.4: (based on Shannon, 1948, [18]) Let p be a non-negative measure on a measure space M . Let 4 : M + [ O , o o ) be a probability density function, i.e.

The entropy of

4 is defined as

where the log stands for the natural logarithm, whenever the integral converges. In his 1957 paper, [9],Hirschman has proven the following theorem and stated the following conjecture.

Theorem 1.5: Let f

E

S(R), with

H(lf 12) Conjecture 1.6: Let f

(Notice that since

;> 1, log(:)

112=

1. T h e n

+ H(lfI2)2 0.

S(R), with

H(lf I?

(4 (b)

E

11 f

11 f

(12=

1. T h e n

+ H(lfI2)2 log(;). > 0.)

The equality holds in (a) i f and only i f f is a constant multiple of a translation and a modulation of a Gaussian e - a x 2 ,a > 0.

As an application of his LP, LQ estimates for the Fourier transform, Beckner proved part (a) of Hirschman’s conjecture.

Theorem 1.7: (Beckner, 1975, [2], [3]) Part (a) of Hirschman’s Conjecture is true. Theorem 1.8: (Ozaydm-Przebinda, 2000, [IS])Part (b) of Hirschman’s Conjecture is true.

Three Uncertainty Principles

5

Let A be a finite Abelian group and let a be a Haar measure on A. Thus a is a positive constant multiple of the counting measure on A. Let ii be the dual Haar measure on the dual group in the sense that for a function f : A 4 @, the Fourier transform and the inverse Fourier transform are given by

a,

JA

(4)

The following theorem is known. For the idea of a proof, various particular cases and generalizations see [9], [14], [13] and [6].

Theorem 1.9: Let f E L2(A,cr),with

(1 f

((2=

1. Then

H(lf 17 + H(l.FI2) L 0. Theorem 1.10: (Ozaydm-Przebinda, 2000, [16]) The equality holds in the above theorem 1.9 i f and only i f f is a constant multiple of a translation and a modulation of the indicator function of a subgroup of A .

Corollary 1.11: (DeBrunner-Ozaydm-Przebinda, 2001, [17]) The discretization of the minimizers f o r the entropy inequality o n R does n o t give minimizers f o r the entropy inequality o n any finite cyclic group A . In other words, it is certain that no discretization of a signal defined on the real line and best concentrated in the time-frequency plane, will lead to a signal defined on a finite cyclic group which is also best concentrated in the corresponding finite time-frequency plane. In a mathematically natural search for the most general theorem, which would generalize all the cases considered above, we arrive at the notion of a locally compact Abelian group, (see [S]).The integration and the notion of the Fourier transform are both well established for such groups. Let A be a locally compact Abelian group. As was explained to the author by Michael Cowling, a result of Ahern and Jeweet [l],together with [8],9.8, imply that A is isomorphic to the direct product of a finite number of copies of R and an Abelian locally compact group B , which contains an open compact subgroup:

A = R " x B. (5) Let be the Pontryagin dual of A . Then = R"x B ,where also contains an open compact subgroup. Let Q be a Haar measure on A and let i 3 be the

a

a

T.Pmebinda

6

A,

Haar measure on dual to (Y,so that the Fourier transform and the inverse Fourier transform are given by the formulas (4). Let V ( L 2 ( A , a )be ) the group of unitary (norm preserving) operators on the Hilbert space L 2 ( A ,a ) , and let G V ( L 2 ( A , a )be ) the group generated by all the translations, all modulations and by the multiplications by complex numbers of absolute value 1. This is the Heisenberg group attached to the Abelian group A .

Theorem 1.12: (Ozaydln-Przebinda, 2000, [16]) For any function f E L 2 ( A , a ) ,with 11 f 112= 1, such that (*)

(I f

Ill< CCJ

and

II f^ Ill<

CCJ,

the following inequality holds (a)

H ( l f 12)

+ H(lf^12)2 n k l ( 5 ) .

The set of functions for which equality occurs an (a) coincides with the union of orbits

G.f,

(b)

where f = g 8 h , g is a Gaussian on R", and h an appropriate constant multiple of the indicator function of a (open-compact) subgroup of B . 2. The support inequality for a finite Abelian group In this section A stands for a finite Abelian group and (Y for the Haar meaa E A. Denote the translations and modulations sure given by a ( a ) =

$7

by

T c : f ( a )+ f (a+ c), M~: f ( a ) 4 b ( a ) f ( a ) .

(6)

Notice that

TbMiT,M?

= ZMieTb+,

(b, c E A ,

6, Li E a),

(7)

where z is a complex number of absolute value 1. Indeed, the action of the operator on the left hand side on a function f ( a ) may be represented as

+ c ) f ( a+ c) 3i(a)e(a+ + c) 3&(a+ b)Li(a + b + c ) f ( a+ b + c) = (6(b)Li(b+ c))(b(a)Li(a))f ( a + ( b + c ) ) .

f ( a ) % e ( a ) f ( a )3?(a

In particular the Heisenberg group attached to A may be written as

G = {zMiT,;

b E A, c E A } .

(8)

T h r e e U n c e r t a i n t y Principles

7

It is esthetically pleasing, and convenient for applications, t o know that an orthonormal basis of a space of functions may be indexed by a group. This is the case for an orthonormal wavelet basis for functions defined on the real line, [5], [15]. The lemma below states that such bases may be chosen from among our minimizers on a finite Abelian group. In contrast, the translations and/or modulations of Gaussians defined on the reals, will not be orthogonal. Lemma 2.1: (see [7], 1171 for the cyclic case) Let B be a subgroup of A and let I B denote the indicator function of B . Let ,B E C be such that )I 112= 1. Then, up to constant multiples of absolute value 1, the orbit G . ,BIB is an orthonormal basis of the Hilbert space L2(A,a ) . Proof: For c E A and 2 E A we have

hf:Tc@,(a)

= ?(a)pIB(U

+

= e(a)pnB-c(a).

C)

Hence, in order to parametrize the orbit, we may choose the 2 modulo B'(= {ii E 6 1 =~ l}),and the c modulo B . Thus the cardinality of the orbit, modulo the multiplications by complex numbers of absolute value 1, is equal to

A;

lA/B'l . IA/BI

=

IBI . IA/BI

=

IAl = d i m L 2 ( A , a ) .

d^ E A.Then,

Let d E A and let

For this quantity to be non-zero, we must have B equal t o

1

IPJ2

B-c

=Ip12

1

E(a)d^(a)da(a)= I/3I2/ ;(a B

-

c = B - d. Then it is

+ c)d^(a+ c) d a ( a )

2(a)d(a)-' d a ( a ) . E(c)d^(c)-',

B

which is non-zero if and only if 22-' E B I . Then, modulo multiplication by the constant e(c)d^(c)-l,the result is

Corollary 2.2: (see [17] for the cyclic case) I n terms of Lemma 2.1, the orbit G . consists of functions such that

T. Przebinda

8

(a)

or equivalently, such that (b)

if and only if (c)

Proof: Since the entropy and the cardinality of the support for f and for f^ are invariant under the action of the Heisenberg group G, i t suffices t o consider f = 81., Then, by a straightforward computation,

f = a(B)pnB.. Hence, Isuppfl = JB'J = J A J / J B and J , since Jsuppf 1 = IBJ,the equivalence of (b) and (c) follows. Furthermore

and similarly,

H ( IPa ( B )b?. I2,

I I I ( B1I

=-P

Q

log ( IP I Ia

(aI ) &!( B ).

Thus (a) is equivalent to

log ( IP I2 , = log ( IP I

I4B)I

ct ( B )6.

1.

Since, by a simple computation,

a(B)&(B') = 1, we see that (a) is equivalent to C L ( B=) ~1. But, by our assumption on a, a ( ~=) so (a) is equivalent to ( c ) .

I~l/JiZii,

For reader's convenience we reproduce here a lemma which is the key t o the proof of theorem 1.2, leaving the complete proof of the theorem as an exercise. (See [17] for an exposition.)

Lemma 2.3: [7]Let f : A -+ C be a non-zero function. Then under the ab identification A = A , &(b)= el*] , the function f cannot have more than lsupp f I consecutive zeros.

9

Three Uncertainty Principles 2T"

Proof: Let supp f = { a l ,a2,a3, ...,a,}.

1

1

1

WUl

wa2

Let w = e - m . Then

.. .. .. ..

1 warn

By Vandermonde, the above matrix is invertible. Thus the vector on the left hand side is non-zero. Since a translation o f f is equivalent t o a modulation of f , which does not change the support of f , the lemma follows. 0 In the remainder of this section we provide an elementary proof of the part of theorem 1.3, which describes the minimizers for the support inequality (which we assume), if A is a finite direct product of groups of order 2: A = ( Z / 2 Z ) N . (See theorem 2.5 below.) This proof is a simplified version of the argument developed in [12]. Lemma 2.4: L e t B a n d C be subgroups of A , s u c h t h a t

A=B@C.

For a f u n c t i o n f : A

4

C let

fc(b)= f(b

+ c)

( b E B, c E C).

Suppose Isuppfl. Isuppfl

=

\A) and

fc

# 0 f o r all c E C.

Then

(a)

I s ~ ~ f. cI S Ul P P ~ C I = IBI

(b)

IS.PPfCl

(c)

l s w f c l = JS~PPfJ

(4

S u P P f c = P(..PPf^)

where P : 2 = B

@ k3 6 + ?

= &lszlppfl

-+

Proof: Clearly, suppf =

(C

(c E (c E

E C),

C),

c>,

(c E

C),

6E B.

u C € C

((S.PPfC)

+ c).

(9)

T. Przebinda

10

Hence,

IsuIPfl =

c

IS'1LPPfCl.

CEC

By the Fourier inversion formula on C we have

(6 E B, c E C ) . Hence, SUPP f c

c_ P ( W P f)

(c E C ) ,

(12)

I lsuppfl

(c E C).

(13)

and therefore

IsuPPfcl Suppose c E C is such that

ICI. IsuPPfcl

< Isumfl.

Then, by (13),

PI.ISUPPfCl. IsuPPfcI < I s W P f l . IsuPPfl

=

14.

Hence, ISUPPfCl.

IS'LlPPfCl

< IBI,

which contradicts the fact that f c # 0. (Here we use the assumption that we have the support inequality for the finite cyclic groups of the form ( Z / 2 Z ) N . )Thus

ICI. ISUPPfCl 2 lsuppfl (cE C). (14) Clearly, (10) and (14) imply (b). Further (b) and (13) imply that for all CE

c,

1S.uPP.fcl. 1SUPP.f

1

I--lS1L?)Pfl. - PI

ISUPP.fCl

1

I * ISUPPfl = IAl/lCl = PI. ICI Thus (a) follows. Part (c) follows from (a) and (b). Part (d) follows from (12) and (c). 5 -1WPf

Theorem 2.5: L e t A = ( Z / 2 Z ) N .Suppose f : A

4

CC is a m i n i m i z e r , i.e.

IsUPPfl. IS'1LPPfl = 14.

T h e n , u p t o a constant multiple, a modulation and a translation, f i s t h e indicator f u n c t i o n of a subgroup of A .

Three Uncertainty Principles

11

Proof: We proceed via the induction on N . It is easy to check that the statement holds for N = 2 . Let f : A --f C be a minimizer. Suppose there is a proper subgroup B A , such that supp f B. Then there is a subgroup C A such that A = B @ C. (In order to have some standard linear algebra a t the disposal, it might be easier here to view A a vector space of dimension N over the field 2/22 of two elements.) In these terms

s

f = (flB)€3 d, where 6 is the Dirac delta a t zero on C. Hence

where 8 is a constant multiple of the function IIc. We see from the above two equations that is a minimizer on B . Thus, by the inductive assumption f l ~ , and hence f has the desired form. Suppose there is a proper subgroup B 5 A and an element c E A such that s u p p f C B + c. Let g(a) = f ( u c), a E A. Then g is a minimizer on A and suppg 2 B. Hence, by the previous argument, g and hence f has the desired form. Let A = B @ C as in Lemma 2.4, with ICI = 2 . By the previous two cases we may assume that

f l ~

+

fc

#0

( C E

C).

By Lemma 2.4 (a), each fc is a minimizer on B. Therefore, by the inductive assumption, for each c E C, supp fc is a coset of a subgroup of By Lemma 2.4 (d) these cosets do not depend on c E C. Thus there is a subgroup D C B and an element 6 E B such that

B.

sup.fC= D

+

60

(C

E C).

Replacing f by an appropriate translation of f we may assume that each jcis a constant on its support. Thus there is a function h : C 4 C such that fC

=

8 h(c)

(c E

C).

Therefore

We see from (15) that h is a minimizer for C. But

T. Prtebinda

12

and our assumption (fc # 0 for all c E C) implies that supp h = C . Hence, J s u p p i J= 1. Thus s u p p i = {&}, for some & E C. Therefore, by (15), suppf

=

(D

+ 60) +

20

=D

+ + (&I

20).

Thus supp f is a coset of a proper subgroup of A. Since f^ is a minimizer, our previous argument implies that and hence f, has the desired property.

f,

0

3. A few words on the notion of the entropy In this section we recall a few basic facts, which indicate that the notion of the entropy is quite natural and useful. For more details we refer the reader to [4] and [18]. Let X be a discrete random variable with the probability distribution function p ( z ) = P ( X = z). Set

H ( X ) = H(P) = - C P ( 4 l O g z ( P ( z ) ) .

(16)

2

Example 3.1: Suppose X has a uniform distribution over 23 = 8 outcomes. Then

c 8

H ( X )= -

x=1

1

1

-logz(-) 8 8 = 3.

This agrees with the number of bits needed to describe X in binary: c(0) = 000, c(1) = 001, c(2) = 010, c(3) = 011, c(4) = 100,

~ ( 5= ) 101, ~ ( 6 = ) 110, ~ ( 7= ) 111.

Example 3.2: Consider a horse race, with eight horses taking part. Suppose the probability of winning the race is distributed as follows: P ( 0 ) = 1/2, P(1) = 1/4, P ( 2 ) = 1/8, P(3) = 1/16, p(4) = p ( 5 ) = p ( 6 ) = p(7) = 1/64. Then H ( X ) = 2. We may encode the horses in binary as follows: c(0) = 0, c(1) = 10, c(2) = 110, c(3) = 1110, c(4) = 111100, c(5) = 111101, ~ ( 6 = ) 111110, ~ ( 7 = ) 111111.

Let l(c(z)) be the length of the code word c(z). Then l(c(0)) = 1, l(c(1)) = 2, l(c(2)) = 3, l(c(3)) = 4, l(c(5)) = ... = l(c(7)) = 6.

Three Uncertainty Principles

13

Hence, the expected value of l ( c ( X ) )is

Thus

H ( X )= E(l(c(X)). This last equality is not a coincidence, as we shall explain below. For details see [4].

A source code for a discrete random variable X is a mapping c from the range of X to (0,I} u {o, I } u~(0,1}3 u ... .

If a codeword c ( x ) belongs t o (0,l } k ,let I ( c ( x ) ) = k denote the length of C ( Z ) . The code is called instantenous if no codeword is a prefix to any other code word. For such codes one can recognize the separate words ~ ( x l )c,( x 2 ) , ... by looking a t the string c ( z ~ ) c (.... z ~For ) instance in our second example, the string 010110111111 is made of the codewords 0, 10, 110,111111.

Theorem 3.3: ( [ 1 8 ]see , also [ 4 ] )For an instantenous code c we have, E ( l ( c ( X ) )2 H ( X ) . Moreover, there is an instantenous code c such that

A theorem of Shannon, which we quote below, gives a characterization of the entropy as a measure of uncertainty of the outcome of an experiment. Let P = { P I , p2,p3, ...,p,} be a finite probability sequence, i.e. p j >_ 0, &p.j = 1. Theorem 3.4: (Shannon, 1948, [18])Suppose H i s a function defined o n finite probability sequences such that (4

(b) H ( $, $, $, ...,

H ( p 1 ,p2,p3, ...,p,) is continuous,

i) is monotonically increasing as a function of n,

(c) if Q j = { q j l , q j 2 , q j 3...}, , j = 1 , 2 , 3 , ..., n, are probability sequences, then H(Uj”=i~ j Q j )= H ( P ) C j ” = l ~ j H ( Q j ) .

+

T. Przebinda

14

Then, up to a constant multiple, n

H ( P )= - X P j W P j ) . j=1

Suppose M is a finite set and p is a positive multiple of the counting measure on M. For a function f : M -+ C and for 1 5 p < 03 we have the LP norm o f f defined by

II f IIp=

(1 If (.iIPWzi) M

lip.

There is a simple and explicit connection between the entropy (see Definition 1.4) and the LP norms, expressed in the lemma below, which may be verified by a straightforward computation (left to the reader). See [20] for more explanations.

Lemma 3.5: Let p

= p(t)=

i, 0 < t 5 1. Then for any function f : M

---f

c, (a) In particular, for f with

(b) and for f with

(b)

4. The entropy inequality for a finite Abelian group In this section we give a proof of theorem 1.10 for a finite Abelian group A. For the general case we refer the reader t o [16]. The proof is based on the following four basic theorems.

Holder’s Inequality 1111:

Plancherel’s Formula 181:

I1 P l12=11 f

112

.

Riesz-Thorin-Young Inequality (1211, Chapter 9, (1.11)): 1 (5 5 t 5 1). I1 lll/(~-t)5ll f JJl/t

f

Three Uncertainty Principles

15

Hopf's Maximum Principle ([ll],Theorem 3.1.6'): Let D 2 CC be an open unit disc, and let u : D -+ R be a harmonic function which extends to a continuous function on the closure B of D , u : D -+ R.Suppose z is a point o n the bounday of D such that u ( z ) 2 u ( z ' ) for all z' E D, and the directional derivative of u at z along the radius which ends at z , is zero. Then u(z) = u(z')

Let f : A

-+

f o r all z' E D.

CC be a minimizer. Consider the following function -

where = 0 outside the support of f , and similarly for f-. A straightIf1 forward application of Holder's inequality and the Riesz-Thorin Theorem shows that for 5 x 5 1, y E R,p = and q defined by the equation 1 ; = 1 (with q = 00 if p = l), we have

i

+

511 l p + i 2 y

l i p . 11 lf122fi2Y

IIp=ll

f

112

. II f

112=

1.

The function F is analytic in the open strip (17) and continuous in the closed strip. A straightforward calculation shows that

Hence, by the Plancherel formula,

Since f is a minimizer, the right hand side of the equation (19) is zero. In particular Re F ( z ) is a real valued harmonic function on the interior of the disc of radius centered a t z = which achieves the maximum at z = !j and has derivative equal t o zero a t this point. Hence the Hopf's Maximum

a

i,

T. Przebinda

16

Principle implies that R e F ( z ) = 1 on the disc. Hence, F ( z ) = 1 on the disc. In particular,

The formula (20) may be rewritten a s

Since,

the equation (21) implies that

Hence,

Thus for ii E s u p p f

Therefore

Im=II f I l l

(6 E

S W f ) ,

(23)

f* Ill

.( E

SzLppf).

(24)

and similarly

l.f(.)I

=lI

The statement (23) implies that the function I f 1 is constant on its support. /~. Since 11 f 112= 1, the constant is equal to & ( s z ~ p p f ) - ~Hence,

H(lfI2)= l o g ( & ( w P f ) ) . Similarly

H(lfI2)= log(+Wf)). Since f is a minimizer,

Three Uncertainty Principles

17

Therefore

a(suppf) &!(suppf) '

=

1.

(25)

We may assume that 0 E suppf^ and 0 E suppf. Then (23) implies

Therefore there is X E C such that f = Xlfl. Hence (23) may be rewritten as

Therefore SUPP

where for a subset S

C A , S'

f^ c

(-SUPP

f)',

(27)

= {Li E A; 21s = 1). Similarly (24) implies

W P f

c (SVP.f)+

(28)

By dualizing (27) and (28) we deduce

But, as is well known and easy to check,

By combining (25), (29) and (30) we see that the inclusions (29) are equalities. In particular suppf is a subgroup of A and f is invariant under the translations by this subgroup. Thus f is a constant multiple of the indicator function of a subgroup of A , as claimed.

Acknowledgement Part of these notes was written during the author's visit t o the Institute for Mathematical Sciences (IMS) at the National University of Singapore. The author thanks IMS for its support and hospitality.

T. Przebinda

18

References 1. P. Ahern and R. Jewett, “Factorization of locally compact Abelian groups”, Illinois Jour. Math., 9 (1965), 23&235. 2. W. Beckner, “Inequalities in Fourier Analysis”, Annals of Math., 102 (1975), 159-182. 3. W. Beckner, “Pitt’s Inequality and the Uncertainty Principle”, Proceedings of the A M S , 123 (1995), 1897-1905. 4. T. Cover and J. Thomas, Elements of information theory, John Wiley & Sons, Inc., New York, 1991. 5. I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1992. 6. A. Dembo, T . M. Cover and J. A. Thomas, “Information Theoretic Inequalities”, E E E Transactions o n Information Theory, 37 (1991), 1501-1518. 7. D. L. Donoho and P. B. Stark, “Uncertainty Principles and Signal Recovery”, S I A M Journal of Applied Mathematics, 49 (1989), 906-931. 8. E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Springer Verlag, 1963. 9. I. I. Hirschman, Jr., “A Note on Entropy”, Amer. Jour. Math., 79 (1957), 152-156. 10. L. Hormander, The Analysis of Linear Partial Differential Operators, I , Springer Verlag, 1983. 11. L. Hormander, Notions of Convezity, Birkhauser, 1994. 12. E. Matusiak, M. Ozaydin and T. Przebinda, “The Donoho - Stark Uncertainty Principle for a Finite Abelian Group”, preprint, available at http://crystal.ou.edu/Ntprzebin/papers.html.

13. H. Maassen, “A discrete entropic uncertainty relation” Quantum Probability and Applications, 5 (1988), 263-266. 14. H. Maassen and J. Uffink, “Generalized Entropic Uncertainty Relations”, Phys. Rev. Lett. , 60 (1988), 1103-1106. 15. M. Ozaydin and T. Przebinda, “Platonic Orthonormal Wavelets”, Applied and Computational Harmonic Analysis, 4 (1997), 351-365. 16. M. Ozaydin and T. Przebinda, “An Entropy-based Uncertainty Principle for a Locally Compact Abelian Group”, preprint, available at http://crystal.ou.edu/-tprzebin/papers.html. 17. T. Przebinda, V. DeBrunner and M.Ozaydin, “The Optimal Transform for the Discrete Hirschman Uncertainty Principle”, I E E E Transactions on Information Theory, 47 (2001), 2086-2090. 18. C. E. Shannon, “A Mathematical Theory of Communication”, The Bell Syst e m Technical Journal”, 27 (1948), 379-656. 19. K. T. Smith, “The Uncertainty Principle on Groups”, S I A M Journal of Applied Mathematics, 50 (1990), 876-882. 20. M. Wickerhauser, Adapted wavelet analysis from theory to software, Cambridge University Press, 1990. 21. A. Zygmund, Trigonometric series, second edition, volumes I and I1 combined, A K Peters, Ltd., Wellesley, MA, 1994.

Lectures on Representations of p-adic Groups

Gordan Savin Department of Mathematics University of Utah Salt Lake City, UT 8.4112, USA E-mail: [email protected]. edu

1. Introduction This text contains an elementary introduction, with exercises, t o representations of p-adic reductive groups. The text is intended t o be more accessible then the standard references, such as the paper of Bernstein and Zelevinsky [I],and (unpublished) notes of Casselman. The material presented here has been known for the past quarter century, and has undergone many modifications in that period. Thus, it would be hard (if not impossible) to acknowledge properly contributions of many people who have worked on the subject. However, it is safe to say that Bernstein, Casselman, HarishChandra, Howe and Jacquet (in alphabetical order) are some of the most prominent contributors to the subject. The text is organized as follows. Sections 2-4 contain definitions and preliminary results on p-adic fields, structure of GL,(F) over a p a d i c field F , and smooth representations. Then, in order to keep the exposition as simplf! as possible, we restrict ourselves to GL2(F). However, the topics and their proofs are chosen so that they easily generalize to GL,(F) and other reductive groups. In Section 5 and 6 we introduce induced and cuspidal representations, and prove that irreducible smooth representations are admissible. Sections 7 and 8 are devoted t o describing the composition factors of induced representations. In section 9 we discuss unitarizable representations, and construct the complementary series for GL2(F).Sections 10 and 11 are devoted to two 19

G. Savin

20

examples. In section 10 we show that the Steinberg representation is square integrable, and in section 11 we construct one cuspidal representation. Finally, in Sections 12 and 13, we go back to GL,(F) and describe the composition factors of (regular) induced representations. This result, due t o Rodier [4], is based on the combinatorics of the root system, and the reduction to the special case of GL2(F) (obtained in Section 7). As such, it gives a good introduction to further, more advanced topics.

Acknowledgments: This material was presented during a week long tutorial in the Institute for Mathematical Sciences at the National University of Singapore. The author would like to thank the people associated with the institute, especially Eng-Chye Tan and Chen-Bo Zhu, for their hospitality and support. Thanks are also due to Jian Shu Li, Allen Moy, and Marko TadiC for useful comments and suggestions at various stages. 2. The field Q, This is a crash course on the field of p a d i c numbers. Let Zdenote, as usual, the ring of integers. Define the p a d i c absolute value on Z as follows. Let x # 0 be an integer. Write x = p"m where m is relatively prime to p. Then

1x1,

=

1

-.

Pa We also put 101, = 0. This absolute value satisfies the usual properties: the multiplicativity, lxlp . lyl, = lxyJp, and the triangle inequality. In fact we have an even stronger property:

1%

+YIP

5 max{l4,1

IYlP).

It follows that d(x,y) = Iz - yl, defines a metric on Z.The ring of p a d i c integers Z,is the completion of Zwith respect to this metric. To understand this ring we can proceed as follows. Let B ( z , r ) be the (closed) ball of radius T 2 0, centered at x. Note that Z can be written as a union of p disjoint balls of radius l/p, centered a t 0,1,. . . ,p - 1: 1

1

1

P

P

P

Z = B(0,-) u B(1, -) u . . . u B(p - 1, -). Note that B ( i ,$) is simply i+pZ, a coset of the maximal ideal (p). Similarly, Z can be written as a union of pn disjoint balls of radius l/pn, centered a t all reminders modulo pn. There are two consequences of this: First, Z is totally bounded, so its completion Z, is compact. Second, the completion

Lectures on Representations of p-adic Groups

21

of Z is the union of completion of individual balls of radius l / p n , which implies that Z/pnZ = Z p / p n Z p .

Exercise. Let x be an integer relatively prime to p . Show that the multiplicative inverse of x exists in Z,.Hint: for every n, there exist integers yn and zn such that x y , pnz, = 1.

+

The ring Z,has ( p ) as unique maximal ideal. The field of fractions Q, is obtained by adjoining l / p . It follows that Qp

= U--m
Next, we shall describe smooth additive characters of Qp. A character i. The minimal such i is called the conductor of 111. We shall now construct a canonical character of conductor 0. Note that Q p / Z pis naturally isomorphic to the ptorsion part of Q/Z. The group Q/Z has a natural additive character given by $J is smooth if it is trivial on pi%, for some some

~ ( x=)ezTZs.

+

The restriction of to Q p / Z pis the additive character of conductor 0. Any other character is of the form $ J a ( x= ) + ( a x ) for an element a in Qp. We shall now say a couple of words about characters of Q; . Note that Q: = ( p ) x Z;, where ( p ) Z is the free group generated by p . The group Zi has a filtration by subgroups 1+ p i Z , with i 2 1. A character of Q; is smooth if it is trivial on some 1 +pZZ,. Moreover, it is called unramified, if it is trivial on Zp”. Every unramified character x is completely determined by its value ~ ( p )In. particular, the group of unramified characters is isomorphic to C X . Any other p a d i c field F is simply a finite extension of Q p (see [3]). The integral closure of Z, will be denoted by 0. It has a unique maximal ideal ( a )which , is the radical of the ideal ( p ) . Let q be the order of the residue field 0/w0.We shall normalize the absolute value on F so that *1

Jwl = -. 4

3. Structure of G L , ( F ) Let F be a p a d i c field, and G = GL,(F). Let K = GL,(0). Then K is a (maximal) open, compact subgroup of G. It has a filtration by the principal

G. Savin

22

congruence subgroups Ki of G defined by

Ki

= {g E G

I g == 1

mod wz}.

These groups are normal in K , and provide a fundamental system of neighborhoods of 1 in G, defining a topology of G. We have the following important decompositions in G: 3.1. Cartan decomposition Let A

E Z"

be the group of diagonal matrices

where ml, . . . , m, are integers. Let A' be the subset of A consisting of diagonal matrices such that ml 2 . . . 2 m,. Then

G = KA'K.

3 . 2 . Bruhat- Tits decomposition Let B denote the group of upper triangular matrices in G, and N the subgroup of B consisting of matrices with 1 on the diagonal. Let W be the group of permutation matrices, which is isomorphic to the permutation group S,. The group W is called the Weyl group of G. Then

G = NWB.

3.3. Iwasawa decomposition

G = BK. Each of the three decompositions given here can be proved by a variant of row-column reduction. This is deceptively simple and somewhat misleading. An entirely different approach is needed for general reductive groups. Thus, we give only a hint here, in form of an exercise.

Exercise. Prove the Cartan decomposition. Hint: Let g be in GL,(F). Let g i j be the entry of g with the largest absolute value. Multiplying g by elementary matrices with coeficients in 0 (so they are in K ) turn all entries

Lectures on Representations of p-adic Groups

23

in the i-th row and j-th column (except g i j ) into 0. Proceed b y induction on n.

A critical difference between real and p-adic groups is that N is a union of open compact subgroups. For example, if G = GL2(F),then N is a union of

As we shall see later, one consequence of this fact is that SL,(F) does not have non-trivial finite dimensional representations. 3.4. Haar Measure Recall that the congruence subgroups Ki give a system of open neighborhoods of 1 which defines the topology of G. Let C F ( G )be the set of locally constant, compactly supported functions on G. In particular, for every function f there exist a sufficiently small congruence subgroup Ki, such that f can be written as a linear combination

where Char(gjK,) is the characteristic function of the right coset gjKi. Now put

This defines a (unique) left G-invariant measure on G such that vol(K)= 1.

Exercise. Show that the measure is right G-invariant. Hint: Let g be in G, and define a left G-invariant measure p, by PLg(f 1 = P(R,(f ))

where R,(f) is the right translate o f f by 9. By uniqueness of the left Ginvariant measure, pg = c(g)pfor some positive constant c(g). Show that c(g) is a character, and then use that [G,GI. Z (Z is the center of G ) is of finite index in G. 4. Smooth representations Let V be a vector space over the field of complex numbers @. A smooth representation of G on V is a homomorphism 7r

: G + GL(V)

G. Sawin

24

such that every vector v in V is fixed by a (sufficiently small) congruence subgroup. We shall refer to V as a (smooth) G-module. For every congruence subgroup Ki, let V K i be the subspace of all vectors fixed by Ki.A representation is admissible if V K zis finite dimensional for every i. It turns out that any irreducible smooth module is in fact admissible. We shall give a proof of this fact for GL2(F).

Proposition 4.1: (Schur's Lemma) Let V be an irreducible smooth representation of G, and A : V + V be a non-trivial homomorphism. Then A = 1-1.I for some ,u in @. (Here I is the identity operator on V ) . Proof: First of all note that V has countable dimension. Indeed, V is spanned by 7r(G)vfor any non-zero vector v in V . Since v is fixed by some Ki, the statement follows, as G/Ki is countable (from the following exercise, for example).

Exercise. Show that for every X in At, the index of Kx = XKX-l

nK

in

K is sandwiched by

-

S(X)-l 5 [K : Kx] 5 [K : K l ] b(X)-'

where b ( X ) =

Hi,,qm*-mJ .

Assume that A # ,u. I for every ,u in C.Then irreducibility of V implies that A - ,u. I is bijective. In particular, R, = 1/(A - ,u. I ) is well defined for every ,u in @. We now need the following:

Exercise. Let v be a non-zero vector in V . Assuming that A - ,u . I are bijective, show that R,v are linearly independent vectors in V. But we know that V has countable dimension. This is a contradiction, and the proposition is proved. 0

Proposition 4.2: Let V be a smooth representation. Then:

If the representation V is finitely generated, then there exists an irreducible quotient. Otherwise, there exits two submodules V" c V' such that V'IV'' is irreducible. Proof: The second statement clearly follows from the first. To prove the first statement, let E be a finite dimensional subspace generating V . If U is a proper submodule, then E n U is a proper subspace of E . Since E is finite dimensional, there exists a maximal subspace Em,, of E which is an

Lectures on Representations of p-adic Groups

25

intersection with a proper invariant subspace. Let U be the family of all invariant subspaces U such that E n U = Em,,. By the Zorn's lemma, the family U has a maximal element U,,. Clearly, V/Um,, is irreducible. The 0 proposition is proved.

Proposition 4.3: Let V be a finitely generated smooth G-module. Then V is also finitely generated over B . Proof: Fix a finite set of generators. Since V is smooth, there exists a congruence subgroup Ki fixing all generators. Since Ki is of finite index in K , the proposition easily follows from the Iwasawa decomposition. 0 An interesting feature of smooth representations it that the trivial representation is the unique finite-dimensional representations of S L , ( F ) .

Proposition 4.4: Let ( 7 r l V ) be a finite dimensional irreducible smooth representation of GL,(F). Then V 2 @, and 7r(g) = X(det) for some character x of F X . Proof: Since V is smooth and finite dimensional, there exists a sufficiently small open compact subgroup Nc c N acting trivially on V . Exercise. Let n be any element in N . Then there exits an element t in T such that tnt-' = n, is in N,. Since 7r(n)w = 7r(t-')7r(nC)7r(t)v= u,it follows that N acts trivially on V . The same argument shows that N acts trivially on on V , and the proposition follows from the following exercise.

Exercise. S L , ( F ) is generated by N and N . Hint: Use row-column reduction. 0 For any smooth G representation V , let V * be the space of all linear functionals on V . Let (u,u*)be the usual pairing between V and V * .Then G acts on V * by

(w,7r*(g)v*)= (7r(g-1)w1 u*).

v

Let be the set of smooth vectors in V * ,and let iidenote the restriction of 7r* t o The representation is called the contragredient dual of V . A priori, it may not be clear that is non-trivial. However, we have the following simple proposition:

v.

v

v

G. Savin

26

vKa

Proposition 4.5: Let V be a smooth representation of G . Then is is adisomorphic to the set of linear functionals on VKx.I n particular, missible, i f and only i f V is so.

v

Proof: Let u be any vector in V. Then u is fixed by Kj for some sufficiently small j which we can assume to be less then i. Define

P(u)=

[Ki : Kj]

Ki / K j

Clearly, P(u) does not depend on the choice of j , and P is a K,-invariant projection from V onto VKa.Now any functional on VK"composed with P defines an element in Conversely, any functional in clearly factors through P . The proposition is proved. 0

vK%.

vIK%

5. Induced representations, case of G L z ( F ) Let x1 and xz be two characters of the multiplicative group F X .Let the character of T , the group of diagonal matrices in G, defined by

x be

We shall now describe the so-called parabolic induction, which produces a smooth representation of G from a character of T . Since BIN = T , the character x can be pulled back to B, and we can consider the smooth induced representation I n d 3 x ) = if : G

(c I

f ( n t g ) = lal/a211/2X(t)f(g)}M,

where here, of course, we have considered only the smooth functions on G. The meaning (and necessity) of the term @I2 = l ~ l / a z 1 ' /will ~ become more clear as we develop more theory. Basically, this factor compensates for the choice of the Bore1 subgroup B containing T , as we could have defined the induction using B,the group of lower triangular matrices. The induced representations are admissible. Indeed, the Iwasawa decomposition implies that

dim I n d z ( ~ ) 5~ [; K : Ki]. Our goal is to understand the composition factors of the induced representations. To do so, we need to introduce an important functor which is dual, in a certain sense, to the parabolic induction.

Lectures on Representations of p-adic Groups

27

5.1. Jacquet Functor We first define a functor from the category of smooth N-modules to the category of vector spaces. Let V be a smooth N-module, and put

VN = V / V ( N ) where V ( N )is the linear span of n(n)v- v for all n E N and v E V . The functor V --+ VN is exact, which will be of fundamental importance:

Theorem 5.1: Let 0

--f

V’ + V

-i

V” -+ 0 be an exact sequence of

N-modules. Then 0 4 vh

4

V N -+

v$

4

0

is exact. Proof: The right exactness is clear. We only need to check that Vh --+ V, is injective. This is equivalent to V‘ n V ( N ) = V ’ ( N ) ,which will follow from the following important characterization of V ( N ) . 0

Lemma 5.2: Let V be a smooth N-module. A vector v is in V ( N ) i f and only i f

k3

n(n)v d n = 0

f o r a sufficiently large open compact subgroup N j . I n particular, V ( N ) = U j e z V ( N j ) , where V ( N j ) is the subspace of all v in V such that the above integral vanishes.

Proof: If v is in V ( N ) ,then v is a finite linear combination of terms 7r(n)u- u. Let Nj be sufficiently large so that it contains all n appearing in that linear combination. Then the integral over Nj will be 0. 0 Exercise. Prove the other direction of the Lemma. Hint: The vector v is invariant under some N i . Now assume that V is a smooth G-module. Then VN is naturally a smooth 2’-module, and V --+ VN is called the Jacquet functor. The Jacquet functor and the parabolic induction are related by the Frobenius reciprocity:

H O ~(V, G I n d g (x))= H O ~( TV , ,6 1 ’ 2 ~ ) Exercise. The Frobenius reciprocity is a tautology. Prove it!

G. Savin

28

Proposition 5.3: Let V be a n irreducible smooth G-module such that VN # 0 . T h e n V i s a submodule of I n d ( X ) f o r some character x of T .

Proof: By Proposition 4.3 we know that V is finitely generated over B. Thus V , is also finitely generated over B , and it has an irreducible quotient by Proposition 4.2. The proposition follows from the Frobenius reciprocity. 0

6. Cuspidal representations, case of GL2 ( F )

We have just seen that a smooth irreducible representation V such that V, # 0, embeds into a principal series representation. Our concern in this section will be the representations such that V, = 0; so called cuspidal representations. Since the center Z of G is not compact, so it will be more convenient to work with a smaller group Go = { g E G L z ( F ) 1 d e t ( g ) E 0 ' ) .

Note that ZG" is of index 2 in G , so representations of Go can be easily related to those of G .

Exercise. Let ( 7 r , V ) be a n irreducible smooth G-module. T h e n the restriction of V t o Go i s either irreducible, o r a direct s u m of two non-isomorphic Gomodules. Moreover, the restriction reduces i f and only if 7r 7r @ y where y i s the unique non-trivial character of G trivial o n ZG". Definition 6.1: A smooth representation V N = 0.

(7r, V

) of Go is called cuspidal if

We shall now show that the matrix coefficients of cuspidal representations are compactly supported. A matrix coefficient of V is a (smooth) function cpu,&7)

where v is in V , and 5 in

= (7Qh

6)

v, the contragredient of V

Proposition 6.2: Let ( T ,V ) be a cuspidal (not necessarily irreducible) representation of G o . T h e n the matrix coe&cients of 7r are compactly sup-

ported. Proof: Let U and 0 be the linear span of 7r(K)vand ii(K)6,respectively. Both spaces are finite dimensional, since the representations are smooth and

29

Lectures on Representations of p-adic Groups

K is compact. Since U is cuspidal, by Lemma 5.2, we can find a sufficiently large i such that such that

s,.

7r(n)ud n = 0

for all u E U , and U is contained in

vK%. Let

Then, for any u in U , and any ii in

u,

(7r(A)u,ii(n)ii)dn

=

(7r(X)7r(n)u7 ii) d n ,

where k = i - 2a. Thus, if a 2 i then the right hand side is 0. Since ii is in the left hand side is equal to (n(X)u,ii), up to a non-zero factor (the measure of N i ) . Therefore, 'pv,c(kXk') = 0 for any two elements k and k' in K . The proposition follows form the Cartan decomposition. 0

vK.,

Proposition 6.3: Let V be an irreducible cuspidal representation of G o , and P : U V a non-trivial homomorphism of smooth Go-modules. Then U the above sequence splits, that is there exists a homomorphism s : V such that P o s ( u ) = u for all v in V . ---f

--f

v

Proof: Let be the contragredient dual of V. Pick u any vector in V. We first define a skew-linear (with respect to the complex conjugation) map i : 4 V by

v

Since pv,?r(h)17= pu,s(h-'g), the map is a skew-linear homomorphism. Since V is irreducible, i must be onto. Finally, since

we see that i is injective, as well. Now pick any u in U such that P ( u ) = v. Define, analogously to i , a skew-linear map from to U by

v

(Here T denotes the action of G on U ) . Now put s = j o i-l. Clearly, s is a linear homomorphism from V to U , and by Schur's Lemma P o s has to be

G. Savin

30

a multiple of the identity operator. Since P ( s ( v ) )= v, for our fixed vector proposition follows. 0

IJ, the

Exercise. The proof of the above proposition shows that smooth irreducible cuspidal representations are admissible. Hint: The map i defines a slcewand V . Now use the fact that v K a is equal to the linear isomorphism of linear dual of V K *(Proposition 4.5).

v

Corollary 6.4: Every irreducible smooth representation V of GL2(F) is admissible. Proof: We have just seen that cuspidal representations are admissible. Otherwise, the representation is a submodule of a principal series representation by Proposition 5.3, so it is again admissible. 0

7. Decomposing principal series

In this section we shall describe the composition series of I n d $ ( X ) . Some critical calculations will be performed in Section 8.

Proposition 7.1: Let V be an irreducible subquotient of I n d s ( X ) . Then VN

# 0.

Proof: If VN = 0, then V is cuspidal, so it has to be a submodule as well. But then by Frobenius reciprocity VN # 0. Contradiction. 0 Since VN # 0 for any subquotient, the following proposition shows that the length of a composition series of I n d $ ( X ) is at most two. The proof is based on the Bruhat-Tits decomposition B U BwN of G, where

Proposition 7.2: Let obtained by conjugating Then

x x

be a character of T , and x" be the character by w, which simply means permuting x1 and x 2 .

0 +. 6 1 / 2 x w

--f

IndE(X)N

-+

d1/2x-+ 0.

Lectures

on

Representations of p-adic Groups

31

Proof: Let I n d g ( & be the B-submodule of functions in I n d g ( X ) supported on the big cell B U N . We shall use the filtration IndE(x)w

c I&(X)

and the exactness of V -+ V ~toJ calculate I n d g ( X ) . Define a functional Q on I n d g ( X ) by

4 f )= f (1). Since

a(7r(nt)f) = 6 1 ' 2 X ( t ) . a ( f ) we have shown that 61/2xis a quotient of I n d g ( X ) N . Clearly, I n d g ( x ) u , is the kernel of a. Define a functional a, on I n @ ( & by

Exercise. Show that the kernel of 01, is precisely I n d : ( & , ( N ) . Hint: Note that I n d g ( & is isomorphic to C F ( N ) ,as a n N-module, and apply Lemma 5.2. An easy calculation shows that

aw(x(nt) f ) = W X " ( t ) .Q w ( f ) . The proposition is proved.

0

We now note a very important consequence of the above result. If x # is called regular) then the above exact sequence splits. This implies that aw extends to I n d $ ( X ) , and thus by the Frobenius reciprocity defines a non-zero intertwining operator

xw (such x

Proposition 7.3: The following gives a complete answer to decomposing principal series representations of G L 2 ( F ) . 0

The induced representation I n d g ( X ) is irreducible if ~ 1 1 x # 2 Otherwise, the composition series is of length two, with one of the subfactors a one-dimensional representation.

G. Savin

32

Proof: The singular case x = xw will be dealt with later. Assume that x # xw . Then the Frobenius reciprocity implies that

Hom(Ind%x),IndS(x)) is one-dimensional, so every intertwining operator is a multiple of the identity I . In particular, it follows that

Aw(X") 0 Aw(X) = c ( x ) . I for some scalar c(x ) (called c-function).

Lemma 7.4: I f x #

x",

then I n d E ( x ) reduces if and only ifc(x) = 0.

To verify this lemma, recall that Ind:(x) can have a t most two subquotients. Let V be a submodule. Then, V is a proper submodule if and only if VN = dl/'x. In that case Indg(X) has a proper quotient U and UN = 6l/'xW. The Frobenius reciprocity implies that A,(x) annihilates V . Likewise, A,(x") annihilates U , which is a submodule of Indg(xW).It follows that Aw(x")o A,(x) = 0. The lemma is proved. The first part of the proposition follows as soon as we calculate c ( x ) . This will be done in the next section. Thus, assume that x1/xz = I . I*'. In fact, since subquotients of Ind:(x) are the same as subquotients of Indg(x"), it suffices t o consider x1/x2 = 1 . I-'. In this case y(det) appears as a submodule, where

y = xll . 11/2 = X ' J . I-.

If y is trivial, then the other (infinite-dimensional) subquotient is called the Steinberg representation. Obviously, for other y, the infinite dimensional subquotient is just the twist of the Steinberg representation by y(det).

8. c-function

The purpose of this section is to obtain an explicit formula for the c-function c ( x ) . We also obtain a result which plays a crucial role in proving that IndS(x) is irreducible for singular x. In order to keep notation simple, we shall restrict ourselves to the family I ( s ) = I n d g ( x ) where

If s

# 0 let A,(s) : I ( s ) 4 I ( - s )

Lectures on Representations of p-adic Groups

33

be the intertwining operator which, via the Frobenius reciprocity, corresponds t o the functional a,. Recall the definition of a,. On the subspace of functions f such that f(1) = 0, it is defined by

and, if s # 0, then a, is the unique extension to I ( s ) . In order t o calculate A,(-s) o A,(s), we need to obtain an explicit formula for a, on the whole I ( s ) . In other words, our task is to regularize this integral for any f in I ( s ) .

To do so, we shall now present some key technical ingredients. Let d x be the Haar measure on F , and for every s in C , let A, be the functional on C r ( F ) defined by

Let r denote the (right) regular representation of F X on C?(Fx). The functional A, is equivariant with respect to r in the sense that As(.)f

=

14-sAs(f).

Proposition 8.1: The functional A, extends to an equivariant functional o n C r ( F ) i f and only i f s # 0 . The extension is unique i f s # 0 .

Proof: Let A, be an extension, if any. Let f be in C r ( F ) . Then the function f - r ( w ) f is in C r ( F X )In . particular,

On the other hand, since As is equivariant, we have A,(f - r ( w ) f )= (1 - Iml-")As(f). Putting things together,

This formula shows that the extension exists and is unique if 1 - qs is not zero. Otherwise, which happens precisely when s = 0, the left hand side is always zero, while the right hand side can be easily arranged to be nonzero (take, for example, the characteristic function of 0).In particular, the functional A0 does not extend. The proposition is proved. We shall now apply this idea to our situation. For any f in I ( s ) define

G. Savin

34

Next note that f - f' vanishes at 1, so it is supported on the big double coset NwB. Arguing exactly as in the proof of the previous proposition we see that an extension of a, to I ( s ) exists if and only if s # 0, in which case it is given by

(f - f ' ) ( w n )d n .

a,(f) = (1- q - S ) - 1 ] N

Corollary 8.2: H o m ~ ( I ( 0I)(,0 ) ) = @. This corollary follows from the Frobenius reciprocity, and the fact that

a, does not extend to I ( 0 ) . Combined with the fact that I ( 0 )is completely decomposable, which will be shown in the next section, it will imply that I ( 0 ) is irreducible.

Proposition 8.3: Let L ( s ) = 1/(1- q-').

In particular, c(s) = 0 i f and only i f

Then

s =fl.

Proof: In order to calculate A, ( - s ) o A, ( s ) , we need to make the formula for a,(f) more explicit. Clearly, f is Ki-invariant for some i, and we shall show that f(1) determines f ( w n ) for all n in N \ N-i+l. Indeed, note the following relation in SLZ(F):

Let j 5 -2, and assume that x is in wjC3 \ wj+lO. Since f is right invariant, and left N-invariant, the relation implies that

In particular, it follows that

Ki-

f - f' is supported on N-i.

Next, note that the Iwasawa decomposition implies that I ( s ) has a unique element fs such that f s ( g k ) = fs(g) for all k in K , and fs(l)= 1. The above calculation can be made very precise for fs: Exercise. Show that aw(fs)

+

= L ( s ) / L ( 1 s).

Lectures o n Representations of p-adac Groups

35

Since A,(f,) is K-invariant, it must be a multiple of f-,. The above exercise shows that the multiple is L ( s ) / L ( l +s). The proposition followa

9. Unitary representations

A smooth representation of G is called unitarizable if it admits a positive definite G-invariant hermitian form. In this section we shall first show that I n d g ( x ) is unitarizable if x is a unitary character (x . k = 1). Combined with Corollary 8.2 this will (finally) show that I ( 0 ) is irreducible. We finish this section by showing that the representations I ( s ) with 0 < s < 1 (the so called complementary series) are also unitarizable. The Iwasawa decomposition implies that each representation I ( s ) has a unique (up to a non-zero constant) K-invariant functional given by

f

J’ f ( k ) d k . K

Since the trivial representation of G is a quotient of 1(1),the K-invariant functional is also G-invariant if s = 1.Now we can easily construct a positive definite G-invariant hermitian form on I n d $ ( x ) . Let f and g be any two functions in IndE(x). If x is unitary, then f . g is in 1(1),so the hermitian form

(fld =

J’ f ( k ) J ( k )dk K

is clearly positive definite and G-invariant.

Proposition 9.1: T h e representation I ( 0 ) i s irreducible. Proof: Since I ( 0 ) is unitarizable, and of finite length, it will decompose as a direct sum of irreducible constituents. In particular, Hom(l(O),I ( 0 ) ) is isomorphic to a direct sum of matrix algebras M n ( @ ) ,one for each irreducible constituent of multiplicity n. On the other hand, we know that Horn(I(O),I ( 0 ) )= @. The proposition follows. 0 Our last topic for G L z ( F )will be a construction of the unitary structure on I ( s ) for -1 < s < 1. We first need to renormalize the intertwining operators Aw(s), so that they are defined at the singular point s = 0 as well. Recall that A,(s) corresponds, via the Frobenius reciprocity, to the functional

G. Savin

36

This is well defined only if s # 0 since L ( s ) has a pole a t s = 0. Let A,(s) be the operator which corresponds to the functional a,L(l + s ) / L ( s ) .Then A,(s) is well defined for every -1 < s < 1, and

A,(s)fs = f - s . We can now easily define a G-invariant hermitian form on I ( s ) as follows. Let f and g be two functions in I ( s). Then A, (s) f is in I ( -s), and as above,

(f,d =

1 K

(Aw(s)f)(k)g(k) dk

defines a G-invariant sesquilinear form on I ( s ) .Apriori, it is not clear that this form is hermitian. However, Schur's Lemma implies that an irreducible smooth G-module can have only one sesquilinear G-invariant form, up to a non-zero scalar. Thus the form ( g , f ) must be proportional to the form (f,g). Since (fa, f s ) = 1, it follows that they are equal, so our form is hermitian, after all. Next, note that A,(O) is the identity operator on I ( 0 ) .Indeed, we know that it is a multiple of the identity operator, but the formula for the spherical vector shows that it is in fact equal to the identity operator. It follows that the form is positive definite on I ( 0 ) . We shall use this observation to prove the following proposition.

Proposition 9.2: If 0 < s < 1, then the G-invariant hermitian f o r m (., .) on I ( s ) is positive definite. Proof: Since I ( s ) is irreducible for -1 < s < 1 the G-invariant form must be non-degenerate (otherwise the kernel of the form would be an invariant subspace). Moreover, as the following exercise shows, the restriction of this form to every finite-dimensional subspace of Ki-invariants is also non-degenerate. Exercise. Let V be a smooth G-module with a non-degenerate G-invariant hermitian f o r m . Then the restriction of the f o r m to V K t is also nondegenerate. Hint: Use the Ki-invariant projection operator P from V onto VKt.

Every element in I ( s ) is completely determined by it restriction to K . In this way, we can identify, in a K-invariant fashion, with the (finitedimensional) space Ei of functions on B n K\K/Ki. By the above exercise, the invariant form on I ( s ) induces a non-degenerate form (., .)s on Ei, which

Lectures on Representations of p-adic G T O U ~ S

37

is positive definite for s = 0. The proposition follows from the following exercise.

Exercise. Let E be a finite dimensional vector space, and (., .)s a family of non degenerate hermitian symmetric forms, depending continuously o n the parameter s in a connected open set in R. If (., . ) s o is positive definite f o r some SO in the open set, then it is positive definite for all s. 0

10. Steinberg representation

By Proposition 7.3 we now know that the principal series I ( s ) decomposes if and only if s = f l , in which case it has a composition series of length 2 . One subquotient is the trivial representation, and the other is the Steinberg representation V . Recall that

v,

= 6.

We shall show that the Steinberg representation is square integrable, which means that the matrix coefficients of V restricted to Go are square integrable. The results of this section are, in large part, due to Casselman. Proposition 10.1: Let V be the Steinberg representation of G. Then the matrix coeficients of V are square integrable, when restricted to Go. Proof: To prove this proposition, we need a result on the asymptotics of matrix coefficients of V . Recall that A,=

( m0a O

).

Lemma 10.2: Let V be the Steinberg representation of G. Then, for every v in V , and V in there exists a positive integer i such that

v,

("(kX,+ilC')V, 6)= 6(Xa)(7r(kXik')W,5). for every non-negative integer a , and any two elements k and k' an K Proof: The proof of this lemma is analogous to the proof of Proposition 6.2. Let U and 0 be the finite dimensional subspaces generated by v and V over K , respectively. Let u be in U , and for every positive integer a , define u, = 7r(Aa)u- b(X,)u.

G. Savin

38

Since V , = 6 it follows that u1 is in V ( N ) .Recall that V ( N )is a union of V ( N j ) ,where V ( N j )is the kernel of the operator

Pj(u) = / N j

7r(n)ud n .

Take i large enough so that every u1 is contained in V ( N - i ) and contained in VKi. This is our choice of i.

0 is

Exercise: Show that 7r(Xl)V(NL)c V(N-i+2).In particular, 7r(X1)V(N-i) is again contained in V ( N - 0 .

Since

an induction argument based on the previous exercise implies that ua is in V ( N - i ) for all positive integers a. Thus, for every u in U ,

and if ii is in

0,

It follows that

ii)

(7r(Xi)ua,

= 0 , as

desired. The lemma is proved.

0

We can now finish the proof of the proposition. Recall the Cartan decomposition

Go = UZoKXaK The measure of KXaK is equal t o the index of K n KXaK in K , which is essentially (see the first exercise in Section 4) &(A,)-' = qZa.This and Lemma 10.2 imply that

for some positive constant C. This series clearly converges, and the proposition is proved.

Lectures o n Representations of p-adic G T O U ~ S

39

11. A cuspidal representation of Go In this section we present an example of a cuspidal representation. To make the example as simple as possible, the trick (which the author learned from Allen Moy) is to restrict to the case p = 2. Thus, in this section only, our groups are defined over the dyadic field Q2. Let

IF2 =

( 0 , l ) be the residual field of Q2. Consider the projective line

P(2) = ( ( 1 1 01, (1,I), (01 1)) over IF2. The action of K/K1 = GLz(2) on P ( 2 ) gives an isomorphism of GLz(2) with 273, the symmetric group on 3 letters. Let E be the usual sign character on Ss. Then one easily checks that

.(;:>

=-1.

Pull E back to K , and define V = ind;’(~). The symbol i n d stands for the induction with compact support. In particular, V consists of compactly supported functions on Go such that f(kg) = E ( k ) f ( g ) for all k in K and g in Go.

Proposition 11.1: The representation V is an irreducible cuspidal representation of Go. Proof: We shall first prove that V, = 0. First of all, note that

~ ( nd n) L

= 0.

K

v

Let e K in be the function supported on K , such that e K ( k ) = ~ ( kfor ) all k in K . The Iwasawa decomposition implies that any element in V is a linear combination of n(b)eK where b is in B. Let Nb = b L 1 ( N n K ) b . The vanishing of the integral above implies that r

It follows that V is cuspidal by Lemma 5.2. It remains to prove that it is irreducible. Let H,(G’//K) be the Hecke algebra of compactly supported functions on Go such that

f ( W ’ )= .(k)f(g)4k’).

G. Suvin

40

for all k and k' in K . As usual, the multiplication in H , ( G o / / K )is defined by convolution of functions (fl

* f2)(g) =

/

f l ( h ) f i ( h - l d dg.

G O

Note that H , ( G o / / K )is contained in V , and equal to V K + the , subspace consisting of all functions f in V such that 7r(k)f = ~ ( k ) ffor all k in K . The element e K defined above is the identity element in H , ( G o / / K ) .

Lemma 11.2: There i s a natural isomorphism Homco(V,V) HE(GO//K).

E

Proof: Let A be in HomGo (V, V ) .Since V is generated by e K , the operator A is completely determined by its value A ( e K ) . Since A ( e K ) lies in VK,', which is the same as H , ( G o / / K ) ,the map

A

H

A(~K)

furnishes a canonical injection from HomGo (V,V) t o H , ( G o / / K ) . Conversely, an element T in H , ( G o / / K ) defines an element A in HomGO (V,V ) by

A ( f ) =T * f. Since A ( e K ) = T * e K = T , we see that A The lemma is proved.

H

A-(eK) is surjective as well. 0

Lemma 11.3: T h e algebra H , ( G o / / K )is one-dimensional. Proof: Recall the Cartan decomposition

Go = Ur=oKA,K where

A,=

("0 2- ) . O

Let f be in H,(Go//K).We shall show that f is a multiple of f is determined by its values at all A,. Let n=

(iy)

andn'=

(it).

eK.

Clearly,

Lectures o n Representations of p-adic Groups

41

If a 2 1, then n is in K1 and ~ ( n=) 1. On the other hand, recall that ~(n= ' ) -1. It follows that

f(&) Thus, f ( A a ) proved.

=

= f(n&L) =

f(W)= -f(U

0 if a >_ 1, and f must be a multiple of e K . The lemma is 0

We can now finish the proof of proposition. Since V is generated by e K , Proposition 4.2 implies that there exists an irreducible quotient V' of V. Let P be the projection from V onto V'. Since V' is cuspidal, by Proposition 6.3 there is a splitting s : V' -+ V such that P o s = I d v f . But s0

PE

HOrnGO

(V,V )= c Idv '

so s o P must be equal to I d v . It follows that V = V', and the proposition is proved. 12. The root system for G L n ( F ) The purpose of this section is to introduce the root system for GL,. It is a combinatorial object which provides us with a language to study representations of GL,(F). (For more information on root systems see [a]). Let IR be the field of real numbers, and consider the space R" with the standard basis e l , . . . , en. With respect to this basis, we shall identify every element IC in IRn with an n-tuple ( ~ 1 ,. .. ,IC,). The group S, acts on R" by permuting the entries of ( X I , . . . ,x,). This action preserves the inner product n

i= 1

Let

a

Note that the group S, preserves R. In fact, it is an irreducible representation. We shall now describe the action of S, by means of Euclidean reflections. Let

9 = {ei - e j I i # j } ,

G. Savin

42

be a finite set in R. Its elements are called roots. Any root a defines a reflection w by W(X)

= 2 - ( a , x ). a.

Exercise. Let a = ei - e j . Show that the corresponding w is simply the permutation of i and j . It follows from the exercise that the reflections w define the representation of S, on 0. Clearly, the fixed points of the reflection corresponding to the root a = ei - ej is the hyperplane { x i = x j } , defined by the equation xi = x j . Consider the open subset obtained by removing all hyperplanes {Xi = X j }

R"

\ (U{Xi

=R

= Zj}).

The connected components of R" are called chambers. We shall point out a particular chamber given by

c+= {.

E

R I 2 1 > . . . > x,}.

Next, note that the entries of any x in R" are pair-wise different, so there exists a unique permutation which puts them in a decreasing order. This means that R" is a disjoint union of w(C+) as w runs thru S,. In particular, every chamber C is equal to

c = {x E R I Xil

> ... > X i n }

for some permutation ( i l l . .. ,in). The choice of C+ gives us a set of generators of S, as follows. Note that the closure of C+ is obtained by adding parts (called walls) of hyperplanes {zi = zi+~}The . corresponding roots

{ e l - e2,. . . , en-1

-

en}

are called simple. Note that the simple roots form a basis of R. Let wibe the (simple) reflection defined by ei - ei+l. We claim that the reflections wi generate S,. Of course, since wi is nothing but the permutation (i, i 1); this is a well known fact. We shall give a proof based on the fact that S, acts simply transitively on the set of chambers. Let w be an element in S,. Pick x+ in C+ and x in w(C+)such that the segment between them avoids the singularities of R \ R". Since those singularities form a codimension 2 subvariety, a generic choice of x+ and x will do. The number of hyperplanes

+

Lectures on Representations of p-adic Groups

43

intersected by the segment is clearly independent of the particular choices, is the length l ( w ) of w. The chambers containing the segment form a gallery

c+, c1,. . . Cqw,= W ( C + ) . Any two consecutive chambers in this gallery share a wall. In particular, there exists a simple reflection wil such that C1 = wil (C+).Next, note that the walls of C1 correspond to reflections wil (the shared wall) and wil wjw;' for all j # il. In particular, there exists i 2 such that C2 = w i l w i z w ~ l ( C 1 ) , which implies C2 = wi2wil(C+).Continuing in this fashion, we can obtain wil l . . . , wit(wlsuch that

. . . . . Wil (C+)= W ( C + ) . Exercise. Make a picture of the root system for GL3(F). The roots form a regular hexagon. Let w be the element in W such that w(C+) = C-. Express w as a product of simple reflections by picking a gallery between C+ and C - . 13. Induced representations for G L , ( F ) Let e l , e2,. . . ,en be the standard basis of F". Let V, be the span of e l , . . . , e,. Consider a partial flag

K, c K, c . . . c K* and let P be its stabilizer in G. Then P is called a parabolic subgroup of G, and it admits the Levi decomposition P = M N , where M is isomorphic to the product of GL(V,/V,-1), and U is the unipotent radical of P . Note that the minimal parabolic subgroup B corresponds to the full flag. If

(7, E )

is a smooth representation of M , then we can induce it to G

by

Indg(7) = { f : G 4 T I f ( u m g ) = 6b'2T(m)f(g)}". The character bp of M is defined so that for every locally constant, compactly supported function h on U we have

I,

h(mum-l) du = S p ( m )

I,

h ( u ) du.

G. Savin

44

As a particular case of interest, consider the minimal parabolic subgroup x of T is given by

B = T U . Note that any smooth character X

( ) .-.

= Xl(a1) . . . . xn(an)

an

for some smooth characters x 1 . . . xn of F X. Assume that x is regular, which means that x # xw for any w in W . In the case of GLn(F) this simply means that xi # x j if i # j. As in the case of GL2(F),the Bruhat-Tits decomposition can be used to show that

@P

Indg(X), =

X " .

WEW

It turns out that V , # 0 for any irreducible subquotient, just as in the case of G L z ( F ) . In particular, each irreducible subquotient is completely determined by V N .Let S be the set of all { i , j } such that x i / x j = I . I*'. Let

R,

=R

\ (US{Zi

= Zj})

where {xi = xj} is the hyperplane in R defined, of course, by the given equation. A result of Rodier states that the irreducible subquotients corresponds to the,connected components of 0,. We shall here give a special case of that result, describing V, for the unique irreducible submodule V . (Since any irreducible subquotient of Indg ( x )is a submodule of Indg (x") for some w in W , the general case easily follows.) Let 0; be the connected component containing the positive chamber Cf. Define W , to be the set of all Weyl group elements w such that

W(C+)c Rxf. Proposition 13.1: Let x be a regular character. Let V be the unique irreducible submodule of Indg ( x ) (normalized induction). Then

v, =

@

P X " .

W€W,

Proof: The proof of this statement is combinatorial, and based on the special case of G L z ( F ) ,proved in Proposition 7.3. Let Pi be the parabolic subgroup corresponding to the flag with V , omitted. Note that its Levi factor Mi has exactly one factor isomorphic t o GL2, and let wi (= w) be

45

Lectures o n Representations of p-adic Groups

the permutation matrix in that factor. Assume that is in W,. Then, by Proposition 7.3

xa/xi+l

# I I*', *

so wi

I n d 2 (x)E In@ (X""). Inducing both representations further up t o G, this gives Indg(X) is a summand of VN.For a general element Inds(XWt). In particular, w of length m in W, we have to consider a gallery x w t

c+,C', . . . , c,

=W(C+)

which is contained in 0: (since 0: is convex), and repeat the above argument m-times. This shows that W€W,

The opposite inclusion will follow from the following lemma: Lemma 13.2: A s s u m e that xa/xI = I . ' 1 for some a < j . If x" is a summand of VN then C+ and w ( C + ) are on the same side of the hyperplane {xa = 2 3 ) .

To be specific, assume that xa/xI= I . I. Let w' be the permutation (i + l , j ) ,and XI = xw' be the character of T obtained by permuting xj and xz+l.Then X:/X:+~ = 1 . I. Inducing in stages (first from B t o Pa),and using the second part of Proposition 7.3 (GLz(F)-reducibility) we get an inclusion 1 4 % (77)

G I c IndB(X 1,

where 77 is a character of M,, which on the GLz(F)-factor is given by composing the determinant with the character

xzl.11/2 = Xz+ll . [-I/? A more general form of the Bruhat-Tits decomposition implies that Indgt(7])N =

@ W(X')W W€W%

+

where W, is the set of all w such that w(i) < w(i 1). This, however, is equivalent to wwI(i) < ww'(j), which happens if and only if C+ and ww'(C+) are on the same side of the hyperplane { z a= xI}. Summarizing, we have 1ndgt(7)N =

@ W(C+)C{Z,>Z~)

61/2Xw.

G. Savin

46

x

be a regular character, and let R1,. . .R, be the connected components of 0,. T h e n I n d g ( X ) has m irreducible subquotients Vl, . . . , V, so that

Corollary 13.3: Let

(K)N =

@ P w ( C + )c a

Exercise. Show that I n d g ( 6 ' / 2 ) has 2"-l special case of GL3( F ).

X " .

subquotients. Hint: Try first the

References 1. J. Bernstein and A. Zelevinsky, Representations of the group G L , ( F ) , where F is a non-archimedean local field, Russian Math. Surveys 31 (1976), 1-68. 2. J. Humphreys, Introduction to Lie Algebras and representation theory, Graduate Texts in Mathematics 9, Springer-Verlag, 1978. 3. N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta Functions, Second Edition, Graduate Texts in Mathematics 58, Springer-Verlag, 1984. 4. F. Rodier, De'composition d e la se'rie principale des groupes re'ductifs p adiques, 408-424, Lecture Notes in Mathematics 880, Springer-Verlag, 1981. 5 . M. Tadid, Representations of classical p-adic groups, 129-204, Pitman Res. Notes Math. Ser. 311, Longman, Harlow 1994.

Lectures on Harmonic Analysis for Reductive p-adic Groups

Stephen DeBacker Department of Mathematics Harvard University Cambridge, M A 02138, U S A E-mail: [email protected]

1. Introduction In his paper T h e Characters of reductive p-adic groups [14] Harish-Chandra outlines his philosophy about harmonic analysis on reductive p a d i c groups. According t o this philosophy, there are two distinguished classes of distributions on the group: orbital integrals and characters. Similarly, there are two classes of distributions on the Lie algebra which are interesting: orbital integrals and their Fourier transforms. The real meat of his philosophy states that we ought to treat orbital integrals on both the group and its Lie algebra in the %&meway” and similarly, we should think of characters and the Fourier transform of orbital integrals in the same way. This philosophy has many manifestations (see, for example, Robert Kottwitz’s excellent article [12]). In this series of lectures, we will examine the various distributions discussed above and discuss one of the deepest connections between them: the Harish-Chandra-Howe local character expansion. Because of requests on the part of participants, I will spend much time reviewing the basics of p a d i c fields and discussing some of the uses of Moy-Prasad filtrations in the representation theory of reductive p a d i c groups. As such, at a small cost in terms of generality, I will concentrate on those techniques which use this theory. These lectures and the notes are meant as an informal and elementary introduction to the material. For complete, rigorous proofs, please see the references. Very little of the material in this set of lectures is original. I have borrowed heavily from the work and lectures of Harish-Chandra, Roger Howe, 47

S. DeBacker

48

Robert Kottwitz, Allen Moy, Gopal Prasad, Paul Sally, Jr., and J.-L. Waldspurger, among others. I thank the organizers of this conference, in particular, Eng-Chye Tan and Chen-Bo Zhu, for inviting me and allowing me to present this series of tutorials. I thank Jeff Adler, Amritanshu Prasad, Joe Rabinoff, Loren Spice, and Chen-Bo Zhu for their helpful comments on earlier drafts of these notes.

2 . Basics 2.1. A n introduction to the p-adics The usual introductory mathematical analysis course proceeds roughly as follows: The class agrees to agree that the set of natural numbers

{1,2,3,…} is very natural and therefore a good place to begin the course. In order to form a group with respect to addition, the additive identity and additive inverses are tossed in to the mix to give us the integers

z:= {. . . , -4,

-3, -2, -1,o, 1 , 2 , 3 , .. .}.

This set does not form a group with respect to multiplication; it is therefore enlarged to form Q,the field of rational numbers. Everything so far has been very natural. At this point, the incompleteness of the rationals is demonstrated by proving that the square root of two is not rational. To compensate for this, the fact that the rationals are ordered (that is, there is a notion of nonpositive and nonnegative) is invoked to define the absolute value, 1.1, of any rational number q: 141 =

q

ifq20

-q

ifq
As usual, the absolute value gives you a metric with respect to which you complete the rationals. We recall that two Cauchy sequences { q n } and (4;) of rational numbers are said to be equivalent with respect to 1.1 provided that {qn - q ; } is a Cauchy sequence which converges to zero. The set of real numbers is then defined to be the set of equivalence classes of Cauchy sequences with respect to 1. 1 , that is, the completion of Q with respect to

Lectures on Harmonic Analysis for Reductive p-adic Groups

49

It turns out, however, that the normal absolute value provides just one of an infinite number of incompatible ways to complete the rational numbers”, and, from the proper perspective, this completion is a somewhat unnatural object. Let p be any prime. If q is a nonzero rational number, then there is a unique integer k such that q = p k . a / b with p a and p b. We can then define the p-adic absolute value, [ . I p , on Q by setting 141, = 0 if q = 0 and )q), = p-k otherwise. The p a d i c absolute value has the following properties. Exercise 2.1.1: If

(1) 17-11, (2) (3)

.

2 0, and

17-1 7-21,

17-1

7-1

17-11,

= 17-11,

and r2 are rational numbers, then = 0 if and only if

7-1

= 0,

. 17-21,, and

+ ral, I max(l7-11,

1

17-21,).

Exercise 2.1.2: In the last item of the previous exercise, show that if (rlIp# lr2(pthen the inequality is an equality. Is the converse of this statement true? From Exercise 2.1.1, it follows that we can define a metric on Q with respect t o the p a d i c absolute value. We define Qpto be the completion of Q with respect to 1., The padic absolute value on Q extends continuously (and uniquely) to a p a d i c absolute value 1., : Q, + { O , p k I k E Z}. We define the valuation up on Q, by 1x1, = p-+(”) for x E Q: and up(0)= 03. Thus u p ( p m )= m for m E Z. We remark that Ostrowski’s theorem tells us that the only nondiscrete, locally compact fields of characteristic zero are the field of real numbers, the field of complex numbers, and the finite extensions of Q,. That is, these are the only fields of characteristic zero on which integration naturally makes sense. 2.2. The structure of Q p and additive characters

If we define Z,to be the set of x E Q, such that (x(,5 1, then Z, is a maximal compact open subring in Q,. As in [33], it can be identified with the completion of Zwith respect to .1, We call Z,the ring of integers in Q,. The subset of Z,consisting of those elements of Z,with p a d i c absolute value less than one forms a maximal ideal ( p ) = p Z , of Z p . The quotient aThe field of real numbers is, up to isomorphism, the unique totally ordered completion of the rationals.

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50

Z,/(p) is isomorphic to IF,, the field with p elements. It follows that we can write

Z,

JJ

=

(i +pZ,).

O
As a slight tangent, we remark that Hensel's lemma (see nearly any book on local fields) tells us that we can lift a generator of IF; to Zp" , the group of units in Z,.Thus, the square root of two is an element of, for example, 0 7 . The set of compact open subrings p"Z, for n 2 0 forms a neighborhood basis of zero. The sets p"Z, also make sense for n < 0, and we note that Qp

=

u

P"&.

nEZ

Since Zp" = Z,\ pZ,, it follows immediately that

From this one can show that for each x E Qg if 1x1, = p-", exist unique coefficients a, E {0,1,2,. . . , ( p - 1)) such that

c

then there

oc)

x =

a, . p n .

n=m This is called the p-adic expansion of x . For example, the 5-adic expansion of 1/3 is

2

If x

E

+ 3 . ( 5 ) + 1 . ( 5 2 ) + 3 . (53) + 1 . (54) + 3 . (55) + . . . .

Q, and 1x1, = P - ~ then , we define the tail of x to be the integer

t ( x )=

iu

En=,a, .pn O

if 1x1, = p-m and m 5 0, and ifxEpZ,.

We then define

A(x)= e

Z.rn.%.t(Z)

p

.

Exercise 2.2.1: Show that A defines an additive character of Q, and that the restriction to p Z , of A is trivial, yet the restriction of A to Z, is not trivial. It is a fact that Q, is isomorphic to the Pontrjagin dual the map a H ( b +-+ A(ba)).

& of Q, via

Lectures on Harmonic Analysis f o r Reductive p-adic Groups

51

2.3. General notation From now on, let k denote a finite extension of Q p l let R denote the ring of integers of k, and let a denote a uniformizer so that 63 = WR where P is the prime ideal. We define Pm := wm ' R for m E Z. We let f := R I P denote the residue field, and we let q = If/. As before, we fix a valuation u such that u(k) = Z. We fix an additive character A of k that is trivial on 63 and not trivial on R. We let G denote the general linear group realized as the set of n x n matrices with entries in k having nonzero determinant. We let g denote the Lie algebra of G. We realize g as M,(k), the set of n x n matrices with entries in the field k , having the usual bracket operation. Let dX denote a Haar measure on g and let dg denote a Haar measure on G. We let B denote the Borel subgroup of G consisting of upper triangular matrices, and we let T denote the maximal split torus consisting of diagonal matrices. The map (X,Y ) H tr(X . Y ) from g x g to k defines a G-invariant, nondegenerate, symmetric, bilinear form on 0. As such, it allows us to identify g with the Pontrjagin dual 5 of g via the map X H (Y H A(tr(X . Y))). A compact, open, R-submodule of g is called a lattice. For example, for each integer i , we can define the standard filtration lattice ti := GJ' * M,(R) of 0.As another example, consider the Iwahori filtration lattices bi/, defined by

bi/, := {Y E g I K

k E

wr*l

. R}.

Note that for all integers i , j , we have wj . ti = ti+j and More concretely, for n = 2 we have

wj

. bi/,

= bj+i.

and

For n = 3 we have

From these examples, it is easy to see that the image of bo in to/tl = M,(f) is a Borel subalgebra with nilradical equal t o the image of 611, in to/tl =

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52

M,(f). (We remark that in all of these examples, the lattice in g is a direct sum of copies of lattices of k.) We have similar filtrations of G by compact open subgroups. The standard fifiltration subgroups are defined by KO= GL,(R) and Ki = 1 + ti for positive integers i. We note that, up to conjugacy, KOis the unique maximal compact open subgroup of G. We also define the Iwahori subgroup Bo = b g and, for positive integers a , we define the Iwahori filtration subgroups B i / , = 1 bi/,. It is easy to check that if 0 k j , then Kj is a normal subgroup of Kk and the lattices ,t (rn E Z)are invariant under the action of Kj. Similar statements apply to the Iwahori filtration lattices and subgroups. If 0 < j k 2 j , then e j / & is an abelian group which is isomorphic via the map induced by H (1 to K j / K k . Similarly, b j / , / b k / , is isomorphic to Bj/n/Bk/n. Given a lattice L in g we define the dual lattice L* of L by

+

< <

< <

+ x)

x

L* = {Y E g I t r ( X . Y ) E P for all X E L } . For example, for any integer i, the dual lattice of ti is t(1-,), and the dual lattice of bi/, is b(l-i),n. We let C r ( g ) denote the set of functions on g which are compactly supported, complex valued, and locally constant. We recall that a function f on g is said to be locally constant if for each X E g there exists a lattice LX such that f ( X !) = f ( X ) for all in L x . If we assume that our f is compactly supported, then this lattice can be chosen uniformly. We let C r ( G ) denote the functions on G which are compactly supported, complex-valued, and locally constant (that is, translation invariant with respect to some compact open subgroup of G). For f E C r ( g ) we define the Fourier transform o f f , by the formula

+

e

f,

f ( X )=

1

f(Y). h ( t r ( X . Y ) )d Y

B

for X in g. For example, if [L]denotes the characteristic function of a lattice L in g, then [L]is [L*]up to a constant. We normalize the measure d X on h

g so that

f ( X ) = f(-X)for all f E C,"(g) and X E g.

Exercise 2.3.1: Show that this normalization of measures implies that for L sufficiently small, the measure of the lattice L with respect to d X is given by one over the square root of the index of L in L*, that is, meas,jx(L) = [L* : L I - ~ / ' .

Lectures on Harmonic Analysis f o r Reductive p-adic GTOUPS

Exercise 2.3.2: Show that the map f

H

f

53

from CF(g) to itself is a

bijection.

+

Finally, we normalize d g so that the map X H (1 X ) takes d X into dg. So, for example, we have that for all j E Z>o, measdx(tj) = measd,(Kj).

3. Moy-Prasad filtrations In this section we describe the Moy-Prasad filtration lattices of g and subgroups of G. 3.1. The apartment of T

Recall that T is the group consisting of diagonal matrices in G. We write t E T as t = ( t l , t 2 , . . . , tn) with t j E k X for 1 < j 5 n. Let a c X * ( T )= Hom(T,kX) denote the set of roots of G with respect to T (that is, the nontrivial eigencharacters for the action of T on 8). More explicitly = {aij 11

< i # j < n and a i j ( t )= t i / t j } .

With respect t o our Bore1 subgroup B we let A denote the set of simple 11 5 i 5 ( n - 1)).We let denote the set of roots; that is, A := {ai(i+l) positive roots in @.We fix a Chevalley basis

{ Z , H ~ , X IP r E A and y E

a}.

Here

(2)kl = bkl and 1

ifk=Z=i

-1

if k = 1 = (i

0

otherwise,

+ 1)

and (Xaij)kZ

bzkajl.

We see that the center of g is k . Z and the Hp together with Z form a basis for t, the Lie algebra of T .

S. DeBacker

54

For example, for GL2(k) we have

Let Z(G) 5 T denote the center of G. For 1 5 k 5 n define Xk E = Hom(kx,T ) by setting

X,(T)

(&(S))aj

=

{

s

ifi=j=k

#k

1

if z = j

0

otherwise,

for s E k x . With respect to our choice of a Chevalley basis, we can identify A = d ( T ) ,the apartmentb corresponding to T , with the real vector space (X*(T) @ R ) / ( X * ( Z ( G ) @ ) N. Let 20 denote the origin in A. The apartment of T is an (n- 1)-dimensional Euclidean space spanned by the set (20 xk 1 1 5 k 5 n}. Note that these spanning vectors satisfy the relation

+

3.2. A simplicia1 structure for the apartment of T We let

9 := { b + n ( 6 E ch and n E Z} denote the set of affine roots of G with respect to T and v. For II,= b+n E Q we let = 6 E a. If II,= S + n E 9 and CXi @ra E X , ( T ) @ R , then we define

4

Here ( , ) denotes the usual pairing between characters and cocharacters. Note that if v~ E X,(T) and vz E X,(Z(G)), then + ( v ~ vz) = II,(vT). Consequently, we can and shall think of an affine root as a function on d(T). For II,E 9, we define

+

= (2 E

d(T) I II,(x) = 0).

In these notes, we are always looking at the reduced building and apartments.

Lectures on Harmonic Analysis for Reductive p-adic Groups

55

The H+ are hyperplanes in d ( T ) and they provide us with a simplicia1 decomposition of d(T). For example, for GL2(k) the apartment d(T) is one-dimensional and it is spanned by the vectors xo XI = 20 - x 2 . See Figure 1.

+

XI

Fig. 1. The standard apartment for GL2(k).

For GL3(k) the apartment A ( ? )is two dimensional, and it is spanned by the vectors 20 XI, 20 XZ, and 20 = xo - (XI &). See, for example Figure 2.

+

Fig. 2.

+

+ x,

The standard apartment for GLs(k)

+

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56

The maximal facets in this simplicia1 decomposition of A(T) are called alcoves or chambers of A(T).For GL2(k), the alcoves are open line segments. For GL3(k), an alcove is the interior of an equilateral triangle. For GL4(k), an alcove is the interior of a regular tetrahedron.

3.3. Some subgroups o f G Note that for all a E

a, the root group

is naturally isomorphic to k as an additive group. From (l),k, regarded as an additive group, has a natural filtration indexed by Zu {fm}. Namely,

woo.R := {0} c . . . c w'. R

c W .R c R =WO R c w *

- *~R

c... c k.

14

Similarly, U, has a natural filtration indexed by ZU {ho} Z {$ E @ = a } U {kco}. The problem is: how do we decide which )I E {y!~ E 0 = a } corresponds to which subgroup of U,? The solution to our problem lies with our choice of a Chevalley basis. The choice of this basis determines the subgroup KO = GLn(R). We define

14

U,+o:= U, n KO. This, along with the requirement U,+l c U,+O , completely determines a natural indexing of subgroups in U, by the set {I) E @ I = a}. For example, for GLB(k), we have

4

Similarly, we have a filtration of gar the a-eigenspace in g, by setting B ~ + O:= M n ( R ) n ga

Note that for all $ E 0 we have an isomorphism of the corresponding additive subgroups g i and U, via the map X H (1 X). The maximal compact open subgroup

+

T o = { ( t l , t z ,. . . , t,) € T J t i € R X } of T also has a filtration indexed by the integers. Because of future considerations, we define, for r E RZO,

T, := {t E To I v(1 - ~ ( t )2)T for all x E X * ( T ) } .

Lectures on Harmonic Analysis for Reductive p-adic Groups

57

We also define

Tr+:= {t E TOI v(1 - ~ ( t )>)T for all

It is a matter of definition to see that Tr i < T 5 (i + l), then

x E X*(T)}.

Trr,, and, for i E Z ~ Oif ,

=

Tr = {(ti&, . . . ,tn) E To I t j E 1 + ZZ(~+') . R for 1 < j 5 TL}. We can define filtrations o f t = Lie(") in a similar fashion. For example, for GL3(k), we have

t5.2 =

(,gR

W'.R

0

0

).

w6.R

3.4. The Moy-Prasad filtrations Suppose r E R and x E d(T). In [27,28], Moy and Prasad define filtration lattices of g according to the formulae

and

For some people, it is easier to process these definitions when they are presented in the following form.

and

Similarly, if r

2 0, then they define

and Gz,r+ := (TT+ , U+){.LE*I + ( z ) > r ) .

These lattices (resp., subgroups) are referred to as the Moy-Prasad filtration lattices of g (resp., subgroups of G).

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58

It follows immediately from the definitions that if X E 82,. and Y E g X + , then X . Y E g x , ( r + s ) . The following exercise is highly recommended; we have previously considered these statements for the congruence and Iwahori filtrations.

Exercise 3.4.1: Suppose s E R and r E R>o. (1) Up to a constant we have [ g x , s ] = [gz,(-s)+]. ( 2 ) If g E G x , r , then ggx,sg-l = g X + and g g ~ ~ , ~ += g-l (3) If 0 < r 5 s 5 2r, then X H (1 X ) induces an abelian group isomorphism of g x , r / g x , s and G x , r / G x + *

+

3.4.1. The special case r = 0 We first attempt to understand these definitions when r = 0. In this case, if x is a point in A(T), then G,,o is called the parahoric subgroup attached to x and Gx,o+is called the pro-unipotent radical of Gx,o.Let F be a facet in A and let x,y E F . It follows from the definitions of both the simplicia1 structure of A and the Moy-Prasad filtrations that GX,o= GY,o,gx,o = gY,o, Gx,o+= G,,o+, and gx,O+ = gy,0+. Thus, it is enough to understand these filtrations on a facet by facet basis, and it is natural t o label the filtrations using facets rather than points. For GL2(k), Figures 3 and 4 describe the subgroups GF,Oand G F , ~for + F a facet in the apartment of T .

Fig. 3.

Parahoric subgroups GF,Ofor GLz(lc)

Note that as we move to the right, that is, in the positive direction as defined by the spherical Weyl chamber corresponding t o B , the positive root spaces “expand” and the negative root spaces (‘shrink’’.This simple observation will play a significant role in what is to come. It is highly recommended that the reader complete the following exercise.

Lectures o n Harmonic Analysis f o r Reductive p-adic Groups

59

Fig. 4. The subgroups GF,O+for G L z ( k )

Exercise 3.4.2: Make diagrams similar to those above, but for the filtration lattices ~ F , Oand g ~ , o + Now, . do the same for GL3(lc). The origin xo is the facet in A defined by the intersection of the hyperplanes H,+o for 0 E @. We see from the above examples (and definitions) that G,,,o = KOand G,,90+ = K1. For GL,(lc), up to conjugation, the facets are indexed by

e

P ( n ) :=

{ ( p i , p z , . . . ,pe) 1 p1

2 p2 2 . . . 2 pe 2 1 and

c p i = n}, i= 1

the set of ordered partitions of n. We briefly describe how this works. We let CObe the alcove in A with vertices { q = xo = xo Cy’l x i , 2r2 = 211 - i 1 , v 3 = wz - x 2 , . . . ,w, = in}. To p E P ( n ) we attach the facet Fp of dimension C with vertices q ,w ~ ( ~ ~ + l ) , . . ., ~ ( ~ , , + ~ , , + . . . + ~ , , - , + 1 ) . The parahoric subgroup G F , , ~can be described as K1 . Q p ( R )where Q p is the “standard” parabolic subgroup of G containing B which corresponds to the partition p = ( P I ,p 2 , . . . ,pg). The pro-unipotent radical GF,,o+ is then the inverse image in G F , , ~of the unipotent radical of the image of GF,,o in GL,(f) Ko/K1. Note that if p , v E P ( n ) with p < v in the usual partial ordering of ordered partitions, then Q p IIQv and similarly for the associated parahoric subgroups. More generally, note that if Fl and F2 are facets in A(T) with F1 c where denotes the closure of F2, then it follows that

+

GFl,O+ c GF2,0+c GF2,O c GF1,O. Moreover, as the examples and exercises above show, Gp2,O/GF1,O+ is a parabolic subgroup of the connected reductive group GF~,O/GF~,O+. The unipotent radical of GFz,o/GFl,o+is GF2,0+/GFl,0+ and GF,,o/GFl,o+has ~ ,of. Levi component isomorphic to G F,o/GF2

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As a tangent, we also observe that if W = NG(T)/T (N&!')nKo)/To denotes the Weyl group, then there exist IWI alcoves in d(T) which contain in their closure. This is because these alcoves correspond to the Bore1 subgroups in GL,(f) = Ko/K1 = G,,,o/G,,,o+ which contain "T(f)".

ICO

3.4.2. The Moy-Prasad filtrations for arbitrary r For arbitrary r , there does not exist such a nice description of what is happening. We first consider GL2(k). In Figure 5 we have identified the T

= (1 - a)(.)

( ;$ ) .... . .. ....

.... . .... ..

( :;)

r ....

....

....

....

. . .... . .. .

.. '..

... . ... ..... ... ..... .....

(; ;) ...........

.

:

:

..............

(; ;)

..........

... .

....

............

.

.

.

( ;;) ...

... .. .. . .. . . .. .

..... .....

.

.

"1)

...............

.... ....

(;

( P2 P P

...

.

.............

.,..::.::.....

...........

... .

1

(8;)

..-.

........

... ....

..... ..... .:

r = ( a + O)(z)

(2 - a)(.)

............

................

(; E)

=

( Pz R P-') R

............. ....

Pi')

'"..... :

..

....'" '.. ..

Fig. 5.

apartment of T with the horizontal axis. The vertical axis measures r . Given a pair ( z , r ) E d(T)x R, we wish to describe the lattice gz,r. Note that the plane has been divided into open convex polygons. The diagonal dotted where $ E 9.These measure lines are the graphs of the equations r = $(). where the root subgroup filtrations "change". The horizontal dotted lines are the graphs of the equations r = n where n E %. These measure where the toral subgroup filtrations "change". Each of the convex polygons is labeled by a lattice: if (5,r ) belongs to a convex polygon, then B,,~ is the lattice so

61

Lectures on Harmonic Analysis for Reductive p-adic Groups

identified. For purposes of assigning lattices to every point in the figure, the convex polygons are “closed at the top”. We note that gr,r = gx,r+ unless the point (x,T ) lies on a dotted line. Finally, note that if we fix T and move to the right, then the positive root space “expands” while the negative root space “shrinks”. Fortunately, even though life is not so nice for arbitrary T , there exists a wonderful result of Moy and PrasadC.Define

0 := { x

E

d(T) I x is the barycenter of a facet}.

An element of 0 is called an optimal pointd.

Lemma 3.4.3: (Moy and Prasad) Suppose z exist points x,y E 0 such that Bx,r C B z , r

E

d(T) and

T

E

R. There

C By,r

and there exist points x‘, y‘ E 0 such that Bzl,r+

C

&,r+

C By’,r+.

A similar pair of statements can be made for the Moy-Prasad filtration subgroups. For example, for GL3(k), the optimal points, up to the action of G (see f + x3, and 3F13+y2+F11. below), are the points

x,

9

Exercise 3.4.4: Check that, up to conjugacy, the points listed above are the optimal points for GL3(k). Describe their associated filtration lattices g x , r and gz,r+.

4. G-domains Recall that the ultimate goal of these notes is to relate certain invariant distributions on G with certain invariant distributions on g. Since the distributions in question are invariant, we will need sets in G and g which are invariant. Such a set can never be compact, so the most we can hope for is t o have sets which are invariant, open, and closed. A set with these properties is called a G-domain. As we show in this section, it is possible to use the Moy-Prasad filtrations to define G-domains which have very nice properties. In particular, we will ‘You will not find the result stated exactly as follows, but it is easy to derive this formulation. dBeware, 0 plays many roles in these notes. However, there is no possibility of confusion.

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define a family of G-domains which form a “neighborhood basis” of n/,the set of nilpotent elements. By using the Cartan decomposition of G (see [33]), in [18] (Lemma 2.4) Howe demonstrates that, for all integers i, the GL,(k)-orbit t of fr, is contained in & N.From, for example, [13] (Lemma 12.2), it is also true that for any compact set w c g, there exists a lattice C c g such that G~ c C N. We analyze these statements from the perspective of MoyPrasad filtration lattices. After doing this, we look at the situation on the group.

+

+

4.1. S o m e comments o n the Bruhat-Tits building of G Before we can continue, we must recall some facts about B, the reduced Bruhat-Tits building of G. To each maximal split torus T’ in G we can attach an apartment A(T’)exactly as we did above. The building of G can then be thought of as the “gluing” together of all these various apartments. In Figure 6 we present a picture of B for GL2(k). An apartment in B is

Fig. 6. A picture of the building of GLz(lc)

the image of any continuous injective map from R which maps integers t o vertices. Just as any two maximal split tori in G are conjugate, there is a natural action of G on B(G) with respect t o which any apartment can be carried into any other. Moreover, a version of the two body problem can be solved: given any two points in the building, there is an apartment which contains both of them. Combining these two facts we have: for any two points x,y in B , there exists a g E G such that gx,gy E d(T). (Here gx denotes the

Lectures on Harmonic Analysis f o r Reductive p-adic Groups

63

image of x under the action of g on B(G).) Finally, the action of G on B is semisimple; that is, for h E GI either there is a point in B which h fixes, or there is a line (i.e., a one-dimensional subspace of some apartment) on which h acts by nontrivial translation. Suppose x E t? and r E R.We define the Moy-Prasad filtration lattices gx,T and gx,r+ as follows. Choose g E G so that gz E A ( T ) .

gx,r := 9-l.&X,r . g and gx,T+ := g-'

' ggX,T+

. 9'

One can check that these definitions make sense. For r 2 0 we define the subgroups Gx,Tand G,,,+ in a similar fashion. Finally, we note that B is endowed with a nontrivial invariant metric, denoted dist. Moreover, with respect to dist, B has nonpositive sectional curvature. 4.2. G-domains f o r the Lie algebra

We begin with a generalization of the result of Howe discussed above; the proof of this result (from [l])nicely illustrates the benefit of working with the Moy-Prasad filtration lattices.

Lemma 4.2.1: Let x,y E

B, and let r

+

E

R. Then g X y Tc

+N.

+

Proof: Since gx,T c N if and only if &X,T c g g y , T N for g E GI we may assume that x and y are elements of A(T). Choose v' E X,(T)@R such that x = y+v'(working modulo X*(Z(G))@ R). Let B' be a Bore1 subgroup determined by 17.That is, B' has a Levi decomposition B' = TN' such that for all roots a E a$,, the set of roots which are positive with respect to N ' , we have ( a , q 2 0. Let B' = TN' denote the parabolic opposite B = TN' and let g = ii' t n' denote the associated Lie algebras. We have

+ + gx,r

= tr @

c

g$

t$€* I Il(z)>rl

In fact, a kind of converse to the above lemma is true. Namely, if X E g belongs to gx,r+ N for all x E B, then there is a point y E B such that

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64

X E gY,,. This result along with the above lemma gives us the following theorem.

Theorem 4.2.2:

In order t o simplify our notation] we define

From its definition] g, is open and invariant. From the above theorem we have that g, is closed. Consequently, g, is a G-domain. We now consider some examples. The set go is usually referred t o as the set of compact or integral elements of g. The set go+ is usually called the set of topologically nilpotent elements in g; it consists of those X E g such that powers of X tend to zero in the p a d i c topology. These G-domains have very natural interpretations; namely, g, = { X E g 1 v ( e ) 2

T

for all eigenvalues e of X }.

(If E is a finite extension of k, then there exists a unique extension of v to E.) For GL,(k) and X E g, the value of .(ex) must lie in the set { k / n I k E Z}. Of course, just as we have Moy-Prasad lattices of the form g X , , + , we also can define gr+ := UxEB g X , , + . These G-domains satisfy the obvious analogues of the results discussed above. From the previous paragraph, we have that g, # gr+ implies that r = k / n for some k E Z.

Exercise 4.2.3: Define the subspace D, of CT(g) by

where the sum is interpreted as follows. A function f belongs t o D, if and only if f can be written as a finite sum f = fi with fi E C c ( g / g x Z , , ) for some xi E 23. We can define D,+ in a similar way. Show that the Fourier transform gives us bijective maps from D,+ to CT(g-,) and from D, to

xi

CT(g(-r)+).

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4.3. G-domains in G . We now turn our attention to the group-side of things. Unlike the Lie algebra, an element of the group can only belong to a Moy-Prasad filtration subgroup if it first belongs to a parahoric. It turns out that the interesting part of the story here rests in proving the equality

u

n

G ~= , ~G , , ~.u

XED

XED

where U denotes the set of unipotent elements in G. Once this result is known to be true, it is very easy to establish the equality Gr =

n

Gx,r . u

Z€B

where T 2 0 and G, = UxEDGx,,.As above, it follows immediately that G, is a G-domain. We show how to prove the equality when r = 0.

Lemma 4.3.1: Go =

n

G, .U.

XED

Proof: We first show that the right-hand side is a subset of the left-hand side. We will argue by contradiction. Suppose that g E Gx,o ‘ U does not belong to Go. Since the action of G on B is semisimple, we either have that there is a point x in B which g fixes or a line e in an apartment A’ of B on which g acts by nontrivial translation. In the first case, we can write g = h . u with h E G,,o and u E U.Since u is unipotent, it must live in G,,o for some y E B. But, from a result of Eugene Kushnirsky this implies that u E Gx,o [lo] (Lemma 4.5.1). In the latter case, there exists a facet F’ in A’ such that F’ n e is open in .! For all z,y E F’ we have G,,o = G,,o. By hypothesis, there exist elements h E Gx,O and u E U such that g = uh. Since u is unipotent, there exists w E B which is fixed by u.We have that for all y E F’ n e

nzEB

dist(w, y) Thus, we have , for Figure 7.

= dist(w, uy) = dist(w, uhy) = dist(w, gy).

IC,

y E F’

n e a picture something like that described in

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S. DeBacker

Fig. 7

However, since I3 has nonpositive sectional curvature, the line segment from x to w must be shorter than the line segments from y and gy to w. Similarly, the segments from gy to w must be shorter than the segments from x and gx to w.Consequently, we have dist(z, w) < dist(gy, w)

< dist(x, w),

a contradiction. We now show that the left-hand side is a subset of the right-hand side. We need to show that for x, y E B,we have G, c U.G,. As in the proof for the Lie algebra, we may assume that x and y both belong to d ( T ) . Let B' = T N ' be a Bore1 subgroup of G so that the (spherical) chamber in d ( T ) determined by N' is invariant under translation by the vector (y -x). Let N' be the unipotent radical of the parabolic opposite B = T N ' . From [3] we can write G = N' . N ' . N' . T . With some work, it follows that we can write

Gx,O= N : . N:. N: .To where NL = N' n G,,o and NJ = N' n G,,o. Because of the way in which N' was chosen, we have NL c G,,o. Thus, if g E G,, then there exist n1, n2 E N,, E E F,, and t E TOsuch that

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4.4. A neighborhood basis f o r the nilpotent cone

We remark that T

and

u =n G T . T

It follows that the G-domains we have defined provide us with a neighborhood basis of the nilpotent cone (resp., unipotent variety) consisting of open, invariant, closed neighborhoods. 5. Nilpatent orbital integrals as distributions

A distribution on g is any element of the linear dual of CF(g). The aim of this section is to convince ourselves tha.t integrating against a nilpotent orbit defines a distribution on g. We follow the argument of Ranga-Rao [29]. 5.1. Orbital integrals Fix an element Y E g. Let Oy = GY := {gYg-' I g E G} denote the G-orbit of Y. We identify the orbit of Y with the homogeneous space G/CG(Y). Thus, the tangent space t o the orbit of Y a t the point Y is identified with

BICB(Y).

We define an alternating bilinear form ( , ) y on g by (A,B ) = tr(Y . [A,B)) for A , B E g. Fix an element B E g. A calculation shows that (A,B ) = 0 for all A E g if and only if B E C,(Y). Since we have a similar statement when we switch the roles of A and B , it follows that ( , ) y induces a nondegenerate alternating form on g/C,(Y). Thus, the dimension of the orbit is even, say 2m. Similarly, for each g Y = gYg-l E G Y ,we have a nondegenerate alternating symmetric form on Tan,y(Oy). Consequently, there exists a nondegenerate, invariant two-form w for Oy and from this we can form a nonzero, invariant volume form w A w A . . . Aw ( mtimes) on oy . Let X I , . . . , Xam be coordinates for g/C,(Y). For each 1 5 i 5 2m, fix a one-form d X i and associated measure IdXiI normalized so that for all f E Cp(g/C,(Y)), we

x:!,

have f(X) = f ( - X ) , where the Fourier transform is taken with respect to the measure IdXiI. There exists a locally convergent power series f ( X 1 ,X 2 , . . . , X z m ) so that (locally)

n:rl

w = f ( X l , X 2 , .. . ,X2,)dXl

A

dX2 A ... A dXzm.

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68

Define 2m

IwI := If(X1,x2,.. ., X2m)I

IdXZl. i=l

Thus, we have an invariant measure on O y . (Since GL,(k) is unimodular, it follows from standard measure theory results that CG(Y) is unimodular as well.) Exercise 5.1.1: Check that this definition is invariant of the various choices we have made.

5.2. A framing of the problem

If Y is a semisimple element of g, then O y is closed in g. Consequently, if f E C r ( g ) , then the restriction of f to O y is an element of Cr(Oy). So it makes sense to define the distribution O y : Cp(g) -+ C by setting O Y ( f ):=

/

f ( V )dg*.

G/CG(Y)

for f E C r (9).Here dg* denotes the invariant measure on G / C G ( Y defined ) above. What if Y is not semisimple? In particular, what if Y is nilpotent? In this case, we need to do some work to show that for arbitrary f E C r ( g ) the integral

makes sense. 5 . 3 . The basic notation associated t o nilpotent orbital integrals

We begin by recalling the familiar parameterization of nilpotent orbits in g. We then discuss various properties of nilpotent orbits which will be important in the sequel. From basic linear algebra we know that the nilpotent orbits in g can be parameterized by P ( n ) ,the set of ordered partitions of n. To p E P(n) we associate the nilpotent element X, E N n gro,O having Jordan canonical form corresponding t o p. That is, if p = ( p l , p 2 , . . . , p k ) , then in the ith

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69

block of size pi x pi we put the matrix with ones on the superdiagonal and zeroes elsewhere. For example, for p = (2,2,1) E P(5) we have

x,=

01000 00000 00010

!:c:j

.

Note that the nilpotent orbits in g z o , ~ / g s o , ~E++ Mn(f) are also indexed by P ( n ) and the map taking X, to the image of X, in M,(f) gives a bijective correspondence between 0(0), the set of nilpotent orbits in 8, and the set of nilpotent orbits in M, (f). We let 0, denote the G-orbit of X,. Note that if p < p' in the usual partial order on P ( n ) ,then 0, c 0,).

+

Exercise 5.3.1: Show that if 0 is a nilpotent orbit such that 0 n (X, # 0, then 0, is contained in the padic closure of 0. Moreover, G 0, n (X, + gzOtr+)= X, for all non-negative T .

gzo,o+)

W ~ Y +

Note that if X E N, and t E k x , then t X E N and in fact, t X and X are GL,(k) conjugate. (In general, it follows from Jacobson-Morosov that t 2 X and X are conjugate, but in GL, we can do better.) So, N and each nilpotent orbit are closed with respect to scaling. We can associate a parabolic subgroup P ( p ) of G to p as follows: For all positive integers j , the matrix X i acts on V = k", and we define V, = ker(X;) c V . Define P ( p ) = {g E G ( g . V, c V, for all positive j ). Then X, lies in the nilradical n(p) of the Lie algebra of P ( p ) , and the P(p)-orbit of X, is dense in n(p) and equal to 0, n n(p). We need most of the basic facts about sl2(k)-triples. For a good reference, see [4].We can complete X, to an slz(lc)-triple (Y,, H,,X,). Explicitly, the ith block of H, is given by the element diag((pi l),(pi - 3), . . . , (1 - pi)) and the ith block of Y, is given by certain entries on the super-subdiagonal. We let A, E X,(G) denote the associated one-parameter subgroup, that is, the ith block of A,(t) looks like diag(t(pi-I), t(pi-'), . . . , t ( ' - , z ) ) . For i E Z,we define g(2) = {X E g

xi

1 x,(t)X= tix}.

We have g = g ( i ) , For j E Z,define g ( 2 j ) := Ci2jg(i). We let p, denote the parabolic subalgebra g(> 0) with Levi subalgebra m, = g(0) and nilradical n, (= g ( 2 1)).We let P, denote the corresponding parabolic

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70

subgroup with Levi decomposition M,N,. The P,-orbit of X , is equal t o M g X , g ( 2 3) and hffiX, is an open and dense subset of g(2). Finally, it is true that CG(X,) c Pp.

+

5.4. A sketch of the proof that nilpotent orbital integrals

define distributions Fix p E P(n).Let f E CT(g). The Iwasawa decomposition (see [33]) tells us that we can write G = G,o,oP,. Fix a Haar measure dk on K and a left Haar measure dip on Pp so that

for all h E CT(G), the space of complex-valued, compactly supported, locally constant functions on G. Because it will be important, we recall that for all po = mono E Pp = M,N, we have de(p0pp;') = (det (rnolnP)l-l dep. Define f E CF(g) by f ( X ) := JGzo,o f ( ' X ) d k . Let us assume for the moment that

/

(* )

f("x)dep*

P/CC(X,)

converges. (Here dep* is the quotient measure.) In this case, we have

(*) =

/

/

f(kpx)

dkdep*

=

/

f(gx,)dg*,

G/Cc(XP)

P/CG(X@L)

and so we can define the invariant distribution 0, by r

f E CT(g). We now sketch the proof of why (*) makes sense. Let dZ be a Haar measure on g(> 2). We'd like to define a function FO on g(>2) so that

for

-1

For po = mono as above, we have d(poZp,') = Idet (rnoJg(z,)) dZ.Consequently, it is sufficient to produce a nonzero function FOwith the property FO(P0Z) = det (pole(')). The map X H (Y H t r ( Y . X ) ) induces an isomorphism of g(1) with g(-1)*. Just as at the beginning of this section, for Z E g(2), we can define

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an alternating bilinear form ( , )z on g(-l). When 2 = X,,the form is also nondegenerate. Consequently, both g( 1) and g ( -1) are even dimensional (of dimension 2n’). Let az be the matrix which represents ( , )z with respect to some fixed basis of g(-1). From [22](Theorem 6.4) there is a polynomial (Pfaffian) Pf of degree n’ on g ( 2 ) so that det(az) = (Pf(Z))2. Moreover, 2 since d e t ( q m z ) )= det (m-llO(-l)) .det(az), the polynomial Pf transforms in the manner we desire. For 2 E g ( 2 2 ) , set F o ( 2 ) = Pf(Z2) where 2 2 denotes the image of Z under the projection map from g ( > 2 ) t o g ( 2 ) .

Example 5.4.1: For the partition ( 3 , 2 ) E P ( 5 ) , we have that the dimension of g(-1) is four. The space g ( 2 ) is spanned by the Chevalley basis elements X,,,, X,,, and X,,,.If 2 = z X a l z yX,,, t X a q 5 ,then

+

+

FO(2)= zy. 5.5. An important calculation

For j

> 0 we calculate 0 , ( [ X ,

+ tj]).

Proposition 5.5.1:

+ 41) = 4(( 1-2j).dim(Og)/2)

0,([X,

Proof: Suppose that m > j is very large. Let 0: := KmX, = {‘X, Km}. It follows from Exercise 5.3.1 that

Ik E

+

c?p([Xp e j ] ) = [Kj : Km ’ CK, ( X p ) ] ’ 0 p ( [ o r ] ) .

Now, by definition, we have 0, ([O,”]) = m e w g *(0,”) = meaSldX1I. IdXz I ...IdXz, I ((ern = [el-, -

+ C,(X,)

[Ce,l-,,

: t,

+ c,(X,) ) / C ,( X J )

+ C0(X,)]-1/2

(X,): C L (X,>1’/2

[t(l-,)

: t,]l/2

On the other hand,

After putting the two pieces together and doing a bit of calculation, we arrive a t

The proposition follows immediately.

0

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5.6. Nilpotent orbital integrals are homogeneous

Suppose t E I c x . For f E Cr(g) we define the dilation t via f , ( X ) = f (tx). For p E P ( n ) we have

ft

E C r ( g ) o f f by

Consequently, since nilpotent orbits are closed with respect to scaling and the invariant measure on a nilpotent orbit is unique up to a constant, O,(ft) and O,(f) must differ by a constant. A calculation shows that

For example, from Proposition 5.5.1 we have

+

oP([m(-2j+l)xpe l + ] )

= 1.

6. The Fourier transforms of invariant distributions 6.1. Basics

A distribution on g is any element of the linear dual of CF(g) (no topological restrictions). We denote the subspace of invariant distributions on g by J ( 8 ) " . Suppose T E J ( g ) . We define the Fourier transform T E J ( g ) of T by

F ( f ) := T(j) for f E CF(g). It is a remarkable fact that 5? is represented by a locally integrable function; that is, there is a function T E L:,,(g) such that for all f E Cy(g),

T ( f )=

I

T ( X ) . f ( X )d X .

Unfortunately, describing this function is beyond our abilities in all but the simplest situations. Example 6.1.1: Consider the trivial nilpotent orbital integral U ( l , ~ , . . . ,El ) J ( g ) . For f E Cr(g) we have

eAccording to Howe, he chose the letter J because and J follows the letter I in the alphabet.

I (for invariant) was already taken,

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and we also have

l)(f)=

6 ( 1 , 1 ,...,

We therefore conclude that

i

f ( X ).6(1, 1,...,l ) ( X ) d X .

6(1,1,...,1) = 1

We let g""."' denote the set of regular semisimple elements in g; this is a dense open subset of g. In our situation, gr's's'consists of those elements of g having distinct eigenvalues. The main idea of this section is t o present a very elegant description of ? ( X ) when T E J ( g ) and X E g""'"~. (From this description, it will follow that T is represented by a locally constant function on gr.".'..)The existence (of a form) of this expression was conjectured by Paul Sally, Jr. and proved by Reid Huntsinger [21]. The proof follows an argument of Harish-Chandra [13]. Originally, Huntsinger and Sally were only interested in studying the behavior of the Fourier transform of a nilpotent orbital integral. We shall temporarily restrict our attention to this situation. Suppose p E P(n).For f E C r ( g ) we have = OJf)

f ( X ). 6 , ( X ) dX = =

J'

f(gx,)dg*

G/CG(X+)

J'

= G/CG(XF)

/

f ( X ) . A(tr(gX, . X ) )dX dg*

g

So, if we could justify the equality in quotation marks, we'd have

6JX)

=

1

G/CG

(x,

A(tr(gX, . X ) ) dg'

= O,(Y w A(tr(Y. X ) ) ) .

This is nearly correct; we now describe what is true.

Theorem 6.1.2: (Reid Huntsinger) Let K be a n y compact open subgroup of G and let dk denote the normalized Haar measure on K . For all X E gr'S'S' we have

6 , ( X ) = O,(Y

H

A(tr(Y. ' X ) ) d k ) .

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S. DeBacker

Remark 6.1.3: For reasons which will become clear later, we remark that the function 6, is canonical. On the face of things, it depends on two choices: the additive character A and the choice of a measure on 0,. However, our choice of the measure on 0, was not arbitrary; it depended on A. As one can verify, this dependence makes 6, canonical. Remark 6.1.4: In our situation, since every nilpotent orbit is Richardson, there is a very nice description, due to Howe, of the function 6,. Namely, 6, can be related to the character of the representation obtained by inducing the trivial representation on P ( p ) up to G.

6.2. A m o r e general statement More generally, following a suggestion of Bob Kottwitz, Reid Huntsinger proved the following statement.

Theorem 6.2.1: (Reid Huntsinger) Fix r E R . If T E J(gr), t h e n T i s represented o n gr.s'"' by

Here, f o r Y E g, q x ( Y ):= J,(A(tr(Y."))) dlc. (As before, K is a compact open subgroup, and dlc is the normalized Haar measure o n K . )

At the heart of the proof of this theorem lies the statement that the map := qx .[gr] is locally constant. Since from g""'"'to Cm(g) sending X to qx,,. the verification of this statement requires some fairly detailed analysis, we shall skip the proof. This statement immediately implies that the function X H T ( Q X ,=~T) ( q x )from gr'S'S'to C is locally constant. ) . need to show Suppose f E C ~ ( g r . " ' " .We

From the previous paragraph, there exists a finite collection { w i } E 1 of compact open disjoint subsets of gr.".'. such that both X t+ T ( q x ) and X ++ f ( X ) are constant on wi and the support of f is contained in u w i .

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Using Fubini’s theorem and the invariance of TI we then havef

m

= T(Y

H

/ / /f(X)

0

= T(Y

H

A(tr(kX.Y)) d k dX)

K

p Y )d k )

= T(f).

7. Characters

In this section we define the character of an admissible representation and discuss some properties of characters. 7.1. Admissible representations

Fix an admissible representation (7rl V ) of G. Recall, from Gordan Savin’s lectures [33], that (7rl V ) is a representation for which (1) for all 21 E V there exists a compact open subgroup K such that 21

E

vK:= (21 E v I n(k)w = 21 for all IC E K )

and (2) for all compact open subgroups K of G we have dime V K < co. The second condition is equivalent to saying that for all compact open subgroups K of G and all irreducible representations 0 of K , the multiplicity of u in 7r is finite.

Example 7.1.1: Suppose a is a cuspidal representation of GL,(f) % Ko/K1. Inflate a to a representation of KO and extend this inflation to a representation (T of Z(G) . KOwhere Z ( G ) denotes the center of G. The ‘This part of the proof supports David Vogan’s adage: “The p-adics: if you can add, you can integrate.”

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representation (xu,Vu) obtained by (compact) induction of r~ from Z(G).Ko up to G is an (irreducible) admissible representation.

Example 7.1.2: Suppose m > 0 and is a character of T/T,. Inflate jj to a character x of T . Since the unipotent radical N of B is normal in B , we may extend x to a character of B . The representation (7rx,Vx) obtained by (compact) parabolic induction of x from B to G is an admissible representation of G. 7.2. T h e character distribution We define the character distribution, O,, associated to the representation ( 7 r , V ) as follows. For f E CF(G), we define ~ ( fE)End(V) by 7r(f)v=

s,

f ( g ) . 4 9 ) vd9

for w E V. Since f is compactly supported and locally constant and ( T , V) is admissible, this integral is really just a finite sum. In fact, if K is a compact open subgroup of G such that f ( k l g k 2 ) = f ( g ) for all k l , k2 E K and g E G, then we have

4f).

= 7r([KI*f

* [Klk = 7 W 1 ) 7 r ( f ) 7 r ( [ K I ) V .

(Here * denotes the usual convolution operation.) Thus, ~ ( fis) a map from V to VK. If eK := measd,(K)-l..rr([K]), then e K is the projection operator from V to V K ,and we have

v = VK @ (1

-

eK)V.

Consequently, since dima:(VK) < m, it follows that ~ ( fis) a finite-rank operator. We can therefore define the character distribution 0,: C r ( G ) 4 C

by sending f to tr(7r(f)). Just as the Fourier transform of an invariant distribution on g is represented on g'.",". by a function in Coo(g'.s.".), so too is the character distribution [15]. We abuse notation and denote this function by 0, E Coo(Gr.S,S.). (Gr.".".denotes the open, dense subset of G consisting of regular semisimple elements, that is, those elements of G whose eigenvalues are distinct.) We call this function the character of 7 r .

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Remark 7.2.1: Fix y E G‘.”.”. and m > 0 large enough so that yKm G‘,”.”..Define fm = rneasdg(Km)-’ . [yK,]. We then have

C

Since the function 0, is locally constant on Gr.‘.’., the right-hand side becomes Q,(y) for all m sufficiently large. Thus, “in the limit” the function 0, looks like a character.

7.3. A calculation Fix j 2 1. Suppose that 7 is an irreducible representation of Kj / K2j. Since Kj / K2j is abelian, 7 is a character. Let denote the corresponding character of Kj; we regard r as an element of CF(G) in the obvious way. Let resKj n denote the restriction of T to K j . Suppose that resKJ n = @,Ezm(a,~ ) a where m(o,T ) denotes the multiplicity of o in resK,

T.

We have

=m ( ~ - lT , ) . measdg(Kj).

In other words, the character picks out the multiplicity of (up to a constant).

7-l

in resK,

T

7.4. Depth

The depth of a representation was introduced in the fundamental papers of Allen Moy and Gopal Prasad [28,27]. Essentially, the depth of a representation is a rational number which measures the first occurrence of fixed

S. DeBacker

78

vectors with respect to all filtration subgroups of G that arise naturally from Bruhat-Tits theory. Recall from $3.4.2 that 0 denotes the set of optimal points. Up to conjugation, the set of optimal points is finite, and so the subset { r E R I g x , r # gx,r+ for some IC E 0 ) is discrete (and in fact, is a subset of Q). For an admissible representation ( T , V ) ,we define p ( ~ )the , depth of ( T , V ) , by

1

p ( ~ := ) min{r E Q>o -

there is an z E 0 such that VGz3r+ # 0).

Proposition 7.4.1: p ( ~ is) the unique rational number satisfying the following statement. If (2, r ) E d(T) x R>o - with V G z 3 ~ #+{0}, then r 2 p ( ~ ) . Proof: Suppose ( x , r ) E d(T) x lR>o - with VGz3r+ # (0). From the group version of Lemma 3.4.3 there exists y E 0 such that GX,,+ 3 GY,r+.Consequently, {O> # c V~Y,V+. 0 v

~

~

~

T

+

All the various things you would want to be true about the depth of a representation are true. For example, if o is an irreducible representation of a parabolic subgroup of G, and T is an irreducible subquotient of the induced representation, then p ( n ) = p ( o ) .

Example 7.4.2: Any representation with Iwahori (that is, Bo) fixed vectors has depth zero. Example 7.4.3:The representation ( T ~V,g )defined in Example 7.1.1 does not have Iwahori fixed vectors, but it does have depth zero. Example 7.4.4: The representation depth (m - 1).

(T,,

V,) defined in Example 7.1.2 has

7 . 5 . Elementary Kirillov theory Fix an irreducible representation ( T , V ) . For nilpotent real groups, Kirillov [24] established that the representations were parameterized by coadjoint orbits in the linear dual of the Lie algebra. In our context, the term Kirillov theory is used to describe the connection between representations of compact open subgroups occurring in T and coadjoint orbits in the linear dual of g. Via our nondegenerate trace form, we have identified g with its dual. The type of result discussed below was first studied, I believe, by Howe [17]. Fix r , s E R>o - and z,y E B.

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The map g H (g - 1) induces an isomorphism of the abelian groups G,,,+ /G,,(zr)+ and B , , ~ + /g,,(2r)+. Moreover, every character of gz,r+/gr,(2r)+ is of the form Y H R(tr(X . Y ) for some E gz,-2r/gz,-r. (That is, the Pontrjagin dual of G,,,+/G,,(zr)+ gz,T+/gI,(2T)+is isomorphic t o Bz,-2rlBz,-r.) Fix a character d of G,,,+/G,,J~~)+and a character 7 of Gy,s+/Gy,(2s)+ Let Xu gI,+ E ~ ~ , - 2 ~ / grepresent ~ , - ~ 5 and X, E represent 7.

+

+

Proposition 7.5.1: If u and r both occur in T , then there exists a g E G such that

"XT

+ gy,--s) n (Xu+ g z , - r )

# 8.

Proof: Let V, c V (resp. V, c V ) denote a one-dimensional subspace of V on which resGz,r+T (resp. resGV,*+ T ) acts by u (resp. 7 ) . Since ( T , V ) is irreducible, there exists a g E G such that the image of T(g)Vuunder the projection of V onto V, is nonzero. This implies that for all h E G,,,+ n gGy,,+, we have

a ( h )= T ( g - ' h g ) . Thus, for all H E B,,~+ n ggy,s+, we have R(tr(X, . H ) ) = A(tr(gX, . H ) ) . This implies that for all H E B,,~+ n ggy,s+,we have tr((X, - gX,) . H ) E P. Consequently, (Xu - gX7) E (g,,.+ n g g y + + ) * = g2,-r ggy,--s. The proposition follows. 0

+

7.6. Understanding the distribution r e s c r ( G p ( . r r ) +0 ),

) be an irreducible admissible representation. From our discussion of depth, we know that we can find an x E A ( T ) such that VG=,p(n)+ # (0). Thus, the trivial representation of Gz,p(n)+ occurs in T ; the associated coset in g is gz,-p(a). Now fix s > p ( ~ ) y, E B, and ? E Gy,s/Gy,s+such that the character T of Gv,s occurs in r. Let X , E gy,(--s)/gy,(--s)+ be the coset corresponding to ?. From our discussion of Kirillov theory, there exists a g E G such that 9 g I , - p ( n ) n ( X , + B~,(-~)+) # 0. However, from Lemma 4.2.1, we have Let

( T ,V

-

+

9

Bz,-p(a)

c By,-p(n)

+N c

By,(--s)+

+N .

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80

Consequently, we can assume that X , is nilpotent! We now use this fact to say something about the distribution we define k E rescr(Gp(n)+)0,. For a function h E Cp(gp(s)+), CT(Gp(,)+)by h ( g ) = h(g - 1). We define the distribution O,g on g by O,,,(f) = O,(.fpc,)) where fp(,) = f . [gp(,)+]. We let Q,denote the Fourier transform of @ .,, From Exercise 4.2.3, we know that the Fourier transform of an elebelongs to Cr(gp(,)+). Suppose z E B and s > P(T). If ment of Lp(,)

f E C ( g r , - s / g z , - p ( x then ) ) , we have & ( f ) = Q,(f). It follows from our discussion above and a few lines of calculation that

&(f) # 0 implies

supp(f) n (g2,(-s)+

+N)# 0.

This last statement is equivalent to the statement

6,(f)# o implies

supp(f) n g(-’)+

# 0.

(3)

7.7. The Harish- Chandra-Howe local character expansion Considerations similar to those above led Roger Howe to make his finiteness conjectures [19] (which we will discuss later) and establish the following remarkable connection between the character of an irreducible representation of G and the Fourier transforms of nilpotent orbital integrals.

Theorem 7.7.1: (Harish-Chandra-Howe local character expansion) If T is an irreducible admissible representation of G , then there exist constants c P ( r ) indexed by p E P(n) such that O,(1

+X ) =

c

CP(7r).

B,(X)

(4)

PEP(,)

for all X E gT.‘.’. suficiently near zero. Remark 7.7.2: Howe [16] proved this result without any restrictions on the characteristic of k. Later, under the assumption that the characteristic of k is zero, Harish-Chandra [13] generalized the proof t o all connected reductive groups. The above theorem and its analogues (local expansions about any semisimple point) play a crucial role in Harish-Chandra’s proof that characters are locally integrable on G just on Gr.’.’.). (The question of integrability is still open for arbitrary groups in positive characteristic. From work of Rodier [30] and Lemaire [25], it is known t o be true for GL,(k) when k has positive characteristic.)

(a

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81

One of the difficulties with this theorem is that it provides no indication of where equation (4) ought to hold. The conjecture of Hales, Moy, and Prasad [28] is a precise statement about this issue:

Statement 7.7.3: Equation (4) ought to be valid on gif;y+. For integral-depth representations this was proved by Waldspurger [35]. With some restrictions on k, it was proved for arbitrary depth representations in [9]. 8. An introduction to Howe’s conjectures and homogeneity In his paper, Two conjectures about reductive p-adic groups, Howe proposed two remarkable finiteness conjectures that now bear his name. Howe’s conjecture for the Lie algebra looks like: dim@rescc:(g/L)J ( w ) < co. Here w c g is any compactly generatedg, invariant, and closed subset of g, J ( w ) denotes the space of invariant distributions supported on w , and L is any lattice in g. For T E J ( w ) , (g/L) T denotes the restriction of T to Cc(g/L). Howe’s conjecture for the group looks very similar: dim@resCc(G/K)J ( w ) < co. Here K is a compact open subgroup of GI and w is a closed, compactly generated, invariant subset of G, and J ( w ) denotes the space of invariant distributions which are supported on w . Of course, although we still refer to these statements as Howe’s conjectures, both are known to be true. Howe’s conjecture for the Lie algebra was proved by Howe [16] in the 1970s for GL, and later by HarishChandra 1131 for general groups (but only in the characteristic zero setting). Waldspurger [37] has also given a proof. Howe’s conjecture for the group was proved in the 1980s by Clozel [6,7] in the characteristic zero setting. In the 199Os, a characteristic free proof of Howe’s conjecture for the group was given by Barbasch and Moy in their beautiful paper [2]. gThat is, w can be realized as the closure of G C for some compact subset C of g.

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82

8.1. Homogeneity Beginning with Waldspurger’s paper [36], much energy has been spent trying to produce optimal versions of Howe’s conjectures. By optimal, we mean that we’d like to choose w and L in such a way so that we can not only describe the dimension of resCc:(g/L)J ( w ) , but we can also find a good basis for this space in terms of distributions with which we are very familiar, namely, nilpotent orbital integrals. These optimal versions of Howe’s conjectures are referred to as homogeneity results. This is an appropriate name, because, according to Webster’s Ninth New Collegiate Dictionary, the word homogeneity means “the state of having identical distribution functions or values”. We restrict our attention to the Lie algebra. Fix T E R.The role of w in Howe’s conjecture will be played by the G-domain gr+ (which is compactly generated). Guided by Harish-Chandra’s philosophy and Theorem 7.7.1, it appears that we want something like the following: For all T E J ( g r + )and for all f E C?(g(+.))

PEP(,)

This last displayed equation is equivalent to requiring

c

T(f”)=

CPL(T).

WP).

PcP(n)

From Exercise 4.2.3 we have f^ E Dr+.So, the role of C?(g/L) will be played by Dr+. Putting it all together, we arrive at the homogeneity statement resoV+J ( g T + ) = r w r + J ( N ) .

+

The fact that gr+ c gz,r+ N for all x E B gives some feeling as to why this homogeneity statement ought to be true. 8 . 2 . From Howe’s conjectures to the Harish- Chandra-Howe local character expansion The homogeneity result discussed in the previous section is actually not strong enough to prove Theorem 7.7.1. As in [16,13,9,35]we need something a bit stronger, but, unfortunately, more complicated. For x E B and s 5 r , define Jz,s,r+

I

:= {T E J(9) for

f

E C(Bs,s/Bs,r+),

if supp(f) n (g,+) = 8, then T ( f )= 0).

Lectures on Harmonic Analysis f o r Reductive p-adic Groups

83

I

c JX+,,+.

Note that J(gr+)

Remark 8.2.1: Although this definition seems a bit unnatural, it arises naturally from our understanding of resC?(G,(,,+) 0,. In fact, from Equa-

tion (3), we have that

6,

E

JX,,,,+if -r > p(.).

The following is the stronger homogeneity statement that needs to be proved in order t o recover Statement 7.7.3:

-

resDT+J,+ = resDT+J ( N ) where

9. Proving homogeneity results

We recall that for

T

E IR we

have

and XEB

We want to show r e s ~ J~( g+r + )

=res~?+ J(N).

(5)

We decided that knowing this, or, in truth, a stronger (but more complicated) statement would tell us that the Harish-Chandra-Howe local character expansion was valid on g;T$+. As we discussed before, g r = gr+ unless T = for some k E Z. Consequently, we only need to verify Equation (5) for r E { Moreover, by taking advantage of the fact that nilpotent orbital integrals are homogeneous, we can further restrict our attention to T of the form with 0 5 k < n. Indeed, pick m E Z such that m r E [0,1).For T E J ( g ) and t E k x , define Tt by Tt(f) = T ( f t ) .Note that

i}.

+

D,+ if and only if

f

E

E

J(g,+) if and only if Tm ,

frn-,,,

E

D(,+,)+

and

T

J(g(,+,)+).

S. DeBacker

84

Consequently, if we know

then for T E J

and f E Dr+ we have

and so we may assume T E [0,1). For the remainder of this section we discuss how to prove Equation (5). We first prove it for GLl(k), we next prove it for GLz(k), and finally we discuss a way to go about giving a general proof. 9.1. A proof f o r GLl(lc)

From the reasoning above, we need to show resCc(k/63)J ( @ )= resC,(k/63) J ( N ) . The right-hand side is one-dimensional and spanned by the distribution f H f(0) for f E C c ( k / 6 3 ) . Moreover, the right-hand side is a vector subspace of the left-hand side. Since GLl(k) is abelian, all distributions are invariant. Note that J(63) consists of linear maps from C T ( k ) to C which are supported on 63. If f E Cc(k/63),then we can write

f=

c

CX.[X+63]

X€k/63 where the c~ are complex numbers which are almost always zero. For T E J ( P ) , we have

Vf)= T(c,s.[@I) = WJI). f(0). Consequently, the left-hand side is also one-dimensional and so the equality is established.

9.2. A proof f o r GLz(lc) This is where things begin to become interesting. Thanks to the remarks a t the beginning of this section, we only need t o verify two statements: reso,,+

J(gO+)

= resDo+ J ( N )

(6)

85

Lectures o n Harmonic Analysis f o r Reductive p-adic Groups

and

We begin by considering the first statement, Equation (6). We note that Do+ may be thought of as an invariant version of Cc(g/f?l), but, for a good way to think about the space Do+. our purposes, this is

9.2.1. Descent and recovery Fix T E J(go+). We wish to show that resDo+T is completely determined by resc(eo/ei)+C(bo/b i / z T . Fix f E Do+.We will demonstrate that T ( f )is completely determined by resC(~o/el)+C(bo/bl/2) T . We write f = f i with f i E CC(g/gzi,o+) for some xi E B. Since T is linear, without loss of generality we may assume for some z E B. We can write that f E Cc(g/gr,o+)

Xi

f=

c

c x . [X+85,0+1

X€0/B,,o+

with the cx complex constants which are almost always zero. Again, since T is linear, without loss of generality we may assume that f = [ X gz,O+]. Now, T ( f )= 0 if the support o f f does not intersect go+. Consequently, since go+ c g2,0+ N , we must have

+

+

(X+Br,O+)nN#@. loss of generality, X E N .

So, without Up t o conjugacy, we have two choices for gz,o+; it is either f?, or b1/2. In what follows, the reader is encouraged t o consult Figure 8 to get a more geometric understanding of what is happening. We first deal with the e l case. Since xo is the only point x where gz,O+= e l , we may suppose that X E N n (gzn,--m gz,,(-m)+) for some m > 0. In other words, X E N n (Lm\ e l p m ) . Since we are free t o conjugate by G,,,o = K O ,we may assume that

with u E R X .For any point y in the chamber CO(that is, a point between xo and x' in Figure (8)),we have that X is "closer" to the origin with respect t o the y filtration than it is in the z filtration. For exampIe, X E g y o , ( 1 p m ) + where yo is the barycenter of CO.The problem is: a t the point y we require local constancy not with respect t o gzo,o+ but with respect to gy,o+ = b1/2.

86

S. DeBacker

Fig. 8.

87

Lectures on Harmonic Analysis for Reductive p-adic Groups

In order to recover the proper type of local constancy, we take advantage of the invariance of T . We write

1

=-

+

. T ( [ X hip]).

4 We have succeeded in writing T ( f )in terms of T evaluated a t a function f’ E Do+ which is supported closer to the origin with respect to some other point in the building. We now examine the b1/2 case. In this case, we are looking a t the coset X + gy,O+ where X E and y is any point in CO.Since we are free t o conjugate by Gy,o= Bo, we may assume that X is either

or

with m

> 0 and u E R X. In the former case, we can write T ( [ X+ By,o+I)

=

c

+ (:

:)

+ BZl,O+l)

@€P/P2 where

21

= d i a g ( w - ’ i l ) ~ ~= Z’is the other vertex of

GO.We have

+

From Figure 8 it is clear that we have expressed T evaluated a t [ X gy,O+] in terms of T evaluated at f ‘ where f ’ E Do+ has support closer t o the origin with respect to the z1 filtration than [ X gy,O+]had with respect to the y filtration.

+

Exercise 9.2.1: Do the analogous analysis for the latter case.

To summarize, the point of descent and recovery is as follows. We begin with a simple function f E C((gZ+ \ gZ,s+)/gZ,o+)for some z E B. From this function, we find a point y E B and a function f’ E C(gy,s+/gy,O+)so that T ( f )= T ( f ’ ) .After a finite number of steps, we will have shown that T (f ) is completely determined by reSC(eo/&1)+C(bo/bl,2) T.

S. DeBacker

88

9.2.2. Counting

We now know the following facts: (1) Thanks t o Harish-Chandra [13], the dimension of the complex vector space reso,+ J ( N ) is equal t o the cardinality of P ( n ) , which, in this case, is 2. (2) From $4.4, we have J ( N ) c J(go+) and so resoo+ J ( N ) c resoo+ J(flo+ 1. (3) From the previous section, we know that dim@reso,,+J ( 8 0 + ) = dim@reSC(eo/el)+C(bo/bl/2)J(80+ ).

Consequently, we need only show that

Since

for any x E ??B, we have that for T E J T is completely determined by

Since

we are done.

Exercise 9.2.2: Using the above proof as a template, prove Equation (7). 9.3. The general approach

In general, the proof is very much like that produced above, only more complicated. 9.3.1. Descent and recovery

Suppose T E J(gr+).The main point is to show that resDr+ T

=0

if and only if r e s p T r+

=0

where XEB

That is, we can find a very small space of functions from which we can choose a dual basis for reso?+ J ( g r + ) .

Lectures on Harmonic Analysis for Reductive p-adic Groups

89

Suppose f E D,+ . As before, without loss of generality we may assume that f = [ X + g x , r + ] for some X E N and some z E B. Under some hypotheses, we can use slz(f)-theory to find a direction in the building in which to move so that, for some y E B in that direction, the support o f f is nearer the origin with respect to y than it was with respect to z. Moreover, if necessary, we can use the basics of slz(f)-representationtheory and the invariance of T to ‘Lbeef-up”the coset gX,,+ so that we arrive a t a function f‘ E C,(g/gY,,+) with the properties: (1) T ( f ’ )= T(f) and (2) the support of f ’ is “closer” to the origin with respect t o y than the support of f with respect to 2.

Remark 9.3.1: Invoking sLz(f)-theoryrequires some restrictions. The theory only works well if the highest weights for the representations one wishes to consider are less than ( p - 3) where p is the characteristic of f.

9.3.2. Counting

This is the real key. In the end, you need some type of correspondence between “suitable” elements of D,‘+ and O(O),the set of nilpotent orbits in 0. When r is zero, the situation is easy to understand. As discussed before, the conjugacy classes of facets in B are in one-to-one correspondence with the elements of P ( n ) .Similarly, the nilpotent orbits are in one-to-one correspondence with the elements of P(n).For p E P ( n ) ,let Fp be the facet described in $3.4.1.The image of x,, in &7,,0/gF,,0+ is distinguished nilpotent (that is, it does not lie in a proper Levi subalgebra of g~,,o/g~,,~+). We let [X,, gF,,o+] E D:+ denote the characteristic function of the coset X , gF,,o+. This is the correspondence between “suitable” elements of D:+ and O(0) alluded to above. Indeed, for T E J ( g O + ) the , restriction of T t o D:+ is zero if and only if

+

+

is zero for all 1-1 E P ( n ) . When r # 0, life is much more complicated; but something beautiful is true. We refer the reader to [lo].

S. DeBacker

90

10. A few comments on the cfi(7r)s Let ( T , V ) denote an irreducible representation of G. We recall that for all regular semisimple X in some neighborhood of zero we have the HarishChandra-Howe local character expansion:

0,(1+ X ) =

c

CJT)

.6,(X).

PEP(,)

In his paper [16], Howe proves that for irreducible supercuspidal representations (that is, those representations that do not occur as subrepresentations of parabolically induced representations) the coefficients occurring in the Harish-Chandra-Howe local character expansion are all integers. In fact, given a bit of additional information, the same proof shows that this is true for all smooth irreducible representations. We first recall what we know about the C ~ ( T ) S ,and we then prove this result of Howe.

10.1. The coefficient c(1,1,...,1)(7r) Suppose that ( T ,V ) is a discrete series representation (that is, the matrix coefficients of ( T , V )are square-integrable mod the center of G). In this case, by using Rogawski [32] one may extend a result of Harish-Chandra [13] to show that

Here deg(T) denotes the formal degree of n,St is the Steinberg representation (see, for example, [5]), and C denotes the semisimple rank of G. On the other hand, if ( T , V) is a tempered representation (that is, it occurs in the Plancherel formula) which is not in the discrete series, then, using results of Kazhdan [23], Huntsinger [20] showed that c(1,1,...,1)(~)= 0 (see also the paper of Schneider and Stuhler 1341). 10.2. The leading coefficient According to Mceglin and Waldspurger [26], the set {P E

v.1

I C P ( 4 # 01

has a unique maximal element; call it ps. Moreover, according to Mceglin and Waldspurger [26] and Rodier [31], the coefficient C ~ , ( T ) is an integer which is equal to the dimension of the degenerate Whittaker model corresponding to ps.

Lectures on

Harmonic Analysis f o r Reductive p-adic

91

Groups

Example 10.2.1: Suppose p = ( n ) E P(n). Recall that X(n) is regular nilpotent. Let B be the Bore1 subgroup consisting of lower triangular matrices in G. Let N denote the unipotent radical of B ;that is, the subgroup of B formed by the set of lower triangular matrices with ones on the diagonal. Define the character 6 of by f i H h(tr(X(,) . (fi - 1)).For an irreducible smooth representation ( T , V) of G, let V ( 6 )denote the subspace of V generated by the set

m

{ ( ~ ( f i-) 6(fi))w I w E V and

fi E

m}.

Define V(n)= V/V(6).Then according t o the results quoted above, we have qn)(T ) = dim@(V(..))= dim@HomG (V, I n d z 8). 10.3.

The remaining coeficients

Not much is known about the remaining coefficients. Following an argument of Roger Howe 1161, we show that they are all integers. We begin by making Remark 6.1.4 precise. If p' E P ( n ) ,then opt

(x)=

@ I ~ ~ G1

WW')

(1

+ x)

for all X E go+. Here Ind$(,,) 1 is the representation of G obtained by inducing the trivial representation of P(p') up t o G. Suppose j > 0, p E P ( n ) ,and T~ is the character of Kj represented by the coset m(1-2j)XP t(l-j) E t l - 2 j / t l - j . From 57.3 we have

+

@,(T~)

= measdg(Kj) . m ( ~ ; l , r )

(8)

and

O I nPb') d ~ l ( ~ P ) = measd,(Kj) . rn(T;',Ind&,)

1).

(9)

From the discussion above, Exercise 5.3.1, and Equation (2) we have

dpt(7,)

= opt(pP)= measdx(t,)

=

.

([-m(1-2j) .

measdx(tj) . m(7;1,1nd&,) measdX(tj)

1)

{o

x P + t (1-j)l)

if p < p', if p = p', and otherwise.

Thus, if j is sufficiently large, then c ~ ( T.m(T;',Ind&) )

m ( ~ i ' , r=) PlP'

1).

92

S. DeBacker

We now proceed by induction. From Rodier [31] we have t h a t c ( ~is) always a n integer. Suppose p E P ( n ) .By induction, if p' > p, then cp(w) E Z.We have c , , / ( w ) . m ( ~ ~ l , I n d ~ (1) , , E)

c p ( w ) . m ( ~ i l , I n d & ) 1) = m(~;',w)-

Z.

PL'>P

Since m ( ~ ; ' Ind&) , 1) = 0,([-a(1-2j) .X ,

+ t2,1-j)])

= 1, we are done.

Acknowledgements The author was partially supported by National Science Foundation Grant

No. 0200542. This paper is based on lectures delivered at the Institute for Mathematical Sciences at the National University of Singapore in 2002. The author thanks the Institute for its support.

References 1. J. Adler and S. DeBacker, Some applications of Bruhat-Tits theory t o harmonic analysis o n the Lie algebra of a reductive p-adic group, Mich. Math. J., 50 (2002), no. 2, pp. 263-286. 2. D. Barbasch and A. Moy, A new proof of the Howe conjecture, J. Amer. Math. SOC.13 (ZOOO), no. 3, pp. 639-650. 3. A. Bore1 and J. Tits, Groupes rdductifs, Inst. Hautes Etudes Sci. Publ. Math., NO. 27, 1965, pp. 55-150. 4. R. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, Reprint of the 1985 original, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. 5. W. Casselman, Introduction t o the theory of admissible representations of p-adic groups, to appear. 6 . L. Clozel, Orbital integrals o n p-adic groups: a proof of the Howe conjecture, Ann. of Math. (2) 129 (1989), no. 2, pp. 237-251. 7. -, Sur une conjecture de Howe. I, Compositio Math. 56 (1985), no. 1, pp. 87-110. 8. S. DeBacker and P. J. Sally, Jr., Germs, characters, and the Fourier transforms of nilpotent orbits, The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998), Proc. Sympos. Pure Math., 68,Amer. Math. SOC.,Providence, RI, 2000, pp. 191-221. 9. S. DeBacker, Homogenefty results for invariant distributions of a reductive p-adic group, Ann. Sci. Ecole Norm. Sup., 35 (2002), no. 3, pp. 391-422. 10. -, Parametrizing nilpotent orbits via Bruhat-Tits theory, Ann. of Math., 156 (2002), no. 1, pp. 295-332. 11. ~, Some applications of Bruhat-Tits theory t o harmonic analysis o n a reductive p-adic group, Mich. Math. J., 50 (2002), no. 2, pp. 241-261.

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12. R. Kottwitz, Harmonic analysis o n semisimple p-adic Lie algebras, Proceedings of the International Congress of Mathematicians, Vol. I1 (Berlin, 1998). Doc. Math. 1998, Extra Vol. 11, pp. 553-562. 13. Harish-Chandra, Admissible invariant distributions o n reductive p-adic groups, Preface and notes by Stephen DeBacker and Paul J. Sally, Jr., University Lecture Series, 16,American Mathematical Society, Providence, RI, 1999. 14. -, T h e characters of reductive p-adic groups, Contributions to algebra (collection of papers dedicated to Ellis Kolchin), Academic Press, New York, 1977, pp. 175-182. 15. -, A submersion principle and its applications, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), no. 2, pp. 95-102. 16. R. Howe, T h e Fourier transform and germs of characters (case of GL over a p-adic field), Math. Ann. 208 (1974), pp. 305-322. Kirillov theory f o r compact p-adic groups, Pacific J. Math. 73 (1977), 17. -, no. 2, pp. 365-381. 18. -, S o m e qualitative results o n the representation theory of G1, over a p-adic field, Pacific J. Math. 73 (1977), no. 2, pp. 479-538. 19. ___ , T w o conjectures about reductive p-adic groups, Harmonic analysis on homogeneous spaces, Proceedings of Symposia in Pure Mathematics, vol. 26, American Mathematical Society, 1973, pp. 377-380. 20. R. Huntsinger, Vanishing of the leading t e r m in Harish-Chandra's local character expansion, Proc. Amer. Math. SOC.124 (1996), no. 7, pp. 2229-2234. 21. -, S o m e aspects of invariant harmonic analysis o n the Lie algebra of a reductive p-adic group, Ph.D. Thesis, The University of Chicago, 1997. 22. N. Jacobson, Basic Algebra, I, second edition, W.H. Freeman and Company, New York, 1985. 23. D. Kazhdan, Cuspidal geometry of p-adic groups, J. Analyse Math. 47 (1986), pp. 1-36. 24. A. A. Kirillov, Unitary representations of nilpotent Lie groups, Russ. Math. Surveys, 17 (1962), pp. 53-104. 25. B. Lemaire, Inte'grabilite' locale des caractkres-distributionsde G L N ( F ) OG F est un corps local non-archime'dien de caracte'ristique quelconque, Compositio Math. 100 (1996), no. 1, pp. 41-75. 26. C. Mceglin and J.-L. Waldspurger, Mod2les de Whittaker de'ge'ne're's pour des groupes p-adiques, Math. Z . 196 (1987), no. 3, pp. 427-452. 27. A. Moy and G. Prasad, Jacquet functors and unrefined minimal K-types, Comment. Math. Helvetici 71 (1996), pp. 98-121. Unrefined minimal K-types forp-adic groups, Inv. Math. 116 (1994), 28. -, pp. 393-408. 29. R. Ranga-Rao, Orbital integrals in reductive groups, Ann. of Math. (2) 96 (1972), pp. 505-510. 30. F. Rodier, Int6grabilit6 locale des caractbres du groupe GL(n,k) ozi k est un corps local de caracte'ristique positive, Duke Math. 3. 52 (1985), no. 3, pp. 771-792. 31. -, M o d d e de Whittaker et caractkres de repre'sentations. Non-

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35. 36. 37.

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commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1974), Lecture Notes in Math., Vol. 466, Springer, Berlin, 1975. pp. 151-171. J. D. Rogawski, An application of the building t o orbital integrals, Compositio Math. 42 (1980/81), no. 3, pp. 417-423. G. Savin, Lectures o n representations of p-adic groups, this volume. P. Schneider and U. Stuhler, Representation theory and sheaves o n the B m h a t - T i t s building, Inst. Hautes tudes Sci. Publ. Math. No. 85 (1997), pp. 97-191. J.-L. Waldspurger, Homoge'nEitE de certaines distributions sur les groupes p-adiques, Inst. Hautes Etudes Sci. Publ. Math. No. 81 (1995), pp. 25-72. -, Quelques resultats de finitude concernant les distributions invariantes sur les algZbres de Lie p-adiques, preprint, 1993. -, Une formule des traces locale pour les algZbres de Lie p-adiques, J. Reine Angew. Math. 465 (1995), pp. 41-99.

On Classification of Some Classes of Irreducible Representations of Classical Groups

Marko TadiL Department of Mathematics University of Zagreb BijeniEka 90, 10000 Zagreb, Croatia E-mail: [email protected] r

Representation theory of reductive p-adic groups, besides its importance for harmonic analyses, is very important for Langlands program. It also gives us often better understanding of representation theory of reductive Lie groups. In these notes we review some parts of representation theory of reductive groups over local fields, in particular over p a d i c fields. We discuss classifications of some families of irreducible representations, which are important for harmonic analysis on these groups. We start with general principles of harmonic analyses on groups (which are given in terms of unitary representations). Then we explain algebraization of the problem of classification of irreducible unitary representations. Langlands classification reduces classification of irreducible representations to tempered representations, which come from square integrable representations by parabolic induction. We present existing classifications of such classes of representations for general linear and classical groups, and discuss connection of this with Langlands correspondences. Special attention is devoted t o classification (modulo cuspidal data) of irreducible square integrable representations of classical padic groups, which implies parameterization of non-unitary duals. This opens possibility t o work on the (very hard) problem of classification of irreducible unitary representation of these group.

Contents

1 Harmonic analysis and unitary duals 2 Non-discrete locally compact fields, classical groups, reductive groups

97 99

3 KO-finite vectors

103

4 Smooth representations

104 95

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5 Parabolically induced representations 6 Jacquet modules 7 Filtrations of Jacquet modules 8 Square integrable and tempered representations 9 Langlands classification 10 Geometric lemma and algebraic structures 11 Square integrable representations of padic general linear groups 12 Two simple examples of square integrable representations of classical padic groups 13 Invariants of square integrable representations of classical padic groups 14 Reduction to cuspidal lines 15 Parameters of D ( p ; a) 16 Integral case 17 Non-integral case 18 Local Langlands correspondences 19 Non-unitary duals of classical padic groups 20 Unitary duals of general linear groups over local fields 21 On the unitariaability problem for classical padic groups References

105 109 111 112 114 120 123 125 127 135 137 138 141 143

145 147 154 158

Introduction In these notes of the lectures given during the special period on representation theory of Lie groups in IMS, NUS, Singapore, we shall discuss the problem of classification of some important series of irreducible representations of general linear and classical groups, having in mind unitary representations. We shall discuss more padic groups, but a part of notes deals also with real groups. One of the main goals of the notes is to give an introduction to the classification modulo cuspidal data, of irreducible square integrable representations of classical padic groups. After that we shall describe unitary duals of general linear groups over local fields, and describe the proof of the classification theorem in the case of complex general linear groups. We shall finish the notes with a series of questions regarding unitary representations of classical padic groups. In these notes, the fields of real and complex numbers are denoted by R and C, and the ring of rational integers is denoted by iz (as usual). Further

z+= {k E Z ; k 2 0) N = {t E z; t 2 1).

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97

We are thankful to the organizers of the special period for providing a very stimulating atmosphere in which we had opportunity to present the lectures.

1. Harmonic analysis and unitary duals 1.1. One can interpret classical harmonic analysis in terms of unitary representations of Rn and (R/Z)n. This point of view opens a possibility of generalizing classical harmonic analysis, and building such a type of theory for a general locally compact group G (in general, neither compact, nor commutative). We shall briefly describe the main problems of harmonic analysis on such a group G. First, we shall introduce a few notions which we shall need for this description. 1.2. A representation (r,V) (or simply r or V) of a group G is a group homomorphism r from the group G to the group of all invertible linear operators on a complex vector space V (there is no requirement on continuity in this definition). A representation r on a non-zero vector space V is called irreducible (or algebraically irreducible) if (0) and V are the only vector subspaces of V which are invariant for all r ( g ) , g E G. A representation (r,H ) is called unitary if H is a Hilbert space and: (1) the mapping (g,v)H r ( g ) u ,

GxH

-+

H

is continuous; (2) each operator .rr(g),g E G is unitary.

If we omit the second requirement in the above definition, then the representation defined in this way will be called continuous (one can consider much more general continuous representations, but we shall not need them in these notes). A unitary (or only continuous) representation ( r , G ) is called irreducible (or topologically irreducible) if (0) and H are the only closed subspaces of H which are invariant for all r ( g ) , g E G. 1.3. Now we can describe the main goals of harmonic analysis on a locally compact group G (which satisfies some technical requirements, which we shall not discuss here, but which are satisfied for the groups that we shall consider in these notes, i.e. for general linear and classical groups over local fields).

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The first problem is to

(1) understand in a convenient way (possibly classify) the set of all the equivalence classes of irreducible unitary representations of G. This set is called the unitary dual of G, and it is denoted by G. The second problem is to

(2) interpret other important unitary representations of G in terms of G. Such important unitary representations are usually given on functional spaces. The most important examples of such representations include representations of G on spaces of square integrable functions (with respect to an invariant measure, assuming that it exists) on a space X where G acts transitively. Then X % H\G for some closed group H of G and G acts by right translations on the space L2(H\G) of the square integrable functions on H\G (in this case H\G carries an invariant measure for right translations of G). The first example of such representation would be when H is the trivial subgroup of G, i.e. the representation of G on the space L2(G)of the square integrable functions on G with respect to right invariant measure on G. This representation is very important. A significant portion of Harish-Chandra’s work is closely related to this representation in the case of semi simple real Lie groups (among others, he described the representation from G necessary for decomposing L2( G ) , and found Plancherel measure by which one decomposes L2(G)in terms of these irreducible representations). In this lectures we shall be more related to the problem (1) of harmonic analysis, although we shall be also related to the problem (2). Irreducible square integrable representations, which are one of the main topics of our notes, are part of both problems, (1) and (2). They are subrepresentations of L2(G)if the center of G is compact.

Remark: Some of the most important parts of the Langlands program can be considered as a kind of problems from harmonic analysis on groups in the above sense. For example. the origin of the Langlands program one can view as a kind of problem of harmonic analysis. The program started as a strategy for proving the Artin’s conjecture that Artin’s L-functions are

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entire. Roughly, Langlands proposed a strategy that irreducible representations of the absolute Galois group of a number field would parameterize irreducible subrepresentations of adelic general linear groups on the spaces of cuspidal automorphic forms (which are unitary representations on functional spaces), in a way that corresponding L-functions match (this can clearly be regarded as a kind of problem of type (2) of harmonic analysis on groups). Realization of this strategy would imply the Artin’s conjecture. The above philosophy has its local counterpart (with corresponding parameterizations). In the local case of the Langlands program, we are more related to the problem of type (1) of harmonic analysis on groups. One can extend the above considerations to other reductive groups, and one can consider also different fields. A question may be, can one do the above mentioned parameterizations in a naturally compatible way. Such a question is related to the functoriality problem. Above we gave only a very rough comments regarding the Langlands program. Details regarding this program can be found in [19] or [36].

2. Non-discrete locally compact fields, classical groups, reductive groups

2.1. Let F be a non-discrete locally compact field. Such field will be called local field. If F is connected, then the field is called archimedean. Otherwise, it is called non-archimedean. Non-archimedean fields are totally disconnected. They contain a basis of neighborhoods of 0 consisting of open compact subrings. If a local field is archimedean, then it is isomorphic to R or C. Let p be a prime integer. Ideals p k Z , k E Z+, define a basis of neighborhoods of 0 in Z. The completion of 2 with respect to this topology (more precisely, uniform structure defined by this topology) is denoted by Z,. The field of fractions of Z,is denoted by Q p . This is the field of padic numbers. We can introduce Q p also as a completion of Q with respect to the absolute value

Any finite extension F of Q, is in a natural way a topological space, and with this topology, F is a local non-archimedean field of characteristic 0. One gets each non-archimedean field of characteristic 0 in this way. Let F,[[X]] be the ring of all formal power series CEO u,Xn over a finite field F, (with q elements), and let F,((X)) be the field of all Laurent

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power series CE-, a,Xn over IF, for which there exists no E Z such that a, = 0 for all n 5 no. Then the powers of the ideal X F , [ [ X ] ]in IF,[[X]] define a basis of neighborhoods of 0 in P,((X)), and therefore a topology on F,((X)).In this way F,( ( X ) )becomes a local non-archimedean field of positive characteristic. One gets each local non-archimedean field of positive characteristic in this way. Topology on a local field can be always defined using an absolute value. Moreover, there exists a unique absolute value 1 JFon a local non-discrete field F such that

for any a E F X and for any continuous, compactly supported function f on G, where dx denotes an invariant (for translations) measure on F . We shall always fix such an absolute value on F . Let us note that for @, this absolute value is a square of the standard one.

2.2. We shall recall now of a definition of the classical groups. A classical group over a local field F is the group of isomorphisms of either symplectic, or orthogonal or unitary space over F (of finite dimension). For the study of representations of classical groups, it is important to understand the representation theory of general linear groups GL(n,F)’s,i.e. of the groups of all isomorphisms of finite dimensional vector spaces over F (soon it will become clear why this is important). In the study of classical groups, we shall use very convenient language of structure theory of reductive groups without going into this theory. In general, we shall try to keep the technicalities as low as possible.

For the simplicity, we shall consider in these lectures two series of classical groups (these series consist of split and connected groups). The reason for this is only t o simplify the notation. 2.3. The first is the series of symplectic groups: Denote 0 0 . . . 0100 ... 10

J, =

: 01 .. . 00

1 0 . . . 00-

E

GL(n,F).

Some Classes of Irreducible Representations

Then the symplectic group is

S

E

GL(2n, F ) ; tS

[

-Jn O

101

"1 s = [ "I). -Jn O 0

Here t S denotes the transposed matrix of S. 2.4. The second series consists of split odd-orthogonal groups:

Denote by Inthe identity matrix in GL(n, F ) . Let SO(2n

+ 1,F )

{ S E SL(2n + 1,F ) ; ' S S = 1zn+l}.

Here S L ( n , F ) = {g E GL(n,F);det(g) = 1) and ' S denotes the transposed matrix of S with respect to the other diagonal. We could work also with O(2n 1,F ) instead of SO(2n 1,F ) .

+

+

In these notes, we shall always deal with matrix forms of classical groups. 2.5. The above groups are connected, split, semi simple algebraic groups over F . They are topological groups in a natural way. In the case when F is a non-archimedean field, these groups are totally disconnected. Then one can write a basis of neighborhoods of identity which consists of open (and closed) compact subgroups. If F is archimedean, then symplectic and odd-orthogonal groups are connected semi simple Lie groups. If G is GL(n,F ) , or Sp(2n,F ) or S0(2n+1, F ) , then we shall denote by Po the subgroup of all the upper triangular matrices in G. Then Po is called standard minimal parabolic subgroup of G. Any subgroup of G containing Pa, is called standard parabolic subgroup of G. There are finitely many of them, and we shall describe them precisely. Any subgroup conjugate to a standard parabolic subgroup is called parabolic subgroup. Let Q

= (n1,...,n k )

be an ordered partitions of n into positive integers. Consider matrices of GL(n,F ) as block matrices with blocks of sizes ni x nj. Let P,"" (resp. M:"), be the upper block-triangular matrices (resp. block-diagonal matrices) in GL(n,F ) . Denote by N z L the (block) matrices in P,"" which have identity matrices on the block-diagonal. Now

is one-to-one mapping of the set of all ordered partitions of n onto the set of all standard parabolic subgroups of GL(n,F ) . We have Levi decomposition

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P,"" = M,""N,"". This means that P,"" is a semi direct product of M,"" and N,"", where N,"" is a normal subgroup in P z L , i.e.

P,""

= M,GL K

N,GL.

This decomposition of P,"" is called the standard Levi decomposition of P:", where M,GL is called the standard Levi factor of P,"" and N:" is called the unipotent radical of P,"".

+

2.6. Standard parabolic subgroups of Sp(2n,F ) and SO(2n 1,F ) are parameterized by ordered partitions a: = (711,. . . ,nk) of integers m, where 0 5 m 5 n. If we consider the group G = Sp(2n,F ) set a' = (721, ...,n k , 2n - 2m, n k , ..., n l ) , while in the case of the group G = SO(2n

a' = (n1,..., n k , 2n

+ 1,F ) set

+ 1 - 2m,nk,...,

n1).

Then a: H Pa = P$" n G

gives a parameterization of standard parabolic subgroups in G. In similar way as in the case of general linear groups, one defines standard Levi decompositions in this case, using standard Levi decompositions of P2". In the sequel, we shall denote by G one of the groups GL(n,F ) , Sp(2n,F ) or SO(2n 1,F ) . We shall denote by A0 the subgroup of all diagonal matrices in G. This is a maximal split (over F ) torus in G (it is also a maximal torus in G).

+

2.7. We shall denote by KOa maximal compact subgroup of G. If F is non-archimedean, let OF =

PF

{x E F ; l x l F 5 I},

= {x E

F ; 1 x 1 ~< 1).

In the non-archimedean case one can take

KO= GL(n,O F )n G. For F = W (resp. F = C ) ,one can define KOin a similar way as above, taking the group O ( n ) (resp. U ( n ) ) of orthogonal matrices in G L ( n , R ) (resp. unitary matrices in GL(n,C)) instead of GL(n,O F ) . It is important to note that KO is an open subgroup if F is nonarchimedean field, which is not the case (in general) in the archimedean case.

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3. KO-finite vectors

3.1. Let ( T ,H) E G. For T E ko denote by m(r : T ) the multiplicity of r in T . The basic property of KO is that it is a large subgroup of G (this was proved by Harish-Chandra in the archimedean case, and by J. Bernstein in the non-archimedean case). It means the that the function T H m(T

:T)

is a bounded function on G , for any fixed I- E KO.This fact has a number of important consequences. Among others, it enables algebraization of the problem of determining of the unitary dual G of G.

3.2. Let

( T , H)

E G. Denote by

H" the set of all vectors w E H such that

dim@spanc T(KO)V < 00. Then H" is a dense KO-invariant vector subspace of H". Suppose that F is non-archimedean. Since for each g E G the group sKog-l n KO has finite index in KO,H" is G-invariant. It follows easily that the following property holds for H": For any u E H" there exists an open subgroup K of G such that T ( k ) w = w for any k E K . This follows from the fact that each continuous representation of KO is trivial on an open subgroup (since open subgroups in KO form a basis of neighborhoods of identity, and GL(n,cC) does not contain small (nontrivial) subgroups).

3.3. In the archimedean case, gKog-l n KO is not (in general) of finite index in K O .Because of this, H" is not (in general) G-invariant. But then one can prove that it is invariant for the natural action of Lie algebra g of G. Moreover, the action of g and KO satisfies a natural condition. Such a structure is called (8, KO)-module. 3.4. At this point usually archimedean and non-archimedean theory continue to develop separately. In the sequel, we shall more discuss the nonarchimedean theory, but a number of topics hold for both theories (these will be examples of Lefschetz principle). We shall usually comment the results which hold in both theories.

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4. Smooth representations

We shall assume in the sequel that F is a local non-archimedean field (if it is not otherwise specified).

4.1.A representation ing condition:

(7r, V

)of G is called smooth if it satisfies the follow-

For any u E V there exists an open subgroup K of G such that 7r(k)v = u for any k E K . Denote by

G the set of all equivalence classes of irreducible smooth representations of G. This set is called non-unitary dual of G, or admissible dual of G. 4.2. The mapping (7r,H

)w

H”);

(Too,

G4G

is injective (here 7r” denotes the restriction of 7r to H”). Therefore, the unitary dual can be identified with a subset of G. We shall assume this identification in further. It can be shown that in this way the unitary dual is identified with the subset of all ( T , V )E G such that on V there exists an inner product which is invariant for the action of G. The problem of classification of G has appeared much more manageable then the problem of classification of G.

4.3.The problem of classification of unitary dual of G now breaks into two parts: problem of classification of G , which is called the problem of nonunitary dual; problem of determining the subset G of G (in other words, the problem of identifying unitarizable classes in G), which is called the unitarizability problem. We shall discuss both problems in these lectures.

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4.4. Regarding the problem of non-unitary duals, let us note that there is Langlands classification of non-unitary duals, which reduces the problem of classification of non-unitary duals to the problem of classification of a special kind of irreducible representations of Levi subgroups, namely to the problem of classification of tempered representations, which will be introduced later. In the moment, let us just note that these tempered representations are unitarizable. The problem of classifying of irreducible tempered representations is very far from being easy. Before we describe Langlands classification, we shall recall of a more simple (and less precise) reduction of the non-unitary duals. We shall need to have a tool by which we shall be able to produce new representations. This tool is provided by parabolic induction, a construction which generalizes in a natural way induction studied already by Schur and Frobenius in the case of finite groups. Further, we shall need a tool for analyzing induced representations. Jacquet modules will be of great help for this.

5 . Parabolically induced representations

5.1. Smooth representations of G and intertwinings form an Abelian category, which will be denoted by Alg(G). Let ( T , V ) be a representation of G. Denote

V” = {Y E V ;there exists an open subgroup K such that 7r(k)v = Y, k E K}. The space V” is a G-subrepresentation of G, and it is called the smooth part of V . For a compact subgroup K of G let

V K = (Y E V ;~ ( l c ) w= Y for any k E K } . This vector space is called the space of K-invariants of V . Further, (T,

V )H V K

is an exact functor on the category Alg(G). 5.2. If ( T , V )is a smooth representation of G, then there is a natural representation d on the space of all linear forms V’ on V defined by (r’(g)v’)(v)= d(7r(gP1)v). The smooth part of this representation is called the contragredient of (7r, V). This representation is denoted by

(*,

V)

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(recall (ii(g)V)(w) = V(n(g-l)w)). Then the mapping

(w,V)

H
v x V -+ c

>=: q w ) ,

is called canonical bilinear form. This form is G-invariant. A function g ++
>

is called a matrix coefficient of n. Further (r,V )

(El V )

extends to a contravariant functor in a natural way. This functor is exact. For a representation ( n I V ) of G, the representation on the complex conjugate vector space 8 of V will be denoted by (ii 8). ,

A smooth representation

( T , V)

will be called Hermitian if -

( T , V)

= (%,V ) .

5.3. A smooth representation (r,V ) of G is called admissible if dim@VK

< 03

for any open compact subgroup K of G. For a smooth representation (n, V ) of G we have always a natural in-

tertwining of V into V . If the representation is admissible] then this is an isomorphism. The converse also holds, i.e. if V and V are isomorphic, then (T] V ) is admissible. It is easy to show that each unitarizable admissible representation of G is Hermitian. 5.4. We shall fix the group G of rational points of a connected reductive

group defined over a local non-archimedean field F . One of the main examples for us are general linear groups and classical groups. We shall fm a maximal split torus AP)in G and a minimal parabolic subgroup P0 of G which contains A @ .Standard parabolic subgroups of G are subgroups of G which contain P0. For a standard parabolic subgroup P of G, a Levi decomposition of P into semi direct product of a reductive subgroup M and a

Some

Classes of Irreducible Representations

107

normal unipotent subgroup N will be called standard if A0 C M. For standard parabolic subgroups we shall always assume that Levi decompositions are standard. Parabolic subgroups and their Levi decompositions one gets from standard parabolic subgroups and their standard decompositions by conjugation with elements of G. We shall fix a maximal compact subgroup KO of G for which Iwasawa decomposition

G = Pa KO holds (such a maximal compact subgroup always exists). 5.5. Let for a moment 0 be a locally compact group. Then there always exists a positive measure which is invariant for right translations. Such a measure will be denoted by d g . Right invariance means that

for any continuous compactly supported function f on Q and any x E 0. This measure is unique up to a multiplication by a constant, and it is called a right Haar measure on B. A right Haar measure does not need to be left invariant (if it is, then the group is called unimodular; reductive groups are unimodular), but there exists a character A, of 6 (which is called the modular function or modular character of G), such that holds

holds for any f and x as above. 5.6. Let us return back to the case of a connected reductive group G over a non-archimedean field F . Fix a parabolic subgroup P of G with a Levi decomposition P = M N (more preciseIy, the group of rational points). Let (a,U ) be a smooth representation of M . Denote by Ir&(a) the space of all functions f G t U which satisfy f(72.V)

= AP(m)1’24m)f ( 9 )

for each m E M , n E N , g E G. Then G acts on In&(a) by right translations (R,f)(rc)= f ( x g ) , x,g E G. The smooth part of the representation Indg(a) is denoted by Indg (D )

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and called a parabolically induced representation of G from P by a. Parabolic induction becomes in an obvious way a functor from Alg(M ) into Alg(G). The functor of parabolic induction is exact. 5.7. If a is unitarizable, then Indg(a) is also unitarizable. The inner product

( , ) on Indg(a) is given by

Further, Indg (a)-Z Indg (8) The canonical bilinear form is given by the same formula as the above inner product:

5.8. Suppose that P = M N is a standard parabolic subgroup of G and P' =

M'N' another standard parabolic subgroup of G (the above decompositions are considered to be standard Levi decompositions). Let

P 2 PI. Then Ind$(a) E Indg,(Ind&L, (a)). This fact is called induction by stages (which gives the same result as the original, direct parabolic induction). It is easy t o prove it (one writes an explicit isomorphism). 5.9. Iwasawa decomposition implies that Indg(a) is an admissible representation if a is admissible. It is less obvious to prove that if (T is a representation of finite length, then Indg(a) is also a representation of finite length (of G).

5.10. Suppose that we have a parabolic subgroup P with Levi decompositions P = M N and P = MINI, which do not need to be the standard one (in the case that really interests us, a t least one Levi decomposition should not be the standard one). Suppose

M=M'.

Some Classes of Irreducible Representations

Let

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109

be a smooth finite length representation of M. Then Indz(a) and Indg, (a) have the same Jordan-Holder series.

This is an important fact, called induction from associate parabolic

subgroups. It is not quite simple to prove it. It relies on the theory of characters. Since we shall not introduce characters in these notes, we shall not comment the proof here. 6. Jacquet modules

In this section we shall introduce a functor which is left adjoint to the functor of parabolic induction.

6.1. Suppose that ( T , V ) is a smooth representation of G and let P = M N be a parabolic subgroup of G (actually, it is enough to assume that ( T , V ) is a smooth representation of P only). Let

V ( N )= spanc { ~ ( n )-ww;n E N , 21 E V } . Since N is normal in P, V ( N )is P-invariant. In particular, it is M-invariant. We have a natural quotient action of M on

r$(V) = V / V ( N ) . We shall consider the action of M on r $ ( V ) which is the quotient action of the action of M (through T ) on V , twisted with A;'". This action will be denoted by

rE ( T ) . The representation (.$(T)l.$(V))

is called the Jacquet module of ( T , V )with respect to P = M N . One defines in a natural way Jacquet functor from A1g(G) into A l g ( M ) . Jacquet functor is exact.

6.2. If P = M N and P I = MINI are standard parabolic subgroup, with standard Levi decompositions, such that

P2

PI,

then

r & ( r g ' ( T ) )E r g ( T ) .

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M. TadiC

This fact will be called transitivity of Jacquet modules.

6.3. The fact that Jacquet functor is left adjoint to the functor of parabolic induction means that we have a natural isomorphism HomG ( T , Indg(a)) 2 HomM ( r $ ( r ) ,a).

The above isomorphism is called Frobenius reciprocity. One constructs this isomorphism using evaluation of f E Homc (T,IndZ(a)) at 1. 6.4. A smooth irreducible representation ( T , V )of G is called cuspidal (or supercuspidal) if all the Jacquet modules for proper parabolic subgroups are trivial modules. It is natural to distinguish these representations, as will become clear very soon. Actually, in the definition of cuspidal representations, it is enough to require triviality Jacquet modules only of proper standard parabolic subgroups. Cuspidal representations are very special representations, as we shall see later. 6.5. One easily sees that if ( T , V ) is a finitely generated smooth representation of G, then r $ ( T ) is finitely generated representation of M . From this

follows that it has an irreducible quotient. Let (n, V) be an irreducible smooth representation of G. Among all the parabolic subgroups P = M N which satisfy T $ ( T ) # {0}, chose a minimal one. Then by the above observation, r Z ( T ) has an irreducible quotient, say a. Minimality of P and transitivity of Jacquet modules (together with exactness) imply that a is cuspidal. Now Frobenius reciprocity implies that n embeds into Indg(a).In this way we have obtained a simple but important

Theorem: An irreducible smooth representation T of G i s cuspidal, o r there exists a proper parabolic subgroup P = M N of G and a n irreducible cuspidal representation a of M such that n is isomorphic t o a subrepresentation of In@ (a). In this way the problem of classification of non-unitary dual C? breaks into two problems. One problem is to classify cuspidal representations of Levi factors, and the other one is t o classify irreducible subrepresentations of representations parabolically induced by cuspidal representations. From the above theorem it is not clear at all how to classify irreducible subrepresentations of representations parabolically induced by cuspidal representations. Langlands classification will provide a strategy for it. There is

S o m e Classes of Irreducible Representations

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another way to describe irreducible cuspidal representations. They can be characterized as those representations which never show up as subquotients of representations parabolically induced from proper parabolic subgroups.

6.6. Remarks: (i) Let G be GL(2,F ) and 7~ be an irreducible representation of G which is not cuspidal. Then Schur lemma and the above theorem imply that 7r is isomorphic to a subrepresentation of Ind$&), for a character x of the Levi factor M0 of 4 (note that M0 is commutative). (ii) There is an archimedean version of the above theorem (see [17]).

6.7.One important property of cuspidal representations is that their matrix coefficients are functions which are compactly supported modulo center. W. Casselman has shown that this property characterizes cuspidal representations. H. Jacquet has proved that each cuspidal representation is admissible (a nice argument for this can be found in [50]).Now 5.9 and Theorem 6.5 imply that each irreducible smooth representation is admissible. This is the reason that G is also called admissible dual. 6.8. Let us note that Jacquet functor carries admissible representations to admissible ones (this is not quite easy to prove). Further, it carries smooth representations of finite length of G to smooth representations of finite length of M . For proofs one can consult [IS]. 7. Filtrations of Jacquet modules

Jacquet modules are very important in analysis of admissible representations, in particular of the induced ones. In general, it is not easy to determine their structure. There is geometric lemma, obtained independently by J. Bernstein and A. V. Zelevinsky, and by W. Casselman, which describes filtrations of Jacquet modules of Ind$(a) in terms of representations parabolically induced by Jacquet modules of a. We shall illustrate this lemma on the example of G = GL(2,F ) . Later we shall describe how one can realize the geometric lemma as a part of an algebraic structure in the case of general linear and classical groups. In this section we assume that

G = GL(2,F ) . 7.1. For G = GL(2,F ) we have P0 = PGL . Irreducible smooth representa(191) tions of M g f ; , are one dimensional, i.e. characters. Since M g f ; , is naturally

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isomorphic to F X x can be written as

F X, each irreducible smooth representations of x 1 8 xz

where x1 and x2 are characters of We shall consider

1

F X.

IndgO(X18 x2). Denote

-

It is not hard to show that the following obvious sequence 0 -+

-

{f E IndgO(xl8 XZ);supp(f) C P0wop0) restriction

IndZO(xl8 XZ)

(flP0; f E IndZO(xlc3 x2)) 0. (1) is exact. Further, considering the mapping f H f ( l ) ,one gets easily that +

G0((fJP0;f E I n d g (Xl 8 x2))) Ei x1 8 xz. It requires a little bit more efforts to show that G T M @ ( {E ~

IndgO(xi8 xz);supp(f) C_ p0wop0))

x2

8 XI.

7.2. Applying Jacquet functor to the exact sequence (1) (recall that the Jacquet functor is exact), we get the following exact sequence 0 4 x2 8 XI

---$

TEO(IndZO(xl 8 XZ))

-+

XI 8 xz

-+

0.

As a consequence of this exact sequence, we can conclude that Indgo(XI @ X Z ) has at most length 2. 8. Square integrable and tempered representations

8.1. Let ( T , V ) be an irreducible smooth representation of G. Then Schur lemma implies that the center Z(G) of G acts by scalars. Corresponding character will be denoted by WT

and called the central character of

T.

113

Some Classes of Irreducible Representations

A smooth representation does not need to be irreducible, but the center can act by scalars. Then we shall say that the representation has a central character.

8.2. An admissible representation (7r,V)of G will be called square integrable (more precisely, square integrable modulo center) if it has central character, if the central character is unitary, and if absolute values of all the matrix coefficients g

H

I < 7r(g)v,V > 1,

21

E

v,v E v,

are square integrable functions modulo center (i.e. square integrable functions on G/Z(G)). In this notes we shall consider only irreducible square integrable represent ations. An admissible representation (7r,V) of G will be called essentially square integrable (or essentially square integrable modulo center) if there exists a character x of G such that x7r is square integrable. 8.3. Each (irreducible) square integrable representation ( 7 r , V ) of G is unitarizable. To see this, take Go E different from 0. Now for u,w E V set

v

One sees directly that this is a G-invariant inner product on V (if irreducible, one proceeds similarly; see [73]).

7r

is not

8.4. Irreducible square integrable representations are very important. First,

they are (very distinguished) elements of the unitary dual. Then, via Langlands classification, they are crucial in the parameterization of non-unitary duals. Further, using matrix coefficients one gets that they are (irreducible) subrepresentations of L2(G) if G has compact center (in the non-compact case, we have similar situation when one fixes central character). Therefore, they are very important for understanding decomposition of L2(G). 8.5. An irreducible smooth (which implies admissible) representation ( 7 r , V ) of G is called tempered if there exists a parabolic subgroup P = M N of G (not necessarily proper) and an irreducible square integrable representation 6 of M such that 7r is isomorphic to a subrepresentation of

Indg (6).

M. Tadic‘

114

An irreducible smooth representation ( T , V )of G is called essentially tempered if there exists a character x of G such that XT is tempered. One usually defines tempered representations without irreducibility requirement, but since we shall work only with irreducible tempered representations in these notes, the above definition is not restrictive for us. 8.6. We shall now introduce notation for general linear groups. The character

will be denoted by

v. If r is an (irreducible) essentially tempered representation of GL(n,F ) , then one easily sees there exists a unique e(T) E

R

and a unique tempered representation rU such that = ye(T)TU.

This requirement uniquely defines e(r).

8.7. There are a very useful criteria of Casselman for checking if an irreducible admissible representation is square irreducible or tempered. We shall explain this criterion on the simplest case, on G = GL(2,F ) . Let ( T , V ) be an irreducible essentially square integrable (resp. essentially tempered) representation of GL(2,F ) . Then for any irreducible subquotient x = x1 @ x 2 of rg:(2’F)(n) ( X I and x2 are characters of F X )we have

4 x 1 )> 4x2)

(resp.

4x1) 2 4x2)).

Moreover, the converse also holds for an irreducible admissible representation ( T , V ) . 9. Langlands classification

9.1. Langlands classification parameterizes representations of 6, by (irreducible) essentially tempered representations of Levi factors of standard

S o m e Classes of Irreducible Representations

115

parabolic subgroups. These essentially tempered representations need to satisfy certain positiveness condition (which will be discussed later). Langlands classification claims the following: For an irreducible essentially tempered representation T of Levi factor A4 of standard parabolic subgroup P of GI which satisfy the above mentioned positiveness condition, the representation Ind$(T) has a unique irreducible quotient. This irreducible quotient is called the Langlands quotient (Ind$ ( T ) is called a standard module of G). Each 7r E G is isomorphic to some Langlands quotient, and moreover 7r determines uniquely the standard parabolic subgroup P and essentially tempered representation T . We shall now explain the positiveness condition for the groups that we consider in these notes. We shall start with general linear groups.

9.2. Each Levi factor of a parabolic subgroup of a general linear group, is a direct product of general linear groups. Because of this, each irreducible essentially tempered representation of a Levi factor of a general linear group is a tensor product of such representations of general linear groups. Therefore, the essentially tempered representations of a Levi factors of a general linear groups are of the form 71

'8 72 '8 . ' ' '8 Tk,

where 71, 7 2 , . . . ,'rk are essentially tempered representations of general ]inear groups. The positiveness condition here is simply e(T1)

> e(7-2) > . . . > e ( n ) .

9.3. We shall see how one gets the Langlands parameterization in the case of the simplest possible example, in the case of G = GL(2,F ) . Let T E G. If 7r is essentially tempered, then it is its own Langlands parameter. Suppose therefore that 7r is not essentially tempered. Then, in particular, it is not cuspidal. Therefore

ii

is a subquotient of

Indga (XI 8 x 2 )

by Theorem 6.5, for some characters x 1 and x 2 of F X (see also Remarks 6.6, (i)). Since 7r is not essentially tempered, ii is also not essentially tempered. Now by 8.7 4x1)

Therefore by 7.2

# 4x2).

M. Tadic'

116

Without lost of generality we can assume

4x1) > 4 x 2 1 7 since IndgO(xl @I x2) and IndgO(x2€3 XI) have the same Jordan-Holder series (one sees this using induction from associate parabolic subgroups; see 5.10). Since i f is not essentially square integrable (recall that 7r it is not essentially tempered), from criterion for essentially square integrability follows that x 2 €3 x1 must be a subquotient of rga(?). Now from (2) we see that there exits a non-trivial homomorphism

rgO(5)

-+

-

x2

63 x1.

Frobenius reciprocity implies 77

Indg(X2 63 X l ) .

Passing to contragredients we get an epimorphism IndZO(x;' 8 x;')

---t

71

Note that e ( x z l ) = -e(xz)

> -e(xl)

= ~(xT').

This implies that we have shown the existence of Langlands parameters for irreducible representations of GL(2,F ) . Their uniqueness follow from the filtration of Jacquet modules (see 7.2). Thus, we have "proved" Langlands classification for GL(2,F ) . This case is too simple to illustrate the proof of the Langlands classification in general, but one can get a t least some idea from this simplest case how proof goes in general. In any case, we see the importance of the Geometric lemma. 9.4. Since we have defined tempered representations by square integrable representations, it is natural to try to express Langlands classification in terms of square integrable representations, if this is possible. In the study the representation theory of general linear groups, it is convenient to use notation that was used by Bernstein and Zelevinsky in their work on the representation theory of general linear groups. We shall now recall of ( a very small part of) it. For smooth representations 7r1 and 7rz of G L ( n l ,F ) and GL(n2,F ) denote 7r1

x

7rz

GL(n1f n 2 ,PI = IndpyL (Tl€3.2). ni.nz)

117

S o m e Classes of Irreducible Representations

Then 7r1 x

(“2

x

“3)

s (“1

x

7T2)

x

(3)

7T3.

This follows from induction by stages. Further, for smooth representation and “2 of finite length,

“1

“1

x

“2

and

7r2

x

7 ~ 1have

the same Jordan-Holder series.

(4)

This follows from induction from associate parabolic subgroups.

Remark: In the case of an archimedean field F , using parabolic induction we define multiplication x between (g, KO)-modules of general linear groups (over F ) in the same way as above. Then (3) and (4) hold also in this case. 9.5. A principal result regarding tempered induction for general linear groups is that this induction is irreducible (this fact holds for all the local fields). This fact has been proved independently at several places, but it seems that the first proof in this setting belongs to H. Jacquet, who proved it is for all the local fields (see [29]). This principal result claims the following:

If “1, “2, . . . ,“k are (unitarizable) irreducible square integrable representations of general linear groups, then irreducible.

“1

x

“2

x

- - .x

%

is

Either from general facts regarding tempered representations, or from A. V. Zelevinsky paper [76], follows that the tempered representation “1 x 7r2 X . . . X “rk determines irreducible square integrable representations “1 ,“2, . . . ,nk up to a permutation. 9.6. Using 9.4 and the fact that for a character

x of F X we have

[(xo d e t ) ~x~ [(x ) o det).irz] ”= (xo det)(7rl x “2) (which one proves directly), we can reformulate the Langlands classification for general linear groups in the following way. Denote by D the set of all the irreducible essentially square integrable representations of GL(n,F)’s for all n 2 1. Let M ( D ) be the set of all finite multisets in D. These are functions from D into Z+with finite supports. We shall write them similar as sets, but repetitions of elements will be allowed. We shall write them as (61,62,.. . ,6k),

where 6i E D.

M. TadiC

118

Take any d = (1,.. . , k} such that

(S1,62,,

. .,Sk)

E

M ( D ) . Take a permutation p of

> e(dp(2))2 . . . 2 4 b p ( k ) ) .

e ( b p ( 1 ) )-

Now the representation Sp(1)

x

bp(2)

x

.. . x

bp(k)r

which will be denoted by

has a unique irreducible quotient (the representation X ( d ) is determined by d E M ( d ) up t o an isomorphism.). This quotient will be denoted by

L(d). In this way one gets bijection between M ( D ) and the set of all the irreducible smooth representations of all general linear groups over F . This is just a reformulation of the Langlands classification for general linear groups. The Langlands classification has a number of natural properties. Let us mention three:

L ( b l , b 2 , .. . , b k ) -

L(&,&, . . . ,&),

x L ( 6 1 , 6 2 , .. . , b k ) 2 L(X61,x s 2 , . . ., X 6 k )

(5)

(7)

( x is a multiplicative character of the field, and further, for a representation of G L ( n ,F ) , X T denotes the representation ( x o det) T ) .

7r

Remark: The Langlands classification for general linear groups holds also if the field is archimedean F . In this case the non-unitary dual GL(n,F ) of G L ( n , F )is the set of all the equivalence classes of irreducible (g,Ko)modules of GL(n,F ) . The irreducible representations (i.e. non-unitary duals) are classified by M ( D ) , where D is the set of all the equivalence classes of irreducible essentially square integrable (8, Ko)-modules of all GL(n,F ) ' s , n 2 1 (if F = C, then D = (C")-, while for F = R we have

D

c ( R x ) - u GL(2,R)-).

9.7.Now we shall describe the Langlands classification for symplectic and odd-orthogonal groups.

S o m e Classes of I n e d u c i b l e R e p r e s e n t a t i o n s

119

It is convenient to introduce for classical groups the following notation, which simplifies notation when one works with parabolically induced representations. Let 7r be a smooth representation of GL(n,F ) and let u be a smooth representation of Sp(2m,F ) (resp. SO(2m 1,F)).Denote

+

7r

x u = Ind S p ( Z ( n + m ) , F ).( '(n)

@ ).

From induction by stages follows 7r1

x

(7r2

x

0)

E (7rl x 7r2) x

Further, for smooth representations 7r

x

(T

and ii x

(T

7r

and

(T.

of finite length,

have the same Jordan-Holder series.

(8)

This follows from induction from associate parabolic subgroups. 9.8. Regarding the Langlands classification for symplectic and oddorthogonal groups, one can first describe it in terms of essentially tempered representations, and after that pass to a description which include only essentially square integrable representations of general linear groups (and tempered representations of symplectic or odd-orthogonal groups), similarly as we did in the case of Langlands classification for general linear groups. Instead of this, we shall skip over the first description and go directly t o the second description. Set

D+ = (6 E D ; e ( 6 )> 0 ) . Denote by T the set of all equivalence classes of tempered representations of all Sp(2m,F ) (resp. SO(2m 1,F ) ) , for all m 2 0. Take

+

((61,62,...,6k),7)

E

M ( D + )x T .

After a renumeration, we can assume

61 2 62 2

" '

26k

Then the representation

61 x

62

x

...x

6k

x 7

has a unique irreducible quotient, which will be denoted by L(61762,. . . , 6 k ;7)

M. TadiC

120

Now the mapping

((61, Jz, . . ., Jk), 7 )

-

L(J1,Jz,

. . .,6kk;7 )

defines a bijection from the set M ( D + ) x T onto the set of all the equivalence classes of irreducible smooth representations of all Sp(2m,F ) (resp. SO(2m 1,F ) ) , m 2 0. This is the Langlands classification for (these) classical groups.

+

Remark: One can define >a also for the case of archimedean fields. The Langlands classification holds here in the same form. 10. Geometric lemma and algebraic structures Geometric lemma, which is a technical result describing filtrations of Jacquet modules of induced representations in terms of representations induced by Jacquet modules of inducing representations, is very important for number of purposes. For general linear and classical groups we can “incorporate” it into an algebraic structures on the representations of these groups.

10.1. Let for a moment G be a reductive group over a non-archimedean field

F . The Grothendieck group of the category of all smooth G-representations of finite length will be denoted by

R(G). This is just a free Z-module over basis G (it is isomorphic to the group of virtual characters of G). For a finite length representation T , let s . s . ( ~ )=

C m(7 :

T)T.

T€C

This is called semi simplification of T . We consider it as an element of R(G). There is a natural order on R ( G ) . We have

R(G1 x Gz)

R(G1)@ R(G2).

(9)

Further, r$ factors in a natural way to a homomorphism from R(G) into R ( M ) , which is denoted again by r z . This is a homomorphism of ordered groups (i.e. it respects also orders). We have analogous definition for the parabolic induction: Indg : R ( M ) -+ R ( G ) ,which is again a morphism of ordered groups.

S o m e Classes of Irreducible R e p r e s e n t a t i o n s

121

10.2. Set

R = & E Z + R ( G L ( ~F,) ) . Then one lifts x to a multiplication on R in a natural way. The mapping x : R x R + R factors in a natural way through a mapping

m : R @R Let

T

+ R.

E GL(n,F)-, We will consider

using the isomorphism (9). Define n

k=O

We can (and will) consider

m*(*)E R @R , since each Rk @ Rn-k

L)

R @ R. We can lift m* to an additive mapping rn*:R

.--)

R @R .

This mapping is called comultiplication. With the multiplication and comultiplication, R is a commutative Hopf algebra (over Z). This algebra was constructed by A. V. Zelevinsky. The most important part of this Hopf algebra structure is the formula m*(7r1x ~

2= ) m*(~1 x )m*(r2),

which explains how to get composition factors of Jacquet modules (for maximal parabolic subgroups) of induced representations, by induction from J acquet modules of inducing representations.

10.3. Suppose (only) in this paragraph that F is an archimedean field. One defines R ( G ) in the same way as in the non-archimedean case, considering the category of (8, K)-modules of finite length (G is a connected reductive (it is isomorphic to group over F ) . This is a free Zmodule over basis the group of virtual characters of G). Now for general linear groups over F one defines R in the same way as in the non-archimedean case. By the same formula as in the non-archimedean case, one defines multiplication x on R (using parabolic induction). In this way R becomes a commutative

M. Tadic‘

122

ring with identity. In the archimedean case, there is no comodule structure on R as in the non-archimedean case.

Remark The Kazhdan-Patterson lifting for G L ( n ,C ) has a very nice and natural description in terms of this algebra (see [64]). 10.4. Assume in this section that F is any local field (archimedean or nonarchimedean). For a = (61,. . . ,bk) E M ( D ) consider S . S ( X ( a ) ) = bl X

. .. X

bk E

R.

Then a simple fact regarding Langlands classification implies that s.s(A(a)),

aE

M(D),

form a Z-basis of R (this simple fact is usually expressed in the following form: characters of standard modules form a Z-basis of R ( G ) ) .In other words, for any local field F holds the following

Proposition: The ring R is a polynomial Z-algebra over

D.

10.5. We shall denote

R ( S ) = & E Z + R ( S PF( )~)~ , if we consider symplectic groups, and

+

R ( S ) = c B ~ , z + R ( S O ( ~1,~F ) ) if we consider odd-orthogonal groups. In this setting, one again lifts x in a natural way to a multiplication R x R ( S )+ R ( S ) .This multiplication factors through a mapping p : R 8 R ( S )4 R ( S ) .

For

7r

E

Sp(2n,F ) - (resp. 7r E SO(2n

+ 1,F)-) set

n

k=O

Consider s.s of

as an element using (9), and consider further

Some Classes of Irreducible Representations

123

Lift p* to an additive mapping p * : R(S) + R @R(S)

,

which will be called comultiplication on R(S). With the above multiplication and complication, R ( S )is a module and a comodule over R. It is not a Hopf module over R, but is also far from this structure as we shall explain now. Define

M * = ( m @ l ) o ( ~ @ m * ) o s o mR*+: R @ R , where 1 denotes the identity mapping, the contragredient mapping and s the transposition mapping C xi @ yi H C yi @ xi. Then N

p*(7r >a

0)= M*(7r) M

p*(o)

(R @ R(S) is a R @ R-module in an obvious way). We say that R(S) is an M*-Hopf module over R. This is again a (combinatorial) formula from which we can again in a simple way get compositions factors of Jacquet modules of parabolically induced representations for classical group. 11. Square integrable representations of p-adic general linear groups 11.1. Denote by

C the set of all equivalence classes of irreducible cuspidal representations of all G L ( n , F ) ,n 2 1. A segment in C is the set of form

A = [PI &I

= { P , v p , . . ., ~ I c P ) ,

where p E C, k E Z+. Denote the set of all such segments by S.

For a segment A = [p, vkp] = { p , v p , . . . , v k p } E S, the representation ukp x

2-1p

x

.. . x

up x p

contains a unique irreducible subrepresentation, which will be denoted by

fi(A).

M. T a d 2

124

Then 6(A) is essentially square integrable representation, and in this way one gets a bijection from S onto D (which is the set of all the irreducible essentially square integrable representations of general linear groups GL(n,F ) , n 2 1).This is one of the consequences of Bernstein and Zelevinsky theory, which is based on Gelfand-Kazhdan theory of derivatives. One can obtain these results also by different methods. In applications of square integrable representations of general linear groups, it is important to know what are Jacquet modules of these representations. This tells the following simple formula

c k

."I>>

m*(6([Pl

=

6([Vi+'P,

&I>

@

S ( [ P , YZPl)

(10)

i=-l

(see 1761).

11.2. As we have seen, each segment of S determines uniquely essentially square integrable representation. Let us explain how t o "read" corresponding segment from an essentially square integrable representation

6

= 6 ( A ) ,A E S .

For this, we shall introduce two natural invariants of 6. There exists exactly two (inequivalent) p 1 , p 2 E C such that p2 x 6 reduce. We can, after a possible renumeration, assume e(Pd

p1

x 6 and

Ie(p2).

Representations p1 and p2 will be called cuspidal reducibilities of 6 the lower one and p 2 the upper one). Then

(p1

Thus cuspidal reducibilities of 6 determine completely the segment in C corresponding to 6.

11.3. Let

61 E

D have cuspidal reducibilities

p1,

and p2. Take 6 2 =-6 ( A 2 )E

D. Then 61

x 62

reduces if and only if

(1) card((LJ1,P 2 ) n A2) = 1; ( 2 ) neither p i nor p 2 is a cuspidal reducibility of d(A2).

Some Classes of Irreducible Representations

125

12. Two simple examples of square integrable representations of classical p-adic groups In the rest of these notes, we shall fix one of the series of classical groups, symplectic or odd-orthogonal, and denote by S, either Sp(2n,F ) or SO(% I, F ) . Before we proceed further with description of general square integrable representations, we shall give two examples of square integrable representations of classical groups. The trivial one-dimensional representation of a group G will be denoted by I G .

+

12.1. Example:In this example we shall describe Steinberg representations

for symplectic groups. Steinberg representation can be constructed for any reductive group. Here we consider the series S, = Sp(2n,F ) . An easy computation of modular character of Pa in S1 = S p ( 2 , F ) = SL(2,F ) implies that

Is,

-

v-l

1 F X

x ls,,

since v-l1FX x Iso contains constant functions. The length of the Jacquet module of v-l 1 F x x IS, for the (standard) minimal parabolic subgroup is 2 (and irreducible subquotients are not isomorphic). This implies that v-' 1 F x x IS is a, length two representation. Further, Frobenius reciprocity implies that v-' I F x x IS, is not completely reducible (i.e. it is not a sum of irreducible subrepresentations). Now passing to contragredients we see that

v

1 F X

x Is,

contains a unique irreducible subrepresentation. This representation will be denoted by Sts, , and called Steinberg representation of S1.We can see (from the algebraic structure of R ( S ) over R ) that the Jacquet module for the minimal parabolic subgroup is v 1 ~ @x Is,. Now Casselman's square integrability criterion in this situation implies that Sts, is square integrable. Define Sts, t o be IS,. Now both v2

~

F

x Sts, and S([v IFX , v 2 1 ~ ~x1Sts, )

X

embed into v2 1FX x u

1Fx

x Sts,.

M. TadiC

126

Analyzing Jacquet modules, we would see that these two subrepresentations have exactly one irreducible subquotient in common, and that this subquotient is square integrable. We shall denote it by Sts,. It is a unique irreducible subrepresentation of u2 1FX x u ~ F Xx Stso. Continuing recursively in the above way, we define the Steinberg representation StSP, for any S,. It is a unique irreducible subrepresentation of un

1p.x

x un-l 1FX x

.. . x

u2 1p.x x u 1FX >a Sts,

(this defines StSp,). It is again easy to write what are the Jacquet modules of these representations: n

p*(Sts,)

= ~ s ( [ u ~ + + l l F x , u ~ l@ FS xt]S)k . k=O

+

12.2. Example: Let we now consider the series S, = SO(2n 1,F ) . An easy computation of modular character of Po in S1 = S O ( 3 , F ) implies that ISl Lf u-l12 1px

x Iso

(since u-l12 lFx x IS, contains constant functions). Consider the representation U1I2

1FX x u-l12 1FX x

Is,.

Here 6([u-1/2 1FX,u112 IFXI)x IS, and u 1 / 21Fx >a IS,

are subrepresentations. Looking a t Jacquet modules, one sees that S ( [ u - 1 / 2 l p x , u 1 / 2lpx])x

Is,

(11)

reduces. This follows from S.S.

>+

( b ( [ v - 1 / 21FX,Y112 1FX])x ISo

g S.S. and

S.S.

(

Y1/'

1 p X

(u112 1FX x

1

>a ISl

u-l12 1FX >a ls,)

Some Classes of Irreducible Representations

127

what one checks using the structure of R ( S ) . The representation (11) is unitarizable, so it is completely reducible. considering the Jacquet module of this representation for P(2,, and applying Frobenius reciprocity, we get that the representation 6 ( [ ~ - l /l~ ~u 1 / 2~I F X, ] )x Iso reduces into a sum of two inequivalent irreducible subrepresentations, say TI and T2. Thus

6([u-1’2 IFx, Y

~ IFx]) / ~x

Iso = Ti @ T2.

Now 6([v-1/2

l F X ,

u 3 / 21FX I) x Iso

L-)

u3’2 1FX x 6([u-1/2 l p x , u1/2 l F X ] ) x Iso = U3l2 1 F x X

(Ti @T2)

Eu

~ 1Fx / x~ Ti @ u ~ 1 F/x x~ T2.

Now the multiplicity of u3l2 l F x @ Ti in corresponding Jacquet module of u3l2 1 ~ xTi x is one. This implies that u3I2 1~~ xTi has a unique irreducible subrepresentation. These two irreducible subrepresentations (for i = 1,2) are square integrable. One can show that u 3 / 2 lFXx Ti are subquotients of corresponding Jacquet module of 6 ( [ ~ - l /I~F x ,v 3 / 2l F x ] ) x 1s0. From this we see that 6([v-1/2 l F X , U 3 I 2

l p ] ) x IS,

has exactly two irreducible subrepresentations. They are inequivalent and square integrable. Next question is how to distinguish these two irreducible square integrable subrepresentations. One can show that they have Jacquet modules of different length. This is one possible way to distinguish them.

13. Invariants of square integrable representations of classical p-adic groups

13.1. Let 7r be an irreducible square integrable representation of a classical group S,. C. Mceglin has attached to it a triple (Jord(.rr),7rcu,p,

Ex).

Each of these three parameters was considered earlier (at least in some form), but C. Mceglin was the first who considered them in this form.

M. Tad2

128

We shall describe each of these parameters. Our goal will be to explain their meaning from the point of harmonic analysis (they have a clear meaning from the point of view of Langlands program, what will be discussed later).

13.2.Jordan block of n:This is probably the most important of the three parameters (this parameter should determine the L-packet in which the representation lies). As we shall see later, the definition of J o r d ( n ) is very natural from the point of harmonic analysis (and can be given completely in terms of harmonic analysis). For p E C and a E N denote

We shall start with the first definition of Jord(n-) (this is a little bit modified definition, to avoid L-functions). Jord(n-) is called the Jordan block of n- and it consists of all (p, a ) E C x N such that (1) p is selfdual (i.e. Z; "= p; then p is unitarizable) and (2) if d / ' p >a Iso is reducible (resp. irreducible), then a is even (resp. odd) and

is irreducible. Although the above definition is simple, the clear meaning and importance of Jordan blocks is not evident from it. This is the reason that we shall give another description of Jordan blocks, from which will be much more clear importance of Jordan blocks for the harmonic analysis.

13.3. As we have mentioned already above, there is a very natural way t o come to Jordan blocks from the point of view of harmonic analysis, and we shall explain it bellow. Besides, because of the importance of Jordan blocks, they deserve to be understood as good as possible. Before we start to explain it, let us note that the classification of irreducible square integrable representation of classical groups is done under a natural assumption, which will be explained in section 15.1. This assumption shows up in proofs, not in the expression of the parameterization of irreducible square integrable representations. We shall assume that it holds in further.

129

S o m e Classes of Irreducible Representations

Now we are going to explain the importance of J o r d ( ~ for ) harmonic analysis. Once we have an irreducible square integrable representation T of a classical group, having in mind classification of the non-unitary dual via the Langlands classification, the first question that arises is: Which irreducible tempered representations can be obtained from this

T.

In other words, we would like to understand how a representation of the form

61 x

62

x

... x

&I,

x

T

(12)

reduces, when 6i are (unitarizable) irreducible square integrable representations of general linear groups. If we would know the answer to this question, we would have a reduction of understanding of irreducible tempered representations of the classical groups (and in this way also of all the irreducible representations) to the problem of understanding of irreducible square integrable representations of the classical groups. Therefore, understanding of such a reduction would be of the first class importance. The theory of R-groups reduces this question to the question when

6xlr

(13)

reduces, for 6 an irreducible (unitarizable) square integrable representation of a general linear group.

Remark: For further discussion of Jordan blocks, one does not need to understand this reduction. But for the classification of the non-unitary duals, one needs it. Therefore, we shall explain the reduction that gives the theory of R-groups, without going deeper in this theory. Consider representation from (12). Denote by e the number of inequivalent 6i among 61,62, . . . ,6k such that 6a x

lr

reduces. Then 61 x 62 x . . . x 6 k x lr is a multiplicity one representation and it reduces into a direct sum of 2j

irreducible (tempered) representations.

M. T a d 2

130

If p is a permutation of {1,2,. . . , k} and e l l € 2 , . . . , Ek E {fl},then representations 61 X 62 x . . ’ X bk x 7r and bitl) X dG:2) X + ’. x b;;, >a 7r are equivalent, where bit.) denotes 6p(i) if E . = 1 and it denotes &(.) if ~i = -1. Let 6; x S$ x . . . x Sk, x 7r‘ be another representation] such that 6; are (unitarizable) irreducible square integrable representations of general linear groups and 7r’ is an irreducible square integrable representation of a classical group Sq,. Suppose that S1 x 62 x . . . x bk >a 7r and 6; x S$ x ... x Sh, XI 7r’ have an irreducible subquotient in common. Then 7r 2 7r’ (and therefore q = q ’ ) ] k = k’ and there exists a permutation p of {II2 , . . . k } and €1, E Z ] . . . ,E k E {fl}such that

s;

E S”

P(%)

for all i = 1 , 2 , . . . , k.

13.4. As we already have mentioned, to understand irreducible tempered representations] we need to understand when representations S x 7r (from (13) reduce. Having in mind the classification of irreducible square integrable representations of general linear groups] one needs to understand when %%a) x

77

reduces, for unitarizable p E C and for a E N. When we fix p , the reducibility of these representations can be described in a very nice way (a crucial role in this is played by Jord(7r)). Frobenius reciprocity implies irreducibility if p is not selfdual. Therefore, it remains to understand the reducibility for selfdual pis. The following two examples are very simple but nice examples, from which one can get an an idea what happens regarding these reducibilities in general.

13.5. Examples: Let S, = Sp(2n, F ) and

7r

= Iso.

(1) Suppose $I is a character of order two of F X = GL(1, F ) . Then

S(+, a ) x Iso is irreducible for all even a; S($I,a) x Is, is reducible for all odd a. We see that understanding of reducibility in this case is very simple. One needs only to know the parity of N for which we have reducibility. Unfortunately, this is not always the case for other square integrable representations.

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S o m e Classes of Irreducible Representations

Now we shall give a simple example of a situation of a slightly different type.

+

(2) We shall consider now instead of the trivial representation GL(1,F ) (on one-dimensional space). Then

~

F

X

of

5(1Fx, a ) x l s o is irreducible for all even a ;

6 ( l F X a, ) x Iso is reducible for all odd a , except for a = 1. Now we shall explain what happens in general regarding such reducibility. Fix selfdual p E C. Then for exactly one parity in

N holds:

(1) b ( p , a ) x x is reducible for all a from that parity, with possibly finitely many exceptions; (2) 6(p,a) x x is irreducible for all a from the other parity. The parity of W for which ( 1 ) holds, will be called the parity of reducibility of p and x (note that for this parity we can have finitely many exceptions of reducibility), and the other parity will be called the parity of irreducibility of p and x (in this parity we have always irreducibility). Therefore, for understanding tempered representations we need to know which is the parity of reducibility for selfdual p E C, and what are exceptions (if there are exceptions, then clearly they determine the parity of reducibility). Therefore, it is very important to know these exceptions. This is just Jord( x):

New definition: Jord(7r) is the set of all exceptions ( p , a ) in (l),when p runs over all selfdual representations in C ( a E N). Suppose that x is an irreducible square integrable representation of S,. C. Mceglin has proved that

where dp is determined by the fact that p is a representation of GL(d,, F ) . The above inequality clearly implies that Jord(x) is finite. The above inequality is expected t o turn to be an equality.

13.6. Partial cuspidal support of x: In general, (conjugacy class of) an irreducible cuspidal representation T of a Levi factor M of a parabolic

M . Tadid

132

subgroup P in a reductive group G is called cuspidal support of 7 r , if 7r is a subquotient of Indg(7). For classical groups, Levi factor M is a direct product of general linear groups and a classical group. Therefore, 7

p1 63 p2 63 . . . 63 pz 63 cl,

where pi E C and CT is an irreducible cuspidal representation of a classical group. Now the definition of partial cuspidal support of 7r is

cusp = CT. We can define partial cuspidal support of 7r also in the following way: an irreducible cuspidal representation CT of a classical groups is called partial cuspidal support of 7r (and denoted by 7rcusp) if there exists a smooth representation 7r' of a general linear group, such that 7r

L)

7r'

x IY.

13.7. Partially defined function E , on Jord(7r):As we could see already from the simple Example 12.2, the construction of irreducible square integrable representations of classical groups involve reducibility of tempered induction, and thus R-groups (which for classical groups are sums of 2/22). This is roughly behind the fact that parameters of irreducible square integrable representations of classical groups will involve functions with values in {fl}. The definition of the domain of partially defined functions E , on Jord(7r) is quite technical. Because of this, we shall not give a complete definition of the partially defined functions (besides, from the general definition of partially defined functions, it is not quite easy to understand what are these functions). Rather, we shall try only to explain the main properties of these functions. One can understand pretty well the classification of irreducible square integrable representation of classical groups without knowing all the details of the definition of partially defined functions E , . Later, we shall give a constructive definition of these functions. Let X be a free Z/2Z-module with basis Jord(7r). We shall denote operation in the module X multiplicatively. Characters of this group are in a natural bijection with functions

Jord(7r)

--$

{fl}.

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Some Classes of Irreducible Representations

We can think of of

E,

as a function on a subset of X, in our case on a subset

Jord(7r) u ( 2 1 2 2 ;

21,Q

E Jord(7r),z1 #

which can be extended to a character of X . Further, for 2 1 , E~ Jord(7r) we shall write also as

22),

~ ( 1 ~ 1 when x 2 ) ~it

is defined,

GI (21 ) E x ( 2 2 )

even if E , ( z ~and ) ~ ~ ( 2 are 2 ) not defined. The fact ~ ( 2 1 x 2= ) 1 (resp. ~ ( 2 1 x 2 = ) -1) will be written also as Gr(Z1) = %(Q)

even if

~ ~ ( 2 and 1 )

(resp.

E,(21)

#452))

E , ( z ~are ) not defined.

13.8. To give an idea of definition of E , , we shall describe one important case. The function E , is always defined on (p,a)(p’,a’) if p = p’, a # a‘ (and both ( p , a ) , ( p ’ , a’) are from Jord(7r)).

< a , and ( p , a ’ ) $ Jord(7r) for any a- < a’ < a.

Suppose ( p , a - ) ,( p , a ) are in Jord(n), a-

Then E , ( ( P , a - ) ( p , a ) ) = ~ , ( p ,a - ) ~ , ( p , a ) is defined and €,(PI

Q-)E7r(P1

a) = 1

if and only if there exists a smooth representation such that 7r

L-)

S([v(”-+1)/2p, v

7r’

( ” - q ] )x

of a classical group

T’.

13.9. In general, ~ , ( p a, ) is not always defined for ( p , a ) E Jord(n). It is always defined if a is even. The definition in this case is the following: Suppose ( p , a ’ ) E Jord(7r) with a’ even. Chose a minimal a such that ( p , a ) E Jord(7r). Then GT(p,a) = 1

if and only if there exists a smooth representation such that 7r

-

7r’

b ( [ v 1 / 2 p ,v ( 4 / 2 p ] )>a 7r’.

of a classical group (14)

M. TadiC

134

If a is odd, then ~ , ( p , a ) is not always defined. It is not defined if and only if (P, b) E

JOT~(T~,,~)

for some b E N. From the above condition for ~ , ( p , a )to be defined in the case of odd a , one can show that if ~ , ( p ,a ) is defined (for odd a ) , then

p

Tcusp

(15)

reduces (this is related to basic assumption under which we consider the classification of irreducible square integrable representations of classical groups; this assumption will be explained later). C. Mceglin has used normalized intertwining operators t o define E , ( P , a ) in this situation. In the case when ~ , ( p , a ) is defined for odd a , as we have mentioned already p x T,,,~ reduces. It reduces into a sum of two inequivalent irreducible subrepresentations. One can chose one of these subrepresentations and attach to it 1, and to the other attach -1. Then one needs t o extend in a natural way this choice to other tempered representations coming from inducing representations including 6 ( p , a ) as a factor. One can do this using intertwining operators, but one can also do it without them. Now we have almost complete definition of 6 , . 13.10. C. Mceglin has shown that for an irreducible square integrable representation 7r of a classical group, the triple

satisfies some technical conditions. The triples that satisfy these technical conditions she called admissible triples. We shall not give in the moment this technical definition. We shall give later a different and more explicit description of admissible triples. Let us just say that admissible triples are combinatorial objects modulo cuspidal data. It will become soon clear what we mean by cuspidal data. C. Mceglin has proved that the mapping attaching an admissible triple to an irreducible square integrable representation of a classical group, is an injective map from the set of all the equivalence classes of irreducible square integrable representations of classical groups (we fix a series of classical groups and a non-archimedean field F ) into the set of all the admissible triples. Jointly, we have proved that this mapping is surjective. This means that we have a bijection between irreducible square integrable representations of classical groups and admissible triples. Since admissible triples

135

S o m e Classes of Irreducible Representations

are combinatorial objects modulo cuspidal data (as we mentioned already above) , we have a classification of irreducible square integrable representations of classical groups modulo cuspidal data.

14. Reduction to cuspidal lines The classification of irreducible square integrable representations of classical groups modulo cuspidal data will be easier to understand if we pass to cuspidal lines. We shall explain it in this section. This reduction could be important also for some other purposes.

14.1. Fix an irreducible cuspidal representation a of a classical group S, and fix inequivalent selfdual irreducible cuspidal representations P I , . . . , p k of general linear groups. Denote by D(Pl,.. . I P k ; 0)

the set of all equivalence classes of irreducible square integrable subquotients of representations ualrl x

v=2r2

x

. . . x Pert

>a

a,

where a, E R,7i E { P I , . . . ,pk}. Then there is a natural bijection from D ( p 1 , . . . , p k ; c) into the Cartesian product

This bijection is given in the following way. Fix T E D ( p 1 , . . . , P k ; a) and 1 5 5 k. Then there exists an irreducible representation r, of some Sn, which is a subquotient of some vplp, x vpzp, x . . . v P ' 3 p, x1 0 , pzE R,and there exists an irreducible representation 0, of a general linear group which is a subquotient of r1 x 7-2 x . . . x rm3with r2E UF=l,tz3 {v"p,; Q E R}, such that T

v 0,

x1 T,

C. Jantzen has proved in 1321 that representations T I , . . . , rk are uniquely determined by T , and that they are square integrable. Further T I---+ k ( T I , . . . , T k ) defines a bijection from D(p1,.. . , P k ; 0 ) onto D(pz;0). Each irreducible square integrable representation of a classical group belongs to some D(p1,.. . , P k ; a). Further, if a o', then

n,=,

D ( P l , . . . i P k ; c )nD(P;,-,Ph')

= @ I

M. TadiC

136

and if

then

v(pl,..., p k ; r ) n ’ D ( p ’ ,,..., ,&;a)=V(& ,..., p ; , , ; ~ ) In this way we obtain a reduction of the classification of irreducible square integrable representations of classical groups, to the problem of classification of sets

( p E C is selfdual).

14.2. Consider the projection OF

OF/PF.

Lift it to the level of groups. In this way one gets a natural homomorphism from the maximal compact subgroup KO in S, to the group S, over the field O F / P F . The preimage in KO of the standard minimal parabolic subgroup in S, over the field OF/pF is denoted by

1 This open compact subgroup is called Iwahori subgroup of S,. The study of irreducible smooth representations with Iwahori fixed vectors has attracted a lot of attention. In this case, for building the representation theory, one does not need non-trivial cuspidal representations (i.e. other than characters). Also, corresponding group algebras for this setting have a nice geometric realization. Using this, one can obtain construction of irreducible representations by geometric methods, what was done by D. Kazhdan and G. Lusztig. For classical groups, to determine irreducible square integrable representations with Iwahori fixed vectors, it is equivalent t o determining of D(1FX

; SO)

v($;lSo),

where +IJ.! is a (unique) character of order 2, which is unramified (i.e. which is trivial on 0 ; ) . Irreducible square integrable representations with Iwahori fixed vectors are parameterized by the Cartesian product 2)(1FX ; Is,) x

V($;l s o ) .

S o m e Classes of Irreducible Representations

137

15. Parameters of D ( p ; a) 15.1. For selfdual p E C and an irreducible cuspidal representation a of S,, A, Silberger has proved there exists a unique Qp,o

20

such that wapj,p

xo

reduces. Now we shall say what is the basic assumption (for p and o),under which V ( p ,a) is classified: (BA) for p and o

ap,a

- Qp,ls, E

z.

This assumption is needed (essentially only) in proofs. F. Shahidi has proved that (BA) holds if a is generic. It is also known that (BA) holds in some other cases. In general, (BA) would follow from the truth of some general Arthur’s conjectures. F. Shahidi has proved that

15.2. We shall fix p and a as above, and assume that (BA) holds for p and a. Denote in the sequel (Y

=

Now since we assume that (BA) holds for p and a, (16) implies CY

Note that for

7r

= ( Y p , o f (1/2)2+.

E V ( p , a),

cusp

= 0.

Therefore, since 0 is fixed, for classification of V ( p ,0)it is enough to consider instead of triples (Jord(7r),7Tcuspr E 7 r )

pairs

(Jord(r) E7r) 7

M. TadiC

138

(which form with g admissible triples). Further, for classification of D(p, 0 ) it is convenient (and enough) to work with

Jord,(n) = { a E W; (p,a) E Jord(n)}. instead of Jord(n). Now pairs

(Jordp(n)E 7 T ) 7

will be parameters of representation in D ( p , o ) . Note that Jord,(n) is a finite subset of either 2N or 2N - 1 and ex is now regarded as a partially defined function on Jord,(n). Because of this, the parameters of D ( p , a ) are now simpler than before. There are two possibilities for a. The first is

a E Z+, which will be called integral case, and the second is Q

E ( ( 1 / 2 P + \ Z+),

which will be called non-integral case. 16. Integral case

We shall suppose in this section that

and describe parameters (Jord,,

z+,

Q

E

E)

of elements of D(p,u ) in this case.

16.1. In the integral case we have always

Jord,

C 2N - 1.

Here partially defined function is defined on elements of Jord, if and only if a = 0. If it is defined on Jord,, then the values on Jord, completely determine the partially defined function. If Q 2 1, then E is defined only on pairs from Jord, (and this partially defined function can be extended to a character of a free Z/2Z-module with basis Jord,). 16.2. In the integral case, Jord, will be called of alternated type if

card(Jord,) = a.

Some Classes of Irreducible Representations

139

Here always exists a unique partially defined function E such that Jord,, and (T form an admissible triple. If cr. = 0, then there is nothing t o define. If a 2 1, then E is not defined on elements on Jord,, but it is defined on pairs. It is completely defined by the following property:

E

For each a _ , a E Jord,, a-

< a , such that

[a-,a] n Jord, = { a - , a } we have 4a-)

# 4a)

(this is where the name alternated type comes from). Now we shall define the representation corresponding to alternated Jord, (and E ) . Write Jord, = { a l ,a2,. . . ,u a } . After a renumeration we can assume

Now the representation

has a unique irreducible subrepresentation, which will be denoted by n(Jordprc,E).

This representation is square integrable. An example of such representations are Steinberg representations for symplectic groups. 16.3. We shall describe now general parameters of elements of D ( p ; a ) . Take Jord, (and E ) of alternated type. Take any a _ , a E 2N - 1, a- < a, such that [ a _ ,a] n Jord, = 0.

Denote

J o r d r ) = Jord, u { a - , a } . Extend

E

in a way that

€ ( a _ )= € ( a ) .

M. Tadid

140

It is easy to see that there are exactly two such extensions. Denote them by €1,€ 2 . Now ( J o r d y ) ,(T,~ i ) i, = 1 , 2 , are (new) admissible triples (in this setting). These triples are no more of alternated type. We can continue this construction, but now starting from J o r d r ) ,(T,~ i . Continuing this process, we construct Jord,(2) , J o r d f ) , . . . (and corresponding partially defined functions). In this way, we shall get all the parameters of 'D(p; 0).

16.4. We shall describe now representations corresponding to these (new) parameters. Let (Jord,,E),a-,a and (Jordr),c,ci)be as in 16.3. To alternated (Jord,,e) we have already attached in 16.2 a square integrable represent ation *(Jord,,U,E).

Now the representation

S( [,-(.--Wp,v(a--1)/2 PI)

>a

~(JO?-dp,U,€)

contains exactly two irreducible subrepresentations. One shows this using the strategy that we have used in Example 12.2. These irreducible subrepresentations are square integrable, and their parameters are (Jord,(1), CT,€1) and (Jord;) , (T,€2). If Jord,, # 0, one determines from 13.8 which subrepresentation corresponds to which ~ i .

16.5. It remains to say which subrepresentation t o attach t o which ~i if Jord, = 0. C. Mmglin has used normalized intertwining operators for this attaching. One possibility would be to proceed in the following way. Suppose

Jord,, Then a

= 0.

= 0.

Write p >a

(T

= 7-1 @ 7 - 1

as a sum of (two inequivalent) irreducible representations. The representations

S( [.-'a-

- W p J", -1)/2 PI)

f J

and 6 ( [ v p ,v(a--1)/2 p ] ) x b ( [ v p ,v(a--1)/2 PI) >a Ti

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Some Classes of Irreducible Representations

have exactly one irreducible subquotient in common (for each i = - l , l ) . Denote it by Ti. Now ,J([v("-+1)/2p, v(a-1)/2p])

Ti

contains a unique irreducible subrepresentation. Denote it by 7 r i . Then 7r1 and 7r-1 are two inequivalent irreducible subrepresentations of ,J( [ v - ( " - - l ) / 2 p, v(a-1)'2Pl) CT (here 7 r ( J O T d p , f l r E ) = .). One natural possibility to distinguish irreducible subrepresentations of

,J([ v - ( a -

-1)/2

p, v ( a - q ) M

c7

is to attach ~i t o 7 r E i ( a ) (for i = - 1 , l ) . Let us note that we have not checked that this choice is the same as the choice that C. Moeglin made using normalized intertwining operators. 16.6. One proceeds further from Jord;) t o J o r d p ) , J o r d p ) to Jord,(3) , . . . recursively in the same way as we did in passing from Jord, to Jord,(1)

(it is even less complicated here, since always Jord;) # 0, J o r d p ) and therefore we do not need to make choices as in 16.5).

# 0,. . .

17. Non-integral case

Now we shall assume that

a E ((1/2)2+\ Z+). 17.1. In this case.

Jord,

2N.

Partially defined functions are defined on elements of Jordp's (and values on Jord,'s completely determine partially defined functions).

17.2. In the non-integral case, Jord, will be called of alternated type if card(Jord,) = (I! f 1/2, i.e. if card(Jord,) = a - 1/2

or

card(Jord,) = a

+ 1/2.

In this case, there exists a unique partially defined function E on Jord, such that Jord,, CT and E form an admissible triple. This partially defined function E is defined (and uniquely determined) by the following conditions: For each a _ , a E Jord,, a- < a, such that [u- , u] n Jord, = { u - , u }

M. TadiC

142

we have

and c(min(Jord,)) =

1

-1

if card(Jord,) = a if card(Jord,)

+ 1/2;

=a -

1/2.

17.3. We shall now describe the representation corresponding to the above alternated Jord,. Let Jord, = { a l ,a 2 , . . . , aa+1/2}.After a renumeration we can assume

Consider first the case card(Jord,)

= CY

-

1/2

Then the representation

has a unique irreducible subrepresentation. This subrepresentation will be denoted by

n(Jord,,u,e). It is a square integrable representation which corresponds to (Jord,, Now assume card(Jord,)

=a

E).

+ 1/2.

Then the representation

has a unique irreducible subrepresentation. We again denote this subrepresentation by

n(JoTdp,urE). This is a square integrable representation which corresponds t o ( Jord,, E ) . We have described above how one attaches square integrable representations t o alternated parameters.

S o m e Classes of Irreducible Representations

143

17.4. Now we define general parameters of D(p;n) in the same way as in the integral case. We also attach square integrable representations in the same way. The only difference which occurs between integral and non-integral case is in passing from Jordp to Jordp) when Jordp = 0. In the non-integral case one uses (14) to determine which irreducible subrepresentation corresponds to which ~ i . 18. Local Langlands correspondences 18.1. Denote by WF the Weil group of F . This is a dense subgroup of the Galois group of the separable algebraic closure of F over F (the topology is not the induced one from the Galois group, but a slightly modified one). Let G be a split connected reductive group over F (as GL(n, F ) ,Sp(2n,F ) or SO(2n 1,F ) ) . By the Langlangs program, there should exist a natural partition of G into finite subsets, called L-packets, which are indexed by (conjugacy classes of) admissible homomorphism of WFx SL(2,C) into the complex dual group LGo of G (admissible here means that the homomorphisms are continuous, that they carry WF into semi simple elements and that they are algebraic on SL(2,C)). For the cases of the groups that we consider, complex dual groups are as follows:

+

L G L ( ~5'),' = G L ( n ,C), LSp(2n,F)' = SO(2n + 1,C), L S 0 ( 2 n+ 1,F)' = S p ( 2 n ,C) (a property of the complex dual group 'Go is that it has the root system dual to the root system of G). We shall now concentrate our attention regarding the above aspect of the Langlands program to irreducible square integrable representations. In this case square integrable L-packets should be parameterized by admissible homomorphisms whose image is not contained in any proper Levi factor. Further, elements of a square integrable L-packet, which is indexed by an admissible homomorphism $, should be parameterized by irreducible representations of the component group of $ (which is the quotient of the centralizer of the image of $ by the connected component). The correspondence that one would get in this way is called local Langlands correspondence for G.

144

M. Tadit

18.2. In the case of general linear groups, by the Langlands program there should be a bijection of irreducible square integrable representations of GL(n,F ) and n-dimensional irreducible representations of WF x SL(2,C) (which are admissible homomorphisms). Here component groups are trivial. In this bijection irreducible cuspidal representations of GL(n,F ) should correspond to irreducible representations of WF The work of Bernstein and Zelevinsky, which gave classification of irreducible square integrable representations of general linear groups modulo cuspidal representations, resulted in a reduction of establishing of local Langlands correspondence to the cuspidal case, i.e. to establishing a correspondence between (classes of) irreducible cuspidal representations of GL(n,F ) and (classes of) irreducible representations of W,. More precisely, suppose that

is such a correspondence for general linear groups between irreducible cuspidal representations of general linear groups and irreducible representations of W, (we consider all general linear groups over F together). From the representation theory of SL(2,C) one knows that for each a E W there exists a unique irreducible algebraic representation

of SL(2,C) on a-dimensional complex vector space (up to an equivalence). Then the formula for local Langlands correspondence on the set of (classes of) irreducible square integrable representations of general linear groups, which we shall denote also by 'p, would be CP(~(a P ), ) = P ( P ) 8 Ea.

Local Langlands conjecture for GL(n,F ) has been recently proved in full generality (by M. Harris and R. Taylor in [25], and by G. Henniart in

[261).

18.3. One may ask does the classification of irreducible square integrable representations of classical groups modulo cuspidal data give also a similar reduction. The natural candidate for the Langlands correspondence @ for classical groups is

Some Classes of Irreducible Representations

145

(p is the local Langlands correspondence for general linear groups, which we have considered before). But it remains a number of facts to prove even to see that it is a good candidate (for the beginning, it is not clear at all that @ ( T ) goes in the right group).

18.4. For classical groups, the centalizers of images of admissible homomorphism of W , x SL(2,cC) into the complex dual group, whose images are not contained in any proper Levi factor, are finite groups which are 2/22modules (these are component groups). Therefore, after choosing a basis, irreducible representations of the component group correspond to functions from the basis to {fl}. Now E~ should give a part of the irreducible representation (i.e. character) of the component group corresponding to 7 r . The rest should come from E ~ , , , ~(once the local Langlands correspondence is established for cuspidal representations of classical groups). Complete discussion regarding this reduction one can find in [41]. 19. Non-unitary duals of classical p-adic groups 19.1. The classification of irreducible square integrable representations of classical groups modulo cuspidal data implies also a classification of all the irreducible smooth representations of classical groups modulo cuspidal data (by cuspidal data we mean irreducible cuspidal representations of general linear and classical groups, and cuspidal reducibilities) . Suppose that a selfdual p E C and irreducible square integrable representation T of a classical group are given. For understanding tempered representations, one needs to know the parity of reducibility. If Jord,(~) # 0, then the parity which shows up in J o r d , ( ~ ) is the parity of reducibility of p and 7 r . If J o r d , ( ~ )= 0, then Jordp(7rcusp)= 0 and then the reducibility of p and 7rcuuSp(and also T ) is at 0 or 1/2. If the reducibility is at 0 (resp 1/2), then the parity of reducibility of p and T is odd (resp. even). 19.2. We can describe also the non-unitary dual by reduction to cuspidal lines. Let be an irreducible cuspidal representation of a classical group S, and let p 1 , . . . ,p k E C be unitarizable such that for z # j , sets { p i , pi} and { p j , & } have no equivalent representations (i.e. pi Pj). p j and pi

146

M. TadiC

Denote by

the set of all equivalence classes of irreducible subquotients of

where

Then by [32] there exists a bijection

similarly as in 14.1 (for complete definition of the bijection one should consult [32]). The classification of the non-unitary duals of classical groups reduces to the classification of the sets Z(pi;a) in a similar way as the classification of the irreducible square integrable representations in 14.1 reduces to cuspidal lines (we shall not write details here, but the reduction is analogous). 19.3. Fix a unitarizable p E C and fix an irreducible cuspidal representation of a classical group. If p is not selfdual, then the tempered induction in Z ( p ; o ) is always irreducible. Now irreducible tempered representations which show up as Langlands parameters of representations in Z(p;g) is easy to write down (using Remark 13.3 to know equivalences among them). Suppose now that p is selfdual. Let the reducibility of p and (T be a = ap,D2 0. Now the parity of reducibility of p and (T (and also each square integrable representation in Z ( p ;0 ) ) is odd (resp. even) if a E Z (resp. a @ Z). Further, one can describe easily irreducible tempered representations which show up as Langlands parameters of representations in Z(p; (T), since we know the parity of reducibility of p and (T (one needs also to use Remark 13.3). (T

S o m e Classes of Irreducible Representations

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20. Unitary duals of general linear groups over local fields

20.1. Denote

D"

= (6 E

D ;e ( b ) = O } .

These are just all the square integrable classes in D. The following theorem describes the unitary duals of general linear groups over any local field (archimedean or non-archimedean).

Theorem: For a representation 6 E D" and m 2 1 denote U ( S ,m ) = L ( v ( " - 1 ) / 2 S ,

v(m-3)/2(5,.

. . ,v - ( m - 1 ) / 2 6 )

For 0 < a < 112 and S and m as above, denote r(u(S,m ) ,a ) = voIu(S,m ) x vPau(S,m).

Let B be the set of all possible u ( b , m ) and n(u(S,m),a)with S,m,a as above. T h e n (i) If rl,7 2 , . . . , r, E B , then the representation r1

x r2 x . . . x

T,

is a n irreducible unitarizable representation of a general linear group. (ii) Let r 1 , r 2 , . . . , r,, ri,ri,. . . ,r;, E B , T h e n

r l x r ~ x ~ ~ ~ x ~ , ~ r i f and only i f n = n' and if one can obtain the sequence ( 7 1 , 7 2 , . . . , r,) f r o m (T;, r;, . . . , T;) by a permutation. (iii) Each irreducible unitary representation of a general linear group is isomorphic to a representation r1

x rz x ... x r,,

forsomer1,r2,...,Tn E B. The above classification theorem is the same for all the local fields. The difference in the form of unitary duals comes from the difference of the sets D" for different fields. 20.2. This theorem is proved in [60] in the non-archimedean case. As we already mentioned, the theorem holds for archimedean fields in the same form (using the notion of (8,K ) modules), with the proof alocg the same

M. TadiC

148

strategy as in the non-archimedean case (see [71]). D. Vogan in [72] has made quite different approach to the classification of unitary duals of general linear groups over archimedean fields. Not to deal all the time with non-archimedean fields, we shall now describe the proof of the above theorem for F = @. Since the proofs in the archimedean and non-archimedean case are along the same strategy, one will be able t o get from this description quite good idea of the proof in the non-archimedean case.

20.3. In the sequel of this section, by a representation we shall mean corresponding (8, KO)-module. Denote by

Irr" = UZroGL(n,C)-. Consider algebra R for complex general linear groups (constructed in 10.3). We shall consider

Irr"

R.

Recall that R is a polynomial ring over D (Proposition 10.7). In particular, R is a factorial ring. Therefore, we can talk about prime elements in R. In the complex case we have

D = GL(1,C)-. Note also that I la: is the square of the usual absolute value in @. We shall now introduce several claims, whose proofs shall be discussed later:

(UO) (Ul) (U2) (U3) (U4)

x T E Irr". c,T E Irr" b E D" and n E N j u(b,n)E Irr". b E D", n E N and 0 < a < 1 / 2 ==+ ~ ( u ( b , n ) , E a )Irru. 6 E D and n E N ==+ u(b,n)is prime in R. a , b E M ( D ) + L ( a ) x L(b) contains L ( a b ) as a subquotient.

+

The addition of multisets (which shows up in (U4)) is defined in obvious way: (21,.

. . , 2") + (Yl,. . . ,YV)

Proposition: Claims (UO) -

= (21, . . . , x u ,Y1, . . . , YV).

(U4)imply Theorem 20.1

Proof: First observe that (Ul), (U2) and (UO) imply (i) of the theorem.

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Some Classes of Ineducable Representations

Further, commutativity of R give implication +== in (ii). The implication + follows from (U3). It remains to prove (iii) (i.e. exhaustion). Suppose 7r E IT?. Then 7r

= L ( Y l , Y 2 , . . .,re)

for some y1,y2,. . . ,-ye E D (see Remark 9.6). Note that 7r is hermitian (since it is unitary). This, together with (5) and (6), implies that 7r

= L(VQ'bl,V-='&,

.. .,Va"k,V-=k6k,6k+l,.

. . , &),

for some ai > 0, 1 _< i 5 k and S j E D", 1 _< j 5 s. We shall use in the sequel the following simple fact

M ( D ) and L ( a l ) , L ( a z )., . . ,L(a,) E then L ( a 1 ) x L ( a 2 ) x . . . x L(a,) = L(a1 + a2 +. . . + a,). If

a1,a2,.

. . ,a,

E

(17)

This follows directly from (U4) and (UO) (by induction). To get an idea of proof of (iii), we shall now give a proof of it in the rigid case, i.e. when all ai E ( 1 / 2 ) 2 . Denote a(&,m ) = (v(,-1)&,

y(,--3)/26,.

. . , v-(m-l)/26).

Then obviously u(S,m) = L(a(6,m ) ) .

Using the fact that ( V a , S .a 1

v-**62)

+ a ( & , 2ai - 1) = a(&, 2a2 + 1)

and (17) (several times), from (Ul) we get that 7r

x u(62, 2a1 - 1) x ' . . x u(&,2 a k U(&,

2a1

-

1)

+ 1) x .. . x U ( b k , 2CYk + 1) x 6k+l x . . . x 6,.

By (U3), on the right hand side we have prime elements (from B ) . Since R is factorial, 7r must be a subproduct of the right hand side (up to a sign). So, 7r must be a product of elements from B . This proves (iii) in the rigid case. The proof of (iii) in the non-rigid case proceeds along a similar idea, but it is slightly technically more complicated in this case. 0 By the above proposition, to prove the theorem, it is enough to prove (UO) - (U4). Now we shall explain how the proofs of each of these claims go.

M. Tadid

150

20.4. ( U l ) : Considering the modular function of the standard minimal parabolic subgroup in G L ( m , C ) ,we get easily that for 6 E D = G L ( l , C ) ^ (i.e. a character of C " ) we have u(S,m ) = 6 o det : GL(m,C ) + C x .

Thus, u(6,m) is unitarizable. Therefore, ( U l ) holds. 20.5. (U2): The restriction of the representation r(u(6,m ) ,a ) (which we consider in (U2)) to SL(2m, C ) ,is a Stein's complementary series representation from [58] (if m > 1; if m = 1, then this is a well known complementary series representation of SL(2, C)). Therefore, it is unitarizable. From this one gets directly that r(u(6,m ) ,a ) is unitarizable as a representations of GL(2m, C ) (since it has unitarizable central character). One can get the unitarizability of representations r(u(6,m ) ,a ) by standard construction of complementary series representations (which are unitarizable). For this, see 20.11 bellow. 20.6. (U3): We shall illustrate the proof of (U3) on the example of u(6,2). Note first that R is a graded ring (by definition). The degree of ~ ( 6 ~ is two. The representation theory of SL(2,C ) implies that

4 4 2 ) = x1 x

x 2-x 3

x

x 4

(18)

for some Xi E D ,i = 1,2,3,4, where all Xi are different. Suppose that u(S,2) is not prime. Since it is primitive (the greatest common divisor of

coefficients is l), it must be a product of homogeneous elements of degree one. Write fz = c1 (4 XI

+ c p x 2 + cpx3+ c y x 4 ,

2 =

fl

and

f2

1,2.

Since X1 x X2 shows up in u(6,2) (see (18)), it follows that cI1) # 0 and cp) # 0 (after possible changing indexes of f1 and f 2 ) . Since X3 x X4 shows up in u(6,2), it follows that cy) # 0 and cf) # 0 (after possible changing indexes of X3 and X4). These observations imply that the total degree of u(6,2) in variables XI and X4 is 2. This obviously contradicts to the expression (18). This contradiction completes the proof that u(b,2) is prime (in R). The proof of (U3) in the general case follows the same strategy and uses only very basic facts about composition series of standard modules, i.e. principal series (which are standard facts of Langlands classification).

S o m e classes of Irreducible Representations

151

20.7. (U4): This claim follows from basic properties of composition series of principal series (which are standard facts of Langlands classification). We shall explain now how it follows. We shall consider s.s.(X(a)) E R for a E M ( D ) . For simplicity, we shall write s.s.(A(a)) as an element of R simply as A(a) E R. Let a1,a2 E M ( D ) . There exists a partial order 5 on M ( D ) (which is quite explicit and which is simple to describe), such that we have

L(a,) = X(a,) +

c

rn$!)A(b(t)),

2

= 1,2,

b ( %
in R (here m$l, E

Z).Now

L(a1) x L(a2) = q a l ) x X(a2)

+

c

rn$j)X(b(l))x

b ( 1 )
+

c b(1)
X(a2)

+

c b ( 2 )
m;;;)X(b(l)) x

m$?,X(b(2)) x

c

X(a1)

mgX(b(2)).

b ( 2 )
We know that L ( a l + a2) is a subquotient of X(a1) x X(a2). Standard properties of the Langlands classification imply that L(a1 a2) is not a subquotient of any of three sums on the right hand side of the above equality. This proves (U4).

+

20.8. (UO): Let P, be the subgroup of all the matrices in GL(n,C)which have bottom row equal t o ( O , O , . . . , O , 1) E Cn.Suppose that we know that

(K)

7r

E GL(n,C)^ jTIP, is irreducible

(in the above claim, we consider 7r as an irreducible unitary representation on a Hilbert space, not as a (8,KO)-module). It has been known for a long time that (K) implies (UO). The implication follows using small Mackey theory. For the implication, for irreducible unitary representations 7r1 and 7r2 of GL(n1,C)and GL(n2,C ) , using (K) one shows that ( T I x 7 r 2 ) 1 Pnlf n z is an induced unitary representation, which is irreducible by small Mackey theory. A complete proof of this implication can be found in [49] (but the proof is implicit already in [22]). M. Baruch proved (K) in [7].

Additional comments

152

M. TadiC

20.9. We shall give here a little bit more explanations regarding Baruch's

proof of (K) and the history of proving of (K). A.A. Kirillov observed in [35] that on a dense open subset of G L ( n ,@), in each GL(n,@)-conjugacyclasses there exists an open dense P,-conjugacy class. This clearly implies that each continuous function on GL(n,@),which is constant on P,-conjugacy classes, is constant on GL(n,@)-conjugacy classes. Further, the last observation implies that if a P,-invariant distribution on GL(n,@) is represented by a continuous function with respect to the Haar measure, then it is an invariant distribution, i.e. invariant for conjugation by elements of GL(n,@) (actually, this conclusion holds for wider class of functions than the continuous ones). Kirillov expected that this property holds for any P,-invariant distribution on G L ( n , @ )(not only for those one which are give by integration against continuous functions). He observed that this property would imply (K) in the following way. Take any T in the commutator of the representation ripn.For proof of (K), by Schur lemma it is enough t o show that T is scalar. Kirillov considered the distribution

AT : 'p H "race(Tr('p)),

(19)

which is Pn-invariant (T is in the commutator of ripn).Now the property that Kirillov expected for general P,-invariant distributions, would imply that the above distribution is invariant for the whole group. Using irreducibility of ir, it is easy t o show that this would imply that T must be a scalar operator. Note that for proving (K), it is enough to prove Kirillov expectation only for P,-invariant eigen-distributions (since AT is an eigen-distribution). At this point, let we recall of the Harish-Chandra regularity theorem for invariant eigen-distributions. He showed that such a distribution is represented by a locally integrable function, which is analytic on regular semi simple elements. If one could prove such a type of result for Pn-invariant eigen-distributions, then the Kirillov's argument for P,-invariant distributions represented by continuous functions could be used to see the invariance for the whole group (and we would prove (K) in this way). Since the geometry of P, and GL(n,@)-conjugacy classes is the same on a big open set in GL(n,C ) , it make sense to try to follow Harish-Chandra's strategy of proof of the regularity theorem, to try to prove such a type of result for Pn-invariant eigen-distributions. As we already have mentioned, M. Baruch proved (K) in [7].The strat-

153

S o m e Classes of In-educable Representations

egy of his proof may be considered as a further development of the ideas that we discussed above (there is a plenty of new moments). 20.10. J. Bernstein proposed the following strategy for proving (K) (and also (UO)). Consider GL(n - 1,C) C GL(n,@)in obvious way. Bernstein asked if each GL(n-1, @)-invariantdistribution on GL(n,@) is invariant for transposition (for our purpose, it is enough to consider only GL(n - 1,C)invariant eigen-distributions). Positive answer to this question would also imply (K). One can consider also other local fields regarding the above question. As far as we remember, in a discussion with D. MiliEi6 we saw that it is an easy exercise to show that the answer to this question is positive for GL(2,R). Unfortunately, this argument cannot be extended to higher GL(n,R).

20.11. Now we shall explain how one can prove (U2), using the standard construction of complementary series representations. For this one needs a simple Lemma: Let y1, . . . ,yuI61, . . . ,6, E D. Suppose that 6i x yj is irreducible 5 u,1 5 j 5 v. Then

f o r all indexes 1 5 i

.

L(Yl1.. ,ru) x

L(61,.. .

16,)

is irreducible. Proof: Using the fact that all 6i x yj are irreducible (which implies bi x yj S x di), and associativity of operation x among representations (see Remark 9.4), we get ~j

X(Y1,.. . ,yu,61,.. . I & )

X(Yl1..

. ,yu) x X ( 6 1 , . .

*

,&I)

(for the definition of A(a), see 9.6). From the above isomorphism, one concludes that L(y1,. . . , yu)x L(61,. . . , 6,) has a unique irreducible quotient, and that this quotient is isomorphic to L(y1,. . . , yu, 61, . . . ,6,). Since 81 x T j are also all irreducible, the above conclusion hold also for them: the representation L(5l , . . . ,5u) -x L(61,-. . . , 6,) has a unique irreducible quotient, which is L(Y1,. . . , TU,61, . . . , &,). Now passing to contragredients (and using ( 5 ) ) , we shall get that L(y1,. . . , yu)x L(61, . . . ,6,) has a unique irreducible subrepresentation, which is isomorphic to the representation L(y1,. . . ,yul61,. . . , 6,).

M. Tadad

154

A basic property of Langlands classification is that the multiplicity of a Langlands quotient in corresponding standard module is one. This implies irreducibility of L(y1,. . . , yu) x L(&,. . . , &,). The above lemma and the representations theory of SL(2,C ) imply that the continuous family of representations T(U(4

m ) ,a ) , 0

I < 1/21

is irreducible (irreducibility a t 0 follows from (UO)). From this, using standard integral intertwining operators we can get on these representations (non-degenerate) hermitian forms, which depend continuously on a , and which make these representations hermitian. Since we have positive definiteness in 0 and we have continuous family of hermitian representations, this implies positive definiteness for all 0 5 a < 1/2 (this easily follows from a finite dimensional argument; see [65]). Therefore, representations n(u(6,m ) ,a ) from (U2) are unitarizable.

21. On the unitarizability problem for classical p-adic groups 21.1. Constructing new irreducible unitarizable representations is a very interesting and puzzling problem. It may be related to a number of other problems. The most interesting (and hardest) question is constructing of irreducible unitarizable representations which show up from “nowhere”, i.e., of isolated irreducible unitarizable representations. Namely, there is a natural topology on unitary duals (defined by approximation of matrix coefficients) and isolated representations are those ones for which { T } is an open set (if the center is not compact, one defines isolated representations modulo center; these representations play the role of isolated representations in this case). As we mentioned above, the construction of isolated irreducible unitarizable representations can be related to a number of other questions. Let us mention some of them: representations in residual spectrum of the group over adels, &correspondences, (conjecturaly) involution (which we mention bellow in 21.2) of square integrable representations. A natural question is: how big portion of isolated representations is in a range of each of these methods, combined with some standard constructions of irreducible unitarizable representations (see section 3. of [65] for standard constructions). We plan to address these questions in the future.

S o m e Classes of Irreducible Representations

155

Now we shall formulate some other (precise) questions regarding the unitarizability problem (the first question is around for a long time and we do not know who posed it for the first time in full generality). These questions may provide a strategy (or may be considered as a part of a general strategy) for attacking unitarizability problem for classical padic groups. It may happen that answers to (at least some of these) questions will be obtained in the same time as we will get the solution of the unitarizability problem for classical padic groups. Nevertheless, as we already mentioned above, some of these questions may be useful guidance on the way to the solution of the unitarizability problem for classical padic groups. This is the reason that we collect them here. 21.2. A.-M. Aubert and also P. Schneider and U. Stuhler ([3], [52]) defined an involution on irreducible representations of connected reductive padic groups. This involution carries irreducible unitarizable representations of general linear groups to the unitarizable ones. This was conjectured by J. Bernstein in [9] (and shown by this author; see for example [SO]). Further, this involution in the case of irreducible representations with Iwahori fixed vectors carries unitarizable representations to the (irreducible) unitarizable ones. This was proved by D. Barbasch and A. Moy in [6]. It is natural to ask if this is the case in general. It would be very important to show this (if this is the case). 21.3. Fix an irreducible cuspidal representation u of a classical group S, and fix unitarizable P I , . . . ,P k E C such that for z # j , sets { p i , p i } and { p j , p j } have no equivalent representations (i.e. pi y pj and pi $f &). We have already observed in 19.2 that there exists a bijection

(for details see [32]) The question is: Is

T

unitarizable if and only if all

21.4. Suppose

TI, ~

. ..

2 , ,*k

are unitarizable?

M. Tadic'

156

where p E C is unitarizable. Take an irreducible unitarizable representation 7-r of a general linear group which is an irreducible subquotients of ualq x

where ai E R , T ~E { p , p } . Then sentation. The question is:

Y02T2 7-r >a

x

'

. . x vatre,

o is an irreducible unitarizable repre-

Does every unitarizable representation in Z(p,a ) come in this way? It is not hard to show that the answer to this question is postive. This gives a reduction of unitarizability problem in Z ( p , u ) to the case of general linear groups (where the unitarizability problem has been solved).

21.5. Suppose that 61,. . . , 6 k E D are unitarizable, and 7-r is an irreducible square integrable representation of a classical group. Let 71 and 72 be irreducible subrepresentations of 61 x

. . . x 61,>a u.

Let h i , . . . ,6: E D+. One may ask the following question: Is L(b',, . . . ,hi, 7 1 ) unitarizable if and only if L(bi,.. . ,S ; , T ~ )is unitar iz able? The answer to this question is negative (the first example that we know, which shows that the answer is negative, is for SO(7,F ) ) .

21.6. Consider Z(p;a ) , with p selfdual. The question is Can the description of unitarizable representations in Z(p; 0) be expressed only in terms of the reducibility point ap,r (similarly as for D(p;o))?

21.7. In the case of general linear groups, solution of the unitarizability problem may be expressed independently of the nature of the (local) field F (see 20.1). This may be viewed as a (very strong) example of Lefschetz principle. The question on the same line is: Can one get also a descriptions of unitary duals for each of series of classical groups independent of the nature of the field?

Some Classes of Irreducible Representations

157

21.8. W. Casselman has proved that in the induced representation from minimal parabolic subgroup, where the Steinberg representation shows up, only two irreducible subquotients are unitarizable, the Steinberg and the trivial one (supposing that the group is not compact modulo center). This fact is important not just for unitarizability problem (see [13]).Let us recall that this fact can be also derived from Howe-Moore theorem on asymptotic behavior of infinite dimensional irreducible unitary representations of reductive groups over local fields. This fact concerns only finitely many irreducible representations regarding unitarizability (for a fixed group). Nevertheless, its importance much overcomes this finite set. Namely, it implies that Steinberg and trivial representation are isolated in unitary dual if the group has compact center and if its split rank is greater than 1. It also implies that the induced representations around the induced representation where the Steinberg and the trivial representation show up, does not have unitarizable subquotients (these regions are determined by certain irreducibility conditions). Using this fact and some simple standard results, one can solve the unitarizability problem for rank two groups (let us note that t o apply this result, one needs t o understand reducibility of parabolically induced representations). Such a fact about exactly two unitarizable subquotients in the whole induced representation where the arbitrary irreducible square integrable representation shows up, holds in the case of general linear groups (the induced representation is always multiplicity one, and further, there is exactly one irreducible square integrable subquotient there). One may ask if this holds for other groups. Already the first example other than general linear group, the example S p ( 4 , F ) , tells that neither one of the nice properties discussed above regarding the whole induced representation, where the arbitrary irreducible square integrable representation shows up hold in this case. Namely, already for Sp(4, F ) there is an example of such induced representation which is not multiplicity one, which contains two inequivalent irreducible square integrable subquotients, which has total length 6, and where all the irreducible subquotients are unitarizable. In general, for classical groups we have quite often plenty of unitarizable subquotients in whole induced representations where the irreducible square integrable representations show up (for the beginning, we can have as many square integrable subquotients as we want). In the moment, we shall ask only the following question: Suppose that p E C is selfdual. Let CT be an irreducible cuspidal repre-

M . TadiC

158

sentation of a classical group. Suppose t h a t v a p A (T reduces for some a Then p+np

p+n-l

p x

’

.’ x

> 0.

uap >a u

is a multiplicity one representation. It contains exactly one irreducible square integrable subquotient (actually] it is a unique subrepresentation in the above representation; such representations we call square integrable representations of St,einberg type). T h e question is: Does the above induced representation have exactly two irreducible subquotients which are unitarizable (a weaker question is, does it have at most two)?

References 1. Arthur, J., “On some problems suggested by the trace formula”, Lie Group Representations 11, Proceedings, University of Maryland 1982-83, 1-49, Lecture Notes in Math. 1041, Springer-Verlag, Berlin, 1984. 2. Arthur, J., “Unipotent automorphic representations: conjectures”, Ast&sque, 171-172(1989), 13-71. 3. Aubert, A.-M., “Dualitk dans le groupe de Grothendieck de la catkgorie des reprksentations lisses de longueur finie d’un groupe reductif padique” , Trans. Amer. Math. SOC.,347 (1995), 2179-2189 (and Erratum, Trans. Amer. Math. SOC.,348 (1996), 4687-4690). 4. Barbasch, D., “The unitary dual for complex classical groups”, Invent. Math., 96 (1989), 103-176. 5 . Barbasch, D., “Unitary spherical spectrum for split classical groups”, preprint, 2002. 6. Barbasch, D. and Moy, A , , “A unitarity criterion for padic groups”, Invent. Math., 98 (1989), 19-37. 7. Baruch, E.M., “A proof of Kirillov’s conjecture”, Ann. of Math., 158(1) (2003), 207-252. 8. Bernstein, J.N., “All reductive padic groups are tame”, Jour. Functional Anal. Appl., 8 (1974), 91-93. 9. Bernstein, J., “P-invariant distributions on G L ( N ) and the classification of unitary representations of G L ( N ) (non-archimedean case)”, Lie Group Representations 11, Proceedings, University of Maryland 1982-83, 50-102, Lecture Notes in Math. 1041, Springer-Verlag, Berlin, 1984. 10. Bernstein, J. (written by K. Rumelhart), “Draft of: Representations of p a d i c groups”, preprint. 11. Bernstein, I. N. and Zelevinsky, A.V., “Representations of the group G L ( n ,F ) , where F is a local non-Archimedean field”, Uspekhi Mat. Nauk., 31 (1976), 5-70. 12. Bernstein, J. and Zelevinsky, A.V., “Induced representations of reductive p adic groups I”, Ann. Sci. Ecole Norm Sup., 10 (1977), 441-472.

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13. Borel, A. and Wallach, N., Continuous cohomology, discrete subgroups, and representations of reductive groups, Princeton University Press, Princeton, 1980. 14. Bushnell, C.J. and Kutzko, P., T h e admissible dual of G L ( N ) via compact open subgroups, Princeton University Press, Princeton, 1993. 15. Cartier, P., “Representations of padic groups; a survey”, Symp. Pure Math. 33, part 1, 111-155, Amer. Math. SOC.,Providence, Rhone Island, 1979. 16. Casselman, W., “Introduction to the theory of admissible representations of p a d i c reductive groups”, preprint. 17. Casselman W. and MiliEiC D., “Asymptotic behavior of matrix coefficients of admissible representations”, Duke Math. J., 49 (No. 4) (1982), 869-930. 18. Clozel L., “Progrks rkcents vers la classification du dual unitaire des groupes rkductifs rkels” , S6minaire Bourbaki, no 681 (1986-87), Aste‘risgue. 19. Gelbart, S.S., “An elementary introduction to the Langlands program” , Bulletin A m e r . Math. SOC.,10 (1984), 177-219. 20. Gelfand, I.M., Graev, M. and Piatetski-Shapiro, Representation theory and automorphic functions, Saunders, Philadelphia, 1969. 21. Gelfand, I.M. and Kazhdan, D.A., “Representations of G L ( n ,k)”, Lie groups and their Representations, 95-118, Halstead Press, Budapest, 1974. 22. Gelfand, I.M. and Naimark, M.A., “Unitare Darstellungen der Klassischen Gruppen”, Akademie Verlag, Berlin, 1957, (German translation). 23. Goldberg, D., “Reducibility of induced representations for S p ( 2 n ) and SO(n)”,A m e r . J . Math., 116 (5) (1994), 1101-1151. 24. Harish-Chandra, Collected papers, Springer-Verlag, Berlin, 1983. 25. Harris, M. and Taylor, R., O n the geometry and cohomology of some simple Shimura varieties, Princeton University Press, Annals of Math. Studies 151, 2001. 26. Henniart, G., “Une preuve simple des conjectures de Langlands pour GL(n) sur un corps padique”, Invent. Math., 139 (2) (2002), 439-455. 27. Howe, R., “0-series and automorphic forms”, Symp. Pure Math. 33, part 1, 275-286, Amer. Math. SOC.,Providence, Rhone Island, 1979. 28. Humpreys, J.E., Linear algebraic groups, Springer-Verlag, New York, 1975. 29. Jacquet, H., “Generic representations”, Non-Commutative Harmonic Analysis, 91-101, Lecture Notes in Math. 587, Spring Verlag, Berlin, 1977. 30. Jacquet, H., “Principal L-functions of the linear group”, Symp. Pure Math. 33, part 2, 63-86, Amer. Math. Soc., Providence, Rhone Island, 1979. 31. Jacquet, H., “On the residual spectrum of GL(n)”, Lie Group Representations 11, 185-208, Proceedings, University of Maryland, 1982-83. Lecture Notes in Math. 1041, Springer-Verlag, Berlin, 1984. 32. Jantzen, C., “On supports of induced representations for symplectic and oddorthogonal groups”, A m e r J. Math., 119 (1997), 1213-1262. 33. Jantzen, C., “On square integrable representations of classical p-adic groups 11” , Representation Theory, 4 (2000), 127-180. 34. Kazhdan, D. and Lusztig, G., “Proof of the Deligne-Langlands conjecture for Hecke algebras”, Invent. Math., 87 (1987), 153-215. 35. Kirillov, A. A . , “Infinite dimensional representations of the general linear

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M. TadiC group”, Dokl. Akad. Nauk SSSR, 114 (1962), 37-39. Soviet Math. Dokl., 3 (1962), 652-655. Knapp, A. W., “Introduction to the Langlands program”, Representation Theory and Automorphic Forms; An Instructional Conference at the International Centre for Mathematical Sciences, Edinburgh, 1996, 245-302, American Mathematical Society, Proceedings of Symposia in Pure Mathematics 61, Providence, Rhode Island, 1997. Langlands, R.P., “On the classification of irreducible representations of real algebraic groups”, Representation Theory and Harmonic Analysis on Semisimple Lie Groups, P.J. Sally, Jr. and D. A. Vogan, Jr. editors, Amer. Math. SOC.,Providence, 1989. Lapid, E., Goran, M. and Tadid, M., “On the generic unitary dual of quasisplit classical groups”, preprint, 2003. MiliEid, D., “On C*-algebras with bounded trace”, Glasnik Mat., 8 ( 2 8 ) (1973), 7-21. Mceglin C., “Normalisation des opkrateurs d’entrelacement et rkductibilitk des induites de cuspidales; le cas des groupes classiques padiques”, Annals of Math., 151 (2) (2000), 817-847. Mceglin, C., “Sur la classification des series discrhtes des groupes classiques p-adiques: paramktres de Langlands et exhaustivitk” , to appear in Journal of the European Mathematical Society. Mceglin, C., “Reprksentations quadratiques unipotentes des groupes classiques p-adiques.”, Duke Math. Jour., 84 (1996), 267-332. Moeglin, C., “Points de rkducibilitk pour les induites de cuspidales” preprint, Institute de mathkmatiques de Jussieu, 2001. Mceglin, C. and TadiC, M., “Construction of discrete series for classical padic 15 (2002), 715-786. groups”, J. Amer. Math. SOC., Mceglin, C., Vignkras, M.-F. and Waldspurger J.-L., Correspondance de Howe sur u n corps p-adique, Lecture Notes in Math 1291, Springer-Verlag, Berlin, 1987. Moy, A., “Representations of GSp(4) over a padic field: parts 1 and 2”, Compositio Math., 66 (1988)) 237-328. Mui6, G., “On generic irreducible representations of S p ( n , F ) and SO(2n 1, F ) ” , Glasnik Mat., 33(53) (1998), 19-31. Rodier, F., “Reprksentations de GL(n,k) oh k est un corps padique”, Skminaire Bourbaki no 587 (1982), Astkrisque, 92-93 (1982), 201-28. Sahi, S., “On Kirillov’s conjecture for archimedean fields”, Compos. Math., 72 (1989), 67-86. Savin, G., “Lectures on representations of padic groups”, in this volume. Sally, P.J. and Tadid, M., “Induced representations and classifications for GSp(2, F ) and Sp(2, F)”, Mkmoires SOC.Math. France, 52 (1993), 75-133. Schneider, P. and Stuhler, U., “Representation theory and sheaves on the Bruhat-Tits building”, Publ. Math. IHES, (1997), 97-191. Shahidi, F., “A proof of Langlands conjecture on Plancherel measures; complementary series for padic groups”, Ann. of Math., 132 (1990), 273-330. Shahidi, F., “Twisted endoscopy and reducibility of induced representations

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for padic groups”, Duke Math. Jour., 66 (1992), 1-41. 55. Silberger, A,, “The Langlands quotient theorem for padic groups”, Math. Ann., 236 (1978), 95-104. 56. Silberger, A , , “Special representations of reductive p-adic groups are not integrable”, Ann. of Math., 111 (1980), 571-587. 57. Speh B., “Unitary representations of G L ( n ,R) with non-trivial (g, K ) cohomology”, Invent. Math., 71 (1983), 443-465. 58. Stein, E.M., “Analysis in matrix spaces and some new representations of SL(N,C)”, Ann. of Math., 86 (1967), 461-490. 59. Steinberg, R., Lectures on Chevalley groups, Yale University, 1968. 60. TadiC, M., “Classification of unitary representations in irreducible representations of general linear group (non-archimedean case)”, Ann. Sci. &ole Norm. SUP., 1 9 (1986), 335-382. 61. TadiC, M., “Topology of unitary dual of non-archimedean GL(n)”, Duke Math. Jour., 55 (1987), 385-422. 62. Tadid, M., “Representations of p-adic symplectic groups”, Compositio Math., 90 (1994), 123-181. 63. TadiC, M., “Structure arising from induction and Jacquet modules of representations of classical padic groups”, Journal of Algebra, 177 (1) (1995), 1-33. 64. Tadid, M., “Correspondence on characters of irreducible unitary representations of G L ( n ,CC)”, Mathematischen Annalen, 305 (1996), 419-438. 65. TadiC, M., “An external approach to unitary representations”, Bulletin Amer. Math. SOC.,28(2) (1993), 215-252. 66. TadiC, M., “Representations of classical p-adic groups”, Representations of Lie groups and quantum groups, 129-204, Pitman Research Notes in Mathematics, Series 311, Longman, Essex, 1994. 67. Tadid, M., “On regular square integrable representations of padic groups”, Amer. J. Math., 120(1) (1998), 159-210. 68. Tadid, M., “On reducibility of parabolic induction”, Israel J . Math., 107 (1998), 29-91. 69. TadiC, M., “Square integrable representations of classical p a d i c groups corresponding to segments”, Representation Theory, 3 (1999), 58-89. 70. TadiC, M., “A family of square integrable representations of classical padic groups”, Glasnik Mat., 37(57) (2002), 21-57. preprint, 2002. 71. Tadid, M., “GL(n,CC)” and GL(n,R)^”, 72. Vogan, D. A., “The unitary dual of G L ( n )over an archimedean field”, Invent. Math., 8 2 (1986), 449-505. 73. Waldspurger, J.-L., “La formule de Plancherel pour les groupes p-adiques, d’apr8s Harish-Chandra”, Journal de l’lnstitut de Math. de Jussieu, 2 (2) (2003), 235-333. 74. Warner, G., Harmonic analysis on Semi-simple Lie Groups I, 11, SpringerVerlag, Berlin, 1972. 75. Weil, A., Basic number theory, Springer-Verlag, New York, 1974. 76. Zelevinsky, A.V., “Induced representations of reductive padic groups 11, On irreducible representations of GL(n)”,A n n . Sci &ole Norm Sup., 13 (1980),

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Dirac Operators in Representation Theory

Jing-Song Huang and Pavle Pandiit Department of Mathematics

The Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong SAR, China Email: [email protected] Department of Mathematics University of Zagreb BijeniEka 30, 10000 Zagreb, Croatia E-mail: pandzicbmath. hr

Vogan’s conjecture on Dirac cohomology reveals an algebraic nature of Dirac operators. In these notes we explain our proof of the Vogan’s conjecture. As applications we describe how to simplify the proof of AtiyahSchmid on geometric construction of the discrete series and sharpen the Langlands-Hotta-Parthasarathy formula on automorphic forms. We also explore the relationship between Dirac cohomoiogy and Lie algebra cohomology. To make these notes more accessible, we include some well known background material on representations and Lie algebra cohomology.

1. ( 8 , K)-modules 1.1. Lie groups and algebras

A Lie group G is a group which is also a smooth manifold, in such a way that the group operations are smooth. In more words, the multiplication map from G x G into G and the inverse map from G into G are required t o be smooth. The Lie algebra g of G consists of left invariant vector fields on G. The left invariance condition means the following: let 1, : G + G be the left translation by g E G, i.e., I,(h) = g h . A vector field X on G is left invariant 163

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if ($l,)hXh = X,, for all g, h E G. It is clear that g is a vector space. It can be identified with the tangent space t o G a t the unit element e. Namely, to any left invariant vector field one can attach its value a t e. Conversely, a tangent vector a t e can be translated to all other points of G to obtain a left invariant vector field. Note that we did not require our vector fields to be smooth; it is however a fact that a left invariant vector field is automatically smooth. The operation making g into a Lie algebra is the bracket of vector fields:

[X, Ylf

=XVf) -Y

( X f1,

for X , Y E G and f a smooth function on G. Here we identify vector fields with derivations of the algebra Cm(G), i.e., think of them as first order differential operators. The operation [ , I satisfies the well known properties of a Lie algebra operation: it is bilinear and anticommutative, and it satisfies the Jacobi identity:

" ~ , ~ 1 , ~ 1 + " ~ , ~ 1 , ~ 1 + " =~ o, ~ 1 , ~ 1 for any X, Y ,Z E g. The main examples are various matrix groups. They fall into several classes. We are primarily interested in the semisimple connected groups, like the group S L ( n ,R) of n x n real matrices with determinant 1. Its Lie algebra is sl(n,R), consisting of the n x n matrices with trace 0. There are also familiar series of compact groups: SU(n), the group of unitary (complex) matrices, with Lie algebra 5 4 7 2 ) consisting of skew Hermitian matrices, and S O ( n ) ,the group of orthogonal (real) matrices with Lie algebra so(n) consisting of antisymmetric matrices. Further examples of matrix groups are the groups S p ( n ) ,S U ( p ,q ) and SO(p,4 ) ; they are all defined as groups of operators preserving certain forms. Other classes of Lie groups one needs to study are solvable groups, like the groups of upper triangular matrices; nilpotent groups like the groups of unipotent matrices and abelian groups like Rn or the groups of diagonal matrices. They are not of our primary interest, but they show up as subgroups of our semisimple groups and therefore have t o be understood. One also often considers reductive groups, which include semisimple groups but are allowed to have a larger center, like GL(n,R)or U ( n ) . The definitions are easier to formulate for Lie algebras g: define Cog = Dog = g, and inductively CZ+lg = [g, Cig],Dis'g = [Dig,Dig]. Then g is nilpotent if Czg = 0 for large i, g is solvable if Dig = 0 for large i, g is

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semisimple if it contains no nonzero solvable ideals, and g is reductive if it contains no nonabelian solvable ideals. Abelian means that [ ,] = 0. Instead of checking that each of the above mentioned groups is a Lie group, on can refer to a theorem of Cartan, which asserts that every closed subgroup of a Lie group is automatically a Lie subgroup in a unique way. Since each of the groups we mentioned is contained in G L ( n , C ) we only need to see that G L ( n , C ) is a Lie group. But G L ( n , C ) is an open subset of Cnz, hence is a manifold, and the matrix multiplication and inverting are clearly smooth. Without using Cartan's theorem, one can apply implicit/inverse function theorems, as each of the above groups is given by certain equations that the matrix coefficients must satisfy. Let us note some common features of all the mentioned examples: 0 g is contained in the matrix algebra M n ( C ) ,and [ X ,Y ]= X Y - Y X , the commutator of matrices; 0 G acts on g by conjugation: Ad(g)X = gXg-'. This is called the adjoint action. The differential of this action with respect to g gives an action of g on itself, a d ( X ) Y = [ X , Y ] ,which is also called the adjoint action. 0 There is an exponential map exp : g -+ G mapping X to ex. This is a local diffeomorphism around 0, i.e., sends a neighborhood of 0 in g diffeomorphically onto a neighborhood of e in G.

1.2. Finite dimensional representations Let V be a complex n-dimensional vector space. A representation of G on V is a continuous homomorphism 7r

:G

4

GL(V).

Any such homomorphism is automatically smooth; this is a version of the already mentioned Cartan's theorem. Given a representation of G as above, we can differentiate it at e and obtain a homomorphism d7r = 7r : g + gI(V)

of Lie algebras. An important special case is the case of a unitary representation. This means V has an inner product such that all the operators 7r(g), g E G , are unitary. Then all .(X), X E g, are skew-hermitian. The main idea of passing from G to g is turning a harder, analytic problem of studying representations of G into an easier, purely algebraic

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(or even combinatorial in some sense) problem of studying representations of g. Actually, since we are only considering complex representations, we can as well complexify g and study representations of gc.

Example: The most basic example and the first one to study is the representations of g = d ( 2 , C). There is an obvious basis for g: take h=

(’

0 -1 O));

e=

(::);

f =

(::)

Then the commutator relations are [h,e] = 2e;

[h, f ] = -2f;

[e,f ] = h.

Let V be an irreducible representation of g of dimension n is a basis {vkIk=-n,-n+2

+ 1. Then there

,..., n - 2 , n )

of V, such that the action of h is diagonal in this basis: n(h)vk

= kvk.

The integers k are called the weights of V. The operator n ( e ) raises the weight by 2, i.e., n(e)vk is proportional to U k + 2 (which is taken t o be 0 if k = n). The operator n(f) lowers the weight by 2. There is an obvious ordering on the set of weights (usual ordering of integers), and n is the highest weight of V ; this corresponds to the fact that n(e)u, = 0. Here is a picture of our representation V:

Finally, the actions of e and f are also completely determined: let us normalize the choice of vk’s by taking v, to be an arbitrary n(h)-eigenvector for eigenvalue n, and then taking un-2j = n(f ) j v n . Then

(*)

n ( e ) v n - ~ j= j ( n - j

+ 1)Vn-Zj+2

All of this is easy to prove; we give an outline here and encourage the reader to complete the details. First, it is a general fact that the action of h diagonalizes in any finite dimensional representation, as h is a semisimple element of g. One can however avoid using this, and just note that the action of h will have an eigenvalue, say X E @, with an eigenvector vx. For any p E @, let us denote by V, the p-eigenspace for n(h); of course, V, will be zero for all but finitely many p.

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From the commutator relations, it is immediate that r ( e ) sends V, to V,+2, while ~ ( fsends ) V, to V,-2 for any p. In particular, the sum of all V, for u , E X + 22 is g-invariant, hence it is all of V by irreducibility. Moreover, the subspace generated by ux will already be everything, so all V, are one-dimensional. We can now replace X by the highest weight in the obvious ordering coming from Z, and fix a highest weight vector vx. Then we can define (as above) ~ x - 2=~ r ( f ) j u x , and prove the analog of the formula (*) with X replacing n, by induction from the commutator relations. R o m (*), finite dimensionality and irreducibility it now follows that X must be a nonnegative integer, say n, and that V is the span of the (nonzero) vectors v,,v,-2,. . . , U v n . Thus we have described all irreducible finite dimensional 5 4 2 , C)modules. Other finite dimensional representations are direct sums of irreducible ones; this is a special case of Weyl’s theorem, which says this is true for any semisimple Lie algebra g. One way t o prove this theorem is the so called unitarian trick of Weyl: one shows that there is a compact group G with the same representations as g. E.g., for d ( 2 , @ ) , G = S U ( 2 ) . Now using invariant integration one shows that every representation of a compact group is unitary. On the other hand, unitary (finite dimensional) representations are easily seen to be direct sums of irreducibles; namely, for every invariant subspace, its orthogonal is an invariant complement. We now pass on to describe finite dimensional representations of a general semisimple Lie algebra g over @. Instead of just one element h t o diagonalize, we can now have a bunch of them. They comprise a Cartan subalgebra b of g, which is by definition a maximal abelian subalgebra consisting of semisimple elements. Elements of b can be simultaneously diagonalized in any (finite dimensional) representation; for each joint eigenspace, the eigenvalues for various X E b are described by a functional X E b*, a weight of the representation under consideration. All possible weights of finite dimensional representations form a lattice in b*, called the weight lattice of g. The nonzero weights of the adjoint representation of g on itself have a prominent role in the theory; they are called the roots of g, and satisfy a number of nice symmetry properties. For example, the roots of 5I(3,C) form a regular hexagon in the plane. The roots of d ( 2 , C) are 2 and -2 (upon identifying b* = (Ch)*with C). One can divide up roots into positive and negative roots, which gives an ordering on b i , the real span of roots, and also a notion of positive

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(“dominant”)weights, The irreducible finite dimensional representations of g are classified by their highest weights. The possible highest weights are

precisely the dominant weights. If we denote by n+ (respectively n-) the subalgebra of g spanned by all positive (respectively negative) root vectors, then we have a triangular decomposition g=n-eben+.

For example, if g = d ( n , C ) , one can take b to be the diagonal matrices in g, n+ the strictly upper triangular matrices and n- the strictly lower triangular matrices. Let V be an irreducible finite dimensional representation with highest weight A. The highest weight vector is unique up t o scalar, and is characterized by being an eigenvector for each X E 4, with eigenvalue X ( X ) , and by being annihilated by all elements of n+. We now address the question of “going back”, i.e., getting representations of G from representations of g which we have just described. This is called “integrating” or “exponentiating” representations. It turns out there is a topological obstacle to integrating representations; this can already be seen in the simplest case of 1-dimensional (abelian) Lie groups. There are two connected 1-dimensional groups: the real line R and the circle group TI.Both have the same Lie algebra R. Consider the one-dimensional representations of the Lie algebra R; each of them is given by t H t X for some X E C (we identify 1 x 1 complex matrices with complex numbers). All of these representations exponentiate to the group R, and give all possible characters of R:

t-

etx,

x E C.

However, of these characters only the periodic ones will be well defined on T1,and etx is periodic if and only if X E 27riZ. In general, when G is connected and simply Connected, then all finite dimensional representations of g integrate to G. Any connected G will be covered by a simply connected 6 , and the representations of G are those representations of G which are trivial on the kernel of the covering. If G is semisimple connected with finite center, then there is a decomposition

G=KP called the Cartan decomposition; here K is the maximal compact subgroup of G I while P is diffeomorphic t o a vector space. For example, if G =

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S L ( n ,R),then K = S O ( n ) ,and P consists of positive matrices in S L ( n ,R); so P is diffeomorphic to the vector space of all symmetric matrices of trace zero via the exponential map. It follows that the topology of G is the same as the topology of K , and a (finite dimensional) representation of g will exponentiate to G if and only if it exponentiates to K. 1.3. Infinite dimensional representations In general, a representation of G is a continuous linear action on a topological vector space 3.1. Some typical choices for 3.1 are: a Hilbert space, a Banach space, or a Frkchet space. The first question would be: can we differentiate a representation to get a representation of g? The answer is: not quite. Actually, g acts, but only on the dense subspace 3.1, of smooth vectors ( a vector w E 3.1 is smooth if the map g H .rr(g)v from G into 3.1 is smooth). For semisimple G with maximal compact K, there is a better choice than 3.1,: we can consider the dense subspace 3 - 1 ~of K-finite vectors in 3-t (a vector w E 3.1 is K-finite if the set .rr(K)wspans a finite dimensional subspace of 3-1). One shows that K-finite vectors are all smooth, and so 3 . 1 ~ becomes a Harish-Chandra module, or a (g, K) module. A vector space V is a (g, K)-module if it has:

(1) an action of g; (2) a finite action of K; (3) the two &actions obtained from (1) and (2) agree. In case we want to allow disconnected groups, we also need (4) the g-action is K-equivariant, i.e., .rr(k).rr(X)w= .rr(Ad(k)X).rr(k)w, for all k E K , X E g and w E V. One usually also puts some finiteness conditions, like finite generation, or admissibility defined below. Now a basic fact about (unitary) representations of compact groups is that they can be decomposed into Hilbert direct sums of irreducibles, which are all finite dimensional. This can be proved using the basic facts about compact operators. Thus our 3.1 decomposes into a Hilbert direct sum of Kirreducibles, and Harish-Chandra modules, being K-finite, decompose into algebraic direct sums of K-irreducibles. For 6 E k , the unitary dual of K , we denote by V ( 6 )the 6-isotypic component of a Harish-Chandra module, i.e., the largest K-submodule of V which is isomorphic to a sum of copies of

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6. We say V is admissible if each V(6) is finite dimensional. In other words, every 6 E k occurs in V with finite multiplicity. Harish-Chandra proved that for each irreducible unitary representation 3-1 of G, the (0, K)-module

3 1 is~ admissible. Going back from (8, K)-modules to representations of G is hard. We call a representation (.,?-I) of G a globalization of a Harish-Chandra module V, if V is isomorphic to 3 1 ~ Every . irreducible V has globalizations. In fact there are many of them; one can choose e.g. a Hilbert space globalization (not necessarily unitary), or a smooth globalization. There are also notions of minimal and maximal globalizations. A few names to mention here are Harish-Chandra, Lepowsky, Rader, Casselman, Wallach and Schmid. We will not need any globalizations, but will from now on work only with (g, K)-modules.

1.4. Infinitesimal characters Recall that a representation of a Lie algebra g on a vector space V is a Lie algebra morphism from g into the Lie algebra End(V) of endomorphisms of V. Now End(V) is actually an associative algebra, which is turned into a Lie algebra by defining [ a ,b] = ab - ba; this can be done for any associative algebra. What we want is to construct an associative algebra U ( g ) containing g, so that representations of g extend to morphisms U ( g ) + End(V) of associative algebras. The construction goes like this: consider first the tensor algebra T(g)of the vector space g. Then define

U ( 8 )= T ( g ) / I , where I is the two-sided ideal of T ( g ) generated by elements X 8 Y Y 8 X - [X, Y ] ,X,Y E g. It is easy to see that U ( g ) satisfies a universal property with respect to maps of g into associative algebras; in particular, representations of g (i.e., g-modules) are the same thing as U(g)-modules. Some further properties are 0 There is a filtration by degree on U ( g ) ,coming from T ( g ) ; 0 The graded algebra associated to the above filtration is the symmetric algebra S(g); 0 One can get a basis for U ( g ) by taking monomials over an ordered basis of g. The last two properties are closely related and are the content of the PoincarB-Birkhoff-Witt theorem. Loosely speaking, one can think of U ( g ) as “noncommutative polynomials over g” , with the commutation laws given

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by the bracket of g. If we think of elements of g as left invariant vector fields on G, then U ( g ) consists of left invariant differential operators on G. Here are some benefits of introducing the algebra U ( g ) .

(1) One can use constructions from the associative algebra setting. For example, there is a well known notion of “extension of scalars”: let B c A be associative algebras and let V be a B-module. One can consider A as a right B-module for the right multiplication and form the vector space A @B V . This vector space is an A-module for the left multiplication in the first factor. So we get a functor from B-modules t o A-modules. Another functor like this is obtained by considering HomB(A, V ) ;now the Hom is taken with respect to the left multiplication action of B on A (and the given action on V ) ,and the (left!) A-action on the space HomB(A, V) is given by right multiplication on A. (2) Since g is semisimple, it has no center. U ( g ) however has a nice center Z(g). It is a finitely generated polynomial algebra (e.g. for g = 5I(n,@) there are n - 1 generators and their degrees are 2 , 3 , . . . , n). The importance of the center follows from a simple observation that is often used in linear algebra: if two operators commute, then an eigenspace for one of them is invariant for the other. This means that we can reduce representations by taking a joint eigenspace for Z(g). Let us examine the center Z(g) and its use in representation theory in more detail. First, there is an element that can easily be written down; it is the simplest and most important element of Z(g) called the Casimir element. To define it we need a little more structure theory. There is an invariant bilinear form on g, the Killing form B. It is defined by

B(X,Y )= tr(ad X ad Y ) , X, Y E g. The invariance condition means that for a Lie group G with Lie algebra g, one has B(Ad(S)X,Ad(S)Y) = B ( X ,Y ) , 9 E G ,

x,y E 0.

The Lie algebra version of this identity is

B ( [ Z XI, , Y )+ B(X, [Z, Y]) = 0, x , Y,zE 8. Furthermore, B is nondegenerate on g; this is actually equivalent to g being semisimple. In many cases like for sl(n) over IR or @, one can instead of B use a simpler form tr(XY), which is equal t o B up to a scalar.

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There is a Cartan decomposition of g: g=tCBp.

t and p can be defined as the eigenspaces for the so called Cartan involution 6 for the eigenvalues 1respectively -1. There is also a Cartan decomposition G = K P on the group level that we already mentioned; here K is the maxinial compact subgroup of G (if G is connected with finite center), and t is the Lie algebra of K . Rather than defining the Cartan involution in general, let us note that for all the matrix examples in 1.1, 6 ( X ) is minus the (conjugate) transpose of X. So e.g. for g = sK(n,IR), t is so(n) and p is the space of symmetric matrices in g. Some further properties of the Cartan decomposition: t is a subalgebra, [t,p] c p and [p, p] c t. The Killing form B is negative definite on t and positive definite on p. To define the Casimir element, we choose orthonormal bases W k for t and Zi for p, so that

B(Wk,Wl) = - 6 k l ;

B ( Z i , Zj) = sij.

The Casimir element is then k

i

It is an element of U ( g ) , and one shows by an easy calculation (which we leave as an exercise) that it commutes with all elements of g and thus is an element of Z(g). Furthermore, it does not depend on the choice of bases w k and Zi. Assume now X is an irreducible (g, K)-module. Then every element of Z(g) acts on X by a scalar. For finite dimensional X it is the well known and obvious Schur’s lemma: let z E Z(g) and take an eigenspace for z in X . This eigenspace is a submodule, hence has t o be all of X. For infinite dimensional X the same argument is applied to a fixed K-type in X; see [as],0.3.19, or [15],Ch. VIII, 53. Now all the scalars coming from the action of Z ( g ) on X put together give a homomorphism

of algebras, which is called the infinitesimal character of X. By a theorem of Harish-Chandra, Z(g) is isomorphic (as an algebra) to S ( ~ J the ) ~ ,Weyl group invariants in the symmetric algebra of a Cartan

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Dirac Operators in Representation Theory

subalgebra 9 in g. (The Weyl group W is a finite reflection group generated by reflections with respect t o roots.) This is obtained by taking the triangular decomposition g = n- CB b @ n+ mentioned before and building a PoincarB-Birkhoff-Witt basis from bases in n-, b and n+. We now get a linear map from U ( g ) into U ( b ) = S(b) by projecting along the span of all monomials that contain a factor which is not in 8. This turns out to be an algebra homomorphism when restricted to Z ( g ) , and this gives the required isomorphism. Now we can identify S(b) with the algebra P(b*)of polynomials on b*, and recall that any algebra homomorphism from P(b*)into C is given by evaluation a t some X E b*. It follows that the homomorphisms from Z(g) into C correspond to W-orbits WX in b*. So, infinitesimal characters are parametrized by the space b*/W.They are important parameters for classifying irreducible (g, K)-modules. It turns out that for every fixed infinitesimal character, there are only finitely many irreducible (8, K)-modules with this infinitesimal character. More details about I-Iarish-Chandra isomorphism can be found e.g. in [16]. To finish, let us describe an example where it is easy to explicitly write down all irreducible (g, K)-modules. This is the case G = SL(2, R), whose representations correspond to (5K(2, C ) ,S0(2))-modules. To see the action of K = SO(2) better, we change basis of d ( 2 , C)and instead of h, e, f used earlier we now use

w=(p;z),

x=-( 1 1 2

Note that W E before:

2

),

i -1

e@. The elements W,X

and

Y = -1( 1 -i 2

)

-i-1

Y satisfy the same relations as

[W,X] = 2x; [W,Y]= -2Y;

[ X , Y ]= W.

Fix X E C and E E ( 0 , l ) . Define an ( d ( 2 , C), S0(2))-module VX,,as follows: 0 a basis of VX,,is given by v,, n E Z,n congruent to E modulo 2; cos8 sin8 v, = einevn; - sin 8 cos 8 0 7r(W)V,= nu,; r ( X ) v n = !(A ( n 1))vn+2; 7r(Y)vn= n ( X - ( n - 1))vn-2. The picture is similar to the one that we had for finite dimensional representations of d ( 2 , C),but now it is infinite:

+ +

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J . 3 . Huang and P. PandiiiC

Also, note that we changed normalization for the VLS; now we do not have a natural place t o start, like a highest weight vector, so it is best to make n ( X ) and n ( Y )as symmetric as possible. The following facts are not very difficult to check: 0 Vx,, is irreducible unless X 1 is an integer congruent t o E modulo 2; 0 The Casimir element R acts by the scalar X2 - 1 on Vx,,; 0 If X = k - 1 where k 2 1 is an integer congruent t o E modulo 2, then V A ,contains ~ two irreducible submodules, one with weights k , k 2 , . . . and the other with weights . . . , -k - 2, -k. If k > 1, these are called discrete series representations, as they occur discretely in the decomposition of the representation L2(G).The quotient of VA,,by the sum of these two submodules is an irreducible module of dimension k - 1. For k = 1 the two submodules are called the limits of discrete series, and their sum is all of

+

+

I.

V0,l.

All this can be found with many more details and proofs in Vogan’s book [28], Chapter 1. Actually, Chapters 0 and 1 of that book contain a lot of material from this section (plus more) and comprise a good introductory reading. Other books where a lot about (g,K)-modules can be found are [15] and [33].

2. Clifford algebras, spinors and Dirac operators 2.1. Clifford algebras From now on we will adopt the convention t o denote real Lie algebras with the subscript 0 and t o refer to complexified Lie algebras with the same letter but no subscript. For example, go = to @ po will denote the Cartan decomposition of the real Lie algebra go, while g = e @ p will be the complexified Cartan decomposition. We saw in 1.4 that Z(g) has an important role: it reduces representations, and defines a parameter (infinitesimal character) for the irreducible representations. We also defined the “smallest” non-constant element of Z(g), the Casimir element R. Let us imagine for a moment that Z ( g ) contains a smaller (degree one) element T , such that T 2 = R. Then T would (in principle) have more eigenvalues than R,as two opposite eigenvalues for T would square to the same eigenvalue for 0. As a consequence, we would get a better reduction of representations, and infinitesimal character would be a finer invariant. Of course, such a T does not exist; degree one elements of U ( g ) are the

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175

elements of g and g has no center being semisimple. The idea is then to twist U ( g ) by a finite dimensional algebra, the Clifford algebra C(p). The algebra U ( g )@I C(p) will contain an element D (the Dirac element) whose square is close to R @ 1. One can define the Clifford algebra C(p) as an associative algebra with unit, generated by an orthonormal basis Zi of p (with respect t o the Killing form B), subject to the relations

zzzj

= -zjzz

(2

z i2 = -1.

#j);

(There are variants obtained by replacing the -1 in the second relation by 1 or by 1/2.) This definition involves choosing a basis, so let us give a nicer one:

C(P) = T(P)/I, where T(p)is the tensor algebra of the vector space p and I is the two-sided ideal of T(p) generated by elements of the form

x @ Y + Y @ X + 2B(X,Y). This definition resembles the definition of U ( g ) ;the similarity extends to an analog of the Poincark-Birkhoff-Witt theorem. Namely, C(p) inherits a filtration by degree from T(p).The associated graded algebra is the exterior algebra A(p). One can obtain a basis for C(p) by taking an (orthonormal) ordered basis Zi for p and forming monomials over it to obtain

za

il<.‘.
Note that no repetitions are allowed here, as 2,” is of lower order. Together with the empty monomial 1, the above monomials form a basis of C(p) which thus has dimension equal to 2 d i m p . Finally, let us note that one can analogously construct a Clifford algebra for any vector space with a symmetric bilinear form.

2.2. Dirac operator Using again our orthonormal basis 2,of p , we define the Dirac operator

D

=

Ca Za

@I

2,

E

U ( g ) C(p).

It is easy to show that D is independent of the choice of basis Zi and K-invariant (for the adjoint action of K on both factors).

J . 3 . Huang and P. PandEaC

176

The adjoint action of PO on po gives a map from Po into so(p0). On the other hand, there is a well known embedding of iio(p0) into C(po), given by

1 2 where Eij denotes the matrix with all matrix entries equal to zero except the i j entry which is 1. One can check directly that this map is a Lie algebra morphism (where C(p0) is considered a Lie algebra in the usual way, by [a,b] = ab - ba). Combining these two maps, we get a map a : Po + C(p0) c C(p), and we use it t o produce a diagonal embedding of PO into U ( 8 )@ C(P), by

Eij

x

H

- Eji

H

--ZiZj,

X @ 1+ 1@ a ( X ) ,

x E Po.

Complexifying this we get a map of P, hence also of U(P) and Z(P) into U ( g )@ C(p). We denote the images by PA, PA) and Z ( ~ A(A ) for diagonal). In particular, we get ReA, the image of the Casimir element Re of Z(P). Since Re = - CkW;, we see =- c ( w k

@ 1 f 1@ ( . ( w k ) ) 2 .

k

Here is now the announced relationship between D2 and the Casimir element R, :

Lemma: (Parthasarathy)

D2 = -0,

@

1 +ReA

+c,

where C is a constant that can be computed explicitly (C = llpc112 - llp112, where p is the half s u m of positive roots and pc i s the half sum of compact positive roots).

We just start the calculation and invite the reader to continue. Using the relations in C(p), we see the left hand side is

iii

i<j

i

On the other hand, the right hand side is k

k

i

One now shows easily that i<j

k

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Dirac Operators in Representation Theory

and somewhat less easily that the one mentioned above).

C kcx(wk)2is a constant

(actually, exactly

2.3. Spinors Let X be a (g,K)-module, so U ( g ) acts on X . We want t o get the Dirac operator to act, so we have to tensor X with a C(p)-module S ; then U ( 8 )8 C ( p )will act on X 8 S , in particular D will act. Since we also want to stay as close as possible to X, we want to take minimal, i.e., simple S. It turns out there are only one or two choices for S , depending on whether dimp is even or odd. A simple C(p)-module is called the spin module, or a space of spinors. Let us first consider the case when dimp is even, say 2r. The construction of S involves taking a maximal isotropic subspace u of p with respect t o B . We can for example construct one such subspace starting from our orthonormal basis Zi of po and dividing it into two groups, 21,. . . , 2, and &+I,. . . , Zzr. Now we can take for u the span of all Z,+iZ,+,, s = 1,.. . ,r. Clearly, ii will be a complementary isotropic subspace (the conjugation is with respect t o po), and we have p=ueii.

Since the restriction of B to both u and ii is zero, it follows that B identifies U with the dual space u* of u. Furthermore, the Clifford algebras over u and ii are equal to the exterior algebras, since B = 0. We take a basis ui of u and the dual basis iii of ii. Let fi, = GI...fir be the corresponding basis

element of Atopii. We define S to be the left ideal in C ( p )generated by ii*. More explicitly, one can identify

s 2 A(u)a*

A(.),

(since ii annihilates &). The action of p is now given by

u . ( u ~A,. . . A u i , ) = u A u i l A . . . A u i s ii.(ui,A . . . A u i s )

=

- 2 C B ( -U , uik)uilA . . . uik . . . A ui, k

for u E u and ii E ii.Namely, u just Clifford multiplies, or equivalently wedge multiplies, from the left, while U. has to commute through, and then eventually gets killed upon meeting fi*. It is quite easy to show that S is a simple C(p)-module, and not too difficult that it is the only one up t o isomorphism. See e.g. [33], 9.2.1.

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J . 3 . Huang and P. PandEiC

+

Let us now briefly examine the case when dimp is odd, say 2r 1. We can still consider maximal isotropic u and ii as before, but now there is an additional element Z E p, orthogonal to both u and U, and we take it t o be of norm 1. We again take S to be Au, and we want t o define an action of 2 on S. From the relations it follows that Z must be scalar on every homogeneous component of Au, and that these scalars must alternate as the degree changes. To see this, start by observing that all elements of u kill the top wedge A' u, and that there are no other elements of A u killed by all elements of u. Since A'u is one dimensional, 2 must act on it by a scalar, Now use anticommuting of Z and ulzf's to see that Z acts on A'-' u by the opposite scalar, etc. From Z2 = -1, the scalars by which Z can act u and -i on are i and -a. We thus have two choices: Z can be i on Aeuen Aoddu, or vice versa. This gives us two nonisomorphic simple C(p)-modules S1 and Sz.One shows by a similar argument as above that these are the only simple C(p)-modules up to isomorphism. 2.4. Spin group

Let us again first assume that dimp is even. Let w E po be of length 1, i.e., B(u,w) = 1. Then u is clearly invertible in C(po), as v-l = -w. Consider now the action of u on po by conjugation in C(p0):

r,(X) = vXu-l=

-wxw, x E po.

If X is orthogonal to w,then Y and X anticommute, and hence r , ( X ) = - X . If X is proportional t o u , then r v ( X ) = X . So we see that r , is minus the reflection with respect t o the hyperplane orthogonal to u. All u as above generate a subgroup of the invertible elements of C(p0) which we denote by Pin(p0). The above discussion shows that we have a map

PNPO)

-

Obo),

which is surjective and has kernel (1, -1). The connected component of Pin(p0) is the spin group Spin(p0); it coincides with the products of an even number of vectors w as above. It is a compact, semisimple group which is a double cover of the group SO(p0). In case dimp is odd, one can do a similar construction. Instead of doing this, let us give a uniform description of S p i n ( p 0 ) valid regardless of the parity of dim p (and also for forms which are not necessarily positive definite); see [17], p. 282. Let a be the antiautomorphism of C(p) given by the identity on p. Then Spin(p0) is the group of all even elements g of C(p0)

Dirac Operators in Representation Theory

179

such that g a ( g ) = 1 and g z a ( g ) E po for all z E po. For g E Spin(po), define T ( g )E GL(po) by T(g)z= g z a ( g ) . Then T : Spin(p0) -+ SO(p0) is a double covering. In particular, Spin(p0) is compact (since B is positive definite on P O ) . Since Spin(p0) is contained in C(p), it acts on any C(p)-module. For dim p even, there is only one such simple module, S. Since S p i n ( p 0 ) consists of even elements of C(p), it preserves the subspaces

s- = /\oddu

s+= /\evenu;

of S. These are actually irreducible, and they are called spin representations. For dimp odd, the two spaces of spinors 5’1 and 5’2 are irreducible for Spin(p0) and equivalent to each other. This representation is also called the spin representation. 2.5. Dirac cohomology

We consider the spin double cover pullback diagram:

K

I K

K of K , constructed from the following

-

Spin(p0)

I S~(PO)

Now if X is a (8, K)-module, then I? acts on X @ S by acting on both factors: on X through K and on S through Spin(p0). Moreover, it is easy to show that X @ S is a (U(g)@C(p),l?)-module. Such modules are defined analogously as (g, K)-modules; here I? acts on U ( g ) @ C(p) through the adjoint action of K on both factors, and the Lie algebra of K embeds into U ( g )@ C(p) as the diagonal t~ described earlier. Dirac operator D acts on X @ S and we define the Dirac cohomology of X to be the I? module

H D ( X )= Ker D /Ker D n Im D.

If we suppose X is a unitary (8, K)-module, then we can define a positive definite hermitian form such that D is symmetric with respect t o this form. For this we use the usual form on S for which all elements of po are skewhermitian (see [33], 9.2.3, or [ 7 ] ) It . now follows that if X is unitary, then Ker D n Im D = 0, and the Dirac cohomology of X is just Ker D. More or less all of the above can be found in any of the following references: [7], (171, [33] or [15] (even case).

J.-S. Huang and P. Pandiii

180

3. Dirac operators and group representations 3.1. Paul Dimc and his operator Paul Dirac (1902-1984) was one of the greatest physicists of the 20th century. His research interests were mainly quantum mechanics and elementary particles. A free particle T in R3 is described by a state function $(t, x) with t E R and x E R3.To understand this function, one needs to understand the square root of the wave operator a2 a2 a2 a2 0= -- -- ax; ax: ax; ax;. In 1928, Dirac found the square root D of 0.It is a matrix valued first

order differential operator, and it is now called the Dirac operator.

3.2. Square root of Laplace operator We first describe the square root of the Laplace operator in Rn:

A = _ -a-2- _ . .a.2_ -

a2

ax: ax;

ax:

'

-&

For n = 1, the Laplacian operator has the form A = and the Dirac operator D = 2% acts on the space of smooth functions f : R -+ @. For n = 2, the Laplacian operator has the form A = a2 - a2 We define

--= w .

D = ( 0 2 ) ~ + ( - l 0o )15 =d7 " & +ar , - , a 't 0 ax aY which acts on the space of smooth functions f : R2 4 C2. It follows from 2

2

y" = yY = -1, TzTy that

D2 = A

(i y)

+

"iy3;:

=0

= AI

For n = 3, the Laplacian operator has the form A We define

a2

Then one has 7: = -1,yiyj

+ yjyi = O ( 2 # j ) ,2, j

a2

= ---

E {1,2,3}.

-

a2 -.

Dirac Operators in Representation Theory

181

Set

which acts on the space of smooth functions f : R3 -+ C2. It follows that

D2=A(kY) = A I

3.3. The original Dirac operator and its generalizations Dirac defined the square root of the wave operator 0 in terms of Pauli matrices:

Then one has

Define 4 x 4 matrices

and

It follows from

that

D2 = 01. Brauer and Weyl afterwards generalized the definition of Dirac operator to arbitrary finite-dimensional (quadratic) space of arbitrary signature. The Dirac operator defined in 2.2 is a much more recent analogue for semisimple Lie algebras; it was introduced by Parthasarathy in 1972.

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J.-S. Huang and P. PandEiC

3.4. Group representations Representations of finite groups were studied by Dedekind, Frobenius, Hurwitz and Schur at the beginning of the 20th century. In 1920’s, the focus of investigations was representation theory of compact Lie groups and its relations to invariant theory. Cartan and Weyl obtained the well-known classification of equivalence classes of irreducible unitary representations of compact Lie groups in terms of highest weights. In 1930’s, Dirac and Wigner started the investigation of infinite-dimensional representations of noncompact Lie groups. Harish-Chandra (1923-1983) was a Ph.D. student at the University of Cambridge under supervision of Dirac during the years 1945-1947. It was Harish-Chandra who began a systematical investigation of infinitedimensional representations of semisimple Lie groups after his Ph.D. thesis. He laid down the foundation for further development of the theory for the last half century. In 1964 and 1965, Harish-Chandra published two papers which gave a complete parametrization of discrete series representations. Later he used this classification to prove the Plancherel formula. This classification of discrete series is also crucial t o Langlands classification of admissible representations. However, Harish-Chandra did not give explicit construction of discrete series. His work was parallel to that of Cartan-Weyl for irreducible unita.ry representations of compact Lie groups. In 1955, Borel and Weil gave explicit realization of irreducible unitary representations of compact Lie groups. In 1957, Bott generalized BorelWeil theorem by considering Dolbeault cohomology of line bundles over G / T , where T is a maximal torus. Soon thereafter, Kostant and Langlands conjectured that discrete series representations are equivalent to certain Dolbeault cohomology of line bundles over G/T for noncompact G. In his 1967 thesis [25], Schmid started a proof of the conjecture of Kostant and Langlands. This work was completed in his Annals of Mathematics paper [261. After Hotta’s work on construction of holomorphic discrete series and Parthasarathy’s construction of most discrete series representations by Dirac operators, Atiyah and Schmid in 1977 proved that all discrete series can be constructed as kernels of the Dirac operator over twisted spinor bundles. The definition of the Dirac operator for a spin bundle was a major accomplishment of Atiyah-Singer, who obtained the celebrated index theorem. The Dirac cohomology is a far reaching generalization of the idea of index

Dirac Operators in Representation Theory

183

theory t o representation theory. In the rest of the sections we see how this simplifies proofs of many classical results such as Bott-Borel-Weil theorem and Atiyah-Schmid theorem and how it sharpens a result of Langlands and Hotta-Parthasarthy on multiplicities of automorphic forms. An important tool in our approach is the theory of A , (A)-modules which we review in the next section. One should bear in mind that this theory was not available at the time when the above mentioned classical results were proved,

4. Introduction to Aq(X)-modules Mackey's construction of induction is based on real analysis, and Zuckerman's construction of induction which is needed for A, (A)-modules is based on complex analysis. It amounts to the geometric construction of representations by using Dolbeault cohomology sections of vector bundles over a noncompact complex homogeneous spaces then passing to the Taylor coefficients.

4.1. 0-stable parabolic subalgerbas

+

Let H = T A be a fundamental Cartan in G. Let l ~ o= to a0 be the corresponding 6'-stable Cartan subalgebra. Then to = bo n to is a Cartan subalgebra of to. As usual we drop the subscript 0 for the complexified Lie algebras. Let X E it0 be such that ad(X) is semisimple with real eigenvalues. We define (i) I to be the zero eigenspace of ad(X), (ii) u to be the sum of positive eigenspaces of ad(X), (iii) q to be the sum of non-negative eigenspaces of ad(X). Then q is a parabolic subalgebra of g and q = I + u is a Levi decomposition. Furthermore, I is the complexification of I0 = q n go. We write L for the connected subgroup of G with Lie algebra Io.Since O(X) = X , K,u and q are all invariant under 6,so q = q net- q np.

In particular, q n €! is a parabolic subalgebra of t with Levi decomposition q n e = In €! +un e.

We call such a q a &stable parabolic subalgebra. Let f c q be any subspace stable under ad(t). Then there is a subset {ai,. . . ,a T }o f t * and subspaces fa, of f such that if y E t and w E fa,, then ad(y)v = ai(y)v.

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J . 3 . Huang and P . PandiaC

We write

A(f,t) = A(f) = { Q l , . . . , Q T ) , the weights or roots o f t in f. Here A(f) is a set with multiplicities, with ai having multiplicity dim f a . . Then if P(f) = P(A(f)) =

1

5

c

Qi E

t*,

ai€A(f)

we have

Fix a system A + ( rn t) of positive roots in the root system A(rnt, t). (Note that we extend the meaning of root system t o include the zero weights.) Then

A+@)= A+(K n t) u A(u n e) is a positive root system for t in t. If Z is an (I,L n K)-module, we write Z# for Z @ AtoPu. We set

Then both pro(Z#) and in@#)

are (8, L n K)-modules.

4.2. Zuckerman and Bernstein functors

I’

Let V be a (8, L n K)-module. The Zuckerman functor r = r;;fnK can be defined as follows (if K is connected): Let r(V)be the sum of all finitedimensional to-invariant subspaces of V such that the to-action can be lifted to K . For a morphism ‘p : V -+ W of (8, L n K)-modules, let r(p)be the restriction of ‘p to I’(V). The functor I? is not exact, but only left exact. Therefore one also needs to study the right derived functors rj. For the study of unitarity, it is useful to consider also a “left analog” of I’,the Bernstein functor IT = II:;fnK and its left derived functors IIj.I2is not so easily defined directly; however, there is a uniform way to describe Zuckerman and Bernstein functors in terms of Hecke algebras, or in terms of relative Lie algebra (co)homology. Let R ( K ) be the space of distributions on K . Let R(g,K ) be the space of left and right K-finite distributions on G with support in K . Then

Dirac Operators in Representation Theory

185

Then one has that the Zuckerman functor on V is

and the Bernstein functor is

We set R j(2) =

rj(pro Z#)and

C j (2) = ITj (ind Z#) .

4.3. Irreducibility and unitarity of cohomologically induced modules The hermitian inner product is not obvious for cohomological parabolic induction. This makes studying unitarity a very difficult problem for cohomologically induced modules. Nevertheless, Vogan proved the following powerful theorem.

Theorem: ([29]) Suppose q is a 8-stable parabolic subalgebra of g and Z is a n (1, L n K ) - m o d u l e with infinitesimal character A. If Z i s weakly good (i.e. Re(X p ( u ) , a ) 2 0 f o r any a E A(u)), then (i) C j ( 2 ) = Rj(2)= 0 f o r j # s (s = dimu n t). (ii) C,(Z)% R"(2). (iii) If Z is irreducible, then C s ( Z ) is irreducible or zero. (iv) If Z is irreducible and in addition is good (i.e. Re(A p(u),a ) > 0 f o r any a E A(u)), then C s ( Z )is irreducible and nonzero. (v) If 2 is unitary, then C,(Z)is unitary.

+

+

Remark: There are two dualities:

where W h is the hermitian dual of W . 4.4. A,(X)-modules

Now we consider Z to be a one-dimensional representation. A : called admissible if it satisfies the following conditions: (i) A is the differential of a unitary character of L (ii) if a E A(u), then ( a , X l t ) 1 0 . Given q and an admissible A, define p(q,A) = representation of

K of highest weight XIt

[ -+

+ 2p(u n p).

Q1 is

(1)

J . 3 . Huang and P. PandiiiC

186

The following theorem is due to Vogan and Zuckerman.

Theorem: ( [ 3 2 ] ,[29])Suppose q is a 8-stable parabolic subalgebra of g and A: I + CC is admissible as defined above. Then there is a unique unitary (9,K)-module A, (A) with the following properties: (i) The restriction ofA,(X) to t contains p(q,X) as defined in (4.1); (ii) Aq(X) has infinitesimal character X p; (iii) If the representation o f t of the highest weight S occurs in Aq(X), then

+

S=p(q,X)+

c

npP

P E A (UmJ)

with np non-negative integers. I n particular, p(q, A) is the lowest K-type of A, (A). We note that the unitarity of Aq(X) in the above theorem was proved in [29].In the context of definition of 8-stable parabolic subalgebras, if we take X to be a regular element, then we obtain a minimal &stable subalgebra b = t, n. We call such a subalgebra b a &stable Bore1 subalgebra. The corresponding representation Ab (A) is called a fundamental series representation. It is the (9,K)-module of a tempered representation of G. If G has a compact Cartan subgroup, then Ab(X) is the (9,K)-module of a discrete series representation of G. Moreover, all (g, K)-modules of discrete series representations of G are of this form; this will be important in Section 6. For the proof of the main result of [12] (Section 5) it is only needed that Ab(X) has infinitesimal character X p and the lowest K-type p(b, A) = X 2pn, where pn = p(n n p). These facts are contained in Theorem 4.4.

+

+

+

4.5. Salamanca-Riba’s classification of the unitary dual

with strongly regular infinitesimal characters

As before, let be the complexification of a fundamental Cartan subalgebra t,o of go. Given any weight A E t,*, fix a choice of positive roots A+(A, t,) for A so that

Set

187

Dirac Operators in Representation Theory

Definition: A weight A E

b* is said to be real if A E it;

+ a;,

and t o be strongly regular if it is real and

Salamanca-Riba [24] proved: Theorem: (Salamanca-Riba) Suppose that X i s an irreducible unitary (8, K)-module with strongly regular infinitesimal character A E h*. Then there exist a 6-stable parabolic subalgebra q = I u and a n admissible char-

+

acter X of L such that X is isomorphic to A4(X). 5. Vogan's conjecture and its proof

In this section we explain Vogan's conjecture on Dirac cohomology and a proof of this conjecture. The presentation mostly follows [12].

5.1. Vogan's conjecture Let T be a maximal torus in K , with Lie algebra to. Let b be the centralizer o f t in g; it is a &stable Cartan subalgebra of g containing t. Since 9 = t@p', we get an embedding o f t * into b*. Therefore any element o f t * determines a character of the center Z(g) of U ( g ) . Here we are using the standard identification Z(g) 2 S(Ij)wvia the Harish-Chandra homomorphism ( W is the Weyl group), by which the characters of Z(g) correspond t o the W-orbits in b*. We fix a positive root system A+(t,t) for t in t; let pc = p(A4(t, t)) be the corresponding half sum of the positive roots. For any finite dimensional irreducible representation (y, E7) of t, we denote its highest weight in t* by y again. Vogan (301 made a conjecture which was proved as the following theorem: Theorem: ([12]) Let X be an irreducible (8, K)-module, such that the Dirac cohomology of X is non-zero. Let y be a K-type contained in the Dirac cohomology. Then the infinitesimal character of X is given by y pc.

+

In view of the remarks in 2.5, in case X is unitarizable, we get the following consequence:

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Corollary: Let X be an irreducible unitarizable (g, K)-module, such that Ker D # 0. Let y be a K-type contained in Ker D. Then the infinitesimal character of X is given b y y p c .

+

In fact, Vogan first conjectured the above corollary and then he saw that the above theorem should be the right generalization t o non-unitary represent at ions.

5.2. An algebraic reduction of the conjecture

Vogan further reduced the claim of his conjecture t o an entirely algebraic statement in the algebra U ( g ) 8 C ( p ) . Let us first recall that in 2.2 we ) Z(~A of f! inside U ( g ) 8 C(p). U ( ~ Aand described a diagonal copy denote the corresponding universal enveloping algebra and its center. It is easy to see that they are also embedded into U ( g ) 8 C(p); namely, if u E U(t) is a PBW monomial, then its image in U ( g )8 C(p) is the sum of u 8 1 and terms of the form w @ a, with w having smaller degree than u. We can now state Vogan's algebraic conjecture that implies the theorem in 5.1. Theorem: Let z E Z(g). Then there is a unique ((2) in the center Z ( ~ of U(tA), and there are K-invariant elements a , b E U ( g )@C(p),such that

z 8 1 = C(z)

+ D a + bD.

To see that this theorem implies the theorem in 5.1, let Z E (X 8 S)(y) be non-zero, such that DZ = 0 and Z @ Im D. Note that both z 8 1 and ( ( z ) act as scalars on Z. The first of these scalars is the infinitesimal character A of X applied to z , and the second is the &infinitesimal character of y applied to C ( z ) , that is, (y p,)(C(z)). On the other hand, since ( z 8 1 - ((2)). = DaZ, and 2 $ I m D , it follows that ( z 8 1 - ( ( z ) ) Z = 0. Thus the above two scalars are the same, i.e.7 = (7 p c ) ( C ( . ) ) . In 5.6 we will show that under identifications Z(g) 2 S(b)w2 P(b*)w and Z ( ~ A ) Z(t) 2 S ( t ) w K 2 P(t*)wK the homomorphism C corresponds to the restriction of polynomials on Q* to t*. Here the already mentioned inclusion oft' into b' is given by extending functionals from t to b, letting them act by 0 on a = p f . It follows that A = y p c , as claimed.

+

+

+

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D i m c Operators in Representation Theory

5.3. A differential complex induced by Dirac operator Let us first note that the Clifford algebra C ( p ) has a natural Z2-gradation into even and odd parts:

W )= C"P)

@ Cl(P).

This gradation induces a Z2-gradation on U ( g )@ C ( p )in an obvious way. We define a map d from U ( g )@ C(p) into itself, as d = do @ d l , where

do : U ( g )€4 C"P)

--f

U ( 8 )€4 C1(P)

is given by

do(.) = Da - aD,

(2)

and

is given by

In other words, if E , denotes the sign of a , that is, 1 for even a and -1 for odd a , then d ( a ) = Da - E,UD(for homogeneous a, i.e., those a which have sign). We will use the formula for D2 from Lemma 2.2, namely

D2=-flg@1+fle,

+c

to prove that our d induces a differential on the K-invariants in U ( g ) @ C ( p ) .

Proposition: Let d be the map defined in (2) and (3). T h e n (i) d is K-equivariant, hence induces a map from ( U ( g )@ C ( P ) into )~ itself. (ii) d2 = 0 o n ( U ( g )@ C ( P ) ) ~ .

Proof: (i) is trivial, since D is K-invariant. Let a E ( U ( g )@ C ( P ) be ) ~even or odd. Then ci2(a)= ~ ( D U - E , U D= ) D~U-ED,DUD-E,(DUD-E,DUD~)

=D ~ U - U D ~ ,

since obviously E,D = E D , = - E , . From the formula for D2 (Lemma 2.2), we see that a will commute with D2 if and only if it commutes with & ! ., If a is K-invariant, then this clearly holds, as a then commutes with all of UP,). 0

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Thus we see that d is a differential on ( U ( g )@ C ( P ) ) of ~ ,degree 1 with respect to the above defined &-gradation. Note that we do not have a Zgradation on ( U ( g )@ C ( P ) )so ~ that d is of degree 1, i.e., this is not a complex in the usual sense. 5 . 4 . Determination of cohomology of the complex

We want to calculate the cohomology of d. Before we state the result, let us note the following:

Proposition: Z(ta) i s in the kernel of d

Proof: Since D is K-invariant, it commutes with t ~and , thus with U(ta) and in particular with Z(ta). Since z ( t ~c)( U ( g )@ C o ( p ) ) Kthe , claim follows. 0 We now state the following theorem which implies Theorem 5.2.

Theorem: Let d be the diflerential on ( U ( g )8 C ( P ) )constructed ~ above. T h e n Kerd = Z(ta)@Imd.In particular, the cohomology of d is isomorphic to z ( t A ) . The proof uses the standard method of filtering the algebra (the filtration comes from the usual filtration on U ( g ) ) ,and then passing t o the graded algebra. This graded algebra is of course S ( g ) @J C ( p ) .The analogue of our theorem in the graded setting is easy; the complex we get is closely related to the standard Koszul complex associated to the vector space p. Namely, the operator d induced by d on S(g) @ C(p) is given by supercommuting with

and one easily calculates that

c k

d(u @ zi,. . . Zi,)= -2

uzz3@J

zi,. . . Zij . . .z2,.

j=1

Upon identifying C(p) and A(p) as vector spaces and writing

(s(P) 8 A(P)L

~ ( 88 ) C(P)= ~ ( t8)

we see that

d = (-2)id@d,,

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191

where d, is the Koszul differential for the vector space p. In particular, 2is a differential with cohomolgy S(e)8 @, which embeds into S(g) @I C(p) by embedding C into S ( p ) 8 A(p) as the constants. Passing t o K-invariants, we see that K e r 2 = S(t)K@ 1 $ I m d on (S(g) 8 C ( P ) ) This ~ . is the graded version of our theorem.

One can now go back to the original setting by an easy induction on the degree of the filtration. The main point is that one can reconstruct an element of Z(t,) from its top term. We refer the reader to our paper [I21 for the details of the above proof. Let us note a consequence, which immediately proves Vogan's conjecture, just put b = a.

Corollary: Let z E Z(g). Then there i s a unique < ( z ) E Z(ta), and there is an a E ( U ( g )@I C ' ( P ) ) such ~ , that z @ 1 = ( ( 2 ) t Da

+ aD.

Proof: This follows a t once from Theorem 5.4, if we just notice that z 8 1 commutes with D (indeed, it is in the center of U ( g )8 C(p)), and being even, it is thus in Ker d. Hence, it is of the form <(z)+d(a)= <(z)+Da+am 5.5. Dirac inequality and unitary representations with nonzero Dirac cohomolog y

We first indicate how to check if a unitarizable X has non-zero Dirac cohomology.

Proposition: Let X be an irreducible unitarizable (9, K)-module with infinitesimal character A. Assume that X @ S contains a K-type y, i e . , ( X @ S)(y) # 0 . Assume further that / ( A / (= (Iy p c ( ( . Then the Dirac cohomology of X , Ker D , contains ( X 8 S ) ( y ) .

+

Proof: This again uses the formula for D2 from Lemma 2.2. The formula implies that D2 acts on (X@ S)(y) by the scalar

-(ll4l2 - llP1I2) + (Ilr +PCIl2

+ (llPcl12 - 11P1I2) = 0.

- llPC1l2>

It follows from self-adjointness of D that D = 0 on (X@I S)(y).

0

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Note that Corollary 5.1 implies the converse of the above proposition: if

X is irreducible unitarizable, with Dirac cohomology containing ( X @ S (y), ) then the infinitesimal character of X is A = y +pc. Hence I J A J= J IJy+pclJ. We note that all irreducible unitary representations with nonzero Dirac cohomology and strongly regular infinitesimal characters were described in [12].They are all Aq(X)-modules. See Proposition 5.6 where this is explained in a special case. Finally, combining the above proposition with Corollary 5.1, we sharpen the Parthasarathy's Dirac inequality:

Theorem: (Extended Dirac Inequality) Let X be an irreducible unitarizable (0, K)-module with infinitesimal character A. If ( X @ S)(y) # 0 , then

ll4l 5 IIY + P c l l . The equality holds i f and only if some W conjugate of A is equal to y + p c . 5.6. Fundamental series and determination of

C

Proposition: Let X be an Ab(X)-module (as in Theorem 4.4) with b a &stable Bore1 subalgebra, i.e., X is a fundamental series representation. Assume that XIa = 0 . Then the Dirac cohomology of X contains a K-type E7 of highest weight = X pn.

+

+

Proof: The lowest K-type p(b, A) has highest weight X 2p,. Since -pn is a weight of S, E-, occurs in p(b,X) @ S, hence in X @ S. Since the infinitesimal character of X is X p = y pc, it follows that E-, is in the kernel of the Dirac operator D ,i.e., in the Dirac cohomology of X.

+

+

Now we can describe the homomorphism

Theorem: The homomorphism gram:

5 explicitly.

4 satisfies the following commutative dia-

Z(B)

1 S(fJ)W

-5 1

2S(t)W"

Here the vertical arrows are the Harish-Chandra homomorphisms, and the map Res corresponds to the restriction of polynomials o n b* to t* under the identifications S(b)w = P(b*)w and S ( t ) W K = P ( t * ) w K . A s before, we can view t* as a subspace of b* by extending functionals from t to b, letting them act by 0 on a.

Dirac Operators in Representation Theory

193

Proof: Let : P(b*)w -+ P(t*)wK be the homomorphism induced by ( under the identifications via Harish-Chandra homomorphisms. Furthermore, let : t*/WK -+ b*/W be the morphism of algebraic varieties inducing the homomorphism We have to show that ( is the restriction map, or alternatively that is given by the inclusion map. We know from the above proposition that the fundamental series representation Ab (A) has the lowest K-type

r

<

t.

p(b, A) =

+ 2~7x7

and infinitesimal character A=X+p. On the other hand, it follows from Proposition 5.5 that if XI, = 0, then the Dirac cohomology of Ab(X) contains the &-type of highest weight y =

+ pn. When proving that Theorem 5 . 2 implies Theorem 5.1, we saw that A(z) = ( y + p c ) ( C ( z ) ) , for all z E Z(g). In our present situation we however have A=

+ p = (A + pn) + pc =

-tpc,

so it follows that A(((z)) = A(z) for all z E Z(g). This means that C(A) = A, for all infinitesimal characters A of the above fundamental series representations. It is clear that when X ranges over all admissible weights in b* such that XI, = 0, then A = X + p form an algebraically dense subset of t*. To see this, it is enough to note that such X span a lattice in t*. Hence ( is indeed the inclusion map. 5 . 7 . Remark on finite-dimensional representations

Both Vogan and Kostant pointed out to us that Theorem 5.6 can also be proved by considering finite-dimensional representations with non-zero Dirac cohomology. These are the representations Vx of highest weight X E t* c b*. See [12], Remark 5.6.

6. Generalized Bott-Borel-Weil Theorem and construction of discrete series 6.1. Kostant cubic Dirac operator

Let G be a compact semisimple Lie group and R be a closed subgroup. Let g and t be the complexifications of the corresponding Lie algebras. Let

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194

g = r e p be the orthogonal decomposition with respect to the Killing form. Choose an orthonormal basis 21,.. . , Z n of p with respect t o the Killing form ( , ). Kostant [18] defines his cubic Dirac operator to be the element

c n

D=

2 2

8

zi + 18

2,

E U ( g )€3 C(p),

i= 1

where

2,

E

C(p) is the image of the fundamental 3-form w

E

A’@*),

under the Chevalley identification A@*) --+ C(p). Kostant’s cubic Dirac operator reduces to the ordinary Dirac operator when (8, t) is a symmetric pair, since w = 0 for the symmetric pair. Kostant ([18],Theorem 2.16) shows that

D2 = R, 8 1 - a,,+ c,

(4)

where C is the constant lip[l2 - (Ipr1[2.This is the generalization of Lemma 2.2. The sign change comes from the fact that Kostant uses a slightly different definition of C(p), requiring Z? to be 1 and not -1. Over C, there is no substantial difference between the two conventions. Now we can define the cohomology of the complex ( U ( g )@ C ( P ) using )~ Kostant’s cubic Dirac operator exactly as in Section 5, i.e., by d ( a ) = Da - e,aD. As before, d2 = 0 on ( U ( g )8 C ( P ) ) Since ~ . the degree of the cubic term is zero in the filtration of U ( g ) 8 C(p) used in Section 5, the proof Theorem 5.4 goes through without change and we get

Theorem: Let d be the differential o n ( U ( g )@ C ( P )defined )~ by Kostant’s cubic Dirac operator as above. T h e n Kerd = I m d @ Z(ta). I n particular, the cohomology of d i s isomorphic to Z(ta). As a consequence we get an analogous homomorphism C : Z(g) + Z(t) for a reductive subalgebra t in a semisimple Lie algebra g and a more general version of Vogan’s conjecture. 6.2. Kostant ’s theorem o n cohomology of homogeneous spaces If we fix a Cartan subalgebra t o f t and extend t to a Cartan subalgebra b of g, then is induced by the Harish-Chandra homomorphism exactly as in Theorem 5.6. This was proved in [19], by constructing a sufficiently large

<

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Dirac Operators in Representation Theory

family of highest weight modules with known infinitesimal characters and nonzero Dirac cohomology. Moreover, the homomorphism C induces the structure of a Z(g)-module on Z(t), which has topological significance. Namely, Kostant has shown that from a well-known theorem of H. Cartan [4], which is by far the most comprehensive result on the real (or complex) cohomology of a homogeneous space, one has

Theorem: [19] There exists an isomorphism Z(t)).

H * ( G / R , C )%

6.3. A generalized W e y l character f o r m u l a We assume that rankR = rankG. Let W1 c W, be the subset of Weyl group elements that map the positive Weyl chamber for g into the positive Weyl chamber for r. If A is a dominant weight for g, then X p(g) lies in the interior of the Weyl chamber for g. It follows that w(A p ( g ) ) lies in the interior of the Weyl chamber for t for any w E W1. Thus, w A = w(X p ( g ) ) - p ( t ) is a dominant weight for t. Let VA denote the irreducible (finite dimensional) representation of g with highest weight A, and let U w . ~ denote the irreducible (finite dimensional) representation of r with highest weight w A.

+ +

+

.

.

Theorem: [8]

v,c3 s+

-

VA

@

s- =

c

(-l)~(W)Uw.A.

WEW'

It follows that

Note that the above formula reduces to the Weyl character formula when R is a maximal torus T . 6.4. A generalized Bott-Borel- Weil Theorem

We assume that U, is an irreducible representation of R (or R ) so that S @ U p is a representation of R. The Dirac operator acts on the smooth

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196

and L2-sections on the twisted spinor bundles over G I R , if we let Zi E g act by differentiating from the right. So we have

D :L2(G) @ R ( S @ U,)

+

L2(G) @ R ( S @ U p ) .

We write this action in another form:

D : HomB(Ui, L2(G) @ S ) + HomR(U;, L2(G) @ S ) . Then D is formally self-adjoint. By Peter-Weyl theorem, one has L2(G) @,EdV, @ V;. It follows that Ker D

=

@ V, @ Ker{D : Homfi(Ui, V:

8 S ) 0).

A&

The proved Vogan's conjecture implies K e r D # 0 if and only if there is some X E such that X p ( g ) is conjugate to p p ( r ) by the Weyl group. Further consideration of the multiplicity results in

+

+

Theorem: One has K e r D = Vw(,+p(r))-p(0) if there exists a w E W, so that w ( p + p ( r ) ) - p(g) is dominant, and Ker D is zero if no such w exists. In the case R = T , a maximal torus, this is a version of Borel-Weil theorem.

Corollary: Consider

D+ : L ~ ( Gg) R

(s+E+ v,)

+

L ~ ( G@R ) (s-@ u,)

and the adjoint

D - : L2(G)@ R (5'-

@I U p )+

L2(G)@ R (5" @ U p ) .

One has IndexD = dim ker D+ - dim Ker D- = (-l)'(w) dim Vw(p+p(t))-

if there exists a w E W, so that w ( p + p(r)) - p(g) is dominant and it is zero
Let G be a linear semisimple noncompact Lie group. Let K be a maximal compact subgroup of G. Assume that rank G = rank K . Let go = to po be the Cartan decomposition of the Lie algebra of G. Then uo = to i p o is a compact real form of g = go @R @. Let U be the compact analytic subgroup in the complexification Ga: of G with Lie algebra UO.

+

+

Dirac Operators in Representation Theory

197

Bore1 showed that there exists a torsion free discrete subgroup r of G so that I'\G and X = r \ G / K are compact smooth manifolds. For any p. E K , the Dirac operator D acts on the smooth sections of the twisted spin bundle in a similar way described in 6.4. A

D : C"(G/K, S 8 E p ) -+ C'(G/K,

S 8 Ep).

Note that the above action of D commutes with the left action of G. So we can consider the elliptic operator

D E ( X ) : C"(I'\G/K,

S+ 8 E p )4 C"(r\G/K, S- 8 E p ) .

The index of D ; ( X ) can be computed by Atiyah-Singer Index Theorem IndexD;(X) =

f(0, dj),

where 0 is the curvature of X and dj is the curvature of the twisted spinor bundle over G / K . By the homogeneity, f(@, a) is a multiple of the volume form depending only on p , i.e., f(0,a) = c(p.)ds. Thus Index D Z ( X ) = c(p)vol(I'\G/K). Let Y = U/K be the compact homogeneous space. By Hirzebruch proportionality principle, the index of

D;(Y): C"(U/K, Sf 8 E p )-i C"(U/K, S- 8 E p ) can be computed in the same way and Index D t ( Y ) = (-l)qc(p)vol(U/K), where q

= d i m G / K = dimU/K.

Index D t ( X ) = (-

It follows that

1)q

vol ( r \ G / K) . Index D t (Y) vol (U/K)

If we normalize the Haar measure so that vol(U) = 1, then Index D: ( X ) = ( -l)qvol(I'\G) Index Dt ( Y ) . Let L2(G) E JE Hj@Hj*dp.(j)be the abstract Plancherel decomposition. Then the L2 sections of the twisted spinor bundles are decomposed as L2(G/K, S 8 E p ) 2

Hj 8 HomK(Hj, S 8 E,)dp(j).

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It follows that the L2-sections of Ker D are decomposed as Ker D

E

Hj

@ Ker{D: Homk(E;L,,Hj* @I 5’) O } d p ( j ) .

By the proved conjecture of Vogan, if Hj occurs in the decomposition of Ker D then it has infinitesimal character p pc. There are a t most finitely many representations with a fixed infinitesimal character. Thus, if Ker D is nonzero, the corresponding H j must be in the discrete spectrum, i.e., a discrete series representation. Namely, the finitely many points must be of nonzero Plancherel measure in order to contribute nontrivially to the direct integral. It follows from Corollary 6.4 and the above discussion of the indices that w ( p -tpc) - p is dominant for some w E W,. Write X for w(p+p,)-p. Then the infinitesimal character X+p of Hj is strongly regular and therefore Hj is an Aq(X)-module [24]. In case X is regular, this Aq(X)module is isomorphic t o Ab(X) with the lowest K-type X 2pn. Therefore, we obtain the following theorem, which is a part of the main results in [l].

+

+

Theorem: (Atiyah-Schmid [l])Let G be a linear group. A s s u m e that there exists a w E Wg so that X = w ( p + p c ) - p is dominant and regular. T h e n the kernel of the Dirac operator D acting o n the L2-sections of the twisted spinor bundle corresponding t o E, is the discrete series representation &,(A) with the lowest K-type + 2pn. Combining this with some of the original arguments in [l]and with results of Harish-Chandra and Hecht-Schmid which say that the discrete series are exactly the &(A) with X dominant, one can obtain a geometric representation of all discrete series representations. We note that Atiyah and Schmid also extended this geometric construction of discrete series to nonlinear groups.

7. Lie algebra cohomology 7.1. Definition Let g be a complex Lie algebra and V a g-module. The space of invariants of V with respect to g is the vector space

vg = {V E VJXV= 0,vx E g}. Note that this definition is consistent with the definition of invariant (fixed) vectors under a group action. Upon differentiation, the condition of being fixed becomes the condition of being annihilated.

Dirac Operators in Representation Theory

199

One also considers invariants not with respect to whole g but with respect to some subalgebra. Let us describe an important example. Let g be semisimple, t) a Cartan subalgebra, A the set of roots for g with respect to t), A+ a set of positive roots, and n = n+ and n- the corresponding subalgebras like in Section 1.2. Then one is interested in the space V" of n-invariants in V , which is now not only a vector space but an t)-module. We already know why this space is interesting; if V is finite dimensional irreducible, then V" is one dimensional (the highest weight subspace), and the 4-action on V " , i.e., the highest weight of V , determines V . If V is still finite dimensional but reducible, V = @iV,, then each V , contributes a weight vector t o V" and V" thus encodes the information about this decomposition. We consider the functor V H V" from the category M ( g ) of g-modules into the category Vectc of complex vector spaces. (Analogously, V H V" would be a functor from M ( g ) into M(t)).)This functor is in general left exact, but not exact. This means that if

o--tu-iv-iw+o is a short exact sequence of g-modules (which is a slightly more precise way to write W = V / U ) , then

0 -i ug

vg -+ wg

i

is exact, but Vg -i Wg is not surjective in general. Thus we define the gcohomology functors to be the right derived functors of the functor V H Vg. To motivate the use of derived functors, let us consider the following simple example. Knowing how one can benefit from the usual duality operation for vector spaces, V * = Homc(V, C ) , one would like t o have something similar for modules, say over Z (for simplicity). If one tries to consider V * = Homz(V,Z),then one notices that for free modules of finite rank it works fine (including double dual giving back the same module), but for say Z2, one has Homz(Z2, Z)= 0. Namely, Z2 is not free, and its generator 1 can not be mapped anywhere but to 0. We can however resolve Zz by free modules:

.% z-i Z2 0. Z 5 Z -+ 0, concentrated 0 4Z

-i

Thus the complex 0 -i in degrees -1 and 0, should replace the module Z2. If we take this resolution and plug it in

200

J.-S. Huang and P. Pandiic'

the functor Homz(-,Z) instead of the module complex

Z2

it resolves, we get the

which is concentrated in degrees 0 and 1 (by contravariance, the arrows changed direction). Its zeroth cohomology is 0, corresponding to the observed fact Homz(Z2,Z) = 0. However, its first cohomology is 252 (this is Ext;(Zz,Z), the first derived functor of Hom). If we now apply duality again, we get back to our resolution of Z2, that is Z2 in the zeroth cohomology. To make the above completely precise, one would need t o pass t o derived category, which is a way of making modules equal (isomorphic) t o their resolutions. However, the point we wanted to make is visible: the functor itself maybe sometimes gives nothing, but considering also the derived functors may give more. We now get back to our covariant, left exact functor V H Vg. In principle, the right derived functors are defined using injective resolutions 0 --+ V -+ I . ; so the i-th g-cohomology space of V would be

Namely, as above, we replace V by its resolution, plug the resolution in the functor, get a complex with induced differential, and then take cohomology. However, this is not very explicit, as injective resolutions are rather complicated to write down. Fortunately we have a better possibility: note that

V g = Hom, (C, V ) , where C is the trivial g-module; simply identify every map q5 : C V with q5(1) E Vg. This means that our g-cohomology H i ( g ; V )is equal to Exte(C, V ) ,and to define it (calculate it) we can resolve C by free modules (or projectives) instead of resolving V by injectives. This is done by the so called standard, or Koszul complex

where the g-action is given by left multiplication in the U(g)-factor, the

201

Dirac Operators in Representation Theory

differential is the deRham differential

[xi, xi A . . . xi...z.j. . . A x k , A

x(-l)i+ju @ xj]A i<j

and E is the augmentation map, given by 1 H 1 and gU(g) H 0. To see that this is indeed a resolution, one considers the graded version, S(g) g~A g and is thus lead to the analogous question of resolving the trivial module C over a polynomial algebra. Now for polynomials in one variable, it is obvious how to do this: 0 + C[X] 5C[X]+ C -+ 0 is clearly the required resolution. To increase the number of variables, this is tensored with itself several times; the introduction of signs which leads to the exterior algebra is forced by the requirement d2 = 0. The reader is invited t o try to construct a resolution for two variables from scratch and see how the exterior algebra appears quite naturally. Now H i ( g ;V) is the i-th cohomology of the complex

HomU(,,(U(g)

@

A'0,V) = Hom&(A'0,V)

with the induced differential

da(XlA...AXk) = xXi(Y(X1A.. .zi...AXk)+x(-l)z+ja([Xi,xj]A...), i<j

a

for a E Hom;-'(A\'g,V).

Of course, H O ( g ;V)

= Vg.

7.2. Properties and basic applications

Here are some properties of these functors: (1) Any short exact sequence

o-+u4v+w-+o of g-modules gives rise to a long exact sequence of cohomology

0 4u g

4

vg

+

4

wo

H1(g;U ) 4Hyg;V) -+

W ) + H y g ; U ) -+ . . .

This is a completely general property of derived functors. It can be used to calculate cohomology in some case, or for various proofs, like the one below.

202

J . 3 . Huang and P. PandiiC

(2) If g is semisimple and if V has nontrivial infinitesimal character, then H i ( g ; V ) = 0 for all i. This can be proved by noting that since any Z E Z(g) with no constant term vanishes on the trivial module @, then by a standard homological argument it follows that the action of Z on the standard complex which is a resolution of C must be homotopic to 0. From this statement one can obtain Weyl’s theorem mentioned in 1.2: any finite dimensional module over a semisimple Lie algebra is a direct sum of irreducibles. This statement is equivalent t o the statement that any short exact sequence

of finite dimensional g modules is split, i.e., there is an s : W 4 V such that p o s is the identity on W . Indeed, such a splitting exhibits V as a direct sum of U and W and we can then decompose modules completely by induction on the dimension. Now to get a splitting, it suffices to know that the sequence Hom,(W,V) %Hom,(W,W)

+O

is exact; then we get our splitting s as a preimage of idw. But the functor Horn, (W,-) is a composition of two exact functors: Homa:(W,-), which is clearly exact, and (-), which is also exact on finite dimensional modules by the long exact sequence and the fact that H1(g;-) vanishes on all finite dimensional modules. The vanishing of H1 has t o be checked separately for the trivial module @; this is an easy calculation. (3) Similarly, Levi’s theorem that any Lie algebra is a semidirect product of the radical and a semisimple subalgebra can be obtained from the fact that the second cohomology of finite dimensional modules over a semisimple Lie algebra vanishes. Namely, by definition the radical r of g is the biggest solvable ideal, so there is an exact sequence

of Lie algebras, with I semisimple. Now the nontrivial extensions of [ by t can be seen to correspond to H 2 (I; r). As the last space is 0, every extension is trivial and hence the above sequence splits.

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7.3. Theorems of Casselman- Osborne and Kostant We next consider a parabolic subalgebra q of g, with a Levi decomposition q = Leu;

this is a slightly different Levi decomposition from the one mentioned above, with u the nilradical (the largest nilpotent ideal) and I a reductive subalgebra. A special case arises in the situation already studied; €or a Cartan subalgebra b and n coming from a choice of positive roots, define b=b@n.

This is a maximal solvable subalgebra of g and such are called Borel subalgebras. A parabolic subalgebra of g can be defined as any subalgebra containing a Borel subalgebra. One should think of Borel subalgebras as algebras of upper triangular matrices, and of parabolic subalgebras as algebras of block upper triangular matrices with some fixed shape of blocks. Then the Levi factor I is the algebra of corresponding block diagonal matrices, and the nilradical u is the algebra of all upper triangular matrices that are zero on blocks. This is for example exactly true (in some complex basis) if g = 5K(n,C). We consider the u-cohomology spaces Hi(u; V )for a g-module V . These spaces are actually [-modules in a natural way; namely, Iacts on the complex defining H i ( u ;V ) by acting both on A u and on V . This action commutes with the differential, hence descends to cohomology. Now Z ( g ) , the center of the enveloping algebra of g, acts on the complex Hom.(Au, V ) and its cohomology Hi(u;V ) in two ways. One action is through the Harish-Chandra homomorphism Z(g) 4 Z(I) and the K (i.e., U(I))action described above, and the other is acting on V only. This second action is well defined only for the center and not for the rest of U ( g ) . Theorem: (Casselman-Osborne) The above two actions of Z(g) agree on cohomology H y u ; V ).

This is a similar statement to Property (2) of 7.2, about the action of the center on the standard complex which was used to get vanishing of gcohomology. A proof can be found in [28], Theorem 3.1.5, or [16], Theorem 4.149. We will also see how to get this in a different way, using Clifford algebra actions and Dirac type operators. Theorem: (Kostant) Let lj be a Cartan subalgebra of g and let b = b @ n be a Borel subalgebra corresponding to a choice of positive roots. Let Vx be

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the irreducible finite dimensional g-module with highest weight A. Then

Hi(n;vX)

a3

= w€

w, 1(w) =i

@w(X+P)-P

as 9-modules. Here Cpis the one dimensional 9-module with weight p, W is the Weyl group, and l ( w ) , the length of w E W , is the smallest number of simple root reflections needed to write w as their product.

A proof of this statement and also its generalization to the case of ucohomology can be found e.g. in [16], theorems 4.135 and 4.139. The original proof is greatly simplified by using the (more recent) Casselman-Osborne theorem to conclude that only the weights w(X p ) - p can appear. Kostant interpreted this theorem as an algebraic version of the BorelWeil-Bott theorem, which realizes finite dimensional g-modules as global sections or cohomology of line bundles on the flag variety B of g. (The flag variety consists of all Borel, i.e., maximal solvable subalgebras of 8). He further used it to obtain an algebraic proof of the Weyl character formula, which explicitly calculates the (global) character of irreducible representations of compact groups.

+

7.4. n-homology and Casselman subrepres entation theorem

Lie algebra homology is defined in a similar way as cohomology. One starts by defining coinvariants

v, = v/gv = v @qg)c of a g-module V; this is the largest quotient of V on which g acts trivially. This is a right exact functor, and Lie algebra homology functors are the left derived functors

The differential is again induced by the deRham differential of the standard complex, which is used to resolve the variable C and thus define the derived functors. To illustrate the importance of (n-)homology, we mention a very standard construction of representations, namely the real parabolic induction. Start with the Cartan decomposition go = t o @ Po

of a real semisimple Lie algebra go. Let a0 c po be a maximal abelian subalgebra (subspace). Then one can consider the (go, a0)-roots, by the

205

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same principle as for in g (the adjoint action of a0 on go diagonalizes etc.) Let no be the sum of positive root spaces (for a choice of positive roots). Then one shows that go = to @ a0 @ no;

this is called the Iwasawa decomposition of go. There is also a group version of this fact: multiplication defines a diffeomorphism

KxAxN

+ KAIV=G

where A and N are the connected subgroups of G corresponding to a0 and no. We denote by L the centralizer of A in G; then L = M A , where M = L n K is the centralizer of A in K . The subgroup

P =MAN is called a minimal (real) parabolic subgroup of G. Note a slight problem with notation: the Lie algebra of P is not p from Cartan decomposition (that one is not a Lie algebra at all). Nevertheless, this is common notation. Here is what is meant by induction from P to G. Let V be a (finite dimensional) representation of P. Consider the principal bundle G + G / P and construct the associated G-equivariant vector bundle G x ~ V

L GIP The space G x p V consists of classes of the equivalence relation on G x V defined by setting (gp,v)

(9,P.V), 9 E GlP E p , v E

v.

The continuous sections of this bundle form a representation Ind,(V) of G called the continuously induced representation. Here G acts on the sections by left translation. (One can also consider smooth sections, or L2 sections of this bundle.) The sections can also be interpreted as functions from G into V with the appropriate transformation property for P ; in this way one can avoid the language of bundles. We will denote by Ind(V) the (8, K)-module of K-finite vectors in Ind, ( V ) . Let us consider the case V=o@X@l, where u is an irreducible finite dimensional representation of M , X is a character of A (which is the same as a character of a), and 1 denotes the

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trivial representation of N . This is basically the only case one needs to study. In this case, Ind(V) is called the principal series representation. Casselman proved the following version of Frobenius reciprocity: for any (8, K)-module W ,

Here p is the Lie algebra of P ; note that M is a maximal compact subgroup of P . Since the action of n on V is trivial, the second Hom-space is further equal to Hom(,,M,(W/nW, V).

If W/nW is not equal to zero, then one can choose V so that the last space is nonzero. It follows that W maps nontrivially into Ind(V), and if W is irreducible this map has to be an embedding. So we get Casselman’s subrepresentation theorem. Namely, W/nW is not equal to zero; in fact, this space contains leading exponents of asymptotic expansions of matrix coefficients of W . See [5]. Let us finish this section by mentioning briefly two more facts. First, there is a version of Poincark duality for Lie algebra cohomology. Second, there is a strong relationship between n-cohomology and BeilinsonBernstein localization theory. In this theory, irreducible (8, K)-modules are realized as global sections (or cohomology) of certain sheaves on the flag variety B of g. The relationship is the following: the geometric fiber of the sheaf corresponding to a (g,K)-module V a t a point b E X is exactly the n-cohomology of V, where n = [b, b] is the nilradical of 6. 8. ( 8 , K)-cohomology

8.1. Definition One can study relative Lie algebra cohomology for the pair (g,t). It has however become more usual to study pairs (8, K ) . The main case we are interested in is the one suggested by our choice of notation: g is a semisimple complex Lie algebra and K is a maximal compact subgroup of a Lie group G with complexified Lie algebra g. This setting can however be generalized; K could be another compact subgroup of G, or a complex reductive subgroup of a complex group with Lie algebra g. In principle K does not even have to be reductive, but then it is not as easy as below to write down resolutions. One can also consider similar pairs (A,K ) where A is an

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associative algebra. A could be U ( g ) ,or a quotient Uo corresponding t o an infinitesimal character (0 is a Weyl group orbit of an element of Q*). For the purpose of this section, let us concentrate on the usual (and the most interesting) case of (8, K ) , the first one mentioned above. Formally, (8, K)-cohomology is analogous to g-cohomology. Namely, one can consider the functor

V

H

Vg9K= {u E V l X v = 0 , kv = w, for all X E g, k E K}

of taking (8, K)-invariants. It is a functor from the category M ( g ,K) of (0,K)-modules into the category of complex vector spaces, which is left exact. The (g, K)-cohomology functors V H H i ( g ,K; V )are the right derived functors of V H W K . As before, one can write V g i K = Hom(,,K)(@,V ) ,

and thus

Hi(%K; V ) = EXt?,,K)(@, V ) As before, rather than resolving V by injectives, we use a projective resolution of the trivial module @. This is the relative standard complex ~ ( L I@u(q ) A'(g/t)

5 @.

-+

0;

Since we are considering compact K, we can replace g/t by the K-invariant direct complement p. The differential d of the above complex and the map E are similar as before: d(u @

xi A

* .

/. . A x k ) = x(-l)i-l~xi 8 xi A .. . xi . . . A x k + i

x ( - l ) i + ' u@

[xi,X j l p A X I A . . . zi . . . zj... A xk,

i<j

for any compact K ; here [ X i , X j l Pdenotes the projection of [ X i , X j ]to p along t. If K is a symmetric subgroup, like in our main case when K is the maximal compact subgroup, then this projection is always zero, so the second sum actually vanishes. E is as before the augmentation map, given by 1 @ 1 H 1 and g U ( 5 ) @ 1 H 0. The relative standard complex is obtained from the standard complex for 5, by taking coinvariants with respect t o the t-action given by right multiplication on U ( g ) and adjoint action on A(g), and also with respect to the action of A(t) on A(5) by (exterior) multiplication. Exactness is

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208

proved using this and exactness of the standard complex for 8. The fact that this is a projective resolution follows from general homological principles: any finite K-module is projective, and the functor W H U ( g ) @ ~ ( e ) W from finite K-modules to (8, K)-modules preserves projectives, as it is left adjoint to the exact functor of forgetting the g-action. Using the above resolution, we can now identify Hi ( g , K; V ) with the ith cohomology of the complex Homi,,K)(U(i~)@'v(c)

A'@),V )= H o m i ( A \ ' ( ~V), ),

with differential

@(xiA . . . A x k ) = x(-l)Z-'xi . f(x1A . . .xi... A x k ) A

i

if K is symmetric, and for other K there is another sum, over i < j. There is also a less often mentioned theory of (g,K)-homology. It is constructed by deriving the functor of (8, K)-coinvariants; so

Hi(g,K; V ) =

V).

which is calculated using the same resolution of C as above.

8.2. Applications

A good reference for learning about various applications of (8, K)cohomology to the theory of automorphic forms is a recent survey article [22]. Let us mention just one very classical application, the Matsushima formula. Let r c G be a cocompact lattice and let E be a finite dimensional representation of G. Then the group cohomology of r with coefficients in E , which is also equal to the cohomology of the space r \ G / K with coefficients in E , can be expressed as

H*(I',E)2

@ m ( n , r ) H * ( g , KH,;

8 E),

,€E where m(n,r) is the multiplicity of the unitary representation (n, H,) of G in L2(r\G). Another application is a construction of derived Zuckerman functors. Namely, let (8, K) be a pair as above, with K complex algebraic (e.g., the complexification of a maximal compact subgroup). Let 7' c K be a closed reductive subgroup. Let R ( K ) be the algebra of regular functions on K . Then one can express the derived Zuckerman modules of a (8, 7')-module V as

rK,T(v)= P ( t ,7';R ( K )@ v).

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209

Here the (t,T) cohomology is taken with respect to the tensor product of the regular action with the given action on V. K acts by right translation on R ( K ) ,and the g-action is obtained by twisting the given action TV on V: if we regard an element of R ( K )@ V as a regular function F : K + V, then for X E g the function 7r(X)F is given by

( 4 X ) F ) ( k )= 7rv(Ad(k)X)(JYk)). There are several versions of this construction, due to Wallach ([33], Chapter 6), Duflo-Vergne, and MiliCiOPandii6.

8.3. The Vogan-Zuckerman classification The central result about (8,K)-cohomology (for K the maximal compact subgroup of G) is the classification of irreducible unitary (0, K)-modules with nonzero (0, K)-cohomology obtained by Vogan and Zuckerman in [32]. This result can be stated as follows. Let V be irreducible unitary, of the same infinitesimal character as a finite dimensional representation F . Note that there is only one possible F for any given V . Then V @I F' has nonzero (0, K)-cohomology if and only if V is an Aq(X) module, as described in previous sections. (We also saw that in this case the lowest K-type of V gives rise to Dirac cohomology.) Note that if V and F do not have the same infinitesimal character then H * ( g ,K ; V @ F ' ) = 0. In case there is cohomology, it is equal to

where L is the Levi subgroup involved in the definition of Aq(X), I is the (complexified) Lie algebra of L and u is the nilradical of q. For more details, see [22] or [32]. 9. Relationship of Dirac cohomology to other kinds of

cohomology In this section we briefly sketch some results from [13], which grew out of the ideas of [31].

9.1. Kostant's cubic Dirac operator In [18],Kostant has constructed a cubic Dirac operator D corresponding to any decomposition g=te5

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J.S. Huang and P. PandiiC

of a semisimple Lie algebra g with r a reductive subalgebra such that the Killing form is non-degenerate on r. The definition is as follows: let Zi be an orthonormal basis for 5 and let Y E A 3 5 correspond t o the alternating trilinear form ( X ,Y,2 ) H B ( [ X Y , ]2 , ) under the isomorphism (A35)' A35 induced by the Killing form. Then

D

=

C Zi €9 Zi + 1€9

Y E

U ( g )€9 C(5).

i

It is easy to see that D is independent of the choice of basis Zi and rinvariant for the adjoint action. Following Kostant, we now use defining relations 2: = 1 instead of 2," = -1 for C(5). Kostant has shown that the formula we had for the Dirac operator corresponding to the Cartan decomposition, i.e.,

still holds in this more general situation. The ingredients of this formula are defined analogously as before. 9.2. Applications to u-cohomology

Let us consider a special case when r 8-stable parabolic subalgebra

= I and 5 =

u @ ii correspond to a

q =leu.

Note that u and ii are both isotropic for B , that B identifies ii with u*, and that ii is complex conjugate to u with respect to the real form go implicit in the above definitions. Let 5' = A' u be a space of spinors for C(5).Since 5 is even-dimensional7 S is unique up to isomorphism. There are two I-actions on S ; one is given by the adjoint action of I on u, and the other is the spin action, coming from [ -+ 5 4 5 ) + C(5).These two actions are related by a twist by pu; see

POI.

Let X be a (8, K)-module. Then the space

x €9 s = A i @ X has an action of D and of the u-homology operator can identify

a. Moreover, since we

A'u €9 X E Hom(A'ii, X )

A' u* E using (A' u)' operator d on X €9 S .

A' it, we also get

an action of the ii-cohomology

Dirac Operators in Representation Theory

211

One checks that

D=d+2d on X 8 S . To see this, take a basis ui for u and a dual basis ur for U; so B(ui,u;)= S i j . Now write

Denote by C (respectively C - ) the element of U ( g ) €3 C(5) composed of the first and the third term (respectively the second and the fourth term) in the above expression for D. Then show that the action of C on X €3 5’ induces the differential d, while the action of C- induces 28. The “half Diracs” C and C- are independent of the choice of basis ui and I-invariant. They however do depend on the choice of u inside 5 , while D does not. Furthermore, both C and C- square to zero, and their supercommutator CC- C-C is equal to D2. It is convenient to introduce another element,

+

This operator acts as a degree operator on X 8 S . It satisfies the following commuting relations: = -c-; [ElD2]= 0. [ElC] = c; [ E ,c-]

It follows that El C, C- and D 2 span a four dimensional superalgebra inside ( U ( g )€3 C(5))‘. This algebra is actually Z-graded, if we set C- to be of degree -1, D2 and E of degree 0 and C of degree 1. This grading is compatible with the obvious grading of X @ S coming from the standard grading of A’ u. This superalgebra was used by physicists under the name supersymmetric algebra. It is denoted by l ( 1 , l ) in Kac’s classification [14]. It is completely solvable and its representation theory is easy but not trivial. As an application of the above facts, let us mention that one can easily prove that the main result of [12]holds for C and C- and in this way get the Casselman-Osborne theorem (see 7.3) as a corollary. Thus the formal analogy between the two results becomes more concrete. There are two questions that arise from the above considerations. The first one, asked by Vogan in [31],is to relate the Dirac cohomology, iicohomology and u-homology of a (g,K)-module X. This could be useful

J.-S. Huang and P. Pandial

212

for example to pass between different choices for u within the same 5 . The second question was implicitly posed by Kostant in [20]. His remark was that X 8 S can be formed not just in the “Levi factor case”, i.e., for t = I, but also for a wider class of t described above. On X 8 S there is always a cubic Dirac operator, and its square is the Laplacian. So it looks natural to try and study this more general setting and see how to make use of that. At this moment, we have some results regarding the first question. Namely, in certain special cases all three (co)homology modules are equal up to appropriate twists. The twist is coming from the already mentioned identification of spin and adjoint actions on A(u), and it is given by pu. 9.3. Hodge decomposition

Let us take X to be unitary, so it carries an invariant positive definite hermitian form. We need to put a similar form on S. There is a natural, [-invariant one, given by the Killing form B on A’ u. This form is however indefinite if u intersects both P and p as will usually be the case. The operators d and 2 8 can be shown to be adjoint with respect to this form, but one can not in general obtain a “Hodge decomposition”. A good case where one can get the “Hodge theory approach” to work is when I = @ so that u c p and B is positive definite on u. This is possible only in the Hermitian symmetric case (i.e., when P has a center). In this case the Dirac operator D is “ordinary” (i.e., has no cubic term) - it is the same D we studied in Section 5 . We know that in this situation D2 is a scalar on each k-type, so in particular, D2 acts (locally) finitely on X 8 S. Note also that D 2is negative semidefinite in the present situation. Namely, the relations in the Clifford algebra are now 2: = 1, not -1, hence D is skew-symmetric and not symmetric as before. So all eigenvalues of D2 are non-positive. Because of these facts, one can use the easy variant of Hodge theory for finite dimensional spaces (see [33], Scholium 9.4.4), and conclude that Ker D Kerd

= Ker D 2= Ker d n Ker 8;

= K e r D @I

Imd;

Ker d = K e r D @ I m a .

In particular, the cohomology of both d and d is equal to the Dirac cohomology of D , Ker D ,as vector spaces. To compare them as [ = @-modules involves the above mentioned twist. The same argument proves an analogous result for X finite dimensional; here one uses the “admissible form” on X , i.e., the one invariant for the

Dirac

Operators in Representation

Theory

213

compact form of 8. This case was known to Vogan and it is also implicit in

POI*

9.4. A counterexample Here is a simple example which shows that the equality of the three cohomology modules does not hold for general ( d ( 2 ,@), S0(2))-modules. Consider the module V which is a nontrivial extension of the discrete series representation of highest weight -2 by the trivial module @: 0 +@+

v-+ w 4 0.

V is a submodule of the module V-l,o from the end of Section 1. The

weights of V (for the basis element

(p

02)

of t) are . . . - 4, -2,o. w e are

considering the case I = t, u is spanned by u = X =

(i -1>

and ii is

spanned by

V@S=V@l@V@u, w i t h d : V @ l - - + V @ u g i v e n b y d ( v @ l ) = u * ~ v @ u , a n dVd@ : u-+V@l givenbyd(w@u)=u.v@l. By an easy direct calculation one sees that the u-homology of V is given by

&(a)

= 0;

H,(a)= cwo 83,

the ii-cohomology of V is given by

and the Dirac cohomology of V is given by HD(V) = Ker D = CVO@ u. So we see

H D ( V )= H . ( a ) # H ' ( d ) .

J . - S . Huang and P . PandiiiC

214

9.5. Dirac cohomology and (8,K)-cohomology

In the rest of this section we make a few comments on the relationship of Dirac cohomology and (8,K)-cohomology. It was proved in [12] that if X is unitary and has (8,K)-cohomology, i.e.,

H * ( g ,K; X €3 F * ) = H*(Homk(A\'p,X €3 F * ) )# 0 for a finite dimensional F (which then necessarily has the same infinitesimal character as X), then X also has Dirac cohomology. In the following we assume that dimp is even. Then we can write p as a direct sum of isotropic vector spaces u and U "- u*. One considers the spinor spaces S = A' u and S* = A' ii; then

S €3 S* 2 A'(.

@ ii) = A'p.

It follows that we can identify the (8,K)-cohomology of X €3 F* with H*(HomX(F€3 S, X €3 S ) ) . There are several possible actions of the Dirac operator D on the above complex; similarly as before, they can be related to the coboundary operator d and the boundary operator d for (g,K)-homology, which also acts on the same complex after appropriate identifications. Now if X is unitary, Wallach has proved that d = 0 (see [33],Proposition 9.4.3, or [a]). Using similar arguments one can analize the above mentioned Dirac actions and the actions of the corresponding "half-Diracs" . In particular, it follows that

H * ( 8 ,K ;

x €3 F*)= HomX(HD(F),H D ( ~ ) ) .

This can be concluded from the fact that the eigenvalues of D2 are of opposite signs on F €3 S and X €3 S ; see [33], 9.4.6. One may hope to generalize some of these facts, either with respect t o X , or with respect to F. We finish by a remark that for nice and natural proofs it would be useful to formalize some constructions in the category of modules over the Clifford algebra, although this may not seem necessary as the category is extremely simple. For example, one can construct a coproduct of C(p) by defining

1 c ( Z )= - ( Z

4

€3 1

+ 1 8 2)

for 2 E p. This coproduct is an algebra morphism in the graded sense. It is not coassociative, but it is cocommutative. There is no counit, but there is

Dirac Operators in Representation Theory

215

an antiautomorphism, the already mentioned a (see 2.4) which can be used instead of an antipode. It seems worthwhile to try to use this structure to clarify notions like tensor products and dual modules.

10. Multiplicities of automorphic forms In this section we prove a formula for multiplicities of automorphic forms which sharpens the result of Langlands and Hotta-Parthasarathy. Let G be a linear semisimple noncompact Lie group. Let K be a maximal compact subgroup of G. Assume that rankG = r a n k K . Let go = PO PO be the Cartan decomposition of the Lie algebra of G. Then uo = to ipo is a compact real form of g = go @R C. Let U be the compact analytic subgroup in the complexification G@of G with Lie algebra UO.

+

+

10.1. Hirzebruch proportionality principle Let r be a torsion free discrete subgroup of G so that r\G and X = r\G/K are compact smooth manifolds. Bore1 showed that such a r always exists. Then the regular representation on r \ G is decomposed discretely with finite multiplicities:

L2(r\G)

2

@ m(r,7r)X,. ,€GI

Let X , be the Harish-Chandra module of X,. For any p E k, we define the Dirac operators D , D,f(X) and D,+(Y) as in 6.5. As we saw in 6.5, if we normalize the Haar measure so that vol(U) = 1, then IndexD:(X)

=

(-l)qvol(r\G) IndexD,f(Y).

On the other hand, m(r,.rr)IndexD:(X,),

IndexDl(X) = 7r€E

where D t ( X , ) : HornR(E;L,X,@S+) --t Homp(E;,X,@S-) is the linear map defined by 4 H D o 4 for any 4 E Homf?-(E;, X, @ Sf).

10.2. Dimension of automorphic forms

If Index D,f(X,) # 0, then the Dirac cohomology Ho(X,) contains Ep*.It follows from the proved Vogan's conjecture that the infinitesimal character of X , is given by p* pc. If we assume that X = w(p* p c ) - p is dominant

+

+

J . 3 . Huang and P. PandiiC

216

for some w E W ,then X , is isomorphic to Aq(X)for some &stable parabolic subalgebra q. If in addition we assume that X is regular with respect to the noncompact roots A+(p), then X , is uniquely determined as a discrete series &(A). Since IndexDL(Ab(X)) = dimDt(Ab(X))-codimDt(Ag(X)) = (-1)Q and IndexD,f(Y) has been calculated in Corollary 6.4., we obtain the following theorem.

Theorem: Let T = &(A) be a discrete series representation with X regular with respect to all noncompact roots. Assume that A is dominant and can be written as X = p - pn for some highest weight p E K. Then h

m ( r ,T ) = vol(I'\G)d,, where d, is the formal degree of

T:

This sharpens the result of Langlands [21] and Hotta-Parthasarathy [1111 who proved the above formula for discrete series representations whose Kfinite matrix coefficients are in L1(G).Trombi-Varadarajan [27] proved that if the K-finite matrix coefficients of the discrete series &(A) are in L1(G), then for all Q E A+(p) and all w E Wg

(A

+ P I 4 > I(wp,a)I.

Hecht-Schmid [9] proved this is also a sufficient condition. Our assumption on the regularity of X with respect to the noncompact roots amounts to the condition that for all a E A+(p)

(A+ P,Q)

> l(P,Q)l.

Therefore] our condition is weaker than that assumed by Langlands and Hotta-Parthasarathy. 10.3. A final remark Bore1 and Wallach [2] proved that for any finite-dimensional representation of G,

~ * (F r) =] @ m ( r , T ) H * ( K, ~ , x, B F ) . ,€E

Dirac Operators in Representation Theory

217

We still assume t h a t rank G = rank K. If the highest weight of F is regular, is uniquely then it follows from a similar argument as in 10.2. t h a t determined as a discrete series Ab(X), and therefore,

x,

d i m H * ( I ‘ , F ) = vol(r\G)d,dimH*(g,K,X,

@ F).

Acknowledgements The research of the first author was partially supported by RGC-CERG grants of Hong Kong SAR and National Nature Science Foundation of China. The research of the second author was partially supported by a grant from the Ministry of Science and Technology of Republic of Croatia. P a r t of these notes was written during authors’ visit t o CNRS, University of Paris VII and IMS at the National University of Singapore. T h e authors thank these institutions for their generous support and hospitality.

References 1 . M. Atiyah and W. Schmid, “A geometric construction of the discrete series for semisimple Lie groups”, Invent. Math., 42 (1977), 1-62, 2. A. Bore1 and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Second edition, Mathematical Surveys and Monographs 67, American Mathematical Society, Providence, RI, 2000. 3. R. Cahn, P. Gilkey and J. Wolf, “Heat equation, proportionality principle, and volume of fundamental domains”, 43-54, Differential Geometry and

4.

5. 6. 7.

Relativity, Mathematical Phys. and Appl. Math., Vol. 3, Reidel, Dordrecht, 1976. H. Cartan, “La transgression dans un groupe de Lie et dans un espace fibre principal”, Colloque de Topologie alge‘brique, C.B.R.M. Bruxelles, (1950), 57-71. W. Casselman and D. MiliEiC, “Asymptotic behavior of matrix coefficients of admissible representations”, Duke Math. Jour. 49 (1982), 869-930. W. Casselman and M. S. Osborne, “The n-cohomology of representations with an infinitesimal character”, Compositio Math., 31 (1975), 219-227. C. Chevalley, The algebraic theory of spinors, Columbia University Press,

1954. 8. B. Gross, B. Kostant, P. Ramond and S. Sternberg, “The Weyl character

formula, the half-spin representations, and equal rank subgroups”, Proc. Nut. Acad. Sci. U.S.A., 95 (1998), 8441-8442. 9. H. Hecht and W. Schmid, “On integrable representations of a semisimple Lie group”, Math. Ann., 220 (1976), 147-149. 10. R. Hotta, “On a realization of the discrete series for semisimple Lie groups”, Jour. Math. SOC. Japan, 23 (1971), 384-407.

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11. R. Hotta and R. Parthasarathy, “A geometric meaning of the multiplicities of integrable discrete classes in L2(r\G)”, Osaka Jour. Math., 10 (1973), 211-234. 12. J.-S. Huang and P. Pandiid, “Dirac cohomology, unitary representations and a proof of a conjecture of Vogan”, J . Amer. Math. SOC.,15 (2002), 185-202. 13. , J.-S. Huang, P. PandiiE and D. Renard, “Dirac operators and ncohomology” , in preparation. 14. V. Kac, “Lie superalgebras”, Adv. in Math., 26 (1977), 8-96. 15. A. W. Knapp, Representation theory of semisimple groups: a n overview based on examples, Princeton University Press, 1986. 16. A. W. Knapp and D. A. Vogan, Jr., Cohomological induction and unitary representations, Princeton University Press, 1995. 17. B. Kostant, “Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, the pdecomposition C(g) = End V, @ C ( P ) ,and the g-module structure of Ag”, Adv. in Math., 125 (1997), 275-350. 18. B. Kostant, “A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups”, Duke Math. Jour., 100 (1999), 447-501. 19. B. Kostant, “Dirac cohomology for the cubic Dirac operator”, Studies in memory of I. Schur, 69-93, Progress in Math. vol. 210, 2003. 20. B. Kostant, “A generalization of the Bott-Borel-Weil theorem and Euler number multiplets of representations”, Lett. Math. Phys., 52 (2000), 61-78. 21. R. Langlands, “The dimension of spaces of automorphic forms”, Amer. Jour. Math., 85 (1963), 99-125. 22. J . 3 . Li and J. Schwermer, “Automorphic representations and cohomology of arithmetic groups” 102-137, Challenges f o r the 21st century, Proceedings of International Conference on Fundamental Sciences: Mathematics and Theoretical Physics (Singapore, 2000) World Scientific Publishing. 23. R. Parthasarathy, “Dirac operator and the discrete series”, Ann. of Math., 96 (1972), 1-30. 24. S . A. Salamanca-Riba, “On the unitary dual of real reductive Lie groups and the A4(X) modules: the strongly regular case”, Duke Math. Jour., 96 (1998), 521-546. 25. W. Schmid, “Homogeneous complex manifolds and representations of semisimple Lie groups” Dissertation] UC Berkeley, 1967, 223-286, in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, Edited by P. Sally and D. Vogan, American Mathematical Society, 1989. 26. W. Schmid, “L2-cohomology and discrete series”, Ann. of Math., 103 (1976), 375-394. 27. P. C.Trombi and V. S. Varadarajan, “Asymptotic behaviour of eigenfunctions on a semisimple Lie group: the discrete spectrum”, Acta. Math., 129 (1972), 237-280. 28. D. A.Vogan, Jr., Representations of real reductive Lie groups, Birkhauser, Boston-Basel-Stuttgart, 1981. 29. D. A. Vogan, Jr., “Unitarizability of certain series of representations”, A n n .

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of Math., 120 (1984), 141-187. 30. D. A. Vogan, Jr., “Dirac operators and unitary representations”, 3 talks at MIT Lie groups seminar, Fall of 1997. 31. D. A. Vogan, Jr., “n-cohomology in representation theory”, a talk at “Functional Analysis V I P , Dubrovnik, Croatia, September 2001. 32. D. A. Vogan, Jr. and G. J. Zuckerman, “Unitary representations with nonzero cohomology”, Compositio Math., 53 (1984), 51-90. 33. N. R. Wallach, Real Reductive Groups, Volume I, Academic Press, 1988.

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On Multiplicity Free Actions

Chal Benson and Gail Ratcliff Department of Mathematics East Carolina University Greenville, NG 27858, U.S.A . Email: bensonfQmai1.ecu. edu, [email protected]. edu

Contents 1 Preliminaries 1.1 Algebraic groups 1.2 Regular functions 1.3 Algebraic groups as Lie groups 1.4 Structure theory 1.5 Rational representations 1.6 Highest weight theory 1.7 The contragredient representation 1.8 Decompositions and multiplicities 1.9 Group actions 1.10 Section 1 notes 2 Multiplicity free actions 2.1 Borel orbits 2.2 Quasi-regular representations 2.3 Maximal unipotent subgroups 2.4 S-varieties 2.5 Spherical pairs 2.6 Section 2 notes 3 Linear multiplicity free actions 3.1 Connectivity of G 3.2 Borel orbits 221

223 223 223 224 224 226 228 229 229 230 231 23 1 232 233 235 236 237 238 239 239 241

222

C. Benson and G. Ratcliff

3.3 Fundamental highest weights for a multiplicity free action 3.4 Section 3 notes 4 Examples of multiplicity free decompositions 4.1 GL(n) @ GL(m) 4.2 S2(GL(n)) 4.3 A2(GL(n)) 4.4 SO(n) x C x 4.5 GL(n)@GL(n) n2(GL(n)) 4.6 Section 4 notes 5 A recursive criterion for multiplicity free actions 5.1 GL(n) 5.2 GL(n) @ GL(n) 5.3 GL(n) @GL(n) A2(GL(n)) 6 The classification of linear multiplicity free actions 6.1 Irreducible multiplicity free actions 6.2 Decomposable actions 6.3 Saturated indecomposable multiplicity free actions 6.4 Non-saturated indecomposable multiplicity free actions 6.5 Completing the classification 6.6 Proof outline 6.7 Section 6 notes 7 Invariant polynomials and differential operators 7.1 Polynomial coefficient differential operators 7.2 Invariants in P’D(V) 7.3 A canonical basis for the invariants 7.4 The fundamental invariants 7.5 The algebra 7.6 Section 7 notes 8 Generalized binomial coefficients 8.1 The polynomials qx 8.2 The generalized binomial coefficients 8.3 Eigenvalues for operators in P’D(V)G 8.4 Examples 8.5 Section 8 notes 9 Eigenvalues for operators in P’D(V)G 9.1 Eigenvalue polynomials 9.2 A Harish-Chandra homomorphism for multiplicity free actions 9.3 Characterizing the eigenvalue polynomials 9.4 GL(n) 8 GL(n) yet again 9.5 Section 9 notes References

245 247 248 248 252 254 257 258 260 262 263 264 264 266 266 268 268 270 270 272 273 273 273 276 278 279 28 1 283 284 284 285 288 289 293 294 294 296 298 299 30 1 301

223

On Multiplicity Free Actions

1. Preliminaries Much of the literature on multiplicity free actions is set in the framework of algebraic groups. We begin by summarizing the basic definitions and results we require concerning such groups and their representations.

1.1. Algebraic groups The general linear group GL(n,C) can be viewed as an algebraic group. Letting gl(n,C) denote the space of n x n complex matrices, the group GL(n,C) can be identified with the zero set for the polynomial function p ( A , w ) = d e t ( A ) w - 1 on gl(n,C) x C.This determines the structure of GL(n,C) as an affine variety. One calls G a reductive complex (linear) algebraic group when 0 0

(linear) G is an algebraic subgroup of GL(n,C),and (reductive) C" is a direct sum of G-irreducible subspaces. The classical examples are

GL(n.C),S L ( n ,C>,O(n,C ) ,SO(n,C>,Sp(2n,C) and direct products of these groups. The torus (C")" is a direct product of copies of GL(1, C ) = Cx . A reductive complex algebraic group is connected (in the Zariski topology) if and only if it is irreducible as an algebraic variety. The classical examples are all connected except for O(n,C),which has two components. W e wall assume that our algebraic groups G are connected unless noted otherwise.

1.2. Regular functions

C [ G ]denotes the ring of regular functions on G. This is the coordinate ring of G as an affine variety. More concretely C[G] is the algebra generated by the matrix.entries of G and det-l.

A function f : G

---t

C is regular if and only if

f is the restriction of a

regular function on GL(n,C). So

@[GI2 C[GL(n,C ) ] / I ( G )where , I(G) = {f E C[GL(n,C ) ] : f(G) = 0). Examples 1.2.1: For g = [aij]E G L ( n , C )let z i j ( g ) = aij. Then 0 0

C[GL(n,C)]= C [ z i j , d e t - l ] , C [ S L ( nC)] , = C[zij], C[(C")"] = C[zll,2;1,. . . ,Znn,z;;].

C. Benson and G. Ratcliff

224

1.3. Algebraic groups as L i e groups As an algebraic group, G carries the Zariski topology. As a set of n x n complex matrices, G also has a subspace topology from g l ( n , C ) . In fact, G is a smooth submanifold of gl(n,C), viewed as a real vector space of dimension ( 2 ~ 2 )In ~ .this way G is seen as a (real) Lie group with Lie algebra g = { A E g l ( n , C ) : etA E G for all

t

E

W}.

Moreover g is closed under multiplication by i and hence is a complex Liesubalgebra of g l ( n , C). Alternatively one can define the (complex) Lie algebra g for G as g = { A E gl(n,C) :

f E I(G)

+ Af E I ( G ) }

where

When G is reductive, g is a complex reductive Lie algebra. This implies that

B = 2(B) e3 9’ where ~ ( g )denotes the center of g and the derived subalgebra 0’ = [g,g] is semi-simple, a direct sum of simple ideals. g’ is the Lie algebra of the commutator subgroup G’ = (GIG ) .

1.4. S t r u c t u r e theory A maximal connected solvable algebraic subgroup B of G is called a Borel subgroup. The following facts concerning such subgroups are well known: a 0

Any two Borel subgroups are conjugate in G. Given any Borel subgroup B , there is an opposite Borel subgroup Bwith the property that B-B is Zariski dense in G and contains an open neighborhood of I . B is the semidirect product B = H N of its commutator subgroup N = ( B ,B ) with a maximal torus H in G. The group N is a maximal unipotent subgroup of G. The Lie algebra b of H is a maximal abelian subalgebra of g. For Q E b*

let ga = {Y E g : [ X , Y ]= a ( X ) Y for all X E b}.

On Multiplicity Free A c t i o n s

225

Then

is the set of roots for g (relative to I)). Each root space g a is one dimensional and

g =b@

@ ,€A

There is a subset A+ of A , called the positive roots, such that N has Lie algebra

Now

0 0

A = A + U (-A+), b = b @ n is the Lie algebra of B, and N - has Lie algebra n- = gPa.

eaEA+

For each a c A+ there are elements which form an sl(2)-triple:

[Ha,X,] = 2X,,

X,

[ H a ,X-,] = -2X-,,

E go,

X-, E

[X,, X-a]

g-,

Ha E b

= Ha.

For each a E A we have a root reflection

sa : b*

-+

b*,

sa(X) =

x - (X,a)a

where

( & a )= X(Ha). The Weyl group W = W(g,q)is the subgroup of GL(b*) generated by { s a : a E A } . It is a finite reflection group that acts by permutations on the set A.

Example 1.4.1: The standard Bore1 subgroup in G = GL(n,C ) is

the group of invertible upper triangular matrices. We have

B, = H , N,

C. Benson and G. Ratclaff

226

where H , denotes the diagonal matrices in G L ( n , C ) and N , denotes the unipotent upper triangular matrices. The opposite Bore1 subgroup for Bn is

B, = H,Ni where B; and N; are the invertible and unipotent lower triangular matrices respectively. The Lie algebra ljn of H, is the set of all diagonal matrices. Letting gi E b* denote the functional Ei(diag(zl,.. . , 2,)) = Zi

one has roots A = {

~ i ~j :

i # j } and positive roots A+ = {

~ i ~j :

i <j).

For a = ~i - ~j E A+ we have

X, = Eij, X-, = Eji, Ha = Eii

- Ejj.

The root reflection s, satisfies s , ( E ~ ) = ~ ~ ( where k ) T E S, is the transposition that interchanges i with j . Thus the Weyl group is isomorphic to sn.

1.5. Rational representations

Definition 1.5.1: Let (a,V ) be a representation of G. (1) (a,V ) is said to be rational (or regular) if it is finite dimensional and its matrix coefficients 9

E(a(g)v),

E E v*,vE

v

all belong to @.[GI. ( 2 ) (u,V )is locally rational (or locally regular) if dim(V) = 00 and for any finite dimensional subspace F of V there is a u(G)-invariant subspace W with F c W c V for which alW is rational. h

We let G denote the set of equivalence classes of irreducible rational representations of G and sometimes write V, for the representation space of aE

e.

Note that subrepresentations of rational (or locally rational) representations are rational (resp. locally rational).

On Multiplicity Free Actions

227

Example 1.5.2: As G = C x is abelian, its irreducible representations are given by characters p : C x 4 C x . For such a character to be rational we require p E C[G] = @[z,l/z]. So p is hoIomorphic on C x and hence determined by its restriction to the unit circle T.One concludes that (CXT={p, : n E Z }

where p,(z) = 2 , . The character

gives a representation of Cx which is not rational. This example illustrates the:

Weyl Unitarian Trick Rational representations of G are determined by holomorphic extension from a maximal compact connected subgroup K of G (viewed as a real Lie group). It now follows from the representation theory for compact Lie groups that rational representations are completely reducible. Moreover, the Unitarian "rick establishes a bijection between and the set of unitary equivalence classes of irreducible unitary representations of K . So one can work entirely in the compact group setting, should one so prefer. Maximal compact subgroups for the classical groups GL(n.C), S L ( n ,C), O(n,C), SO(n,C), Sp(2n,C) are

k

U ( n ) ,sU(n),o(n,R), SO(n,a), Sp(2n) = U ( 2 n )n sp(an,C ) . Each rational representation (a,V )of G is smooth. That is, t H a(etx)v is a smooth map R -+ V for each u E V , X E 0. Thus we obtain a derived representation

of the Lie algebra 0 on V . (Note that we are using the same notation for the representation CT on both the Lie group and the Lie algebra.) When there is no ambiguity, we will denote the action of G (or g) on V by o ( g ) u = g . u (or a ( X ) u = X

. v).

C. Benson and G. Ratclaff

228

1.6. Highest weight theory Let B = H N be a Borel subgroup in G and (a,V) be a rational representation. By the Lie-Kolchin Theorem, there are non-zero u(B)-eigenvectors. That is, there are vectors v # 0 in V such that

u(b)v = $(b)v for all b E B , where ?I, : B and hence 0 0

--f

C x is a regular character. As N = (B, B ) , we have

$IN

=1

{v E V : v is a B-eigenvector} = V N ,the N-fixed vectors in V, and $ is determined by $ 1 ~E @.

Highest weight theory asserts that u is irreducible

d i m ( V N ) = 1.

h

For each u E G, there is a non-zero B-eigenvector v, E V,, unique up to scalar multiples. This is the highest weight vector for u. The corresponding character $ : B -+ Cx can be differentiated to give a functional X on the Lie algebra 6, with X(n) = 0. We have X . v , = X(X)v, for all X E 6. As noted above, X is determined by its value on the Lie algebra b of H . The functional X in b* is the highest weight for u. We can extend X to b (or 6-) by taking X(n) = 0 (resp. X(n-) = 0). Thus v, is, up to scalars, the unique vector in V with

X . v,

= X(X)v, for all

X

E

6.

Highest weight theory asserts, moreover, that is determined up to equivalence by its highest weight. Given a representation of G with highest weight A, we will denote the corresponding representation as V,.(Keep in mind that this all depends on the initial choice of a Borel subgroup.) For an element b in the subgroup B, we will denote the corresponding character by b H b X . The highest weights for G L ( n , C ) (with respect to the standard Borel subgroup) are n.

(diag(h1,. . . ,h,)

dihi : d l , . . . ,d, E Z with d l >_ dz 2 . . . >_ d,}.

H

i= 1

On Multiplicity Free Actions

229

1.7. The contragredient representation Given a representation of G on V, we define the contragredient representation of G on V' by: g . ((v) = ((g-'

. w) for 5 E V*, w E V .

If V has highest weight vector v with highest weight X with respect to some Borel subgroup B , then V* has a highest weight vector w* with weight -A with respect t o the opposite Borel subgroup B-. 1.8. Decompositions and multiplicities Let (p, W ) be a rational representation of G and u E sum decomposition

e. One has a direct

W=$W" U€G

of W into a-isotypic components

W" = c { V : V is a p(G)-invariant subspace of W with plV N u ) = C { T ( V u ):

T : V,

-+

W intertwines u with p}.

Then

W"

21 m(u,p)V,=

v, @ . . . €3 V" m(a>P)

as G-modules where the multiplicity m(a,p) of u in p is given by m(o,p) = dim(W")/dim(V,)= dim(HOm~(Vu1 W)).

(Homc(V",W ) is the space of linear maps V, P.

+

W intertwining u with

1

The multiplicity m(c1p ) can also be characterized using highest weight theory. Let B be a Borel subgroup of G and X be the highest weight for u. Then

m(a,p) = dim ( W B > A ) where wBJ = {W E

W

:

b.w

= b'w}

is the space of weight vectors in W with weight A.

230

C. Benson and G. Ratclaff

1.9. Group actions

We use the notation G:X to indicate that there is a rational action of G on a variety GxX This means G x X

-+ X

4

X,

X,

(g,x) Hg . x .

is a morphism of algebraic varieties satisfying

( g h ). IC = g . ( h . x) and e .x = x. The variety X may be affine or, more generally, quasi-projective. That is, the intersection of a closed with an open set in CP". It follows that i f f E @ [ X I the , regular functions on X ,then 2 H f(g-1

. x)

is a regular function. Thus we obtain a representation p of G on @ [ X I ,

( p ( g ) f ) ( z )= ( 9 . f)(.)

=

fb-' . .).

Lemma 1.9.1: T h e representation p of G o n C[X] is locally rational. Remark 1.9.2: Linear actions G : X will be our main concern. That is, X will usually be a vector space on which G acts by a rational representation. Then @[XIis the algebra of polynomials on the vector space X and one can give an easy proof of Lemma 1.9.1. Let ? k ( X ) denote the space of polynomials on X homogeneous of degree k. For any finite dimensional subspace F of C [ X ] we , have F c W where n

for n sufficiently large. Now W is a finite dimensional, G-invariant subspace and we have a rational representation. Now for

0

E

define the a-isotypic component C [ X ] "as:

@[XI"= C { T ( V u ): T

E

Homc(Vm, @ [ X I ) } .

Lemma 1.9.3: W e have a n algebraic direct sum

C [ X ]= @ C[X]". v€l3

231

O n Multiplicity Free Actions

Thus

C[X]

= @ m(a,@[X])Vg a&

as G-modules, where the (possibly infinite) multiplicity

C[X]) of

m(g,

(T

in

@[XIis m(a,@.[XI)= d i m ( H o m ~ ( V C[X])) a, = dim (C[X]'*'), where X is the highest weight for the representation a. We can now introduce our principal objects of study.

Definition 1.9.4: G : X is a multiplicity free action if m(u,@[XI)5 1 for each a 6

e.

Given an action G : X and an algebraic subgroup H of G, one has

H : X multiplicity free

G : X multiplicity free.

(1)

Indeed, if G : X fails to be multiplicity free, then some representation a E occurs in @[XIwith multiplicity greater than one. Thus all irreducible constituents of a l occur ~ with multiplicity greater than one. So H : X also fails to be multiplicity free.

1.10. Section 1 notes The material in this section is standard and there are many excellent references. One is the book [17] by Goodman and Wallach. See Chapter 1 and Appendix D in [17] for further details on foundational material concerning algebraic groups and Lie groups. See Section 11.3 of [17] for proofs of the assertions concerning Bore1 subgroups. Our treatment of the isotypic decomposition for @[XIfollows Section 12.1 in [17].

2. Multiplicity free actions This work is mainly devoted to the study of linear multiplicity free actions. In this section, however, we consider actions G : X in the more general context of algebraic varieties. Our main purpose is to describe some noteworthy non-linear examples.

232

C. Benson and G. Ratclaff

2.1. Borel orbits There is a simple criterion for multiplicity free actions.

Theorem 2.1.1: If a Borel subgroup B in G has a (Zariski) dense orbit in X then G : X i s multiplicity free. Proof: Suppose that B . IC, is dense in X. Let CT E occur in @[XI(that is m(a,C[X]) > 0) and let A be the highest weight for CT. Let f 1 , f i E @[XI be two B-highest weight vectors with weight A. One has

f j is regular and B . xo dense in X, we see that f j is completely determined by the value fj(x,). In particular, fj(x0) # 0 (as f j # 0) and we can write

As

So the space of A-highest weight vectors in hence m(a,@[XI)= 1.

C[X] is one

dimensional and 0

Since any two Borel subgroups are conjugate in G, the criterion in Theorem 2.1.1 does not depend on the choice of Borel subgroup B . Suppose that B . xo is dense in X. Since B is connected, it follows that X must be an irreducible variety and that B . x , is a Zariski open set. Thus X\(B.x,) is a closed set which contains no open subsets in X. We conclude that there is only one dense open B-orbit in X . Conversely, if G : X is a multiplicity free action with X an irreducible a f i n e variety then X contains an open (hence dense) B-orbit. We will prove this result for linear multiplicity free actions in the following section. (See Theorem 3.2.8 below.) One special case of the converse admits, however, a direct proof. This is the case where G is an algebraic torus G = A 2 (C")". In this case the Borel subgroup is A itself and one has the following.

Proposition 2.1.2: Let X be a n irreducible a f i n e variety and A be a torus. If A : X is multiplicity free then there is a n open (hence dense) A-orbit in X. Proof: One can choose weight vectors f l , . . . , f, in @[XIthat generate C[X] as an algebra. Since A : X is multiplicity free the weights { A i , . . . , A,}

O n Multiplicity Free Actions

233

for f 1 , . . . , fr must be linearly independent. Choose a point xo E X for which f j ( x o )# 0 for 1 5 j 5 r. Define a map p : @[XI+ @[A]by ( p f ) ( a ) = ( a . f ) ( x o )= f(a-l

We claim that p is injective. Indeed, for f = f

'

xo).

=f y '

'

. .f y r

one has

( p f ) ( a ) = am'X' . . . U r n F X , f ( X o ) = a r n X f ( x 0 ) , and hence (pf)(a) =

c

CmamXfm(xo)

m

for f = Cmcmfm E @[XI.If ( p f ) ( a ) = 0 for all a E A , then linear independence of the X j ' s implies that c, = 0 for all m. Thus every regular function on X is determined by its restriction to A . x o . It follows that A . xo is open and dense in X . 0 2.2. Quasi-regular representations

We continue to let G denote a reductive complex linear algebraic group. The left and right actions of G on X = G

-

0

g x = gx, and g.x=xg- 1

give rise to the left and right regular representations

.

L ( g ) f ( x )= f ( g - l x > ,and R ( g ) f ( 5 )= f (x g ) respectively

of G on @[GI.For any algebraic subgroup H of G we let

@ [ G / H= ] @ [ G ] R ( H )@[H\G] , = C[G]L(H) and define the left (resp. right) quasi-regular representation of G as the restriction of L to C [ G / H ](resp. R to @[H\G]).The representations L and R of G on @ [ G / Hand ] @[H\G] are equivalent via the intertwining operator

T : C [ G / H ]4 @[H\G], T ( f )= f, ( f ( x )= f ( x - l ) ) . In fact the homogeneous space G / H is a smooth quasi-projective variety with coordinate ring @[GIN]. If H is a reductive or normal subgroup then G / H is an affine variety. We remark that one can have @ [ G / H = ] @ even when d i m ( G / H ) > 0. This situation occurs whenever G / H is a projective variety, in particular when H is a Bore1 subgroup of G. In any case, the

C. Benson and G. Ratclzff

234

left action G : (G/H) is rational and gives rise t o the left quasi-regular representation. Similar remarks apply for H\G and the right quasi-regular representation. The isotypic decomposition for the quasi-regular representations is given by Frobenius Reciprocity:

Theorem 2.2.1: A s a G-module we have @[G/H] % @ dim(V$)V,. U€Gc

I n particular, C[G] "- @,dim(V,)V,.

Proof: Lemma 1.9.3 applies here since G : (G/H)is a rational action. It % V 3 . For this, one verifies that suffices t o show that Homc(V,, @[G/H]) @ :

v?

+

Homc(Vu,@[G/H]), @ ( E ) ( v ) ( g=) E(c(g-')v)

is an isomorphism with inverse

Corollary 2.2.2: The action of G o n G/H (or o n H\G free if and only i f dim(V/) 5 1 for all a E

e.

) is multiplicity

Note that if H1 and H2 are algebraic subgroups of G with H1 c H2 then V$ c V$. Thus if G : (G/H1)is a multiplicity free action then so is G : (G/H2). The proof of Theorem 2.2.1 shows that the a-isotypic component in @[GIfor the left regular representation is = {T(v) :

2,

E V,,T E Homc(V,,C[G])}

= { Q ( E ) ( v ) : v E VU,E E V,.} = : 21 E V,,[ E v;},

where mc,v(g)= <(a(g)v) is a matrix coefficient and mi,,(g) = mc,v(g-l). The right regular representation preserves C[G]">L, since L and R commute. As a module for GxG, C[G]"Jis isomorphic to V,@V,* = V,@V,*.Indeed, the map

@ : V, 8 V,* -+ C[G]"!L,@(v 8 E ) = mi,v intertwines

@ c* with

the left-right regular representation

(LR)(g1,g2)f(x) =f ( g & 7 2 )

On Multiplicity Free Actions

235

of G x G on @[GI.This gives the Peter- Weyl Theorem:

Theorem 2.2.3: The left-right action of G x G on G is multiplicity free. As a G x G-module we have @[GIC

@ u 18 u*. oEG^

2.3. Maximal unipotent subgroups Let B = HN be a Borel subgroup in G. For each u E we have dim(V/) = 1,by highest weight theory. Theorem 2.2.1 now shows:

Theorem 2.3.1: The action of G on G I N is multiplicity free. Moreover, each irreducible representation u E 6 occurs exactly once in @ [ G I N ] . The Borel- Weil Theorem provides an explicit model for the irreducible representation with specified highest weight X E rj* inside @[GIN-].

Theorem 2.3.2: The irreducible representation with highest weight X E rj* is given by the left regular representation of G on

Rx

= { f E @[GI : f ( g b ) = b-'f

( 9 ) f o r all b E B-}.

Moreover, a highest weight vector in Rx is given on the dense subset NHNof G b y fx(nhn-) = h-'.

Proof: First observe that Rx is a L(G)-invariant subspace of @[GIN-]. If f E Rx is a B-highest weight vector then f(nhn-) = f ( h )= (h-l . f ) ( e ) and also f ( h ) = h e x f ( e ) . Thus f has weight X and f = f ( e )fx. So Rx is L(G)-irreducible with highest weight X and highest weight vector fx.

Corollary 2.3.3: The span of RxR, is Rx+,. Proof: The span of RxR, is a G-invariant subspace of Rx+,.

0

Remark 2.3.4: It follows from Theorem 2.1.1 that G : (G/B) is also multiplicity free. But V," = (0) unless u E 6 is the trivial representation. So C[G/B] = C and this is an uninteresting example. Alternatively, one can note that G I B is a flag manifold, hence projective, hence compact. So every regular function on G I B is constant.

C. Benson and G. Ratclzff

236

2.4. 5'-varieties Let B = H N be a Borel subgroup in G with opposite Borel subgroup B- = H N - = N - H . For i = l l . . . k l let C J ~ E G act on the space V,, and let vi E V , be a B--highest weight vector with weight -Ai. Let v = v1 + . . . vk E VI @ . . @ vk . Then A

+

-

X=G.V, the Zariski-closure of the orbit of v in Vl @ . . . @ V I ,is a G-invariant subvariety, called an S-variety. Since N- . v = {v}, we see that

B

.V = N H N -

.v

is dense in G . v and hence also in X . It follows from Theorem 2.1.1 that G : X is a multiplicity free action. In the present context, however, we will show directly that G : X is multiplicity free by exhibiting the decomposition of @[XI.

Theorem 2.4.1: The multiplicity free decomposition of @[XIis

@[XIE! @ R x , XEA

where A = (alX1

+ . . + akXk

: aj E

N}.

(Throughout, N denotes the non-negative integers, including zero.)

Proof: We can lift an element E of v,' to V, and define a function fc on G by f t ( g ) = E(g . v) = ((g . vi). Then for b E B-, fc(gb) = f ( g b . v,) = b-XtE(g. vi) = b-'. fc(v). Thus fc is in the G-irreducible space Rx, defined above. Hence {Elx : E v,'} is equivalent to the G-irreducible Rx,. Note that @[XIis generated by the restriction of elements of each V,. to X . So by Corollary 2.3.3, we conclude that the irreducible components are products of the subspaces Rx, . 0

<

There is another characterization of S-varieties:

Theorem 2.4.2: Let X be an irreducible a f i n e G-variety with a G-open orbit, such that the isotropy subgroup of any point in the open orbit contains a maximal unipotent subgroup. Then X is a n S-variety.

Proof: One can find a (rational) representation embedding of X into V,. (See [44].)

CJ

of G and a G-equivariant

237

On Multiplicity Free Actions

So assume that X c V, and let v E X be any point in the open G-orbit. Choose a Bore1 subgroup B with N c B such that N c Stabc(v). Then we can write v = v1+ ' . ' v k , where each vi is a weight vector with weight Xi, and the XLs are distinct. Since v is stabilized by N , each vi is a highest weight vector. Thus our variety X is the closure (in V )of G . v , where v = v1 +..+vk.

+

0

2.5. Spherical pairs Suppose that H is a reductive algebraic subgroup of G. We say that (G, H) is a spherical pair if dim(Vf) 5 1 for all u E Equivalently, the actions of G on G / H and H \ G are multiplicity free. In this section we will summarize some results concerning spherical pairs, without proof. Let U and K denote maximal compact connected subgroups of G and H. These are compact real Lie groups. Recall that the irreducible rational representations of G and H correspond to irreducible unitary representations of U and K via the Unitarian Trick. So

e.

(G, N)is spherical

d i m ( V f )_< 1 for all

E

6.

We say that (U,K ) is a compact Gelfand pair in this case. It is known that (U,K ) is a Gelfand pair if and only if the algebra L ' ( U / / K ) of integrable K-hi-invariant functions on U is abelian with respect to convolution. Table 1: The classical compact Riemannian symmetric spaces -U

IK

1 2 3 4 5 6 7 8 9

ia The preceding remarks show that the spherical pairs (G, H) are precisely the complexifications of the compact Gelfand pairs ( U ,K ) . It is well known

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238

that (U, K ) is a Gelfand pair whenever U / K is a Riemannian symmetric space. The classification of irreducible symmetric spaces of compact type produces ten families of examples] listed in Table 1, together with seventeen exceptional cases. In entries 8 and 9, U ( n ) is embedded in SO(2n,R) and Sp(2n) via

A+iBH

.I?[

-B A

In each entry of Table 1, G is a simple complex algebraic group (and U is a simple compact Lie group). This means that the adjoint representation for G on its Lie algebra g is irreducible. Hence d i m ( g H ) _< 1 for spherical pairs ( G , H ) with G simple. In particular] the center of H is a t most one dimensional. The spherical pairs (G, H ) with G simple were classified by Kramer. In addition to the examples that arise from symmetric spaces, there are six further families and six exceptional cases. These are listed in Table 2. The compact form (U,K ) for each entry in Table 2 is a compact Gelfand pair for which U / K is not a symmetric space. Table 2: Spherical pairs with G simple, not arising from symmetric spaces G

1 2 3 4 5 6 7 8

H

SL(P + 4, @) (P # 4 ) SL(P1

SL(2n+11C) SL(2n+11C) Sp(2n,C) SO(2n+11C) S0(4n+2,@) SO(1O1C) SO(9,C) 9 SO(8,C) 10 SO(7,C) 11 E t 12 G:

x SL(ql @)

CX x Sp(2n1C)

Sp(2n, C) Sp(2n - 2, C)x SO(2, C, GL(n1 C) S L ( 2 n + 1,C) Spin(7,C) x SO(2,C) Spin(7,C)

G: G; Spin(l0,C) SL(3,@)

2.6. Section 2 notes

Theorem 2.1.1 is due to Servedio, [49]. The converse was proved by Vinberg in [52].

On Multiplicity Free Actions

239

The structure of GIH as a quasi-projective algebraic variety is discussed in Section 11.2.1 of [17]. The classical Peter-Weyl Theorem concerns the decomposition of L2( K ) under the action of K x K for compact Lie groups K . Let &, denote the One has a space of matrix coefficients rnc,,(k) = c(u(k)v)for u E Hilbert space direct sum

d.

P ( K ) = @ €, UEZ

with K x K acting on &, by a copy of u* 8 u. Theorem 2.2.3 yields the algebraic content of this theorem, via the Unitarian Trick. A complete proof also requires two analytic facts: 0 0

Schur Orthogonality: I, I&: in L 2 ( K )for a UuEg€, is dense in L 2 ( K ) .

+ a’;and

See, for example, Section 5.2 in [15] for proofs. We refer the reader to Chapter IV in [21] or Chapter 1 in [16] for information concerning Gelfand pairs (U,K ) . In particular, for the fact that Riemannnian symmetric spaces yield Gelfand pairs. The classification of Riemannian symmetric spaces, including the exceptional cases, can be found in Chapter X of [20]. In [17],Section 12.3, it is shown that symmetric spaces yield spherical pairs (G,H ) by showing that there is an open Bore1 orbit in G I H . This approach involves a complex version of the Iwasawa decomposition. Kramer’s classification of spherical pairs (G,H ) with G simple appeared in [36]. This classification was pushed further by Brion [8] and by Mikityuk [40] t o encompass all spherical pairs (G,H ) with G semi-simple. This results in eight additional families of examples. The classification of spherical pairs can also be found in Vinberg’s recent survey article [53]. S-varieties were introduced by Vinberg and Popov [54].

3. Linear multiplicity free actions We now restrict our attention t o linear actions G : V. That is, G is a reductive algebraic group acting on a complex vector space V by some rational representation.

3.1. Connectivity of G We generally assume that our groups G are connected unless stated otherwise. Proposition 3.1.3 below shows that this entails no great loss of gen-

C. Benson and G. Ratcliff

240

erality. For Lemma 3.1.1 and Proposition 3.1.2 we assume G is connected and B denotes a Bore1 subgroup.

Lemma 3.1.1: Let h E @ [ V ]be a highest weight vector for B with p r i m e decomposition

T h e n each irreducible factor p j i s a B-highest weight vector.

+ :B

Proofi Let

-+

CX be the weight for h. Then for b E B ,

+(b)h=b.h= (b.pl)ml...(b.pk)mk and each b p j is an irreducible polynomial. By uniqueness of the prime decomposition for h we conclude that, for each j, b . p j is a non-zero multiple of one of the prime factors. That is, b . p j E U f = l C x p ~ .But B acts continuously on this space, and B is connected, so B must act by a scalar on each p j . 0 +

Proposition 3.1.2: If G : V i s not a multiplicity free action t h e n the multiplicities { m ( a , C [ V ] ): a E are unbounded.

e}

Proof: As G : V is not multiplicity free we can find a pair of linearly independent B-highest weight vectors h l , h z in @[V] with common weight $ : B -+ (Cx . In view of Lemma 3.1.1 we can remove any common irreducible factors and assume h l , h z are relatively prime. For each N and 0 5 k 5 N , hFh;-' has weight I+!I~.Moreover, {h?hF-k : 0 5 k 5 N } is a linearly independent set in @ [ V ] .For otherwise we could express some hFoh;-ko as a linear combination N

k=k,+I

and conclude that hl divides hFPko.This contradicts the fact that hl, h2 are relatively prime. Thus the irreducible representation with highest weight GN occurs in @(V] with multiplicity a t least N. Proposition 3.1.3: Let G be multiply-connected with identity component Go. T h e n G V i s a multiplicity free action if and only if Go : V i s a multiplicity free action.

241

On Multiplicity Free Actions

Proof: If Go : V is multiplicity free then so is G : V , in view of (1). Conversely, suppose that G : V is a multiplicity free action but that Go : V is not multiplicity free. Let

@[VI = @ P A x

denote the decomposition of C [ V ]into distinct G-irreducible subspaces and { g l , . . . , ge) be a complete set of coset representatives for Go in G. Proposition 3.1.2 ensures that for any N there is some 0 E Go with m(a,C [ V ] 2 ) N12e. As the Px’s are Go-invariant, each copy of (T in C [ V ]is contained in some PA. Let W denote a subspace of some PA on which Go acts by a copy of u. Since Go is normal in G, gi . W c PA is Go-invariant with Go acting via a copy of ui E Go, where h

h

g i ( g ) = a(giggil).

As PA is G-irreducible, we must have

for some subset J of (1,.. . , e } . Thus PA is equivalent to CjEJaj as a Go-module. Equation 2 contains at most C factors, so at least N2e distinct PA’Smust contain copies of 0 . As there are only 2e possibilities for J , at least N of these PA’Smust be equivalent as Go-modules. Thus we have shown that for each N one can find N distinct irreducible G-modules that are equivalent as Go-modules. This is impossible since G/Go is a finite group. 0 3.2. Borel orbits

We will prove the converse of Theorem 2.1.1 in the context of linear actions G : V . We use the notation introduced in Section 1.4: Choose a Borel subgroup B = H N in G and let Af c g* be the associated set of positive roots. Now suppose that h E C [ V ]is a B-highest weight vector with weight X E b*. Let P=Ph={gEG : g.hECXh}.

(3)

This is a parabolic subgroup of G that contains B. We have X . h = X ( X ) h for all X E p, the Lie algebra of P.

C. B e n s o n and G. Ratcliff

242

Let L and U be the Levi component and unipotent part of P , so P =

LU. On the Lie algebra level we can write where Ap = { a E A

p = l ~ @ga,

I (X,a)L 0 ) .

aEAp

Now setting

AL = { a E A (A, a ) = O}, A ~ J = { c Y € IA(X,CY) > O } = { a € A I -~r$!Ap}, one has

c

[=?J@

u=

gal

a€AL

c

ga.

aEAu

Letting u- = CaEAu g P a we have u c n, u-

c n-

and

g=peu- =I@u@u-.

We will make extensive use of the following map (which depends on h ) . Let

V, = { Z

E

V

:

h ( z )# 0},

a principal open set in V , and define

x = X h V,

4

g*

via

Lemma 3.2.1:

x is P-equivariant.

Proof: Note that the action of P on V preserves V,.Also P cts on * via the coadjoint action ( 9 .N X ) = f ( A d ( g - l ) X ) .

For g

E

P one has

Thus X(g-'

. z ) ( X )= (9-l . x ( z ) ) ( X )so x is P-equivariant.

0

On Multiplicity FTee Actions

243

Lemma 3.2.2: The stabilizer of X in P is Stabp(X) = L . Moreover, Stab,y(X) = { e } . Proof: Here X E Q* is regarded as a functional on all of g with X(n) = (0) = X(n-). For X E p one has X . X = 0 if and only if X[X,g]= ( 0 ) . Clearly g stabilizes X because [b, g] c n n-. Also for any a: E AL and any

+

y

=

+Cpe*

c p x , E 9, X([X,, Y ] )= c-,X(H,)

= c-,(X,

a ) = 0.

+ xaEaL

Thus X , stabilizes X for each a: E AL. So I= b g, stabilizes A. We have shown that L c Stabp(X). For the reverse inclusion, let

stabilize A. Then for

P E A,,

+,

X-pl = b p X ( H p ) = b p ( X , P).

But (X,P) > 0, and hence b p = 0. Thus Stabp(X) c L and the stabilizer of 0 X in U is trivial.

Proposition 3.2.3: The image of in g*.

x : V, -+g*

is X

+ p l , a single P-orbit

Proof: For all z E V , and X E p ,

Thus x ( z ) - X annihilates p and x(Vo)c X+p'. Note that dirn(X +p') = dirn(u).As U is unipotent and acts without stabilizer on X we conclude that U.XisbothopenandclosedinX+p'andhenceU.X=X+p'.As P = LU and L stabilizes X we also have P . X = X + p'. As x is P-equivariant we must have P . X = x(Vo). 0

Corollary 3.2.4: X = x ( z o )f o r some zo E V,. Let

c = Ch = x-'(X)

= (2 E

The group L acts on C because

v, :

x ( z ) = A}.

x is P-equivariant

Lemma 3.2.5: U x C E V, via ( g , z ) H g . z .

(4)

and L stabilizes A.

244

C. Benson and G. Ratcliff

Proof: Given z E V, one has x ( z ) = g . X for some g E U in view of Proposition 3.2.3. Thus X = g-l . x ( z ) = x ( g - l . z ) , so 9-l . z E C. Now (9,g-I . z ) E U x C maps to z . To see that ( 9 ,z ) H g . z is injective, suppose that g . z = 9’. z’ for some g , g’ E U , z , z’ E C. Applying x gives g . X = g . x ( z ) = 9’. x(z’) = g’ . x and thus g-lg‘ stabilizes A. As that g = g‘ and thus also z = z’.

U acts without stabilizer on X it follows 0

Lemma 3.2.6: There is a unique parabolic subgroup P = Ph of lowest possible dimension. Moreover, P c Ph’ f o r all B-highest weight vectors h’ E C [ V ] .

Proof: Suppose that h E C [ V ]is a B-highest weight vector with prime decomposition h =pyl ...pyre The proof of Lemma 3.1.1 shows that Ph acts by a character on each irreducible factor p j . Thus

Ph = Ppl n . . . n pP,. Now assume that h, h’ E C [ V ]are two highest weight vectors and that Ph has minimal dimension. Let P I , . . . , p , be the prime factors of h and 41, . . . ,qs the prime factors of h’. Letting h” = p l . . .p,ql . . . qs one has

PhjJ = Ppl n ” ’ n Ppr n Pql n “ ’ n Pq8 = Ph n Ph,.

<

If Ph Ph‘ then Ph,, is a proper subset of Ph and hence dim(Ph,,) < dirn(Ph),a contradiction. Thus ph C Ph’ as claimed. Moreover if Ph’ also has minimal dimension then Ph! C Ph and now Ph = Pht. This shows uniqueness.

0

Lemma 3.2.7: Let P = Ph = L u be the unique parabolic subgroup of minimal dimension. Then ( L ,L ) acts trivially o n C = ch.

Proof: First note that as P acts on h by a character, ( L ,L ) c (P,P ) acts trivially on h. Suppose, however, that ( L ,L ) does not act trivially on C. It follows that there is some highest weight vector h’ E C[C] for the action L : C that is not fixed by ( L ,L ) . Recall that B = H N with N c U and that V, = U ’ C. Thus we can extend h’ to a B-semi-invariant function on V,.

On Multiplicity

Free

Actions

245

For N sufficiently large, hNh' is a regular function on V . Now hNh' E C [ V ] is a B-highest weight vector and P C PhNh' by Lemma 3.2.6. Thus P acts by a character on hNh' and hence ( L , L ) acts trivially on hNh'. As (L, I;) fixes both h and hNh', it must fix h', a contradiction. 0 Lemma 3.2.7 implies that for the minimal parabolic P = Ph = LU, the action of L on c = ch is a torus action. Lemma 3.2.5 implies that the variety C is affine and irreducible.

Theorem 3.2.8: If G : V is multiplicity free then there is an open B-orbit in V . Proof: Suppose that there is no open B-orbit in V . Let P = Ph = LU be the parabolic subgroup of minimal dimension and c = Ch. We claim that H : C is not multiplicity free. Indeed, if H : C were multiplicity free then there would be an open H-orbit in C, H . v, say, by Proposition 2.1.2. Now as U . C = V,, one has U H .vo open in V, and hence open in V . As U c N , then B . vo = H N .u0 is open in V , a contradiction. As H : C is not multiplicity free we can find a pair of linearly independent highest weight vectors hl, hz E C[C] with a common weight for the action L : C. As in the proof of Lemma 3.2.7, one can extend hl and h2 to B-semi-invariant functions on V,. Now for N sufficiently large, hNhl and hNhz are regular on V and share a common weight. Hence the action G : V fails to be multiplicity free. 0 Note that B . vo is open in V if and only if dirn(b . v,) = dim(B . u,) = dirn(V).Thus B . v, is open if and only if b . vo = V . We obtain an infinitesimal version of Theorems 2.1.1 and 3.2.8:

Corollary 3.2.9: G : V is multiplicity free if and only if b . v, = V for some point u, E V . We will apply Corollary 3.2.9 to the study of examples in Section 4.

3.3. Fundamental highest weights f o r a multiplicity free action Let G : V be a linear multiplicity free action, B = H N a Bore1 subgroup open in V. Let of G and v, a point in V with B ' IJ,

AcB"

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246

denote the set of highest weights X for representations ox E 6 that occur C x given by X E A will be denoted b H b’. in @[If]. The character B For X E A, let PA c C[V] be the irreducible subspace of C[V] on which G acts by a copy of q.Then ---f

is the multiplicity free decomposition of C[V] under the action of G. As G preserves the subspaces z?,( V ) of polynomials homogeneous of degree m, each PA is contained in some Pm(V). Each PA contains a B-highest weight vector hx, which is unique up to scalar multiples. As shown in the proof of Theorem 2.1.1, we must have hA(wo)# 0 and we can normalize our choice of hx by the condition

hx(vo)= 1. For A, p E A,

Hence

X

+p E A

and

hi+, = hxh,

In particular, A is an additive semigroup in Next suppose X E A and let

ti*.

be the prime decomposition for hx. Lemma 3.1.1 shows that each p j is a B-highest weight vector. Suitably normalizing the pj’s, we can now say that p j = hx, for some X j E A and X = mlXl .. ~I,XI,. Now let

+. +

A’ = { A E A : hx is an irreducible polynomial}. We know that A‘ # @ and that A’ generates the semigroup A. We will show that A’ is a Q-linearly independent subset of ti*. Indeed, suppose that XI, . . . , XI, E A’ satisfy a non-trivial linear dependence relation over Q. Clearing denominators we obtain a non-trivial linear dependence alXl

+ .. . + a k X k = 0

with integer coefficients a j . Let

L = { j ..

aj

2 0)

247

O n Multiplicity Free Actions

and set

mj

= ]ajl for all 1

5 j 5 k. Then

and hence

since these are both highest weight vectors with a common weight and take value 1 a t the point vo. But this is impossible, since the hx,’s are distinct irreducibles and @[V] is a unique factorization domain. It now follows that A’ is a finite set with a t most dim(t)) elements. In summary, we have proved the following.

Proposition 3.3.1: A‘ = {A E A : hx is irreducible} is a Q-linearly independent subset of b* with at most d i m ( € € ) elements. The additive semigroup A is freely generated by A’. Writing

A’ = { X i , . . . ,Xr}, the decomposition for C [ V ]can be written a=[V]= @ P A x

+.

where the sum is taken over all N-linear combinations X = mlXl . . mrXr. The highest weight vector in the irreducible subspace PA is hx = hy . . . h? . Definition 3.3.2: The number r is the rank of the multiplicity free action G : V , {XI,.. . , A r } are the fundamental highest weights and {hl = hxl, . . . , h, = hx,} are the fundamental highest weight vectors. 3.4. Section 3 notes

Many results in this section are due to Roger Howe. Proposition 3.3.1 is from [23]. Proposition 3.1.3 appeared in [3] but the proof was shown to the authors by Howe. The proof given for Theorem 3.2.8 is due t o Friedrich Knop [31] and based on ideas from [lo]. The more standard proof (see [52]) is shorter but requires use of a hard result of Rosenlicht. For background on parabolic subgroups see Section V.7 in [29]. A linear action G : V is called a prehomogeneous vector space when there is an open G-orbit in V . We refer the reader t o [28] for a survey of

C. Benson and G. Ratcliff

248

this subject, including classification and applications to analysis. Theorem 3.2.8 implies that each linear multiplicity free action is, in particular, a prehomogeneous vector space. Proposition 3.1.2 shows that linear actions G : V are of three types: (1) G : V is multiplicity free. (2) Some o E occurs in C [ V ]with infinite multiplicity. ( 3 ) Each o E has m ( a , C [ V ] )< 00 but these multiplicities are unbounded.

2 2

It is known that (2) is equivalent t o the existence of a non-constant Ginvariant in C [ V ] (See . Theorem 5.5 in [3].)In this case, every o E 6 which occurs in C [ V ]does so with infinite multiplicity.

4. Examples of multiplicity free decompositions 4.1. GL(n) @ GL(m) Here we consider the action of G = GL(n,C)x GL(m,C)on C"@C" via the the outer tensor product of the defining representations for the two factors. of complex n x m-matrices Identifying C" @ Cm with the space (via ei @ ej ++ E+) one has (9,g') . v

= g4g')t

(6)

In fact it will prove convenient to twist the action by composing with the automorphisms g H (g-l)t on both factors. This gives

and the associated representation on C[Mn,m(C)]becomes (9,g') . P ( V ) = p(gtw').

Of course the decompositions for C[Mn,m(C)]under the two actions (6) and (7) are the same. Twisting by g H ( g - l ) t has the effect of interchanging representations with their contragredients. So decomposing C[Mn,m(C)]with respect to (7) amounts to decomposing the symmetric algebra S(Mn,m(C))2 C[Mn,,(C)*] with respect t o (6). Recall that the upper triangular matrices in GL(L,C) give the standard Bore1 subgroup Bk with Lie algebra bk. The diagonal matrices Hk in GL(lc,C)give the maximal torus with Lie algebra b k Ck.We let

B = B, x Bm,

b = 6, x bm,

H = Hn x Hm,

= bn x

bm,

O n Multzplicity Free Actions

~j

E

4; be

249

Ej(diag(h1,.. . , h,)) = hj and likewise for

E$

E

Qh.We some-

times use the shorthand ( p 1 ~= ) (~1,...11~n;~11...1~)

for the weight X = p ~ ~ ~ + . ~ ~ + p , ~ , + u ~in~b*.~ The + ~dominant ~ ~ + u , weights have p E Z", u E Z" with p1 2 . . . 2 p, and u1 2 . . . 2 vm. We say that X is non-negative and write X 2 0 when pj 2 0 and uj 2 0 for all j.

For any (h,h') = (diag(h1,.. . ,hn),diag(h',. . . ,h L ) ) E H one has

where zij : + C is the (z,j)-th entry function. So zi,j is a weight ' :2 is a weight vector with vector with weight E ~ + E $ . Hence also z a = weight

nij

i,j

As the z a ' s form a basis for CIMn,m(C)]lall weight vectors in C[Mn,m(C)] are non-negative. Thus the highest weights X = ( p , u ) that occur in C[Mn,m(C)]all have p1

2 ... 2 pn 2 0 ,

u1

2 . . . 2 urn 2 0.

This explains why we are using the action (7) in place of (6) Let us assume that n 2 m and set

A typical element X in the Lie algebra b for B has the form

where A, D are m x m upper triangular, C is ( n - m) x ( n - m) upper triangular and B is an arbitrary m x ( n - m)-matrix. The derived action for (7) gives

I$[ [3[+I

x . 'uo = -

-

At + D

D

=-

.

(9)

Here At, D are arbitrary m x m lower and upper triangular matrices respectively and Bt is an arbitrary matrix of size ( n- m) x m. We conclude that

C. Benson and G. Ratclaff

250

b . wo = Mn,,(C). Hence B . w o is open in Mn,,(C), so (7) is a multiplicity free action. The element X in Equation 8 belongs to g when B = 0 and A, C, D are diagonal,

A = diag(al,. . . ,u,),

C = diag(cl, . . . ,cn-,),

D

= diag(d1,. . . ,d,)

-

say. For such X ,Equation 9 shows X uo = 0 if and only if D = -A. So the stabilizer go of wo in g is

go

c1

,

... -am

Cn-m

\

If

["

am

=

is the highest weight for a representation that occurs in That is, then we must have

for all aj, cj. Thus we must have

We have now shown that the onlly candidates for highest weights of have the form representations occurring in

is equivalent Note that a representation with huighest weight the outer tensor product of the irreducible representations of to GL(n,C) and GL(m,C) with highest u. We will show that all such weights do occur in C[M(C)] by exhibiting a l-highest weight vector in C[Mn,m]. For this, let zll

Zlk

" *

(k(z)=

1

zkl

* ' *

1

%kk

For

the leading minor determinant for one computes

251

On Multiplicity Free Actions

Hence ( k is a highest weight vector with weight (&I ( l k l, k ) .For any p1 2 ... 2 p m 2 0

cP =

(Yl-P2($-P3

+. . . + & k ,

E:

t

. . . (kyy1-Pm(Km

has weight X = ( p ,p ) and degree

+ . . . + prn.

1 ~ =1 In summary, we have proved:

Theorem 4.1.1: The action of G = GL(n,C ) x GL(m,C ) on C" @ C" 2 Mn,"(C) is multiplicity free. For n 2 m one has the decomposition

@p p

@ i ~ n , m ( ~= > I

P

where the sum is over all p E N" with p1

2 . . . 2 p m 2 0. The polynomial

cP given by (10) is a highest weight vector in PP with weight ( p , p ) and PP 2 UK

@ U&

as a G-module. Moreover

pk(Mn,m(c))=

@ pp. IPl=k

We observe the following points regarding this example. 0

0

0

0

The action has rank m = min(n,m).Here { ( l k , l k ): 1 5 k 5 m } are the fundamental highest weights and (1 . . . ,J m are the fundamental highest weight vectors. Indeed, the determinant of a matrix is an irreducible polynomial in the matrix entries. For the more conventional (untwisted) action (6), the decomposition of @[Mn,m(C)] is as in Theorem 4.1.1, but now (, is a highest weight vector for the opposite Borel with weight ( - p , -/I). The group G L ( n , C ) x G L ( m , C ) can be replaced by S L ( n , C ) x G L ( m , C )or S L ( n , C ) x S L ( m , C ) x C x . If we restrict to S L ( n , C ) x S L ( m , C ) then the action remains multiplicity free provided n # m. The Borel subalgebra 6' consists of matrices X as in (8) where t r ( A ) t r ( C ) = 0 = t r ( D ) . Equation 9 shows 6' . v, = Mn,m(C)when n > m. When n = m, the action SL(n) @I SL(n) fails to be multiplicity free. Indeed, det : M,,,(C) -+ C is a non-constant ( S L ( n , C )x S L ( n , C ) ) invariant polynomial. So C[Mn,,(cC)] contains two copies of the trivial representation of S L ( n ,C) x S L ( n ,C ) .

+

C. Benson and G. Ratcliff

252

0

When m = 1 this example reduces to the action of G L ( n , C ) on C" by the defining representation (or its twisted form, the contragredient representation on ( C " ) * ) . In this case the decomposition in Theorem 4.1.1 reduces to M

C [ z l , .. . , zn] = @pk(Cn). k=O

The action has rank one with fundamental highest weight (11)and zf is a highest weight vector in Z)k(Cn).

4.2. S2(GL(n)) Next we consider the action of GL(n,C) on the symmetric 2-tensors S2(C2) via the symmetric square of the defining representation. Identifying S2(Cn) with the complex n x n-symmetric matrices

Sym(n,C) = { A E Ad,,,(@)

:

At = A}

our action reads

g . u = g v g t.

(11)

As in the preceding example we prefer to twist the action by g This gives

H

(g-l)t.

g'u = (g-l)tug-l,

g.p(u) = p(gtug) for u E S y m ( n , C ) ,p E C [ S y m ( n , ~ ) (12) As for GL(n) @ GL(m),twisting ensures that all weights X = (XI,. . . ,A,) that occur in C [ S y m ( n C , ) ] are non-negative. Again we let

y 2 . . .<,-IA n - 1 - A n EnAn for dominant weights A 1 2 ... 2 A, 2 0, where determinant restricted to Sym(n,C ) c Ad,,,(C). EA

=E

(k

is a leading minor

Theorem 4.2.1: The action of GL(n,C ) on S y m ( n , C ) is multiplicity free. W e have the decomposition @[Sym(n,C)l = @ P A A

where the sum is over all X E N" with X1 2 . . . 2 A, 2 0. The polynomial is a highest weight vector in PA with weight 2X. Moreover, %(Sym(n,@)) =

@ PA. IAI=k

O n Multiplicity Free Actions

253

Proof: The proof parallels that for Theorem 4.1.1. First we note that for the derived action of gl(n, C) one has

X . I = -(Xt+X). We obtain all nxn-symmetric matrices as X ranges over all upper triangular matrices. Thus 6, . I = Syrn(n,@) and hence vo = I has an open Borel orbit. This proves that our action is multiplicity free. Suppose that f E C[Syrn(n,(C)]is a highest weight vector with weight p. For h = diag(h1,. . . , h,) E H,, f(h2) = ( h . f)(I) = h’”f(I)= hY1 . . * h ? f ( I ) . Now for h = diag(f1,. . . fl) we have h2 = 1 and hence f ( I ) = h p f ( 1 ) for all such h. As B, . I is open in Sym(n,C), f is determined by the value f ( 1 ) and we must have f ( I ) # 0. We conclude that h’” = 1 for all h = diag(f1,. . . I f l ) and thus each pj must be even. (Note that H, = (diag(f1,. . . I f l ) }is the stabilizer of v, = I in H,.) Since all weights that occur in C[Syrn(n,C ) ] are non-negative we conclude that all dominant weights p that occur have the form p = 2x = (2x1,.

. . 2X,) ]

for some X 1 2 ... 2 A, 2 0. To complete the proof, one checks that the polynomial Ex is a weight vector with weight 2X. Indeed for h E H, and liIc
so & is a (2k)-weight vector.

0

As regards this example we note: 0

0

0

The action has rank n with fundamental highest weights (Zk)and fundamental highest weight vectors & ] 1 k 5 n. It is known that the determinant of a symmetric matrix is irreducible as a polynomial in the entries zij with i 5 j. For the untwisted action (ll),the decomposition of @[Sym(n,C)] is as in Theorem 4.2.1 but now E x is a highest weight vector for the opposite Borel with weight -2X. S’(SL(n)) is not a multiplicity free action because det E @[Sym(n,C)] is a non-constant SL(n,C)-invariant.

<

254

C. Benson and G. Ratclzff

4.3. A2(GL(n)) We now turn to the action of G L ( n , C ) on A2(Cn),the skew-symmetric 2-tensors. Identifying A2(C") with

S k e w ( n , C ) = { A E Mn,n(C)

:

At

=

-A},

the action is given by Equation 11. As in the preceding examples, we will employ the twisted version, as in Equation 12. The leading minor determinant & vanishes on S k e w ( n , C) when j is odd and is a perfect square when j is even. We let uj E C [ S k e w ( n , C ) ]denote the Pfaffian polynomial:

.j(.)

=Pf

[

Zl,l

. ..

21,Zj

...

Z2j,2j

i ZZj,l

i

]

7

the square root of the principal 2 j x 2 j subdeterminant. So deg(vj) = j and u; = J2j for j = 1,.. . I [n/2j. The polynomials uj are given explicitly by the formula

Let m = [n/2J and write

Theorem 4.3.1: The action of G L ( n , C ) on S k e w ( n , C ) is multiplicity free. W e have the decomposition CJ[S/cew(n,c)]= @ PA x

where the sum is over all X E N" with A 1 2 ' . ' v~ is a highest weight vector in PA with weight

-

= (A17

2 ,A _> 0. The polynomial

X I , X2, X Z , . . .,Am, Am).

Moreover, P k ( S k e w ( n , C ) )=

@ PA. IAI=k

On Multiplicity Free Actions

[ -;i]

Proof: Let J =

255

and let

7be the n x n-matrix

J = J @ . . . @ J for n = 2m even and I

m-times

7= J @ . . . @ J @O for n = 2m + 1 odd. m-times

7

We claim that has an open B,-orbit in Skew(n,C).Hence our action is multiplicity free. In fact, for X E gl(n,C ) ,

x .T = -(XtT+

TX)= -(7X - ( 7 X ) t ) .

First suppose that n = 2m and let X E 6,. Decompose X into 2 x 2-blocks X i j , 1 5 i,j _< m. Now TX has 2 x 2-blocks J X i j , which are arbitrary 00 for i < j . Taking Xii = gives J X i i = This shows that an 0 ca arbitrary strictly upper triangular matrix can be written in the form FX for some X E b,. So 6,. J = Skew(n,C)when n is even. When n = 2 m + 1 the same argument shows that for suitable entries in the first n - 1 rows and columns of X E b, 7X contains any desired strictly upper triangular matrix in its first n - 1 rows and columns. As the last column of 7X is

[ ]

[0 :].

7=

Skew(n,C)when n is odd. we conclude that 6, . Suppose that p is a highest weight that occurs in @ [ S k e w ( n , C )and ] that f is a p-weight vector. Since all weights in C [ S k e w ( n , @ )are ] nonnegative, we have p1 2 . . . 2 p n 2 0. Moreover f(7)# 0 as B, . J is open. For h E H, we have

-

h p f ( Y )= ( h . f ) ( T )= f ( h T h ) .

But h?h =

[

hlh2 J h3h4J

C. Benson and G. Ratcliff

256

so that h?h = ? for elements h = diag(h1,. . . , h,) of the form

h={

for n = 2 m diag(h1, h l l , hz, h z l , . . . ,hm, hm) diag(h1,h;', hz, h;', . . . ,h,, h,, hzm+l) for n = 2m

+1

'

It follows that hp = 1 for all such h and thus p1

= p 2 , p3 = p 4 , . . . ,p2nz-1 = pZnz and p, = 0 when n is odd.

Thus all highest weights occurring in C [ S k e w ( n ,C)] have the form = ( X l , x1, x 2 , x 2 , .

. . ,,,x

xm)

for some A1 _> . . . 2 A , 2 0. To complete the proof one just needs to check that vx has weight This reduces to the calculation

x.

which shows that uj is a weight vector with weight ( 1 2 j ) .

0

Regarding the action A2(GL(n)) we note: 0

0

0

The Pfaffian of a skew symmetric matrix is irreducible as a polynomial in the entries zij with i < j . Thus A2(GL(n)) has rank m = Ln/2] with fundamental highest weights (1") and fundamental highest weight vectors vj. The untwisted action has the same decomposition but now ux is a lowest weight vector with weight -A. For the trace zero upper triangular matrices b;, one sees, as in the proof of Theorem 4.3.1, that 6; . = S k e w ( n , C) when n is odd. So h2(SL(2m 1)) is a multiplicity free action. On the other hand, this fails for n = 2 m even. Indeed, det E C[Slcew(2m,C)] is a non-constant S L ( 2 m ,C)-invariant so A2(SL(2m)) is not multiplicity free.

-

+

7

O n Multiplicity Free Actions

4.4. SO(n)

257

x CX

The group G = SO(n, C) x C x acts on V = cCn in the usual way: ( g , t ) . v = tgv.

The decomposition for @[V] = C [ Z .~. ,. , zn] is given by the classical theory of spherical harmonics. Let E(Z)

= z1"

+ .+Z i * '

and

A = &(a)= a? + . . . + a:, the constant coefficient differential operator with Wick symbol p ( z , w) = E ( w ) . Since E is an SO(n,C)-invariant polynomial, A is an SO(n,@)invariant operator. It follows that the space of harmonic polynomials

'H = K e r ( A ) = { p E @[V]: Ap = 0) is SO(n,@)-invariant, as is

H ',

= 'FI n ;D,(V),

the harmonic polynomials homogeneous of degree m. It is well known that F ' I, is an irreducible module for SO(n,@) and

Pm(V)= H ' , a3 EPm-2(V). This gives, in particular,

(13)

(

dim('H,) = dim(P,) - dim(Pm-a)= m + n - 1 ) - ( m + n - 3 ) m m-2 for m 2 2. Now Equation 13 leads inductively to the decomposition

T m ( v )=

@

Pk,l

(pk,l =HkEe)

(14)

k+2l=m

of P,(V) into irreducible SO(n,@)-modules.The modules { P k , [ : k + 2 l = m} appearing in (14) are clearly inequivalent because their dimensions are distinct. Equation 14 now gives a decomposition

of C[V] into irreducibles for G = SO(n,C). This is not, however, multiplicity free because x k , ( 2 Ek,eI as SO(n,c)-modules for e # e'. w e use the scalars CX to repair this defect.

C. Benson and G. Ratclifl

258

The action of Cx on C [ V ]by scalars commutes with A and hence preserves the ‘HFlm’s and Pk,e’s. Now C x acts on Pk,e by the character t H t-(k+2e).Hence Pk,(and Pk,$!are inequivalent as CX-moduleswhen C # P. Thus (15) is multiplicity free as a decomposition for the group G = SO(n,C) x ex. A highest weight vector in P k , e is given by (21+ Z Z ~ ) ~ E ( Z ) ~ for , a suitably chosen Bore1 subgroup in G = SO(n,C)x C x . The multiplicity free action G : C” has rank 2 with fundamental highest weight vectors 21 iz2 and

+

4%). 4.5. GL(n) @ G L ( ~ )A2(GL(n)>

The group G = GL(n, C) acts diagonally on

V = V1@V2 = C”

@

A2(Cn) 2 C”

@ Skew(n, C).

For consistency with Sections 4.1 and 4.3 we twist the action of G on both V1 and V2. We have seen how t o decompose C[Vl]and C[V2]under the action of G. Writing u p = cg for the irreducible representation of G with highest weight p 2 0 we have

C[VI]N

@[V2] =

@CP

x as G-modules. Here the second sum is over all X = (XI 2 . . . (m = Ln/2J) and X = (Xl,X1,. . .,A, .A), Thus we can write k

2 ,A _> 0)

I

C [ V ]= C[K] €9 C[V2]N

1d k ) d. @I

k,x

Tensor product representations of GL(n,C) can be decomposed using the Littlewood-Richardson rules. To apply this technique, one identifies each highest weight p 2 0 with a Young’s diagram consisting-of pj boxes on row j . The Littlewood-Richardson rules ensure that d k ) @Idhas a multiplicity free decomposition. Moreover, the representations u p that occur in the decomposition are given by diagrams that can be obtained from that for by adding Ic boxes, no two of which fall in the same column. For example, when -X = (3,2), and k = 3, we must add three boxes t o the Young’s diagram for X = (3,3,2,2). For n 2 5 this produces five diagrams:

p

259

O n Multiplicity Free Actions

The three boxes added to the diagram for ( 3 , 3 , 2 , 2 ) have been marked with 1's in each case. This exercise with the Littlewood-Richardson rules shows (T(3)

g

-

0 ( 3 , 3 , 2 , 2 ) - g(6,3,2,2)

g(5,3,3,2)

@

a(5,3,2,2,1)

@

u(4,3,3,2,1)

@

g(3,3,3,2,2)

In general, the Littlewood-Richardson rules yield

P

where the sum is over all highest weights p 2 0 of the form

p = (A1

+ j l , X l , A2 + j Z , X 2 ] .

. .), j l

+ ...+ j ,

= k,

ji

I Xk-I - x i .

Note that X can be recovered from p by extracting every - other row. T h k shows that the representations (TP that occur in dk) @gA and in dk') 80'' are distinct - for A' # A. Moreover] the irreducibles # in dk) 8 ~7' and dk') @ d are clearly distinct when k # k' as these satisfy JpJ = 21x1 k and lpl = 21x1 k' respectively. This shows that C [ V ]has a multiplicity free decomposition under the action of G. Note that an arbitrary highest weight p 2 0 can be written in the form "1-1 = (XI + j 1 , X1, A2 + j 2 , X Z , . . .)" by letting j e = p 2 e - 1 - p 2 e . We have proved the following.

+

+

Theorem 4.5.1: The diagonal action ofGL(n,C) on C"@A2(C") is multiplicity free. Moreover C[C" @ A2(@")] -N @ o p P B

as a GL(n,C)-module. That is, every non-negative highest weight occurs in

C[C"@ A2(Cn)]with multiplicity one. There is another viewpoint on this example. One can identify V = S k e w ( n 1,C)by regarding the first row (or column) of an ( n 1) x ( n 1) skew symmetric matrix as an element of C n , and the remaining entries as an element of A'(@") F Skew(n, C).For z E Skew(n + 1,C), we write z' for the element of Skew(n,C) obtained by removing the first row and column of z . Under this identification] the diagonal action of GL(n,C) on V1 @ V, is realized on Skew(n 1,C) by restricting the action of GL(n 1,C) to the subgroup GL(n,C) c G L ( n 1,C), embedded as

-+

@ Vz = C" @ A2(Cn) with

+

+

+

+

+

260

C. Benson and G. Ratcliff

The fundamental highest weight vectors arising from the separate actions of G L ( n , C ) on V1 and VZ are 212 and v i ( z ) for k = 1 , . . . , where vi(z) is the Pfaffian of the first 2k rows and columns of the n x n matrix z'. There are additional fundamental highest weight vectors, v k ( z ) for k = 1,.. . , [ ( n 1)/2]. These are the Pfaffians of the first 2k rows and columns of the ( n 1) x ( n 1) matrix z . Note that v1(z) = z12, so our fundamental highest weight vectors are the uk's together with the vi's.For a highest weight p 2 0, a p-highest weight vector in @[V]is

+ +

+

vP1-/k2 ( , i ) P Z - P 3

1

P3-bl

v2

. . . yLAn-l-Pn m

(4l)"-

when n = 2m is even, and

+

when n = 2m 1 is odd. In particular, this is a rank n multiplicity free action with fundamental highest weights {(lk) : 1 5 k 5 n}. 4.6. Section

4

notes

The decomposition for action GL(n) @ GL(m) is called G L ( n ) - G L ( m ) duality. This decomposition and those for the actions S2(GL(n)) and A2(GL(n))were popularized by Howe in [22]. Reference [23] explains how these results were rediscovered independently by various mathematicians and are implicit in earlier work of Weyl and Schur. G L ( n )- G L ( m )duality is equivalent t o Schur duality, which gives the decomposition for (Cn)*m under the action of GL(n,C)x S,. In fact, as explained in [23],both dualities can be derived from the First Fundamental Theorem (FFT) of Invariant Theory. We refer the reader to Section B.2.6 in [17] for the theory of Pfaffian polynomials. The irreducibility of determinants and Pfaffians on S y m ( n ,C) and Skew(n,C)is proved in Section B.2.7 of [17]. The theory of spherical harmonics can be found in Section 5.2.3 in [17] as well as many other sources. The Littlewood-Richardson rules are discussed in Section 1.9 of [39l. Compact

forms for the actions Gl(n) @ GL(m), S2(GL(n)), A2(GL(n))and SO(n) x C x arise in connection with Hermitian symmetric spaces. In fact, let G / K be an irreducible Hermitian symmetric space of non-compact type. Here G is a semi-simple real Lie group and K is a compact Lie subgroup. The symmetric space structure gives a Cartan decomposition JJ = t + p for the Lie algebra. The complex structure on

O n Multiplicity Free Actions

261

T,(G/K) is viewed as an R-linear map J : p 4 p with J 2 = -I. Now Pc=P+@Pwhere p* are the (fi)-eigenspaces for J on the complexification of p. The complexified adjoint action of K on gc preserves ph. For the classical irreducible Hermitian symmetric spaces this construction leads to the following actions. (See [20).) 0

0

Type A 111: G / K = S U ( n , m ) / S ( U ( n )x U(m)).The action of K = S ( U ( n ) x U ( m ) )on p+ can be identified with the action of K on Ad,,,(@) via (k,k’) . u = ku(k’)*. The group K has a one dimensional center and we essentially have a compact form of the multiplicity free action SL(n) 8 GL(m). More precisely, the action here agrees with = on the second factor. action (6) twisted by k’ H Type C I: G / K = S p ( 2 n , W ) / U ( n ) .The action of K = V ( n )on p+ can be identified with that of K on S y m ( n , C) via k v = k v k t . So this is a compact form of S2(GL(n)). Type D 111: G / K = S O * ( 2 n ) / U ( n ) . The action of K = U ( n ) on p+ can be identified with that of K on S k e w ( n , C) via k . u = kukt. This is a compact form of the multiplicity free action A2(GL(n)). Type BD I: G / K = S O O ( n , 2 ) / S O ( n )x S O ( 2 ) . The action of K = SO(n,R) x T on p+ can be identified with the action of K on C n via (k,t ) . z = t k z . This is a compact form of SO(n) x C x . +

0

The classification of irreducible Hermitian symmetric spaces includes two exceptional cases, in addition to the four families described above. 0

0

Type E 111: In this case K = S p i n ( l 0 ) x T and G has Lie algebra e6(-14), a certain real form for the complex Lie algebra e6. One can identify p+ with Aeven(C5)s @I6 and S p i n ( l 0 ) acts by the positive half-spin representation. It is known that (with the scalars included) this action in multiplicity free. Type E VII: Here K = E 6 x T and G has Lie algebra e7(-25), a real form of e 7 . In this case p+ can be identified with an exceptional Jordan algebra 3 of dimension 27. The representation of & on J is described in [ll].This action (with the scalars included) is multiplicity free.

Thus one has:

Theorem 4.6.1: 0251) The linear action K : p+ associated to any irreducible H e m i t i a n symmetric space of non-compact type is multiplicity free.

262

C. Benson and G. Ratclzff

5. A recursive criterion for multiplicity free actions

In this section we present a recursive criterion for multiplicity free actions due to Knop [31]. Given a linear action G : V we let

9 = 9 ( V )c g* be the set of all weights for the representation of H on V , listed with multiplicities. As usual, B = H N is a Bore1 subgroup and A = A + u ( - A + ) are the roots for G, as in Section 1.4. For a highest weight X E 9 let Sx = { a E A+ : (X,a)

> 0).

If z E V is a A-highest weight vector then g . t = @z if and only if Sx = 0. If Sx = 0 for all highest weights X E a, then G acts by scalars on every weight space in V . In this case, we essentially have a torus action. In particular, G : V is multiplicity free if and only if H : V is multiplicity free. This happens if and only if the set 9 of weights for H : V is linearly independent. Suppose that X E 9 is a highest weight with Sx # 0. Let zo E V be a A-highest weight vector and let fo E V* be the corresponding (-A)-lowest weight vector normalized so that fo(zo)= 1. Let P = Pfoand C = Cfo be as in Equations 3 and 4. Now P- = LU- is the opposite parabolic subgroup to P = LU and P- = Pzo. From the definition of C one has that z E C if and only if f o ( z )# 0 and ( X . fo)(z)= - X ( X ) f o ( z ) for all X E g = p u-. But X . fo = - X ( X ) f o for X E p, so

+

C = { z E V : f o ( z )# 0 and (u- . fo)(z)= O}. Hence C is a open set in the subspace

w = (u-

. fo) I

of V . Recall that C is invariant under the action of the Levi component L of P . As U . C = V, = { z E V : f o ( t ) # 0} by Lemma 3.2.5, we see that there is an open B-orbit in V if and only if there is an open L n B-orbit in C. Equivalently, G : V is multiplicity free if and only if L : W is multiplicity free. We know, moreover, that I = I) CacA(L) ga where

+

A(L) = { a E A : (X,CY) = 0)

= A\Sx.

The positive roots for L are A+(L) = A+\Sx. Each of the root vectors { X - a : a E Sx} c u- acts non-trivially on fo, so the set of weights in

O n Multiplicity Free Actions

263

+

u- . fo is {-A a : a E Sx}. Thus the set of weights in W = (u- . ')of is *\(A - Sx). In summary, we have a recursive algorithm that begins with the pair

(A: = A f l 90= 9) where 0

0

A+ is the set of positive roots for G, and

9 c b* is the set of all weights for the representation of G on V , listed with multiplicity.

Given a pair (A:, 0 0

0

do the following:

For each highest weight X E 9 nlet SX = { a E A: : ( & a )> O}; If Sx = 8 for all highest weights X E then G : V is a multiplicity free action if and only if Q n is linearly independent; Otherwise, choose a highest weight X E Q n with Sx # 8 and apply the above steps to the pair (A:+ll 9n+l)= (A$\Sx, !Pn\(X - Sx)).

an,

Eventually all the SX'Sare empty and the algorithm terminates at the second step above. To illustrate this method, we revisit some of the examples described in Section 4. 5.1. GL(n)

Here G

= G L ( n , C ) acts

A$ = A+ = {

on V = Cn as usual. One has

~ i ~j

:

i <j } ,

90= 9 = ( ~ 1 , .. . , E ~ } .

We know e l = (II0, . . . I 0) is a highest weight vector in V with weight and so

s,, = {El

-

E2,.

.., E l

~ 1 ,

- En}.

The coordinate vector z1 E V * dual to el has parabolic subgroup P = Pz, with Lie algebra p spanned by {Ell}U {Eij : i > 2). The Levi component is L = GL(1,C) x GL(n - 1,C). The nilpotent u- = Span{Elj : j 2 2) gives u-21 = Span(z2,. . . zn} and hence W = (u-21)' = Cel. Thus we have

AT = A$\S,, = {

~ i ~j

: 2

5 i <j},

91 = Q O \ { E ~ ~ .

..,

E ~ = } (~1).

Since S,, = 8, the process terminates. One concludes that G : V is multiplicity free since 91 is a linearly independent set.

C.Benson

264

and G. Ratcliff

In practice, it is not necessary to identify L and W at each stage in the induction. We illustrate this in the next example. 5.2. GL(n) C3 GL(n) Here G = GL(n,C) x GL(n,C) acts on V = Cn @ C n . Now

A , + = A + = { E ~ - E: ~I ~ ~ < ~ < ~ ) U { E : - E :; i ~ i < j i ~ ) and : 1 Ii I n , 1 Ij I n } .

90=9= {Ei+E; X

= ~1

+ E:

is the only highest weight in @o. One has

SEIfE; = { E l - E 2 , . . . , E l - E n }

u {&; - E'2,. . . , &'1 - E k } .

Thus

A ; = { E ~ - E ~:

~ < Z < ~ < ~ } U ( E : - E ;

:

25i<j
and

91 = { E l + E i } U

{&z

+E;

:

2

< i 5 n,2 Ij I n}.

From these roots and weights we can see that L : W is equivalent to the action of C x x GL(n - 1,C) x Cx x GL(n - 1,C) on C @ (Cn--l @ C n W 1 ) . At the next stage of the algorithm one takes X = ~2 + E; and obtains

A;

: 3

={ ~ i ~j

9 2 = {El +E;,E2

5 i < j } u {E:

- E;

+Eh}U {Ei +E;

: 3

: 3

Ii <j } ,

Ii,j}.

After n steps, the process terminates with

an = ( ~ +1 E ; , . . . , E , +EL}. L 2 (ex)" x (ex)"and S A = 0 for all X A:

= 8,

At this point linearly independent, G : V is multiplicity free.

E

9,. As @, is

5.3. GL(n) @ G L ( ~ A2(GL(n)) ) For the diagonal action of G = GL(n,C) on V = C 2 @ A2(Cn)one has

A:

= A+ = { ~ i - ~: j i

Choose X

A;

= ~ 1 so ,

<j},

that SX = (

={ ~ i ~j

: 2

Q o = 9 = ( ~ 1 , ... , ~ , } U { & i + & j : i ~ 1 ~j


: 2

< j } and

91 = ( ~ 1 )U { E i

+

Ej

Z

<j}.

<j}.

On Multiplicity Free Actions

Next choose X = ~1

+ ~2 t o obtain Sx = { ~ -2

A i = { ~ i - ~ :j 3 I i < j } ,

~j

265

: 3

5j

} and : 25i<j}.

* Z = { E ~ , E ~ + E ~ ) U { E ~ + E ~

After n steps, the process terminates with A? = 0 and

As this is a linearly independent set, one concludes G : V is a multiplicity free action. Alternatively, one could begin with X = ~1 ~2 in the first step. This gives Sx = {EI - ~j : 3 5 j } U {Q - E~ : 3 5 j } and

+

Next one could choose a highest weight in 91 that was not available in V, namely X = E~ ~ 4 This . gives SX = (63 - ~ j ~4, - ~j : 5 5 j } and

+

={El

. . . ,E " } u { E l

+

E 2 , E3

+

Ed}

u {Ei

+

Ej

: 5

We could then continue in this manner (choosing X = X = ~ 1The . latter choice gives Sx = { E ~ E ~ } ,

A; = {

u

~ 3 ~ q } { ~ i ~j

: 5

5 i <j}.

~5

+ ~ g )or take

5 i < j } , and

For n even, the algorithm could terminate with

This example illustrates how different choices for a highest weight X E 9 k can yield different paths through the algorithm and result in different terminal sets.

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6. The classification of linear multiplicity free actions

Let (n, V ) be a rational representation of some reductive algebraic group G and consider the question of whether or not this gives a multiplicity free action. The answer depends only on the algebraic subgroup n ( G )of G L ( V ) . One says that two rational representations ( T I ,Vl),(n2,V2) of groups G I , G2 are geometrically equivalent if irl(G1) coincides with nz(G2) under some isomorphism Vl 4 V2. For example, ( r , V )is geometrically equivalent to (no cp, V ) for any cp E Aut(G).Taking y ( g ) = (St)-' on G = GL(n,C ) , we see that any rational representation of GL(n,C)is geometrically equivalent to its contragredient. Linear multiplicity free actions have been completely classified up to geometric equivalence. In this section we present the results of this classification.

6.1. Irreducible multiplicity free actions The multiplicity free actions G : V where G acts irreducibly on V were classified by Victor Kac in [26], building on earlier work including that of Sato and Kimura [47]. The open Bore1 orbit criterion provided the main technique used to achieve this classification. First suppose that the image G of G in G L ( V ) contains a copy of the scalars. In this case, G is reductive but not semisimple. The group G coincides with the image of GI x C x , where GI = (G,G) is the commutator subgroup in G, a semisimple group with finite center. The possibilites for GI, the image of GI in G L ( V ) ,are listed in Table 3, up to geometric equivalence. That is, for each such G', the joint action of G' and Cx on V is multiplicity free. Bold faced type is used in Table 3 to indicate a subgroup of G L ( V ) , the notation indicating the representation involved. The first three entries denote the defining representations of S L ( n ,C), SO(n,C) and Sp(2n,C) on C", C" and C2n respectively. S2(SL(n))and A2(SL(n))denote the images in GL(S2(Cn))and GL(R2(Cn))of the symmetric and skewsymmetric squares of the defining representation of S L ( n ) .SL(n) @ SL(m) denotes the image of the representation of S L ( n ) x SL(m) on C" @ C" (outer tensor product of the two defining representations) and similarly for SL(n) @ Sp(2m). Spin(7), Spin(9) denote the image of the spin representations of Spin(7,C), Spin(9,C) on C8 and C16 respectively. Spin(l0) indicates the positive spin representation of Spin(l0,C) on el6.G2 and Ee denote actions on C7 and C2' respectively. The conditions on n and m

On Multiplicity

Free Actions

267

Table 3: Irreducible multiplicity free actions G’ x C x : V , G’ semisimple Degrees of fundamental Rank Semisimple Group G’ (G’ x Cx : V is multiplicity free) highest weight vectors 1 1 SL(n) .( 2 1) 112 2 SO(n) .( 2 3) 1 1 Sp(2n) .( 2 2) 1,2,. . . ,n n S2(SL(n))( n 2 2) A2(sL(n))( n 2 4) 1,2,. . . > L V P l LnPJ 1,2,. . . , min(n, m ) min(n, m) SL(n) @ SL(m) (n,m 2 2) 1,272 3 Sp(2n) @ SL(2) ( n L 2) 1,2,2,3,3,4 6 Sp(2n) 8 SL(3) ( n 2 2) 1,2,2,3,4,4 6 SP(4) 8 SL(n) .( 2 4) 172 2 Spin(7) 172, 2 3 Spin(9) Spin(10) 172 2 1,2 2 G Z 1,273 3 E6

in Table 3 are imposed to eliminate redundancies caused by isomorphisms in low dimensions. Note that the actions SO(n), S2(SL(n)),A2(SL(n)) and SL(n) 8 SL(m) were discussed in Section 4. Table 4: Irreducible multiplicity free actions G’ : V , G’ semisimple

The scalars @” act on P,(V) by the character t H tPm.Thus G’ x CX : V is a multiplicity free action if and only if the representations of G’ on each P,(V) are multiplicity free. If we remove the scalars, then multiplicities can arise across different degrees of homogeneity in C[V]. This happens, for example, when G = S O ( n , C ) x C x , as discussed in Section 4.4. In fact, most of the actions from Table 3 fail to be multiplicity free when the

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C. Benson and G. Ratcliff

scalars are removed. Table 4 lists the irreducible multiplicity free actions for semisimple groups up to geometric equivalence. 6.2. Decomposable actions

Given linear actions GI : V1, G2 : VZ,one can form the product action G1 x Gz : Vl eV2. If G1 : V1, Gz : Vz are both multiplicity free with decompositions C[V,]= @A,-Aj PA,then C[V1@ VZ]2 C[V1]8 C[V2]decomposes in a multiplicity free fashion

@[KeVz]=

@

P A 8 P A ~

AEAI,A'EA~

under the action of G1 x Gz. Thus products of multiplicity free actions are multiplicity free. One says that an action G : V is decomposable if it is geometrically equivalent to a product action G1 x Gz : V1 @ V2 with non-zero V,. Otherwise, we say G : V is indecomposable. Each irreducible multiplicity free action is indecomposable, but the converse is far from true. The action GL(n) @ G L ( ~ A2(GL(n)) ) described in Section 4.5 provides one example of an indecomposable multiplicity free action that is not irreducible. 6.3. Saturated indecomposable multiplicity free actions The classification of linear multiplicity free actions was completed independently by the authors [5] and Andrew Leahy [38]. Given a non-irreducible linear action G : V , one decomposes V as a direct sum of G-irreducible subspaces

v = v-ff3 . . . @ v,. G : V is said to be saturated if the image G of G in G L ( V )contains a full torus (C")". That is, the dimension of the center in G equals the number m of irreducible summands. Given G : V one can always form a saturated action G' x (C")" : V . This action is multiplicity free if and only if the representation of G' on

Cpk,(Vl)@ . . . @ P k , ( K L ) is multiplicity free for each ( k l , . . . , k m ) . The saturated indecomposable multiplicity free actions consist of the irreducible actions in Table 3 together with the actions listed in Table 5. Each entry in Table 5 denotes the image G' of the semisimple part G' of G

On Multiplicity Free Actions

269

Table 5: Indecomposable non-irreducible saturated multiplicity free actions

Semisimple Group G’ (G’ x (C”)’ : V is multiplicity free)

Degrees of fundamental highest weight vectors (rank)

172,. . ., LnPJ7 172,. . . 9 l(. 11/21 (n) 172, ’ . ‘ I LnPJ9 172,. . . , l(n - 1)/2J ( n 1) 1,2,. . . ,min(n,m), 1,2,. . . ,min(n, m 1)

+

+

(min(n,m) + min(n, m + 1)) 1,2,. . . , min(n,m), 1,2,. . . , min(n, m + 1) (min(n, m)

+ min(n, m + 1))

in GL(V).In each case, V has two irreducible summands, V = V1 @ Vz, and one simple factor in G’ is acting diagonally. Thus, for example, SL(n) @ S L ( ~(SL(n) ) 8 SL(m))denotes the image of S L ( n ,C)x SL(rn,C) under the representation on V = V1 @ VZ = (C”) @ (C”8 C”) where S L ( n ,C ) acts diagonally on V1 and V,. Spin(8) @spin(8) S O ( 8 ) denotes the image of the action of Spin(8,C) on C8 @ C8 via the direct sum of the positive spin representation with the natural representation (via SO(8,C)). The notation SL(n)*denotes the contragredient to the defining representation. For each group G’ in Table 5 the saturated action G’ x (Cx)z : V is multiplicity free. Together, Tables 3 and 5 classify all saturated indecom-

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C. Benson and G. Ratcliff

posable multiplicity free actions up to geometric equivalence. In particular, the classification shows that a saturated indecomposable multiplicity free action can have at most two irreducible summands.

6.4. Non-saturated indecomposable multiplicity free actions Only one entry in Table 5 remains multiplicity free when the torus (Cx)2 is removed, namely

(SL(n)8 SL(2)) @ s L ( ~ ) (SL(2) 8 SL(m)) for n, m

23

In addition, for each group G‘ in Table 5 one can consider the joint action of G’ x C x where C x acts on V = V1 @ Vz via

t . (211, 212)

=

(tau1, t%z)

for some integers a , b. Such actions are multiplicity free in the cases listed in Table 6. Table 6: Non-saturated multiplicity free actions

6 . 5 . Completing the classification Let G be a connected complex algebraic reductive group acting on V via some rational representation 7r. The commutator subgroup G‘ of G is semisimple and, by lifting to a finite covering if necessary, we can suppose that

On Multiplicity Free Actions

271

G = G’ x A where A is some algebraic torus. Write V as a direct sum

V

= W1@w 2 @.

. . @ W,

of 7r(G)-invariant subspaces Wj which are indecomposable under the action of G‘. Letting 7rj denote the action of G on Wj, we have 7r(G‘) = GI’ x Gz’ x - .. x G,’ acting on V via the product action, where Gj’ = 7rj(G’).

Theorem 6.5.1: 051) G : V is a multiplicity free action if and only if A contains a direct product of the form A1 x A2 x . . . x A, where each Aj is a torus of dimension at most 2 and the actions Gj’ x Aj : Wj are multiplicity free f o r j = 1,.. . , r . One can choose the Aj’s in Theorem 6.5.1 to be minimal, in the sense that the action of Gj’ x B on Wj fails to be multiplicity free for all proper subgroups B of Aj. Some factors Aj can be trivial and A1 x . . . x A, need not act on W1 @ . . . @ W, via a product action. That is, the Aj’s can act diagonally on the indecomposable GI-summands Wj .

Example 6.5.2: As an example, consider G’ = S L ( n ) x S L ( n ) for n 2 2 acting on V = W1 @ W 2 = (C” @ C”) @ (C” @ C”) via (g,h) (xl,Yl,z2?Y2)= (gxl,gYl,hx2,hYz) ’

for g , h E S L ( n ) and x j , y j E C”. Here GI’ = SL(n) @ s L ( ~ )SL(n) = Gz’ appear in Table 5. Let A = (Cx)3 act on V via

(tlrt2,t3)’ ( 5 1 3 Y l r Z Z , Y 2 ) = ( t l t 2 2 l , t ~ t 2 Y l , t l t 2 x 2 , t l t ~ Y 2 )

for tj E c x .

The joint action of G’ x A on V is multiplicity free. Indeed, if we let A1 = Cx x (1) x (1) and A2 = (1) x C x x (1) then we see that the actions Gj’ x Aj : Wj appear in Table 6. Here, however, one can’t find subtori Aj as in Theorem 6.5.1 that act independently on W1 and W2. Theorem 6.5.1 completes the classification of multiplicity free actions because we have exhibited all of the possibilities for the groups Gj’ c GL(Wj) and for the actions of the Aj’s on the Wj’s. More precisely, if G : V is multiplicity free then for each 1 5 j 5 r we must have either: (1) Wj is Gj’ irreducible and (a) 7rj(Aj) = Cx and Gj’ appears in Table 3, or (b) Aj = (1) and Gj’ appears in Table 4. (2)

Wj

is a sum of two Gj’-irreducible subspaces and

(a) 7rj(Aj)= ( C x ) 2and Gj’ appears in Table 5, or

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C. Benson and G. Ratcliff

(b) Gj’ appears together with 7rj(Aj)2 Cx in Table 6, or (c) A j = (1) and Gj’ = (SL(n) @I SL(2)) @ s L ( ~ )(SL(2) @ SL(m)) with n,m 2 3. 6.6. Proof outline

The saturated indecomposable multiplicity free actions in Table 5 are obtained from the irreducible actions in Table 3 by extensive case-by-case analysis. Suppose that G = G‘ is semisimple and that (7r,V) is an indecomposable action with V = V1 @ V2, a direct sum of two G-irreducible subspaces. Let 7rj = 7rlV, and Gj = 7rj(G) c GL(V,) for j = 1 , 2 . If G x ( C x ) 2 : V is multiplicity free then it is clear that both G x Cx : V1 and G x Cx : V2 must be multiplicity free. Thus both GI and G2 appear in Table 3. Define normal subgroups K1 and K2 of GI and G2 by

K1 := ~l(Ker(7r2)) and K2 := 7r2(Ker(7r1)).

(16)

F : G1/K1+ G2/K2

(17)

The map

given by F(7r1(g)K1):= 7r2(g)K2 is a well-defined group isomorphism. If K1 = GI then it follows that K2 = Gz and we have T(G) = G1 x G2. In this case, our action decomposes as a direct product of the multiplicity free actions Gj x ( C ” ) . Next suppose that the Kj’s are proper subgroups of the Gj’s. Note that Kj need not be connected. We write Kj” for the identity component in Kj. Since Kj is a normal subgroup of Gj, so is KY. As Gj appears in Table 3, Gj is either a simple group or a product of two simple factors. Thus we can write Gj = KjOHj, where either Hj = Gj (when Kj” = {e}) or Hj is one of two simple factors in Gj. As Hj 2 Gj/Kj” covers Gj/Kj, the derivative of F yields an isomorphism between the Lie algebras of H I and Ha. In fact, Table 3 shows that Hj is simply connected except when Hj = SO(n). Thus, we can realize F as a group isomorphism HI 2 Ha or as a covering of one of the Hj’s by the other. We write F : H1 + H2 for this map after interchanging the roles of G1 and G2 when Ha covers H I . If we define a new group L by L = K i x H1 x KS and a representation of L on V = V i e V2 by

~(h hi,,k2)

:= (kihi, F(hi)k2) E GL(Vi) x GL(V2)

273

On Multiplicity Free Actions

then we see that r ( G ) = o ( L ) = GI @ H Ga. ~ That is, Hi acts diagonally on VI and Vz via F . In summary, we have shown that any saturated indecomposable multiplicity free action with two irreducible summands is obtained from two entries in Table 3 via diagonalization along simple factors. Combining pairs of entries in Table 3 which share a common simple factor yields a large number of indecomposable actions. It is necessary to examine each of these in turn to determine which are multiplicity free. Various techniques can be applied in each case. One can look for an open Borel orbit. (Many candidates can be eliminated easily because the underlying vector space has dimension greater than that of a Borel subgroup.) One can apply the Littlewood-Richardson rules (and variants for the classical groups SO(n,C),Sp(2n,C)) or use the recursive criterion from Section 5. Table 5 is the end result of this analysis. If V = V, @ ... @ Vr is a sum of T irreducible subspaces, then the above reasoning applies to each pair of groups Gi = r i ( G ) , Gj = r i ( G ) . For the action to be indecomposable, at least one simple factor in each Gi must act diagonally on at least one V, with j # i. Diagonalization from Tables 3 and 5 produces indecomposable actions with more than two irreducible summands. The analysis shows, however, that no such actions are multiplicity free. 6.7. Section 6 notes

The articles [7]and [31] contain formulas for the fundamental highest weight vectors and further detailed information for each action in the classification. For the irreducible actions, much of this is due to Howe and Umeda [24]. 7. Invariant polynomials and differential operators 7.1. Polynomial coefficient differential operators

For each f E C [ V ]we have the multiplication operator M f E End(C[V]),

M f ( h )= fh. Let P(V)= { M f : f E @ [ V ] }so , P ( V ) 2 @[V]. For each w E V we have the directional derivative 8, E End(C[V]),

-

The algebra D ( V ) generated by {av : w E V } is the algebra of constant coefficient differential operators. The embedding V D ( V ) ,w H 8, extends

C. Benson and G. Ratcliff

274

t o an isomorphism from the symmetric algebra S(V) t o D(V).Thus

D(V)E S ( V )E @[V*] as algebras. We let PD(V)be the subalgebra of End(@[V]) generated by P ( V )and D(V). This is the algebra of polynomial coefficient differential operators. The product rule for derivatives shows that the map

P ( V )8 D(V)-+ PD(V), p 8 L

pL

given by multiplication is a vector space isomorphism. Composing this with the algebra isomorphisms

=

@[V@ V * ] @[V] @3 a=[V*] E P ( V )8 D(V) produces a vector space isomorphism

6 : @[VCB V'] + PD(V), p

H

p ( z , a)

We say that p E @ [ V @ V " ]is the W i c k symbol for the operator p ( z , a). The inverse map c7

: PD(V) + @[V63 V * ]

of 6 is called the polarized symbol map. The above discussion can be made more concrete by introducing coordinates. Let ( ~ 1 , ... , z,) be coordinates on V with respect t o some basis { e l , . . . ,e n } and (w1,. . . ,w n )be coordinates on V*with respect t o the dual basis { e; , . . . , e:}. We have

@[V] = @[z1,.. . , z n ] , @[VCB V * ]= @ [ Z l , . . . , z,,

w1,. . . ,w,].

The monomials

z a = zH1 . . . z E n form a basis for @[V] and the differential operators

form a basis for D(V) as a = (a1,. . . , a,) ranges over all multi-indices a E N". The operator p ( z ,a) with Wick symbol

p ( z , w)=

c

c,,pzawP.

On Multiplicity Free Actions

is

p(.,

a) =

c

275

c,,pz"dP.

a,P

The symbol mapping is not an algebra isomorphism because C [ V @ V *is] abelian whereas P D ( V )is not. In fact, one has the Heisenberg commutation relations

[a,,Z j ] = 6i,j in P D ( V ) .To obtain an algebra isomorphism, one can pass to the associated graded algebras. The algebra C[V 69 V " ]is filtered by

C(""T/ @ v']=

pa(v)(8pb(v*). a+b
Then

PD("(V) = 6(C("[V @ V " ] ) gives a filtration of the algebra P D ( V ) . That is

u 00

C = pZ)(O)c pD(1) c . . . c P D ( k ) ( V )c . . . ,

P D @ ) ( V )= P D ( V ) ,

k=O

and

PD("(V)PD(e)(V)c PD("t)(V) in view of the commutation relations. In terms of our basis,

PD("(V) = Span{zQaP

:

la\ + IpI 5 k}

+ +

where la1 = a1 . . . an. The map 6 : C[V 69 V'] + P D ( V ) induces an algebra isomorphism from gr(@[V69 V " ] )E C[V @ V " ]t o 00

g r ( P D ( V ) )=

CPD('+~)(v)/PD(~)(v), k=O

an abelian algebra that is canonically isomorphic to P D ( V ) as a vector space. Lemma 7.1.1: P D ( V ) i s strongly dense in E n d ( @ [ V ] )That . is,

PD(V)JX = H o m ( X , @[V]) for any finite dimensional subspace X of (C[V].

276

C. Benson and G. Ratcliff

Proof: Let X c C [ V ]= C [ Z ~. ,. ., tn]be a finite dimensional subspace and T E H o m ( X , @ [ V ]Let ) . { P I , . . . ,p,} be a basis for X ,ordered so that deg(p1) 5 . . . 5 deg(p,). By taking suitable linear combinations, one can ensure that p k contains a monomial mk which 0

Let

has degree d e g ( p k ) and p l , . . . ,pk-1 contain no non-zero multiples of fj = T ( p j ) for

1 5 j 5 m, and let 1

D1

has

D1 ( P l ) = f l

=fl-aa a!

ml

mk.

= z a . Then the operator

(a!= f f l ! . . . c y n ! )

.

Assume inductively that there is some Dk E P D ( V ) with D k ( p j ) = fj for 1 5 j 5 k. Let mk+l = zp. Then dPp1 = . . . = dPpk = 0, and dPpk+l = P! NOW =

Dk+l

(fk+l

- DkPk+l)'a'

+

P!

-k

Dk

satisfies D k + l ( p j ) = fj for 1 5 j 5 k 1. By induction we conclude that there is an operator D E P D ( V ) with DIX = T . 0

7.2. Invariants in P D ( V ) Now suppose that a reductive algebraic group G is acting linearly on V . For the moment, we do not assume that G : V is a multiplicity free action. The group G acts on P D ( V ) via conjugation: ( 9 . D)(f) = 9 . D(L7-l

. f).

In terms of coordinates one calculates:

e

e It follows that the polarized symbol mapping intertwines the action of G on PD(V) with its action on C [ V @ V " ]= @ [ z l l .. ., z n , W l l . . . ,w,] via 9

' P(.l

'UI)

=p(g-'z,

dw).

This formula agrees with that for the representation of G on @[V@ V " ] arising from G : V and its contragredient G : V " , 9

. (211 E )

Thus we have shown:

= (9 '

*, 9 . E ) = ( 9 . E 0 9 - l ) . *l

On Multiplicity Free A c t i o n s

277

Lemma 7.2.1: The G-invariants in P V ( V ) are

P D ( V )= ~ PD(V) n EndG(@[V]) = { p ( z ,8)

: p E

C[V EB v * ] ~ ) .

Let K be a maximal compact connected subgroup in G. The Unitarian Trick ensures that an operator D E P D ( V ) is G-invariant if and only if it is K-invariant. Now we define the K-average of D via

Db = k ( k . D)dk where dk denotes normalized Haar measure on K . The operator D belongs to a finite dimensional subspace PD(m)(V)for some m, and the integral converges in PD(”)(v). SO ~h is a polynomial coefficient differential operator. A change of variables and unimodularity of K shows that k , .Db = Dh for each k , E K . Thus Dh E P’D(V)G.

Lemma 7.2.2: P D ( V ) G acts irreducibly on @[VlB>’for all dominant weights A. Proof: For

D E P V ( V ) G ,b E B

and h a A-highest weight vector, we have

b . ( D h ) = D ( b . h ) = D(b’h)

= bXDh.

Thus PD(V)Gpreserves the space C[V]”>X of A-highest weight vectors. and consider Next let h l , h2 be two A-highest weight vectors in @[V] the finite dimensional G-invariant space

X

= Span(G . h i )

+ Span( G . h ~ ) .

Since S p a n ( G . h l ) and S p a n ( G . hz) are equivalent as G-modules, there is some T E E n d c ( X ) with T(h1)= hz. By Lemma 7.1.1 there is an operator D E P D ( V ) with D]X= T . Now Dh E P D ( V ) G and

D‘hl =

L

IC. D(k-’ . h i ) d k

=L

k . T ( K 1. h l ) d k

=

J,T ( h 1 ) d k

(as k-l . hl E X )

(as T is G-invariant)

= T(h1) = hz.

This shows that PD(V)Gacts irreducibly on C[VIByX

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C. Benson and G. Ratclzff

Theorem 7.2.3: G : V is a multiplicity free action if and only i f PD(V)G is abelian. Proof: Let G : V be a multiplicity free action and

@[VI= @ P A X€A

be the decomposition of C[V]into pair-wise inequivalent G-irreducibles. Schur’s Lemma ensures that any operator D E PD(V)G must preserve each PA and acts on PA as multiplication by some scalar. It follows that any two operators in PD(V)Gcommute. Conversely, suppose that PD(V)Gis abelian. As PD(V)G acts irreducibly on C[VIBgXwe must have dirn(C[V]BiX) 5 1 for all dominant weights A. Hence G : V is multiplicity free. 0 Thus when G : V is multiplicity free, both algebras @[V@ V*IGand PD(V)Gare abelian. Although S and o are not algebra maps, they induce algebra isomorphisms

@[VI3 V*]G2 gr(@[VG3 V*]G)2 gr(PD(V)G) between the associated graded algebras. Concretely, this means that although one generally has p ( z , a)&, a) # ( p q ) ( z ,a), the operators p ( z , a ) q ( z , a ) and ( p q ) ( z , d ) have the same top degree terms. Here “degree” in PD(V) is defined using the filtration PD(‘)(V) from Section 7.1. In particular, zadP has degree la1 [PI.

+

7.3. A canonical basis for the invariants Suppose that G : V is a (linear) multiplicity free action. The trivial representation of G occurs in @[V]on Po(V) = @, the constant polynomials. As the representation of G on @[V] is multiplicity free, it follows that @[VIG= C. That is, there are no non-constant G-invariants in @[V]. Because of the connection with G-invariant differential operators it is, however, of interest to study the G-invariants in @[V@ V * ] . Let

@[V] = @PA XE A

denote the multiplicity free decomposition of @[V] under the action of G. Then

@[V@ V ” ]= @[V] @ @[V]* = @ PA X,X’EA

P;,,

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O n Multiplicity Free Actions

where the subspaces

PA8 P:, are G-invariant in @[V@ V * ] Thus .

@[V@ V*]C=

@

(PA @ PX*,)".

X,X'EA

But

H ( p +-+< ( p )f). by Schur's Lemma. The first isomporphism is given by f The element in PA 8 Pi that corresponds to IpA under the isomorphism PA 8 Pi "= Horn(P~, PA)is

c dx

px

=

fj

8 fj'

(18)

j=1

where dx = dirn(Px) and {fj : 1 5 j 5 d x } is any basis for PA with dual basis {f;}. Thus we have shown that

@[V@ V*]G= $(PA 8 P;)" = @@px. XEA

XEA

So {FA1 X E A} is a basis for @[V@V*IG. As Equation 18 does not depend on the choice of basis { fj} for PA,the basis {pX I X E A} for @[V@ V*IG is canonical. Applying Wick quantization we obtain a canonical basis for PV(V)G. We call the polynomials pX the unnormalized canonical invariants. To achieve some simplification in formulae to be derived below, we also introduce the (normalized) canonical invariants 1(dx = d i m ( P x ) ) . dX In summary we have proved the following. PA = -PA,

Proposition 7.3.1: { p ~: X E A} and { p ~ ( z , a ): X E A} are canonical vector space bases f o r @[V@ V*IGand PD(V)G respectively.

7.4. The fundamental invariants Now let r be the rank of the multiplicity free action G : V and

A' = { A i , . . . , A T ) be the set of fundamental highest weights. Recall that A mrAT I rn E W}. (See Proposition 3.3.1.)

= (rnlX1

+ .. . +

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Definition 7.4.1: The fundamental invariants for G : V are (71,. . . , r,.} where rj = PA,.

For X E A let 1x1 E N denote the degree of homogeneity of PA. That is, PA c Plxl(V). Then the canonical invariant p x is homogeneous of degree 214. For any weights p , v E b* we will write

when v - p is a sum of positive roots.

Lemma 7.4.2: For any A, p E A there are values c,

= CA,,,,

f o r which

PAP, = c c u p w , U

where the sum is over all v E A with 1v1 = (XI # 0.

+ 1pl and v 3 X+p.

Moreover,

CA+,

Proof: The product pxp, is G-invariant and belongs t o P l ~ l + l ~ l ( V8) PIAl+l,l(V*). As the p,’s form a homogeneous basis for CIV @ V*IG we conclude that PAP, =

c

CUP,

l4=I4+IP

for some values c,. Let { fj},”L, and {h}j$’l be bases of weight vectors for PA and Pp so that f l , hl are highest weight vectors. We know that all other weights in an irreducible representation space precede the highest weight in the partial ordering defined above. We have

The @[V]-components f i h j in this sum are weight vectors with weights X i + p j 3 X+p. It follows that c, = 0 unless v + X+p. Moreover, the term f l h l @ f,’h; contains the (A + p)-highest weight vector f l h l . We conclude that CA+, # 0. 0

Theorem 7.4.3: C[V @ V*IG = C [ y l , . .. ,r,.]. That is, @[V@ V*IG is a polynomial ring freely generated b y the fundamental invariants.

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O n Multiplicity Free Actions

Proof: Given m E N', let X = mlXl+. . . + mrXr E A. Lemma 7.4.2 shows that

y" = yl"'

...

y r T

= P?

-P?

= axpx

+CaYPY Y+X

for some coefficients ay with ax # 0. As { P A } is a basis for @[V@ V*IG,we 0 conclude that {y" I m E N ' } is also a basis for @[VCB V*IG.

Corollary 7.4.4: PD(V)G is a polynomial ring freely generated by ( 0 3 = yj(z,d) : 15 j 5 r}. Proof: From Theorem 7.4.3 we see that { y " ( z , d ) : m E W} is a basis for the vector space PD(V)G.Also, given m = ( m l ,. . . , m y ) ,

D"

=

07' ...OFT= ~ 1 ( ~ , d ) ".'. . y r ( ~ , 6 ' ) " '

+

differs from ym(z,d) by an element of PD(21xl-1)(V), where X = mlX1 . . . m,.X,.. By induction on degree in PD(V) we conclude that {D" : m E W} is a vector space basis for PD(V)G.Thus PD(V)G= @ [ D 1 ,.. . , D,].0 7.5. The algebra

@[VR]~

An alternative viewpoint on @[V@ V*IGwill prove useful. Let K denote a maximal compact connected Lie subgroup of G and (., .) be any K-invariant positive definite Hermitian inner product on V . The conjugate-linear vector space isomorphism

v + v*,

w H w* = ( . , 2 1 )

is K-equivariant (but not G-equivariant). In view of the Unitarian Trick we obtain an algebra isomorphism

@[VR]K = C[V @3 V ] K E @[V@ V*]K= @[V@ V*]G.

(19)

v

Here denotes V with the conjugate complex structure and VR is the underlying real vector space for V . Introducing coordinates ( 2 1 , . . . , zn) on V with respect t o an orthonorma1 basis, one has C [ V ]= @ [ z l , .. . , zn]. This polynomial ring also carries an inner product, namely (P, 43 = ( P ( W (0) = ( m p ) (011

the so-called Fischer i n n e r product. Here p ( d ) = p ( & , . . . , an) for p = C , c,P, ij(z) = C , c z " .

p ( z 1 , . . . , zn) and for q ( z ) =

C. Benson and G. Ratclzff

282

Thus

( z a ,zp)F = &,,pa! = 6%P a l ! .. . a,! for multi-indices a = (a1,.. . ,an),/3 = (PI,. . . ,p,).

Lemma 7.5.1: The subspaces {PA : X E A} in @[V] are pair-wise orthogonal with respect to the Fischer inner product. Proof: This follows from the fact that K c U ( V ) and U ( V ) preserves (., .)F.This can be seen using an alternative formula for the Fisher inner product: 1 ( p , q)F = /p(z)~e-~’lZdz&. (20) In ( 2 0 ) , n = d i m @ ( V ) ,1 . ~ 1 =~ ( z , z ) and “dz&” denotes Lebesgue measure on Vw normalized using (., .). We see that (k . p , k . q)F = ( p ,q)F for k E V ( V )via a change of variables in ( 2 0 ) , since both 1 . ~ 1 and ~ dzdZ are V ( V ) invariant. To establish (20) it suffices to verify that

J

z a ~e-lz12 ’ d z o= ~ Ir, 6,,, a!.

For this, use polar coordinates

zj

The integral in 0, is zero unless

= rjez’,

aj

to write

= a;, in which case one has

Using isomorphism ( 1 9 ) we can regard the canonical invariants p~ as elements of @[VjIK = @[zl,.. . , z,, z1,. . . ,Z,lK. We have

where { e j } is any orthonormal basis for PA (with respect to (., .)F).Note that each p~ E @[VwlK is real valued and non-negative. We later require the formula

On Multiplicity Free Actions

283

Indeed C,x,=k dxpx(z,Z) = C, le(z)I2 where e ranges over an orthonorma1 basis for ? k ( v ) obtained by concatenation of orthonormal bases for {PA : 1x1 = k}. The sum is, however, independent of the basis and we can use {za/&J : la1 = k} to compute

as stated. On @[Vi]= product

@[z1,.

. . , zn,Z1,. . . ,Zn]

we consider the Hermitian inner

(P, 4 ) . = ( P P , 3if)( 0 ) = (.(a, B ) P ) (0). This “doubled Fischer inner product” is determined by (zaZ@,za’z@’)*= ba,a’bp,p’a!p!.

Proposition 7.5.2: {px I X E A} is a n orthogonal basis f o r @[ViIKwith respect t o the inner product (., .)*. Moreover ( p x , p x ) * = l/dx. Proof: Let {ej}, {fj} be (., .)F-orthonormal bases for PA, P,. Using Lemma 7.5.1 we compute (PhPP)* = ( P x ( a , @ , ( z , m O )

7.6. Section 7 notes

Theorems 7.2.3, 7.4.3 and Corollary 7.4.4 are from [24]. An action whose invariants form a polynomial ring is said to be coregular. The coregular actions for simple groups are classified in [27] and [48].

284

C. B e n s o n and G. Ratcliff

The Fischer inner product is also called the Fock inner product, especially in connection with Equation 20. Fock space 3 is the Hilbert space This can be identified as the completion of C[V] with respect to (.,.),. space of holomorphic functions on V square integrable with respect to the Gaussian measure e-IZ12dz& [14].The inner product (., .)* can be regarded as the restriction of (., .),@(., .)pfrom F@F*to C [ V @ v ]2 C[V]@(C[V]*. One can identify F @ F with the space of Hilbert-Schmidt operators on F. Now (., )., induces the Hilbert-Schmidt norm. There is also a connection with the Berezin star product, as explained in [I]. 8. Generalized binomial coefficients

We continue to assume that G : V is a multiplicity free action. As in the previous section, {px I X E A} are the canonical invariants. We view these as living in cc [Vw]K .

8.1. The polynomials qx Let A = a

= dl81

+ . . . + anan and consider the operator T : C[Vi]+

@[VwI, A

( T p ) ( z , ~=) ( e p ) ( z ,-z) = e-

A

( p ( z ,-z)).

Note that T is an involutive automorphism. Indeed, writing ( M p ) ( z , t )= p ( z , -F) one has

T

= M o e a = e P A o M = T-’.

(22)

Definition 8.1.1: Let qx = T(px)= (-l)lxle-Apx for each X E A. The two formulas in the definition for qx agree because p x ( z , -F) = (-1)’xlpx(z,F).

Lemma 8.1.2: { q x : X E A} is a vector space basis for CC[VwIK.

Proof: First note that qx E (C[Vw]is K-invariant because px is K-invariant and A is a U(V)-invariant operator. Moreover qx = (-1)”lpx

+ rx

where px E Pzlxl(Vw) and rx is of lower degree. As { p x } is a basis for WQI~, so is { q x } . 0

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285

8.2. The generalized binomial coefficients

As qx belongs to @[Vi]",it can be written as a finite linear combination of the canonical invariants p,.

Definition 8.2.1: The generalized binomial coefficients A, u E A via

[t]are defined for

The proof of Lemma 8.1.2 shows that

So in fact qx =

c

[;I.,

(-l)lVl

l,lllxl

=

(-l)lx'px

+

c

(-l)IYI[t]P,.

I,I
The terminology "generalized binomial coefficient" is motivated by Example 8.4.1 below. Our immediate goal is to develop some combinatorial properties of these coefficients and to relate them to eigenvalues for operators in P D ( V ) G .We begin with the following.

Proposition 8.2.2:

Proof: First note that as qx = T(px) and T 2 = I one has

an interesting formula in its own right. But now

and the result follows from the linear independence of { q x : X E A}.

C. B e n s o n and G. Ratcliff

286

Proposition 8.2.3: For X E A and k E

N

and

Proof: We have

Equating homogeneous components of degree 2(lXl - k) on both sides of this equation yields (24). Applying the operator T to both sides of (24) yields (25) since

The next theorem is the key to all subsequent results in this section.

Theorem 8.2.4: (Yan's Pieri Formula) For v E A, k E

N,

Proof: First note that for polynomials p , q E ( ~ [ V one W ] has

O n Multiplicity Free Actions

and hence 24,

(IZ~~P,

287

q)* = ( p , Aq)*.Now using Proposition 7.5.2 and Equation

Corollary 8.2.5: For

1x1 = IvJ + k

where the s u m i s over all

(€1,.

one has

. . , ~ k - l ) with

( ~ j=(

(vI

+j.

Proposition 8.2.6: The generalized binomial coeficients are non-negative real numbers.

[t]

Proof: Corollary 8.2.5 shows that it suffices to prove 2 0 for = 1 . ~ 1 + 1. Let {fl, . . , ,fd,} be an orthonormal basis for P, with respect to the Fischer inner product. Since z i f j E Pl,,l+l(V) = @ l A l = l v l + l PA,we can write

where f~(i,j) E PA.Hence also

The sum ~lxl=lx,l=lv,+l PA @ is direct in C[Vj]= C [ V ]8 C[v] and each PA@ PA!is a K-invariant subspace. Since IzJ2d,p, is a K-invariant polynomial, it follows that n

d,.

2aa

is K-invariant for each

C. Benson and G. Ratcliff

1x1 = IvI + 1 = IX’I.

But

via Schur’s Lemma, as in Section 7.3. We conclude that CZ1C;’’, f~(i,j) f ~ t ( z , j ) = 0 for X # A’ and that Cy=lC;L, Ifx(i,j)I2 = c ~ p x for some CA E @. As PA and C;=,C::, If~(z,j))~ are both non-negative real valued polynomials on VR, we must have CA 2 0. Hence

for some values CA 2 0. Theorem 8.2.4 now implies

[:]

= c ~ / d x2 0.

8.3. Eigenvalues f o r operators in P’D(V)G The operators p y ( z , d ) in the canonical basis for PD(V)G act by scalars on each subspace PA c @[V]. This is a consequence of Schur’s Lemma, G-invariance of pV(z,a)and the fact that the decomposition @[V] = @PA is multiplicity free.

Definition 8.3.1: For v,X E A let &(A) E C denote the eigenvalue of pV(z,8) on PA.That is, p V ( z ,a)(, = &(X)Ip,. Proposition 8.3.2: d,p2(X) =

[i] for all A,v E A.

Proof: Note that

Now using Theorem 8.2.4,

289

On Multiplicity Free Actions

But we can also use Equation 21 to write 4

12k

Thus

dypy(z,a)elz12= C d , d x ~ ~ ~ ) p x ( z , ~ ) . x

Comparing these two expressions for d,p,,(z, d)elz12 gives d,p?,(X) = claimed.

[t] as 0

Together Propositions 8.2.6 and 8.3.2 yield: Corollary 8.3.3: The eagenvalues &(A)

are non-negative real numbers.

8.4. Examples

Here are two examples to illustrate the circle of ideas developed above. Even the most basic example is of interest in this context. Example 8.4.1: GL(n): Consider the usual action of G = G L ( n , C )on V = Cn. We have maximal compact subgroup K = U ( n ) and the standard inner product ( z , z ' ) = z .Z' is K-invariant. C [ V ]decomposes as

C [ V ]= c [ z ~ ... , z,] =

@ 'Pm(V) mEN

under the actions of G and of K. Using the orthonormal basis {P/& : la/ = m } for 'Pm(V) (with respect to (.,.)F)we compute the canonical invariant p m ( z ,z)E @[ViIK:

Substituting d, = dim(P,(V)) =

('":+ '> -

one obtains Pm(Z,T) =

( n - l)!

( m + n - I)!

I Z p .

The fundamental invariant is y = pl = 1zI2/n and the above computation ] ~C[y], illustrating Theorem 7.4.3. shows that ( C [ V R=

C. Benson and G. Ratcliff

290

Next we compute Iz]2(m--k)

( m - k)!

dkpk =

It12(m-k)

H2k= (") k m!

(m - k)! k!

2m =

(T)dmpm.

Now Theorem 8.2.4 implies

for k , m E N. This fact motivates the terminology "generalized binomial coefficient". The polynomials {qm I m E N} are given by

where

is the generalized Laguerre polynomial of order r and degree m, normalized to have value 1 at z = 0.

Now according to Proposition 8.3.2 the operator d k p k ( z , a) E PD(V)G has eigenvalue

on Pm(V).One can see this directly because

for m 2 k. For this example Proposition 8.2.2 asserts that

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291

Equivalently, the lower triangular matrix A with entries Am,k = (-l)(T) (0 5 m, k 5 N say) is self inverting: A2 = I . One can verify ( 2 6 ) directly as follows.

Example 8.4.2: GL(n) @ GL(n): Recall the (twisted) action (7) of G = G L ( n , C ) x G L ( n , C ) on V = Cn @ @" 2 A4n,n(C). Restricting to the maximal compact subgroup K = U ( n ) x U ( n ) we have (kl,kz) . z = Iclwk;

where 'k = Et = k-l. The inner product (z,w) = t r ( z w * ) on V is Kinvariant. The decomposition w%L,n(@)I

=

@ PA X€A

under the action of K , given in Theorem 4.1.1, is indexed by partitions A = (A, 2 . ' . 2 A, 2 0 ) . Let C c V denote the set of matrices of the form h = diag(d1,. . . , d,) with d j E I%+. Proposition 8.4.3: 061) For A E A, px E C [ V R ]is~ determined by its restriction to C via the formula

p x ( h ) = cxsx(df,. . . , d:) where sx is a Schur polynomial in n variables and cx is a positive constant. The Schur polynomial sx arises as the character of the representation g," of GL(n,C) with highest weight A, sx(z1,.

. . ,z,)

= tr(cr;(diag(zl,.

. ., z n ) ) ,

and is given explicitly by the determinantal formula

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C. Benson and G. Ratcliff

The Schur function ~ ( ~ is k )the k’th elementary symmetric function e k . Thus the fundamental invariants ~k = p ( l k ) E C[VwIKare determined, up to normalization, by ~ k ( h )= ckek(d?, . . . ,d i )

on the cross-section C. It is useful to identify partitions X E A with their Young’s diagrams. Given two diagrams A, u E A, we write v c A when v is a sub-diagram of A. That is, vj 5 X j for all j . If 1x1 = Jut k then CxV will denote the number of sequences ( E O , . . . , ~ k of ) Young’s diagrams such that

+

0 0

u = €0 C J EI ~= IvI

€1

C . . . C Ek-1 C

&k = A,

+ j for j = 1,. . . ,k.

and

That is, ~j is obtained from ~ j - 1 by adding a single box 0 to some row. Note that CxV = 0 if v A. When Y c A, CxV is the number of standard tableaux of shape X - u. That is, the number of ways to assign the values 1 , 2 , . . . , k to the boxes of the skew-diagram X - u so that values increase as we move along rows from left to right and as we move down columns.

c

Proposition 8.4.4: T h e generalized binomial coeficients can be expressed as

[C]

for

=

CXV

1x1 = )u1 + k.

Proof: Note that the polynomial y(z) = ( z I 2 = t r ( z z * ) is given on C by

y ( h ) = df Consider the case where yield

+ . . . + d i = s ( l ) ( d ? ,. . . ,d i ) .

1x1 = IuI + 1. Theorem 8.2.4 and Proposition 8.4.3

On the other hand, the classical Pieri formula asserts:

Comparing these expressions gives

On Multiplicity Free Actions

293

An application of Corollary 8.2.5 now yields the result for ( A ( = Iv(

+ k. 0

The dimensions d, = dim(P,) are d, = dim(^^,")^, in view of Theorem 4.1.1. A classical formula for these dimensions gives 2

d,=

k’i-7:’-i] .

Remark 8.4.5: This example motivates the name “Yan’s Pieri Formula” for Theorem 8.2.4. We will return to this example below in Section 9.4. As a byproduct of this subsequent analysis, one sees that the normalization constants cx appearing in Propositions 8.4.3 and 8.4.4 are given by cx = l / H ( X ) ,where H ( X ) is as in Proposition 9.4.1.

8.5. Section 8 notes The polynomials yx play a role in connection with analysis on the Heisenberg group. Let HV = V x IIB with product (Z,t)(Z’,t‘)

=

(z+z’,t+t’-

1

-Im(z,z’)).

2 Any compact Lie subgroup K of U(V) act by automorphisms on HV via k . ( z , t )= (kZ,t).

It is known that ( K

H v , K ) is a Gelfand pair if and only if K : V is a multiplicity free action. A generic set of spherical functions for such a K

Gelfand pair is completely determined by the yx-polynomials. In particular,

+ ( z ,t ) = Y x ( z ) e

-JZ12/2eit

is one such spherical function. We refer the reader to [4] concerning this connection. using EquaOne can extend the inner product (., .)F from @[V]to C[Vi] tion 20. The q x ’ s are then orthogonal with respect to (., .)F.In fact, they can be obtained via Gram-Schmidt orthogonalization using (., .)F from the p X ’ s [41. Generalized binomial coefficients were first introduced in the setting of Hermitian symmetric spaces. See [12], [37] and [13]. Z. Yan subsequently defined generalized binomial coefficients in the more general context of multiplicity free actions in his unpublished manuscript [56]. This contains the first proofs of Theorem 8.2.4 and Proposition 8.3.2. Further combinatorial

C. Benson and G. Ratclaff

294

identities concerning the generalized binomial coefficients can be found in [6]. Both [56] and [6] use representation theory for the Heisenberg group and exploit the connection with spherical functions outlined above. The treatment given here achieves some simplification. In [7]it is shown that the generalized binomial coefficients are (nonnegative) rational numbers. Thus the same holds for the eigenvalues FV(X). The proof involves extensive case-by-case analysis working from the classification for multiplicity free actions presented in Section 6 . For background on Schur functions, including the Pieri formula, we refer the reader t o Chapter I in [39]. 9. Eigenvalues for operators in 'PD(V)G

Recall that for a multiplicity free action G : V each D E P D ( V ) G acts by scalars on the irreducible subspaces {PA : X E A} in the decomposition . v E A, &(A) denotes the eigenvalue of pV(z,a) on PA. This is of C [ V ] For dvFv(X) = We will see that the map p^, : A 4 CC extends in a natural way to a polynomial function on the subspace Spanc(A) of fj*.

.I:[

9.1. Eigenvalue polynomials

We say that an operator D E P D ( V ) has order m if

D E 6(@[V] 8

C Pj(V*)). jlm

Here 6 : @[V@ V * ] --+ P D ( V ) is the map given by Wick quantization, p H p(.z,d). In terms of coordinates, D has order m when D E

Span{zadP

:

[PI ~ m } .

Proposition 9.1.1: Let

D

E

P D ( V ) and f = f:' ...f,". E C [ V ] .T h e n

Of(.) = P D (2; a1 . . . 1

1

a ~ ) f2)(

where PD E C [ V ] [ ,~. .T. ~ , f ; l ] [ u ~ , . ,a,]. .. Proof: The proof is by induction on the order of D. For operators of order 1, it is enough to consider D = d,, a directional derivative. In this case

so

=

Ciaj (a, fj)/ fj in this case.

On Multiplicity Free Actions

295

+Ci

For D of order greater than 1, write D = g Xi& where g E C [ V ] and the Xi’s have order 1. Now Ei f ( 2 ) = @i(z; a l , . . . , a r )f ( z ) for suitable pi by the induction hypothesis. As

the result follows.

0

Corollary 9.1.2: (1) As a polynomial in a = ( a l , . . , , a,.), the degree of @ o ( z a; ) i s order(D). (2) As polynomials in a , the homogeneous components of highest degree in ~ D ( z ; u and ) ~ D ( z y, ( z ) ) agree. Here CJD E C[V@ V * ]denotes the W i c k symbol of D and

df = a1-dfl -

f

fl

+ . . . +a,--.df r f T

Proof: (1) follows by induction from the proof of Proposition 9.1.1. For (2), we first examine of(.). For D = 8, one has

Given any polynomial b ( a ) in a = (a1 . . . , a r ) we will write top(b) for the homogeneous component of highest degree. If we look closely a t the XiEi, we see that induction step, with D = g

+ xi

By the induction hypothesis, top(@,) = t o p ( a E J z , df / f ) ) ,and we have just seen that t o p ( o x , ) ( z , df / f ) = a,Xz( f , ) / f,. The result now follows since toP(OX,E,) = top(OX,OE$). 0

x,

Now suppose that G : V is a multiplicity free action with, as usual, (1) A

c 6’ the set of highest weights for the representations that occur in

@.[VI, (2) A’ = { A l , . . . , A T } the set of fundamental highest weights, and (3) hj = h x , (1 5 j 5 T ) the fundamental highest weight vectors.

C. Benson and G . Ratcliff

296

Then h = hi"' . . . hfp is a highest weight vector in PA for X = alX1 arXT E A. Thus for any u E A,

p^,(X)h(z) =pV(zla)h(z)

+ . .. +

= pV(z;al,...luT)h(z)

with pV = ppy(z,q as in Proposition 9.1.1. It follows that p y ( z ;a ) = &(A) for all z . Thus pv = p V ( u )is a polynomial in the parameters a , independent of t. Since Fv(a1h

we see that Spanc(A) in

b*.

+ . . . + a,A,)

= P v ( a 1 , . . . 1 a,)

extends in a natural way to a polynomial function on Moreover, by Corollary 9.1.2 we have

Note that although both arguments on the right hand side of Equation 28 depend on z , the result is independent of z , and gives a polynomial function of the parameters al,. . . ,a,. 9.2. A Harish-Chandra homomorphism for multiplicity free

actions Let B = H N be a Bore1 subgroup in G with A+ C positive roots. Let W denote the We$ group and p=:

b* the associated set of

ca

a€A+

be half the sum of the positive roots. U ( g )is the universal enveloping algebra for g with center 2 U ( g ) . The Harish-Chandra homomorphism is an algebra isomorphism

H : 2 U ( g ) + cC[(l*]W such that

2 E 2 U ( g ) acts on VAby the scalar H ( Z ) ( X

+ p) for all highest weights A.

Let G : V be a multiplicity free action. Knop constructs a map on

PD(V)Ganalogous to the Harish-Chandra homomorphism. Let a* = Spanc(A) = Spanc(A1,. . . ,A,}

c b*,

A = a*

+p

and define a map

+

h : PD(V)G-+ @[A], h(D)(X p) = P D ( ~ ) .

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297

(As before, D acts on PA by the scalar Po(X) when X E A.) Then the following diagram commutes:

The map T is induced by the representation of G on V, and r is the restriction map.

Theorem 9.2.1: 0311) h : P’D(V)c E C[AIWo as algebras where W, C Stabw(a* + p ) is a finite reflection group. The proof of Theorem 9.2.1 is given in Lemmas 9.2.2 through 9.2.6 below:

Lemma 9.2.2: Let N = Stabw(A). Then I m ( r ) c @[AINand these two rings have the same field of fractions. Proof: From (30) it is clear that I m ( r ) c C[AIN and r factors through

Im(r): C[5*Iw -+ I m ( r ) -+ C[AIN. Letting b*//W, V and A / / N denote the varieties with coordinate rings @[$*Iw, I m ( r ) and @[AINrespectively, we have

b*//W t--‘ V

ft

A//N.

We see that r* : A / / N -+ b*//W fails to be one-to-one whenever 0 E W , cr 6 N but a ( A )n A # 8. In this case, a ( A ) # A, so a ( A )n A is a lower-dimensional subvariety in A. Thus r* fails to be one-to-one on u,,w\N(a(A) n A ) , whose complement is an open set in A . Thus the map A / / N + Y is onto and generically one-to-one, so these varieties have the same field of rational functions. 0

Lemma 9.2.3: C[AINis the integral closure of I m ( r ) in its field of fractions. Proof: Since @[A] is a polynomials ring, it is integrally closed in its quotient field @ ( A ) Thus . if f E @ ( A ) Nis integral over @[AINit is integral This shows that (C[AIN over C[A]and hence f E @ [ A ]n @ ( A ) N= @[AIN.

is integrally closed in @ ( A ) N .

298

C. Benson and G . Ratclzff

It is known that @[t)*] is integral over C[9*lW.(See Lemma 4.1.2 in [50].) Hence the homomorphic image @[A] of @[t)*] is integral over I m ( r ) and in particular @[AINis integral over I m ( r ) . Using Lemma 9.2.2, we conclude that @[AINis the integral closure of I m ( r ) in its quotient field. Lemma 9.2.4: @[AINc I m ( h ) c @[A]. Proof: The map h is injective since each D E PD(V)G is completely de: X E A}. As PD(V)Gis a polynomial termined by its eigenvalues {P,(X) ring (by Corollary 7.4.4),so is Im(h). We have I m ( r ) c I m ( h ) c @[A]. Given f E (C[AIN,we know that f is integral over I m ( r ) ,hence over I m ( h ) . Thus f E I m ( h ) . 0 Lemma 9.2.5: I m ( h ) = @[AlWofor some subgroup W, c N . Proof: Apply Galois theory to the fraction fields of the rings @[AIN c I m ( h ) c @[A]. 0 Lemma 9.2.6: W, is a finite reflection group. Proof: This follows from the fact that @[AlWo= I m ( h ) is a polynomial ring. (See Theorem 4.2.5 in [50].) 0 9.3. Characterizing the eigenvalue polynomials Recall that for v E A, PZ, can be regarded as a polynomial function on a* = Span@(R).We now shift d,F, to obtain a polynomial e, on A = a*+p: eu(X

+ P ) = d,Fu(X)

for X E a*. Note that

eu = h(d,p,(z, 8 ) )

+

where h is given by (29). Moreover, e,(X p ) = [;] when X E A, in view of Proposition 8.3.2. These polynomials have the following remarkable property.

Theorem 9.3.1: 0311) For v E A the polynomial e, is the unique polynomial such that: ( I ) e, is W,-invariant. (2) dedeu) I 14.

299

On Multiplicity Free Actions

(3) e,(X

(4)

e,(y

+ p ) = 0 for all X E A with (XI 5 IvI, X # v. + p ) = 1.

+

Proof: Property (1) holds because e , belongs to I m ( h ) = @[a* plW0. Property (2) holds by Corollary 9.1.2. Properties (3) and (4) are basic = e,(X p ) , as facts concerning the generalized binomial coefficients explained following Definition 8.2.1. It remains t o show uniqueness of e,. Let d = IvJ and set

[t]

Ad = {A E A :

+

1x1 5 d } .

The map h restricts t o yield an isomorphism

+

h : 'PD(d)(V)G-+dd)[a* p]wO

where P D ( d ) ( V ) G = P D ( d ) ( Vn)P D ( V ) Gare the G-invariants in P D ( d ) ( V ) andC(d)[a*+p]wo= (Cm5dPm(a*+p))nCC[a*+p]Wo.As { p x ( z , a ) : E Ad} is a basis for P D ( d ) ( V ) Gwe , have that {ex : X E Ad} is a basis for d d ) [ a * plWo. Now consider the linear map

+

E

+

: C(d)[a* plW0

---f

C A d , ~ ( f=) (f(X

+ p)

: X E Ad).

In particular we have

As

[i] = 1 and [t] = 0 whenever 1v1 > 1x1 or lvl = 1x1 but v # A, we see that

: v E A d } is a basis for C A d .Thus E is a vector space isomorphism. Now i f f E @[a*+ p ] satisfies properties (1)-(4) then f E C(d)[a*+plw0 and 0 ~ ( f=) E(e,). Hence f = e, as desired. {E(e,)

9.4. GL(n) @ GL(n) yet again

Recall the action of G = GL(n, C) x GL(n, C) on V = CC" @Cn Mn,n(C). Here 4 = 4, x and the Weyl group W is isomorphic to Sn x S,. The highest weights that occur in C [ V ]have the form (X;X) for X E 4: nonnegative and dominant. Thus we identify A with the set of all partitions X = (XI 2 ... 2 A, 2 0) and a* = Span@(A) = Span@{(lk;lk) : 1 5 Ic 5 n} with b* C". We can take p = ( n- 1 , n - 2 , . . ., 1 , 0 )

under this identification. W, = Stabw(a* + p ) is the diagonal subgroup in W which we identify with Sn acting as usual on Cn: n.(z1,...14

= (zD(l),.->zD(")).

C. Benson and G. Ratclzff

300

Thus @[a* + plW0 is identified with

A*(n)= { p E C[z1,.. . ,zn) : p ( z

+ p ) = p ( a . z + p ) for all

(T

E

Sn},

the algebra of shifted symmetric polynomials in n variables. Now given v E A the shifted Schur polynomial $ ( z ) is defined by det[(zi-tn - i 1 vi + n - j ) ] ’ det[(zi+ n - i 1 n - j ) ] where ( y 1 k ) is the falling factorial sZ(z1,.

. ., z n ) =

(y 1 k ) = y(y

-

1) . . . (y - k

+ 1).

It is shown in [42] that s: is a well defined polynomial function. The reader should compare Equation (31) with the determinantal formula (27) for Schur polynomials. Recall that the canonical invariants p , for this example are given, up to normalization, by the Schur polynomials s., (See Proposition 8.4.3.) The eigenvalue polynomials e,, , which interpolate the generalized binomial coefficients, are, up to normalization, the shifted Schur polynomials. Proposition 9.4.1: The eigenvalue polynomial e, for partition u is given by 1 e,(X p ) = -.$(A)

+

H(v)

where

H(v)=

+ n - i)! - vj

-2

+j)

is the product of the hook-lengths for v.

Proof: We apply the characterization Theorem 9.3.1. The definition of s: shows that sT/is shifted symmetric with degree ( u ( . Suppose that X # v is a partition with 1x1 5 lvl. We have .:(A) =0 as required by property (3) in Theorem 9.3.1. Indeed, Xe < ve for some f!. Thus for all j 5 .t 5 i one has X i 5 Xe < ve 5 vj and hence (Xi n - i 1 vj n - j) = 0. That is, the ( i , j ) ’ t h entry in the determinant from the numerator in (31) vanishes for j 5 f! 5 i. It follows that s:(A) = 0 as claimed. Finally we check that s:(u) = H ( u ) . First note that (ui + n - i 1 ui n - j ) = 0 for i > j. Hence

+

+

+

+

det[(vi n - i 1 vj

+n -j ) ]= n ( v i+n i

-

i)!

On Multiplicity Free Actions

301

The denominator in st(v) is the Vandermonde determinant in the variables v + p. This gives det[(vi

+ n - i 1 n - j)]= n ( v i - vj - i + j ) , i<j

so st(v) = H ( v ) as claimed.

0

9.5. Section 9 notes

The Harish-Chandra isomorphism first appeared in [19]. The reader can find a modern treatment in Section V.5 of [29]. One reference for facts concerning integrality and integral closures, used in the proof of Theorem 9.2.1, is the text [a] by Atiyah and Macdonald. The results in this section are due to Knop. A considerably more general version Theorem 9.2.1 was proved in [30]. This asserts that the center of the ring of invariant differential operators for any smooth aEne G-variety is a polynomial ring, canonically isomorphic t o the ring of invariants for a finite reflection group. Knop calls W, the little Weyl group. This group is given explicitly in [31] for each saturated indecomposable multiplicity free action in the classification from Section 6. In most cases, W, coincides with N = Stab,(A). Theorem 9.3.1 was conjectured by Sahi, who proved a special case in [46]. Shifted Schur polynomials are due to Okounkov and Olshanski [42,43]. The proof of vanishing is taken from [42], which includes many remarkable properties for these functions. Recent work of Knop yields the eigenvalue polynomials e, for many other multiplicity free actions. For the actions GL(n,@): S2(@"),GL(n,@): R2((Cn),SO(n,@)x CX : @" and E6 x Cx : @27 this results in shafted Jack polynomials with various parameters. See [32,33]. A multiplicity free action G : V is said to be a Capelli action when the map 7r : 2 U ( g ) --f PD(V)Gin (30) is surjective. In [24] it is shown that the irreducible multiplicity free actions G' x ex : V in Table 3 are all Capelli actions except for

G' = Sp(2n) 8 SL(3),Spin(9),Eg References 1. D. Arnal, 0. Boukary Baoua, C. Benson, and G. Ratcliff, Invariant theory f o r the orthogonal group via star products, J. Lie Theory 11 (2001), 441-458. 2. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, Mass., 1969.

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3. C. Benson, J. Jenkins, R. Lipsman, and G. Ratcliff, A geometric criterion for Gelfand pairs associated with the Heisenberg group, Pacific J. Math. 178 (1997), 1-36. 4. c. Benson, J. Jenkins, and G. Ratcliff, Bounded K-spherical functions o n Heisenberg groups, J. F'unct. Anal. 105 (1992), 409-443. 5. C. Benson and G. Ratcliff, A classification for multiplicity free actions, J. Algebra 181 (1996), 152-186. 6. -, Combinatorics and spherical functions o n the Heisenberg group, Representation Theory 2 (1998), 79-105. 7. __ , Rationality of the generalized binomial coefficients f o r a multiplicity free action, J. Austrl. Math. SOC.(Series A) 68 (2000), 387-410. 8. M. Brion, Classification des espaces homogbnes sphe'riques, Compos. Math. 63 (1987), 189-208. Spherical varieties: a n introduction, Prog. Math., vol. 80, pp. 11-26, 9. -, Birkhauser, Basel, 1989. 10. M. Brion, D. Luna, and T. Vust, Espaces homogbnes shpe'riques, Invent. Math. 84 (1986), 565-619. 11. C. Chevalley and R. Shafer, The ezceptional Lie algebras F4 and EG,Proc. Nat. Acad. Sci. Amer. 36 (1950), 137-141. 12. H. Dib, Fonctions de Bessel sur une algbbre de Jordan, J. Math. Pures Appl. 69 (1990), 403-448. 13. J. Faraut and A. Koranyi, Analysis o n symmetric cones, Oxford University Press, New York, 1994. 14. G. Folland, Harmonic analysis in phase space, Princeton University Press, New Jersey, 1989. 15. -, A course in abstract harmonic analysis, CRC Press, Boca Raton, 1995. 16. R. Gangolli and V.S. Varadarajan, Harmonic analysis of spherical functions o n real reductive groups, Springer-Verlag, New York, 1988. 17. R. Goodman and N. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, New York, 1998. 18. V. Guillemin and S. Sternberg, Multiplicity free spaces, J. Differential Geom. 19 (1984), 31-56. 19. Harish-Chandra, O n some applications of the enveloping algebra of a semisimple Lie algebra, Trans. Amer. Math. SOC.70 (1951), 185-243. 20. S. Helgason, Diflerential geometry, Lie groups, and symmetric spaces, Academic Press, New York, 1978. 21. ___ , Groups and geometric analysis, Academic Press, New York, 1984. 22. R. Howe, Remarks o n classical invariant theory, Trans. Amer. Math. SOC. 313 (1989), 539-570. 23. -, Perspectives o n invariant theory: Schur duality, multiplicity-free actions and beyond, Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995. 24. R. Howe and T. Umeda, T h e Capelli identity, the double commutant theorem and multiplicity-free actions, Math. Ann. 290 (1991), 565-619.

O n Multiplicity Free Actions

303

25. K. Johnson, O n a ring of invariant polynomials o n a hermitian symmetric space, J. Algebra 62 (1980), 72-81. 26. V. Kac, S o m e remarks o n nilpotent orbits, J. Algebra 64 (1980), 190-213. 27. V. Kac, V. L. Popov, and E. B. Vinberg, S u r les groupes lin aires alg briques dont l'alg bre des invariants est libre, C. R. Acad. Sci. Paris S r. A-B 283 (1976), A875-A878. 28. T. Kimura, Introduction t o prehomogeneous vector spaces, Transl. Math. Mono., vol. 215, Amer. Math. SOC.,Providence, Rhode Island, 2003. 29. A. Knapp, Lie groups beyond a n introduction, Progress in Math., vol. 140, Birkhauser , Boston, 1996. 30. F. Knop, A Harish-Chandra homomorphism for reductive group actions, Annals of Math. 140 (1994), 253-289. 31. ___, S o m e remarks o n multiplicity free spaces, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 514, pp. 301-317, Kluwer Acad. Publ., Dordrecht, 1998. 32. ___ , Construction of commuting difference operators for multiplicity free spaces, Sel. Math., New ser. 6 (2000), 443-470. 33. -, Semisymmetric polynomials and the invariant theory of matrix vector pairs, Representation Theory 5 (2001), 224-266. 34. F. Knop and S. Sahi, Difference equations and symmetric polynomials defined by their zeroes, International Math. Research Notes 10 (1996), 473-486. 35. B. Kostant and S. Sahi, The Capelli identity, tube domains and the generalized Laplace transform, Advances in Math. 87 (1991), 71-92. 36. M. Kramer, Spharische untergruppen in kompakten zusammenhangenden Liegruppen, Compos. Math. 38 (1979), 129-153. 37. M. Lassalle, Une formule de binhme ge'ne'ralise'e pour les polynhmes de Jack, C . R. Acad. Sci. Paris, SQrieI 310 (1990), 253-256. 38. A. Leahy, A classification of multiplicity free representations, J. Lie Theory 8 (1998), 367-391. 39. I. G. Macdonald, Symmetric functions and Hall polynomials, second edition, Clarendon Press, Oxford, 1995. 40. I. V. Mikityuk, O n the integrability of invariant Hamiltonian systems with homogeneous configuration spaces, Math USSR-Sb. 57 (1987), 527-546. 41. A. Okounkov and G. Olshanski, Shifted Jack polynomials, binomial formula, and applications, Math. Res. Letters 4 (1997), 69-78. 42. -, Shifted Schur functions, St. Petersburg Math. 9 (1998), 239-300. Shifted Schur functions II. The binomial formula for characters of 43. -, classical groups and its applications, Amer. Math. SOC.Transl. Ser 2 181 (1998), 245-271. 44. V. L. Popov, Stability of the action of a n algebraic group o n a n algebraic variety, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 367-379. 45. B. Brsted and G. Zhang, Weyl quantization and tensor products of Fock and Bergman spaces, Indiana Math. Journal 43 (1994), 551-583. 46. S. Sahi, T h e spectrum of certain invariant differential operators associated t o Hermitian symmetric spaces, Lie Theory and Geometry (J. L. Brylinski, ed.), Progress in Math., vol. 123, Birkhauser, Boston, 1994, pp. 569-576.

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47. M. Sat0 and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1-155. 48. G. Schwarz, Representations of simple Lie groups with regular rings of invariants, Invent. Math. 49 (1978), 167-191. 49. F. Servedio, Prehomogeneous vector spaces and varieties, Trans. Amer. Math. SOC.176 (1973), 421-444. 50. T.A. Springer, Invariant theory, Lecture Notes in Math., vol. 585, Springer Verlag, New York, 1977. 51. R. Stanley, Some combinatorial properties of Jack symmetric functions, Advances in Math. 77 (1989), 76-115. 52. E. B. Vinberg, Complexity of actions of reductive Lie groups, h n c t . Anal. and Appl. 20 (1986), 1-11. 53. ___ , Commutative homogeneous spaces and co-isotropic symplectic actions, Russian Math. Surveys 56 (2001), 1-60. 54. E. B. Vinberg and V. L. Popov, O n a class of quasihomogeneous a f i n e varieties, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 749-764. 55. H. Weyl, T h e classical groups, their invariants and representations, Princeton University Press, Princeton, N.J., 1946. 56. Z. Yan, Special functions associated with multiplicity-free representations, unpublished preprint.

Multiplicity-Free Spaces and Schur-Weyl-Howe Duality

Roe Goodman Rutgers University Department of Mathematics 110 Relinghuysen Rd Piscataway N J 08854.8019. USA E-mail: [email protected]

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Representations and duality . . . . . . . . . . . . . . . . . . . . . . . 1.1. Representations of algebraic groups . . . . . . . . . . . . . . . 1.2. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Reductive groups and isotypic decompositions . . . . . . . . . 1.4. Multiplicities and duality . . . . . . . . . . . . . . . . . . . . . 2 . Proof of duality theorem and examples . . . . . . . . . . . . . . . . . 2.1. Densitylemmas . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Proof of duality theorem . . . . . . . . . . . . . . . . . . . . . 2.3. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Schur-Weyl duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Commutant of GL(n) action on tensors . . . . . . . . . . . . . 3.2. Highest weight theory . . . . . . . . . . . . . . . . . . . . . . . 3.3. Duality and N-fixed vectors . . . . . . . . . . . . . . . . . . . 4 . Commutant character formulas . . . . . . . . . . . . . . . . . . . . . 4.1. Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Frobenius and determinant character formulas . . . . . . . . . 4.3. Proof of Frobenius character formula . . . . . . . . . . . . . . 4.4. Proof of determinant character formula . . . . . . . . . . . . . 5 . Character formulas for Schur-Weyl duality . . . . . . . . . . . . . . . 5.1. Frobenius formula for 6 k characters . . . . . . . . . . . . . . . 5.2. Determinant formula for 6 k characters . . . . . . . . . . . . . 5.3. Schur-Weyl duality and GL(k)-GL(n) duality . . . . . . . . . 305

307 307 307 308 309 311 313 313 315 316 320 320 321 326 329 329 329 330 331 332 332 334 336

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R . Goodman

6 . Polynomial invariants and FFT . . . . . . . . . . . . . . . . . . . . . 6.1. Invariant polynomials . . . . . . . . . . . . . . . . . . . . . . . 6.2. Invariants of vectors and covectors . . . . . . . . . . . . . . . . 6.3. Polynomial FFT for GL(n) . . . . . . . . . . . . . . . . . . . . 6.4. Polynomial FFT for the orthogonal group . . . . . . . . . . . 6.5. Polynomial FFT for the symplectic group . . . . . . . . . . . . 7. Tensor invariants and proof of FFT . . . . . . . . . . . . . . . . . . . 7.1. Tensor invariants for GL(V) . . . . . . . . . . . . . . . . . . . 7.2. Proof of polynomial FFT for GL(V) . . . . . . . . . . . . . . . 7.3. Tensor invariants for orthogonal and symplectic groups . . . . 7.4. Proof of polynomial FFT for orthogonal and symplectic groups . . . . . . . . . . . . . . . . . . . . . . . . . 8. Weyl algebra and Howe duality . . . . . . . . . . . . . . . . . . . . . 8.1. Duality in the Weyl algebra . . . . . . . . . . . . . . . . . . . 8.2. Howe duality for orthogonal/symplectic groups . . . . . . . . . 8.3. Howe duality for GL(k) . . . . . . . . . . . . . . . . . . . . . . 9. Harmonic duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Harmonic polynomials . . . . . . . . . . . . . . . . . . . . . . 9.2. Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Decomposition of harmonic polynomials . . . . . . . . . . . . . . . . 10.1. O ( k ) Harmonics ( k odd) . . . . . . . . . . . . . . . . . . . . . 10.2. O ( k ) Harmonics ( k even) . . . . . . . . . . . . . . . . . . . . . 10.3. Examples of harmonic decompositions . . . . . . . . . . . . . . 11. Symplectic group and oscillator representation . . . . . . . . . . . . . 11.1. Real symplectic group . . . . . . . . . . . . . . . . . . . . . . . 11.2. Holomorphic (coherent-state) model for oscillator represent at ion . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3. Bargmann-Segal transform . . . . . . . . . . . . . . . . . . . . 11.4. Real (oscillatory-wave) model for oscillator representation . . . 11.5. Analytic vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Dual pair S p ( n , R)-O(k) . . . . . . . . . . . . . . . . . . . . . . . . . 12.1. Decomposition of H 2 ( M n x k ) under Mp(n, W) x O ( k ) . . . . . 12.2. Square-integrable representations of Sp(n,R) . . . . . . . . . . 13. Brauer algebra and tensor harmonics . . . . . . . . . . . . . . . . . . 13.1. Centralizer algebra and Brauer diagrams . . . . . . . . . . . . 13.2. Generators for the centralizer algebra . . . . . . . . . . . . . . 13.3. Relations in the centralizer algebra . . . . . . . . . . . . . . . 13.4. Harmonic tensors . . . . . . . . . . . . . . . . . . . . . . . . . 13.5. Decomposition of harmonic tensors for Sp(V) . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

338 338 339 341 341 343 344 344 345 347 350 351 351 355 357 358 358 360 364 365 371 374 375 376 383 387 389 391 394 394 399 402 402 405 407 409 410 414

Multiplicity-Bee Spaces and Schur- Weyl-Howe Duality

307

Dedicated to the memory of Irving E. Segal, who introduced m e to the beauties and mysteries of representation theory. Introduction The unifying theme of these lectures is the duality between the irreducible representations occuring in a linear group action and irreducible representations of the commuting algebra relative to this action. This notion of duality in representation theory was introduced by Schur a century ago, and it has developed into an important tool with many applications. In keeping with the tutorial aspect, I have tried to tell the story starting from the beginning and including complete proofs of all the major results (at several points I refer to the lectures of Benson-Ratcliff in the present volume for details). Of course, this limits the scope of the lectures to the more classical parts of the theory: Schur-Weyl-Brauer duality for finitedimensional representations, and Howe duality between finite-dimensional and infinite-dimensional highest-weight representations. Substantial parts of these lectures are based on joint work with Nolan Wallach and I would like to acknowledge his contributions to my understanding of representation theory. I would also like to thank Eng-Chye Tan and Chen-Bo Zhu for inviting me to give these lectures and for their wonderful hospitality.

1. Representations and duality 1.1. Representations of algebraic groups

Assume that G c GL(n, C ) is an algebraic group (defined by a set of polynomial equations in the matrix entry functions). We denote by Aff (G) the commutative algebra of regular functions on G (the restrictions to G of polynomials in the matrix entry functions x i j and det-l). Let ( p , L ) be a representation of G on a complex vector space L. If L is finite-dimensional, then we say that p is regular (rational) if the representative functions g H tr(p(g)E), for E E End(L), are regular. Every regular function on G arises as such a representative function. When L is infinite dimensional, we say that p is locally regular if for all x E L there is finite-dimensional G-invariant subspace M containing x so that ( p (M , M ) is a regular representation. The most fundamental tool in representation theory is Schur’s Lemma: I f E and F are irreducible, finite-dimensional representations of

R. Goodman

308

a group

G, then dim HomG (E,F ) =

c

1 i f E r F 0 ifEFF

(HomG(E,F) denotes the space of linear transformations T : E F that intertwine the G actions on the two spaces). To prove Schur's Lemma, observe that the null space and range of T are G-invariant subspaces, so T must be either zero or bijective, with the first case holding if E F. When E F and S,T are two nonzero intertwining maps, take X t o be an eigenvalue of S P I T .Since S-IT - X I commutes with the action of G on E and has a nonzero null space, it must be zero. --f

1.2, Examples (1) Let ( T , V ) be any regular (finite-dimensional) representation of G. We denote by P ( V )the algebra of complex-valued polynomial functions on V . Define a representation of G on P ( V ) by p(g)f(v) = f(.rr(g)-'v)

for

f E P ( V )and g

E

G.

Since the G action is linear, it commutes with the (Cx action on V by scalar multiplication, and we have the direct-sum decomposition into finitedimensional G-invariant subspaces

P ( V )= @ P W )

7

k>O

where P k ( V )is the space of homogeneous polynomials of degree k. The action of G on each of these spaces is regular, so the representation p is locally regular. Furthermore, the G action preserves the multiplication on

P(V). (2) With (n,V ) as above, we can take the full tensor algebra

I ( V )= @ v @ k k>O

with G action p ( g ) ( v l 8 . . . 8 vk) = T(g)vl 8 .. . @ T(g)vk. Since G leaves invariant each subspace V@'", the representation p is locally regular. As in the previous example, the G action preserves the (noncommutative) multiplication on I ( V ) .

(3) Let X c Cm be an affine algebraic set (the zero set of a family of polynomials) and suppose that there is a regular G action on X

GxX

--t

X,

( g , X )H g . z .

Multiplicity-Free Spaces and Schvr- Weyl-Howe Duality

309

Set L = Aff (X) (the restriction to X of the polynomial functions on C"). Let G act on L by p(g)f(z) = f(g-l . x). We can prove that this representation is locally regular as follows. Given f E Aff(X),set Vf = Span{p(g)f : g E G}. The function (9, z) H f(g-' . x) on G x X is regular, and Aff(G x X) = Aff(G) 8 Aff(X), there are regular functions qhi on G and Qi on X so that

k=l In particular, V f C Span{$k} is finite-dimensional, so we can choose 91,. . . , gr in G such that the functions fi = p(gi)f give a basis for V f . Now choose points x1 , . . . , xk in X so that the evaluation functionals d,, are a basis for V f *Since . r

M g ) f i , ~ z j )= P(ggi)f(xj) = C 4 k ( g g i M ( x j ) , k= 1

we see that the representation of G on V f is regular. Thus ( p , L ) is locally regular.

1.3. R e d u c t i v e groups a n d isotypic decompositions

A complex algebraic group G is called reductive if every finite dimensional regular representation decomposes as a direct sum of irreducible representations (this property is equivalent to every G-invariant subspace of a regular representation having a G-invariant complementary subspace). The classical groups are reductive: 0 0

0

0

the general linear group GL(n, C ) of invertible n x n complex matrices the special linear group SL(n,C ) of n x n complex matrices of determinant one the orthogonal group O ( C n , u ) of n x n matrices preserving a nondegenerate symmetric bilinear form u ( z , y ) = xtBy on C", where B is a symmetric invertible n x n matrix (defining equation gtBg = B ) the special orthogonal group SO(@", w)of orthogonal matrices of determinant one the symplectic group Sp(Canlw)of 2n x 2n matrices preserving a nondegenerate skew-symmetric bilinear form w ( x ,y) = zt J y on Can, where J is a skew-symmetric invertible 2n x 2n matrix (defining equation gtJg = J )

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Finite groups are shown to be reductive by the method of averaging over the group. The proof that classical groups are reductive can be carried out analytically by integrating over a compact real form (Weyl’s unitary trick see [16, Theorem 2.4.7]), or algebraically by using a Casimir operator (see [16, Theorem 2.4.51). Direct products of reductive groups are reductive, and the quotient of a reductive group by a closed normal subgroup is reductive (this is obvious). An algebraic group is reductive if and only if its identity component is reductive. Assume G is reductive, and let G be the equivalence classes of irreducible finite-dimensional regular representations of G. For each A E 6 fix F A )in the class A. Let A* be the equivalence class of a representation (rA, the contragredient representation on the dual space ( F A ) * . Given a locally regular representation ( p , L ) of G, set h

L ( A )= X V

(sum of all V

c L such that plv

S

FA).

Call L(x) the A-isotypic component of L. Define Spec(p) = {A E L(x) # 0) (the G-spectrum of ( p , L ) ) .

Proposition 1.1: L = $AESpec(L) L(A)

:

(algebraic direct s u m ) .

Proof: We first verify that the sum is direct. Suppose, for the sake of contradiction, that L ( A )n L ( p )# 0 for some A , p E 6 with A # p. Then there exists a G-invariant subspace W # 0 so that W c L ( x )n L ( p )and dim W < 00. Since G is reductive, W = Vl @ . . . @ V,, where each V, is an irreducible G-module. Hence V, ? F A and also 5 2 F P , a contradiction. To see that L is the sum of its isotypic components, set LO= L ( x ) .If LO# L , then there exists a nonzero x E L\Lo. But x is contained in a finite dimensional G invariant subspace W that is the direct sum of irreducible G-invariant subspaces. Hence W c Lo, a contradiction. 0

Corollary 1.2: There is a linear projection x H xb f r o m L onto the space LG of G-fixed vectors. We now turn to the G-module structure of the isotypic components of a representation L. Denote by Homc(FX,L ) the vector space of all linear maps T : FA---t L that intertwine the G actions on these spaces. This is the space of covariants of type A.

Theorem 1.3: If ( p , L ) i s a locally regular representation of a complex reductive algebraic group G , then

~r

@I E

X S p 4 p )

~

F

~

,

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where E X = HomG(FX,L ) and G acts by 18 p on each summand. In particular, the multiplicity of X in p is the dimension of the space of covariants of type A. Proof: Let T E HomG(FX,L ) be a nonzero intertwining operator. Then T is injective, by Schur’s lemma. Conversely, if W c L ( x )is a G invariant irreducible subspace, then there is an intertwining map T so that W = T(FX).This implies that the map T 8 v H Tv from E X 8 F A to L ( x )is surjective. It remains to prove that the map E X@ F A4 L ( x )is injective. Suppose v3 E FAand Tj E E X satisfy C j T j v j = 0. We may assume that {vj} is linearly independent. Fix a decomposition

L ( x )= @ Fi ,

Fi

S

FA

a

This defines G-invariant projections

Pi : L ( x )+ FA,and by assumption

By Schur’s lemma, PiTj = cij I for some cij E CC, so we conclude that cij = 0 for all i, j , by the linear independence of {vj}. Hence Tj = 0 for all j . 0

1.4. Multiplicities and duality One says that L is multiplicity-free as a G module if dim E X = 1 for all X E Spec(p). In this case L is uniquely determined as a G-module by its spectrum. For a detailed analysis of such representations when L = P ( X ) and X is a vector space or afine variety with regular G action see the lectures by Benson-Ratcliff in this volume. In these lectures we will study representations (p, L ) that are not multiplicity free. We want to determine 0 0

c,

The spectrum Spec(p) c The multiplicities rnx = dim E X , Explicit models for the multiplicity spaces E X

Let EndG(L) be the algebra of linear transformations on L that commute with the G action. There is a natural representation of this algebra on each multiplicity space EX.Indeed, if A E EndG(L) and T E E X ,then the linear map A o T : FA-+ L also commutes with the G action on L , and

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hence is an element of E X .Following the ideas of I. Schur, H. Weyl and R. Howe, the unifying theme in our approach will be

Hidden Symmetry: Study the spaces E X as modules for good subalgebras of EndG (L). The term hidden symmetry comes from applications of representation theory to quantum mechanics in cases where the geometric symmetries such as rotation invariance do not suffice to explain the multiplicities in the energy spectrum. In some cases, one can find a larger symmetry group containing G and extend the representation of G to a representation of this larger group on L that is multiplicity free. In other cases the hidden symmetries are given by a Lie algebra of differential operators commuting with the G action (see [30]). When L is infinite-dimension (for example, when L = Aff (X) with X an affine G variety), then End(L) is too big to deal with purely algebraically. In the context of unitary representations on a Hilbert space, one uses the von Neumann algebra of bounded operators that commute with G. In our algebraic setting we shall assume that L is of countable dimension and that we have a subalgebra R c End(L) that satisfies (i) R acts irreducibly on L (ii) R is invariant under G, relative to the action Ad(g)T = p(g)Tp(g)-', and the representation Ad of G on R is locally regular

In case dim L < 00 we take R = End(L) L 8 L* and these conditions are always satisfied. When L = P ( X ) with X a smooth afine G variety, we take R = D ( X ) , the algebraic differential operators on X (see Agricola [l]). In particular, if X is a vector space with linear G action, then D(X) is the Weyl algebra P D ( X ) of differential operators with polynomial coefficients, which we will examine in detail in Section 8. Fix R satisfying the conditions (i) and (ii) and let

RG = {T E R : Ad(g)T = T for all g E G} (the commutant of p(G) in R).

Theorem 1.4: Each multiplicity space E X is an irreducible RG module. Furthermore, if A , p E Spec(p) and E X 2 EP as an RG module, then

x = p.

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In the next section we will prove this theorem.a At this point we derive some consequences. The following corollary plays a fundamental role in our approach to Howe duality.

Corollary 1.5: Let u be the representation of RG on L. Then ( a l L ) is a semisimple RG module, and each irreducible submodule E X occurs with finite multiplicity dim FA. When L is finite-dimensional then R = End(L), and from the inequivalence of the representations E X together with Schur's lemma we obtain the classical Double Commutant Theorem.

Corollary 1.6: If dim L < co and B = EndG(L), then Span{p(G)} consists of all linear transformations on L that commute with B. Corollary 1.7 (Duality Correspondence): Let Spec(a) denote the set of equivalence classes of the irreducible representations of the algebra RG that occur in L. Then the map F A -+ E X sets up a bijection between Spec(p) and Spec(u).

2. Proof of duality theorem and examples 2.1. Density lemmas Lemma 2.1 (Dixmier-Schur): Let L be a vector space over C of countable dimension. Let R c End(L) be a subalgebra that acts irreducibly on L . Suppose A E End(L) commutes with R. Then A = XI for some X E C. Proof: Suppose that A is not a multiple of the identity. Since R acts irreducibly] Schurls lemma implies that A - X I is invertible for all X E C. Hence for every nonzero polynomial p ( x ) in one variable the operator p ( A ) is invertible (factor p(x) into linear factors). Thus there is an algebra homomorphism from the field C ( x )of rational functions in one variable into End(L) given by p ( x ) / q ( z )H p(A)q(A)-'.Fix a nonzero vector v E L. Then the linear map r(x) ++ r(A)v is injective from C(z) to L. But C ( x ) has uncountable dimension as a vector space over C , since the functions {(x - A)-' : X E C} are linearly independent, a contradiction. 0 Assume now that L has countable dimension as a complex vector space and that R c End(L) is a subalgebra that acts irreducibly on L. "See [16,Theorem 4.5.121 for the case that presented here is due to Agricola [l].

R is a graded algebra; the generalization

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Lemma 2.2 (Jacobson): Let X be any finite-dimensional subspace of L. Then every f E Hom(X, L) is of the fonn rIx for some r E R.

Proof: Let (51,. . . , z},

be a basis for X . Define n copies

n copies

L e t R a c t o n L ( " ) b y r . [ y l , . . . , y , ] = [ ryl, ..., r y , ] € o r r ~ R a n d y i E L , and extend f to a linear map f(") : X ( n ) -+ L(") by

f ( n ) [ ~..l ,

YnI =

. )

[ ~ ( Y I.).,. )f ( ~ n ).I

Denote by M = R . z(n)the cyclic R submodule generated by d"). Define Li c L(n) to be the vectors that have arbitrary entries from L in the ith place and are zero in the other positions. Pick a maximal subset I c { 1,.. . ,n} with the property that the sum N=M+CLi iEI

is direct. Then N is an R submodule of L("). Since R acts irreducibly on L, the R modules N n L j are either zero or L j for each j . But if N n L j = 0, then the sum N Lj would be direct, contradicting the choice of I . Hence N= proving that M has an R-invariant complement. Thus there is a projection P : L(n) + M that commutes with the action of R.We can write

+

1

n

L j=1 wherepij E End(L). Since P commutes with R on L(n),the transformations pij all commute with R on L . Hence pij E CI by Lemma 2.1 and [pij] is a matrix of scalars. Now calculate

[

n

1

f ( " ) P [ Y l , . . . , ~ n l= C P l , f ( y j ) , . . . l ~ P n j f ( Y j )= ~ f ( n ) [ Y l l . - , Y n l . j=1

j=1

Hence f(") commutes with P . Since dn)E A4 we have f(n),(n)

= f( n ) p , ( n ) = p f ( n ) & )

E

M.

Thus there exists T E R so that f ( n ) z ( n = ) r d n ) . Since basis for X , this implies that f = T I X .

,.. .

(21

z,}

is a 0

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

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Corollary 2.3 (Burnside): If dim L < KI then R = End(L). Now let ( p , L ) be a locally regular representation of G with dim L countable. Assume that R c End(L) satisfies conditions (i) and (ii) stated before Theorem 1.4. Lemma 2.4: Let X c L be a finite-dimensional G invariant subspace. Then RGlx = HOmG(X,L).

Proof: Let T E HomG(X, L). Then by Lemma 2 . 2 there exists r E R such that T I X = T . Since G is reductive, condition (ii) implies that there is a projection r H rh from R -+ RG.But the map R -+ Hom(X,L) given by y H ylx intertwines the G actions, since X is G-invariant. Hence T = T b= ~hlx. 2.2. Proof of duality theorem

Take X E Spec(p) and let Zx c L(x) be any irreducible G-submodule. Given f E L, we denote by U f = RGf the cyclic RG module generated by f . We write @[GIfor the group algebra of G (the formal finite linear combinations of the elements of G). (a) If 0 # M

c L(x) is an RG-module,then M n Zx # 0.

To verify this, take 0 # m E M and set X = Span{p(G)m}. Then d i m X < 00 and X c L(x). Hence there exists T E HomG(X, Zx) with T m # 0. By Lemma 2.4 there exists r E RG with T [ X = T . Then r m = T m E M n Zx.

(b) If 0 # f E ZA then U f rl Zx

= Cf.

Take u = r f E U f n Zx. Since Zx = Span{p(G)f}, we have

rZx = Span{p(G)rf} = Span{p(G)u} c Zx Thus rlzX E EndG(Zx) = C I by Schur’s Lemma. So u = r . f E C f , proving (b). (c) I f f E Zx is nonzero, then U f is an irreducible RG-module. Indeed, if 0 # M C U f is an RG-submodule, then 0 # M n Zx and (b). Thus f E M and hence M = U f , which proves (c).

c Cf by (a)

(d) Let f i , . . . , f d be a basis of Zx. Set Mi = U f i . Then the sum is direct and Mi S Mj as RG modules.

cld_lMi

We have Span{p(g)lz, : g E G} = End(Zx) by Corollary 2.3. Thus for each i there exists an element ui E @[GIsuch that p ( u i ) f j = 6ij fj. Suppose

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mi

E Mi and

xi

m i = 0.

There exist

ri

E

i

i

RG SO that

m i = Ti f i .

Hence

i

for j = 1,.. , , d . This proves the first statement of (d). For the second, apply Corollary 2.3 again to obtain uji E @[GI such that p ( u j i )fi = fj. Since Mi and Mj are irreducible by (c), the map p ( u j i ) : Mi 4 Mj is an RG-module isomorphism, by Schur’s Lemma. (e) Let

Mi

d

be as in (d). Then L(x) = @i=l M i .

Recall that L(x)is the sum of all irreducible G-submodules of L that are in the class A. Thus it is enough to show that if Wx is such a submodule then d

i=l

Take a G isomorphism T : ZA -+ Wx. Then Lemma 2.4 furnishes r E RG such that r = T on Zx.Hence Wx satisfies (l),which proves (e). The first assertion of Theorem 1.4 now follows from (c), (d), and (e). To prove the second assertion, it suffices to prove the following.

(f) Let f A and f, be nonzero vectors in irreducible G subspaces Zx and 2,. Suppose U f , 2 U f , as RG-modules. Then A = p . Let T : UfA -+ U f , be an RG-module isomorphism. Let X be a finitedimensional G-invariant subspace containing f x and Tfx. There is a projection operator PA : X -+ L(x) onto the A-isotypic component of L , and Lemma 2.4 furnishes r E RG such that T I X = PA. Thus r . fx = fx so we have

T fx = T r fx = rT fx = PATfx E L(x). Since T is an RG module isomorphism, it follows that U f , f, E L ( x ) ,and so we conclude that p = A.

c L(x).Hence O

2.3. Examples (1) (Product groups) Let H and K be reductive complex algebraic groups, and let G = H x K be the direct product algebraic group, where Aff (G) Aff (H)@Aff(K) under the natural pointwise multiplication map. We can use the duality theorem to prove that = x k : Every irreducible regular representation ( L , p ) of G is given by

L

=M @N

,

p(h, Ic) = a ( h )8 ~ ( k )for h E H and Ic E K

(2)

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317

where (a,M ) is an irreducible representation of H and (7,N ) is an irreducible representation K. To prove this, suppose first that (p,L) is defined by (2). Then Corollary 2.3 implies that End(L) is spanned by the transformations { p ( h , Ic) : h E H , k E K } and hence E n d c ( L ) = @ I ,showing that L is irreducible. Conversely, given an irreducible regular representation ( p , L ) of G, use Theorem 1.4 (with R = End(L)) to decompose L as a K-module:

@ EX@PFX. (3) X€i2 Set a ( h ) = p ( h , l ) and ~ ( k =) p(1,Ic). Since a ( h ) commutes with ~ ( k )H, acts on each E X by some representation p X .We claim that EXis irreducible under H . To prove this, note that L=

@ End(EX)@ I . (4) X€R Given T E EndK(L), we know by Corollary 2.3 that T is a linear combination of the transformations a(h)T(k).Under the isomorphism (4) the K-invariant transformations only act on EX.This proves that EndK(L) is spanned by { a ( h ) : h E H } , and hence EXis irreducible under H by Theorem 1.4. Thus each summand in (3) is an irreducible G module, by the earlier argument, so there can be only one summand. EndK(L)

(2) (Multiplicity-free representations of product groups) Suppose ( p , L ) is any locally regular representation of G that is multiplicity-free. By Example (1) the isotypic decomposition of L under H x K is of the form

L=

@ E"@F@

(5)

P)EA

(0%

where A c E x 2 and E" is the irreducible H-module of type a , while F P is the irreducible K-module of type p. Set a = p l and ~ 7 = p l ~ . Then Spec(a) is the projection A + G, whereas Spec(.r) is the projection A k .In general A is not determined by these projections. If both of these projections are injective, we say that the representation p sets up a duality correspondence between Spec(a) and Spec(-r). Clearly such representations of G must be very special, and in these lectures they will play an important role. The next example is the most familiar of them. ---f

( 3 ) (Two-sided group action) Let K be any reductive complex algebraic group. Set G = K x K and L = Aff(K). Define the representation p of G on L by P ( Z , y)f(k) = f(.-lW

for k , 2 , y E K .

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= From Example (1) we know that x I?. Consider Aff(K) as a K-module relative to the right translation action p(1,Ic) and apply Theorem 1.3:

Aff(K) =

@ E X@ F A X€il

Here K acts on E X = HomK(FX,Aff(K)) by p ( k , 1) o T,where T : F A -+ Aff (K) intertwines the action of K on F A with the right translation action of K on Aff(K). We claim that E X 2 F A * . To prove this, define a map E X 4 FA* (a special case of Frobenius reciprocity) by

T

+-+

T^ E F ~ ,'

@,u) = ( ~ v ) ( i ) for

EFA.

This map obviously intertwines the action of K. It is injective, since (Tv)(l) = 0 for all IJ E F A implies

(Tv)(lc)= (T7rX(Ic)v)(l)= 0 for all Ic E K , and hence T = 0. The map is surjective, since w* E FA* defines T E E X by

(Tv)(Ic)= ( V * , 7 r X ( I c ) W ) . A

Clearly T = v*.Thus the decomposition (6), relative to the action of K x K , is Aff(K) 2

@ FA*8 F A Z @ End(FX). A&

XE

-

ii-

This shows that Aff(K) is multiplicity free as a representation of K x K and there is a duality correspondence X X*. (4) (Harmonics on the zero-sphere) Let G = 0(1) = {fl} acting on C , and take L = P(C). In this case

E = {F+, F - }

(trivial, signum).

The G-isotypic decomposition of L is thus

L = L+ @I L-

(even polynomials

@

odd polynomials)

and each component has infinite multiplicity. We apply the duality philosophy to explain the multiplicities by finding operators on L that commute with G. Let P D ( C ) be the polynomial coefficient differential operators on P(C). Then one has

Multiplicity-Free Spaces and Schur- Weyl-Houe Duality

319

+

(a) The operators A = ( d / d ~ ) multiplication ~, by x 2 , and x ( d / d x ) 1 / 2 (shifled Euler operator) commute with G and span a Lie algebra 0‘ E d(2,c ) in P D ( c ) . (b) The Lie algebra 0’ generates the commutant PP(@)Gof G. The proof of (a) is an easy calculation. The proof of (b) follows by considering the symbol f (x,<) of a differential operator and using the fact that the algebra of G-invariant polynomials in (2, E ) is generated by the quadratic polynomials x2, x€,, and C2. We define the G-harmonic polynomials ‘FI = Ker(A) = (C 1)@ (C. x) . Since A commutes with G, we have G . ‘FI = ‘FI. Also ‘FI is multiplicity-free as a G-module. Let

z= P(C)G = C[x2] (the G-invariant polynomials). Then we have the Invariant-Harmonic Decomposition:

P ( C ) = E+ CB E- 2 Z@’FI where E+ = @[x2].I,E - = C [ x 2 .x. ] We view this decomposition from the perspective of duality as follows:

E f is an irreducible g’ module generated by 1. E - is an irreducible 0’ module generated by x. 0

P(C)is multiplicity-free as module for 0’ x

G:

P ( C ) = (E+@ F + ) @ ( E -

@F-)

From the algebraic point of view, we now have a complete picture of

P ( C ) as a module for G and 8’. However, there is much more that can be seen on the analytical side. There is a pre-Hilbert space structure on P ( C ) given by the Fischer inner product:

(where dp(x) is normalized Gaussian measure on C ) . We define the Bargmann-Fock space H2 as the completion of P ( C ) in this norm. The elements of H2 are holomorphic functions on C that are square-integrable relative to Gaussian measure. Let gb =

{X E g’

:

X is skew-Hermitian relative to (. I .)} .

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Then gb is a real form of the Lie algebra g1 and is isomorphic to 5I(2,R).Let G' = SL(2,W) and let (? be the two-fold cover of GI. The analytic duality correspondence between G and is the following.

Theorem 2.5: The representation ofgb on P(C) integrates to a unitary representation of 5 o n H2 (the oscillator or metaplectic representation). I t decomposes under the action of 2;' x G as a direct sum of irreducible Halbert spaces

(H:

@

Ft)CB (H? @ F - )

(multiplicity-free)

This is a special case of Howe duality for unitary highest-weight representations. We will study it in full generality in later sections. 3. Schur-Weyl duality

3.1. Commutant of GL(n) action on tensors

Consider the action of G L ( n ,C ) on its defining representation:

BkC"

by the lcth tensor power

pk(g)(vl 8 . . . @ w )= g q @ . . . @ gvk

The symmetric group B k acts on

for vi E C"

pk

of

.

BkCn by permuting the tensor positions:

Ok(S)(VI @ * . . @ V k )

= Vs-'(l) @ . . . @ V v , - l ( k )

(the vector in position i is moved to the vector in position s ( i ) ) .It is clear that a k ( s ) p k ( g )= plc(g)ak(s)for all g E G L ( n , C ) and s E Gk.

Proposition 3.1 (Schur): A n y linear transformation B on B k C n that commutes with c k ( B k ) is a linear combination of the transformations p k ( g ) , g E GL(n,@). Proof: Let { e i } be the standard basis for Cn.Then

eI

= ei, @

. . . 8 eik , where I

BkC"

has basis

= (il, . . . , ik) with 1 5 ij

5 n.

For s E B we have alc(s)el = e s . I , where s . I = ( i ~ - ~ ( ~ ~ , . . . , i ~ ( s moves the positions 1,.. . , k of the indices; it does not permute their values 1 , .. . ,n). If we write B e J = b:eI, then the condition that B commute with 6 k is expressed by

zI

b: = b:::

for all s E 6 k and all indices I , J .

(7)

Thus if we denote by 2 = { ( I ,J ) } (all pairs of indices), then B k acts diagonally on E and the matrix for B is an invariant function on Z.

- l

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

321

Set B = E n d s k ( @ k @ n )and let T ++ Tb be the projection from End(@k@") onto B. The bilinear form ( X , Y ) = tr(XY) on B is nondegenerate, since ( T ,Y ) = (Tb,Y ) for T E E n d ( B k and Y E B. Write

en)

130 = Span{pk(g) : g E GL(n,@)).

Then to show ,130 = B,it suffices to show that if B E B and ( B , p k ( g ) ) = 0 for all g E GL(n, C), then B = 0. Define F ( g ) = ( B ,p k ( g ) ) . Since g H F ( g ) is a polynomial function on the space Ad,(@) of n x n matrices and F vanishes on GL(n,@), it is identically zero. Hence {b:} satisfy the linear equations

in addition to the invariance condition (7), where z1J = ziljl . . . x i k j , . It is easy to verify that z1J = XI/ as functions on kfn(C)if and only if I = s . I / and J = s J' for some s E 6 k . Let r be a cross-section for the orbits of 6 k on E.Then the set of monomials {xY : y E I?} are all distinct, and hence linearly independent functions on M n ( C ) . Since b: is constant on 6,+orbits by (7), equation (8) can be written as J?

IGk . y/b,x, = 0

for all x E M n ( @ ) .

7Er

0

Thus b, = 0 for all y.

Applying Proposition 3.1 and Theorem 1.4, we obtain a preliminary version of Schur- Weyl duality: Corollary 3.2: There are irreducible, mutually inequivalent 6 k modules E X and irreducible, mutually inequivalent GL(n,C) modules FAso that @ C ~ E

EXBFX €Spec ( P k )

as a representation of 6 k x GL(n, C).The representation E X uniquely determines F A and conversely. 3.2. Highest weight theory

To make Schur-Weyl duality an effective tool, we will construct the irreducible regular representations of GL(n,C) by the Theorem of the Highest Weight. We give details for GL(n, C); analogous results hold for any complex reductive algebraic group (see [16, Chap. 51). The starting point is the

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Gauss decomposition. Let H be the subgroup of diagonal matrices, N the subgroup of upper-triangular unipotent matrices (all diagonal entries 1), and ft the subgroup of lower triangular unipotent matrices. Then R H N is a Zariski-dense open subset of G, and a generic element g E G has a unique factorization g = iihn.b Thus a regular representation of G is completely determined by its restriction to the subgroups N,H , and N . The subgroup H is a maximal algebraic torus in G. In particular, it is a reductive complex algebraic group. The irreducible representations of H are one-dimensional and given by h = diag[xl,. . . ,z],

++

hp = 5 m 1 1 . . . z z n, where p

=

[ml,.. . ,m,] E

Zn.

Thus we may identify fi with Z".If ( p , V ) is a regular representation of G, then the restriction of p to H decomposes into weight spaces:

V=

@

V ( p ),

where V ( p )# 0 and p(h)v = h p v for v E V ( p ) .

PEQW)

We call @ ( V c ) the set of weights of V . Let NormG(H) be the nomnalizer of H in G ( H g H = g H ) , and W = Norm c ( H ) / H the Weyl group of G. The elements of W permute the weight spaces and the weights of V . In this case, W enmay be identified with the group of permutation matrices in G , and the action of W on H and is by the usual permutation of coordinates. Every W orbit in 2 contains a unique dominant weight

fi

p = [ m 1 , . . . ,m,],

We denote by

rnl L r n 2 L ' . . L r n n .

Zn++ the set of all such p

E Zn.

Examples (1) Let V = C n be the defining representation of G. Then

@ ( V= ) (

..

~ 1 , . ,E,}

,

where ~ ( h=) xi for h = diag[zl,. . . , x n ]

Here @ ( V = ) W . ~1 is a single W orbit with dominant weight ~ 1 .

(2) Let V = BkCn. The basis { e l } used in the proof of Proposition 3.1 diagonalizes pk(H). For an index I = [ill.. . , Zk], with 1 5 i j 5 n, define p ~ = [ p. ~ ..,p,],

wherepp=#{j : i j = p } .

bThe precise condition from linear algebra is that the principal minors Ai(g) # 0 for i = 1 , 2 , . . . , 71.

Multiplicity-fie Spaces and Schur- Weyl-Howe Duality

323

Then pk(h)er = hprer for h E H . Hence for X E H , V(A) = Span(e1 :

= A}.

In particular, V(X) # 0 if and only if X i 2 0 for i = 1,.. . , n and 1x1 = Ic, where 1x1 = Xl+...+X, . Thus @(@’C@”)= W.Par(Ic,n), wherePar(Ic,n) is the set of all partitions of Ic with at most n parts. Each such partition defines a dominant weight p. of H such that h H h p is a polynomial function on H (no negative powers of the coordinates xi).

(3) Let g = Lie(G) = M n ( C ) be the Lie algebra of G, and let Ad(g)z = gzg-’ be the adjoint representation. The weights are 0 and { ~ i- ~j : 1 5 i # j 5 n } . We call the nonzero weights the roots of fj on g. The corresponding root spaces are go =

= Lie(H) ,

= CEij

where Eij is the usual elementary matrix with 1 in position ( i l j )and zero elsewhere. If a = ~i - ~ j then , we say a > 0 if i < j (so Eij is upper triangular) and a < 0 if i > j. We denote the set of positive roots by @+ and the set of negative roots by @-. Thus n=Lie(N) =

@ g,

i i = ~ i e ( i V )=

@

ga.

a€@-

,€@+

The Lie algebra (additive) version of the Gauss decomposition is the triangular decomposition g = ii@ fj @ n. If ( p , V) is any regular representation of G, then there is an associated Lie algebra representation d p of g defined by d dp(X)v = -p(exptX)v dt

IL0

Clearly p(g)dp(X)v = dp(Ad(g)X)p(g)v. Hence if h E H , E , E ga and v E V(p), then

p(h)dp(E,)v

= dp(Ad(h)E,)p(h)v = h”+’”dp(E,)v.

+

This shows that d p ( g c r ) V ( p )C V(p. a ) . Thus dp(n)V(p.) c

@

V(4.

(9)

X€P+Q+

We call p E @(V) an N-extreme weight if p.

+ a # @(V) for all a E a+.

R. Goodman

324

Theorem 3.3: Let (p,V) be an irreducible regular representation of G. Then there is a unique N-extreme weight po E @ ( V ) This . weight is dominant, the weight space V(p0) = VN (the N-fied vectors in V), and dimVN = 1. Proof: Define the dominance order on by u 4 p if p # u and p - u is a nonnegative integer combination of positive roots. Since @(V)is a finite set, it contains an element po that is maximal relative to this partial order. Any maximal element must be dominant, since if po had a pair of coordinates mi < mj for some i < j , then

v = 110

+ (mj -

mi)(Ei

-~

j =) sijpo

E @(V),

where s i j is the permutation i H j . This would contradict the maximality of po. Since no element of po @+ can be a weight of V , we see from (9) that dp(n)V(po) = 0. But N = expn, so we conclude that V ( p 0 ) c VN. Take any nonzero vector uo E V(p0). Then by irreducibility and the Gauss Decomposition,

+

V = Spanp(G)vo = Spanp(N)vo = dp(U(ii))wo

~

where U(ii)is the universal enveloping algebra of ii.But the nonzero weights of H on U(ii) are positive integer combinations of negative roots, and the zero weight space is Cl. Hence

v = C U O + @ V(X). A-WO

It follows that po is the unique maximal weight and V ( p 0 ) = V N .

0

We call po the highest weight of the representation ( p , V ) .We next show that po determines (p,V) uniquely up to isomorphism. Let SO E W be the permutation [l,2,. . . ,n]+-+ [n,. . . , 2 , 1 ] . Then Ad(s0)N = N and so.@+ = W (this last property uniquely characterizes SO). Since the weights and weight multiplicities of ( p , V) are invariant under the Weyl group, we see from Theorem 3.3 that @(V)has a unique minimal element -sop0 (the lowest weight), and V ( - s o p o ) = V N .The natural bilinear form on V x I/* is invariant under H , so its restriction to V(X) x V*(-A) is nondegenerate. Thus -po is the lowest weight of V*.

Theorem 3.4: Let ( p , V ) and (a,U ) be irreducible regular representations of G with the same highest weight A. If vo E V and uo E U are highest weight vectors, then there exists a unique G-isomorphism T : U + V such that Tuo = V O . Thus ( p , V) is uniquely determined by its highest weight.

Multiplicity-Bee Spaces and Schur- Weyl-Howe Duality

Proof: We can take vo E V N and v: E V*N so that cp(g) = (p(g)vo,v;)

325

(v0,vg)= 1. Set

(generating function) .

Then cp E Aff(G) and cp(fihn) = (p(h)vo,p*(fi)v:)= hX for all fihn E N H N . This equation uniquely determines cp in terms of the highest weight A, and using U and uowould give the same generating function. Define TO: V --+ Aff(G) by Tov(g) = (p(g)v,v;).Then TOintertwines the representation p on V with the right translation representation on Aff(G). Since Tow0 = 'p, the map TO is nonzero. Hence it is injective, by Schur's lemma, and V 2 To(V).But V = Span{p(G)v}, so To(V) is spanned by the right translates of the function cp. Let TI : U --t Aff(G) be 0 likewise defined. Then T = Tr'Tl is the desired intertwining map.

For applications to duality we will need the following sharpening of Theorem 3.3. Theorem 3.5: Let ( p , L ) be any regular representation of G. Suppose that 0 # vo E I ~ ( X ) ~ Then . the subspace V = Span{p(G)vo} of L is an irreducible G-module with highest weight A. Proof: From the proof of Theorem 3.3 we have a weight space decomposition

v = cwo @ @ V ( p ) . /L<X

Let Uo c V be a proper G-invariant subspace. Then vo @ UO,so Uo = Uo(p). Since G is reductive, there is a complementary G-invariant subspace U1 so that V = UO@ U1. Since Ul is the direct sum of its weight 0 spaces, we have vo E Ul and hence Ul = V, proving irreducibility.

Corollary 3.6: For every dominant weight X of H there exists an irreducible representation (d, F A ) with highest weight A. Proof: Let A k ( g ) be the kth principal minor of g E GL(n,@),for k = 1,.. . ,n. If X = [ml,.. . , m,], set Since X is dominant, we have ml 2 m2 . . . 2 m,. Also A,(g) = det(g) # 0 for g E G. Hence fX is a regular function on G. Also fx(fihn) = hX for

iihn E NHN. Let F A c Aff(G) be the span of the right translates of fx, and let 7rX be the restriction to FAof the right translation representation of G on

R. Goodman

326

Aff(G). Then d i m F X < 00 since f~ is a regular function. Furthermore, 7rX is irreducible with highest weight X by Theorem 3.5, since f x is N-fixed of weight A. 0 We shall refer to Theorems 3.3-3.5 and Corollary 3.6 collectively as the Theorem of the Highest Weight. 3.3. Duality and N-fixed vectors

Let ( p , L) be any regular G-module. By the Theorem of the Highest Weight we can identify Spec(p) with the set of dominant weights X such that L(X)N # 0. For X E Spec(p) set E X = L(X)N.This space is invariant under the commuting algebra EndG (L).

Theorem 3.7: Under the joint action of G and EndG(L) the space L decomposes as

Furthermore, E X is an irreducible module for EndG(L), and distinct values of X give inequivalent modules for EndG(L). Proof: Let X E Spec(p) and fix a highest weight vector vx E F A . Define a linear map HOmG(FX,L)

+ L(X)N

,

T

w

TUX.

This map is injective by the irreducibility of F A , surjective by Theorem 3.4, and it obviously intertwines the action of EndG(L). The theorem now follows from Theorem 1.4. 0 We can now give a more precise version of Schur-Weyl duality (Corollary 3.2). A regular representation T of G is said to be polynomial if the matrix entries of 7r are polynomial functions on G (with no negative powers of det(g)). When T = 7rX is irreducible, it is a polynomial representation if and only if A, 2 0. In this case X corresponds to a partition of 1x1 with at most n parts.

Theorem 3.8 (Schur-Weyl Duality): For X E Par(k, n) let (aX,E X )be the representation of 6 k on the space of N-fixed k-tensors of weight A, where N is the upper triangular unipotent subgroup of GL(n, C).Let (d, F h , ) be the irreducible representation of GL(n,C) with highest weight A. Then E X

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

327

is an irreducible 6 k module. Under the action of 61, x GL(n, C), the space of k-tensors over Cn decomposes as

X €Par(Ic, n )

The representations E X are mutually inequivalent, and when n 2 k they give all the irreducible representations Of 6 k . Furthermore, every irreducible polynomial representation of GL(n, C) occurs in ( C n ) B k for some k. Proof: From Example (2) of Section 3.2 the dominant weights of ( C n ) B k are in Par(lc, n).To verify that for every X E Par(k, n)there exists a nonzero N-fixed k-tensor of weight X, we define

+ ... + &i,

mi = &1

for i

u,= el A . . . A ei

=

1 , 2 , .. . ,n.

Ai

Then P, is the highest weight of GL(n, C ) on C", and ui is the corresponding highest weight vector. If we set ci = mi - mi+l (with mn+l = 0), then c1, . . . , cn are nonnegative integers and X = c l a l . . . c n a n . The tensor

+ +

has weight X and is N-fixed, since G acts as automorphisms of the tensor algebra over Cn. It follows that Spec(pk) = Par(k,n) (we label elements of 6 by their highest weights). The assertions of the theorem now all follow from Corollary 3.2 and the Theorem of the Highest Weight, except for the fact that all irreducible representations of 6 k occur when n k. To prove this, recall that the number of irreducible representations of a finite group is the same as the number of conjugacy classes (this common number is the dimension of the center of the group algebra). In the case of 6 k , each conjugacy class corresponds to a cycle decomposition, and the lengths of the cycles determine a unique partition of k . Thus 6 k has #Par(k) irreducible representations, where Par(k) denotes the set of all partitions of k . Since a partition of k has a t most k parts, every partition occurs in Par(k,n) when n 2 k , and hence (C")@'" contains all irreducible representations of 61,in this case. 0

>

Examples

(1) The group Gk has two one-dimensional representations: the trivial representation and the sign representation. The corresponding subspaces of @."C" are the symmetric tensors S k ( C n )and (if n 2 k ) the skew symmetric tensors C". Hence these subspaces must be irreducible GL(n, C )

Ak

R. Goodman

328

modules, by Schur-Weyl duality (this is also easy to verify directly). The symmetric tensor eFk is N-fixed with weight kal , while the skew-symmetric . . &k (when k 5 n).Thus tensor el A ' . ' A ek is N-fixed with weight in the duality correspondence,

--

(trivial, C) = (Jk1,dk1)

(sgn, C) = ( a [ l k ]E,[ l k ] )

+

(Jk1,s'(c")) ,[lk],

AkC")

if n 2 k .

When k = 2 and n 2 2 this gives the complete decomposition of B 2 C n : under 6 2 x G L ( n , @ ) :

a3C" for n 2 3. There are three partitions of 3, giving the

(2) Consider decomposition

B3 2 { E[3,01 @"

@

> {,9[2,11 Fl2j11> {E [ ' I ~ , ~A]

S3((cn)

@

@

@

@

under 6 3 x G L ( n , @ ) .Here the representation E[2i1]of dimensional standard representation on C3/C[l, 1,1].

6 3

3Q.n}

is the two-

We can view Schur-Weyl Duality as a method to construct representations of G' = 6 k from representations of G = G L ( n , @ )via the Theorem of the Highest Weight. Here we take the representations of G as the known objects, and the representations of G' as the unknown objects.c The relative size of n (the rank of G) and k then determines which representations of GI we get this way.

n 2 k: All partitions of k have a t most k parts, so all representations of 6 k occur in B~C" in this case. n 5 k: Only those representations Of 6 k occur in @" that correspond to partitions of k with a t most n parts.

mk

To make this method effective, we will develop character formulas for the representations of 6 k in the next two sections, based on the celebrated Wegl character formula for G L ( n ,C ) . CTherepresentations of 6 k can be constructed directly by group-theoretic and combinatorial methods. Special elements of the group algebra C [ G k ] (Young symrnetrizers) project tensor space onto irreducible representations of GL(n, C) - the so-called Weyl modules - see [16,Sec. 9.31.

Multiplicity-Ree Spaces and Schur- Weyl-Howe Duality

329

4. Commutant character formulas 4.1. Characters Let G be a connected complex reductive algebraic group. Then G contains a maximal algebraic torus H and a maximal connected solvable subgroup H N (semidirect product), where N is the unipotent radical of H N . In fact, one can always embed G into GL(n,C) so that H consists of the diagonal matrices in G, and N the upper-triangular unipotent matrices in G, just as in the case of GL(n, C ) treated in Section 3. Let l~ = Lie(N) and n = Lie(N). The irreducible representations of the torus H are given by h H h X , where X is in the weight lattice P c b* of H . By the Theorem of the Highest Weight (which is proved for G along the same lines as in Section 3 for GL(n, C ) ) , the irreducible regular representations of G are parameterized by the set P++ of dominant weights determined by the choice of N . For X E P++ let (xX,FA)be the irreducible representation of G with highest weight A. Let ( T , V) be a finite-dimensional rational representation of G. Set B = Endc(V). From Theorem 3.7 V decomposes under the joint action of G and B into a multiplicity-free direct sum

Here g E G acts by 1@ d ( g ) and b E B acts by d ( b ) @I 1 on the summands in (11).We may take E X = V(X)N(the space of N-fixed vectors of weight X in V ) with d ( b ) the natural action of b E B on this space. Finding the spaces V ( X ) Nexplicitly is usually difficult. An easier problem is to calculate characters. For X E P++ we write

4.2. Frobenius and determinant character formulas

We now obtain two formulas for the characters xx that only involve the full H-weight spaces in V . Let be the weights of Ad(H) on n and p = $ CaEa+ a. Let W = NormG(H)/H be the Weyl group of (G, H). Set

D ( h )=

sgn(s) hS’P

for h E H

SEW

(the Weyl denominator). Here s signum character on W .

H

sgn(s) = det(Ad(s)Ir,) is the usual

R. Goodman

330

Theorem 4.1 (Generalized Frobenius Formula): For X E P++ and b E B one has

xx(b)

=

coeficient of hx+p in ~ ( htrv(x(h)b) )

(12)

(where h E H ) . Theorem 4.2 (Generalized Determinant Formula): For X E P++ and b E 23 one has

xx(b) =

c

sgn(s) trV(X+p-s.p) ( b ) .

(13)

SEW

I n particular,

4.3. Proof of Frobenius character f o r m u l a

For X E P++ we write xx(g) = tr(xx(g)) for g E G . (the character of the representation with highest weight A). We note from (11) that trv(x(g)b) =

c

xX(g)Xx(b)

for g E G and b E

B.

(15)

XESpec(?r)

By the Weyl character formula (WCF), we have

D(h)x x ( h ) =

c

sgn(s) hS’(’+”)

for h E H .

SEW

Using the WCF in (15) we can write

D ( h )t r v ( 4 h ) b ) =

c c

(16)

Sgn(s) ~ x ( b h) S ’ ( x + p ) .

XESpec(.rr) sE W

Due to the shift by p, the map (s, A) H s . (A + p ) from W x P++ P is injective.d Hence the character h H hX+p only occurs once in (16), and has 0 coefficient xx ( b ) , as claimed. --f

In the case of G L ( n ,C ) the partition X

+ p has all parts of dzfierent sizes.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

4.4. Proof of

331

determinant character formula

For the proof of Theorem 4.2, we need the following consequence of the WCF (which is, in fact, equivalent to the WCF).

Lemma 4.3: Let m x ( p ) = dim F X ( p )for p E P and X E P++ (the multiplicity of the weight p in F A ) . Then for X,p E P++ one has

C sgn(s) m , ( ~+ p - s . p ) = 6Xp . SEW

Proof: Write the Weyl denominator as an alternating sum over W of the characters h H hs'p. Multiplying this sum by x p , we get

D(h)X'l(h)=

c{

SEW

sgn(s)m,(v) hvfS'P}

for h E H .

YEP

In the inner sum make the substitution v the right becomes

--f

v

+ p - s .p; then the sum on

On the other hand, the WCF asserts that the coefficient of h X f p in 0 D ( h ) f ( h )is d ~ ,when Alp E P++.

Proposition 4.4 (Outer Multiplicity Formula): Let L be any regular G module. For X E P++ let rnultL(X) be the multiplicity of the representation F A . Then multr,(X) =

sgn(s) d i m L ( X + p - s . p ) . SEW

Proof: For v E P we have dimL(v) = right side of (17) is

PEP++

(17)

cPEp++ multr,(p) m,(v). Hence the

SEW

But the inner sum is b ~ by , Lemma 4.3, which proves (17).

0

Proof of Theorem 4.2: Let b E B. Then b has a Jordan decomposition b = bs b,, where b, is semisimple, b, is nilpotent, and be is a polynomial in b. Hence b, E B and X x ( b ) = X x ( b , ) . So we may assume b is semisimple. For E C and X E P++ define

+

<

V, = {V E V : bV = [v},

E$ = {V E E X : bV = CV}.

R. Goodman

332

These spaces are G-invariant and

ys @

E ; ~ F ~

XEP++

as a G-module. In particular, the multiplicity of F A in Vc is dim Et. Now apply Proposition 4.4 to the G-module L = Vc to get

J

SEW

=

c

sgn(s) trV(X+p-s.p)(b)

SEW

which proves Theorem 4.2. 5. Character formulas for Schur-Weyl duality 5.1. Frobenius formula f o r 6 k characters

We now apply the commutant character formulas to the Schur-Weyl duality between G = GL(n, C ) and 6 k , both acting on V = C=".Recall that the conjugacy classes in 6 k are described by cycle lengths. We denote by C(lr12T2 . . . k r k ) the class of elements with rj cycles of length j , where 7-1 27-2 37-3 . . = k. A permutation s in this class has r1 fixed points, 7-2 transpositions, and so on. To apply Theorem 4.1 in this context, we need to calculate the polynomial

ak

+ + +.

h H trV(pk(h)ck(s)),

where h = diag[zl,. . . , xn] .

Recall that the tensors {el} give a basis for V . The action of h E H and s E Gk is pk(h)el = zp(')eI,

ck(s)el = es.l

where p ( I ) = [ P I , .. . , pn] with pj = # { p : i, = j } and s . I = [is-1(q,.

. . ,is-'@)].

Since g ( s ) permutes the basis {el} and each follows that

eI

is a weight vector for H , it

for h E H , (18) trv(pk(h)ck(s)) = trpS(pk(h)) where F, = Span{er : s . I = I } . Let = Span{e,@j , e @ 2 j,... , e @ n j} c @ j C".

4

M u l t i p l i c i t y - h e Spaces and Schur- Weyl-Howe Duality

333

Lemma 5.1: If s E C(1T12p2- .krk) then +

as an H-module. Proof: We may assume the cycle decomposition of s contains the integers 1,.. . , k in increasing order, since replacing s by a conjugate element in 61, does not change the H-module structure of F,. In this case the condition s I = I for an index I means that

I = [ a1 > . . * I aT, b l , b l .~. . 3

1

b,Z

I bT, I

cl> c1 I

C l l ' ' ' I cT3

"

\-

r1 singles

pairs

~2

r3

> cr3 >

cT3 I ' '

'1

triples

where a,, b,, cZr. . . range from 1 to n. Hence

er = e a 1 8 . . . @ e a r l @ e r @ . . . @ e Frz2@ e r @ . . . @ e F r t 8 . . .

0

The lemma follows. For h = diag[xl,. . . , xn] define

p , (x)= tr% ( p k ( h ) ) = xi

+ . + x; s .

( j t h power sum).

Then Lemma 5.1 implies that k

k

trF,(Pk(h)) = n t r V . ( p k ( h ) ) " = n p ~ ( x ) .' ~ ,=1

j=l

Hence from Theorem 4.1 and (18) we obtain the Frobenius character formula:

Theorem 5.2: Let s E C(lT12T2 . . . I c r k ) and X E Par(k, n ) . Then

c1 }

x x ( s ) = coeficient ofx'+p[nl in D ~ ( ~ Hc p)j ( x l T j where

,qn] = [n- 1,n - 2,.

,,

,1,0] and D n ( x ) = n,,,<jln(xi

-

xj).

Examples

+

(1) Suppose s = ( 1 , 2 , . . . , m ) ( m 1 ) .. . ( I c ) is a single m-cycle with k - m fixed points. Then

x x ( s ) = coefficient of

xx+plnl

in

(21

+ . . . + Z,)~-~(Z;" + . . . + xr)

R. Goodman

334

We call a monomial x;' ...x? strictly dominant if a1 > a2 > . . . > a,. For partitions X with two parts and cycles of maximum length m = Ic, the strictly dominant terms in this formula are z : + l - zfz2. Hence for s = (1,2 ,.", Ic),

x x ( s )=

{ -:

for X = [k - 1,1], for X = [k - j , j ] with j

> 1.

'

(2) Consider the group 6 3 , which has three conjugacy classes: C(13) = {identity), C(112') = {(12), (13),(23)) and C(3l) = {(123), (132)). As we noted at the end of Section 3, the three representations of (553 are (the trivial representation), a[2y1](the two-dimensional standard representation), and o [ l ~ l ~(the l l signum representation). To calculate the character of a[2)11 by the Frobenius formula, we let z = [ X I , 521 and expand the polynomials ~ 2 ( x ) p(x13 l =

x;'+ 223x2 + . . .

Dz(x)p1(x)pz(z)= z;' + * . * DZ(X)p3(X)

3 = 214 - 21x2

+

' ' '

where . . . indicates non-dominant terms. By Theorem 4.1 the coefficients of the dominant terms in these formulas furnish the entries in the character table for 6 3 . We k i t e x x for the character of the representation a x . For example, when X = [2,1] we have X + p = [3,1],so the coefficient of x;z2 in D 2 ( ~ ) p 3 (gives ~ ) the value of X [ ~ , J on ] the conjugacy class C(3l). Table 1 gives all the characters, where the top row indicates the number of elements in each conjugacy class, and the rows in the table give the character values for each irreducible representation. Table 1. Character table of

5.2. Determinant formula for

6 k

63.

characters

We next apply Theorem 4.2 to obtain an alternating sum formula for the characters of 6 k . For this, we need to identify the weight spaces V ( Y )as

Multiplicity-Fkee Spaces and Schur- Weyl-Howe Duality

6k-modules. Here v = [vl,. . . ,v,] with vi 2 0 and v1 have already seen that V ( v )= Span{er : p(1) = v}.

Lemma 5.3: Let 6, = G,, x a 6k-module.

+ . . . + u,

335

=

k. We

. . . x G,, c 6 k . Then V ( v )cx e[6,/6:,]as

Proof: If I is a multi-index such that p(1) = u, then there is s that

---

E 6 k so

s . I = [ l ,..., 1 , 2, . . . ,2 , 3,.", 3 ,...1 . v1

m

v3

0

Since uk(s)el = es.r, the lemma follows.

From the lemma we see that ~ ( s acts ) as a permutation matrix on V ( v ) ,and hence trV(y)(uk(s))= #{fixed points of s on 6 k / G y } . The Weyl group for G is 6, and acts on the weight lattice P as permutations of the coordinates of the weights. Applying Theorem 4.2 and using Lemma 5.3, we obtain the following character formula.

Theorem 5.4: Let X

E Par(k,n)

and s E 6 k . Then

x x ( s ) = C s g n ( t ) #{fixed points of s on

6k/6X+pjnl--t.p,,l).

t

Here the sum is over all t E 6,such that all the coordinates of X+p[,]-t.p[,l are nonnegative. In particular,

In Theorem 5.4 44 = [n- 1, n - 2 , . . . , 1 , 0 ] and

is the multinomial coefficient (with the usual convention that it is zero if any entry in u is negative). The dimension formula can be written as a determinant and then reduced to Vandermonde form. This gives the following product formula for the dimension of the representation E X that is analogous to the Weyl dimension formula for the representation FA.

Corollary 5.5: Let X E Par(k,n). Then d i m E X= &Dn(X

+ p[,l).

R. Goodman

336

A partition X = [XI,. . . , An] can be represented in terms of its Ferrers diagram: a left-justified array of boxes, with X i boxes in the ith row (counting from the top down). Each box in the diagram has a hook length: the total number of boxes to the right and below the given box (including the box itself). We can then fill each box with its hook length. For example, X = [4,3,1] E Par(8,3) has Ferrers diagram and hook lengths

F’rom Corollary 5.5 one obtains (by induction on the number of columns of A) the Hook Length Formula

where hij(X) is the hook length of the i j box in the Ferrers diagram of A. By way of comparison, the Weyl Dimension Formula for GL(n, C)can be written as

(see [16, Sec. 9.1.4, Example #9]). For example, for on B 8 C 3 we have p[31 = [2,1,0]and

(58

x GL(3,C) acting

@I F/;j”.’l is an irreducible subspace of @*C3 of dimension Thus E[4,3i1] (70) * (15).

5.3. Schur-Weyl duality and GL(k)-GL(n) duality There is another model for the irreducible representations of 61, that comes from the identification of (5k with the Weyl group of GL(k,C). Let X = Mk,, (k x n complex matrices) and let GL(lc, C)x GL(n, C ) act on P ( X ) by p(gi, g z ) f ( x ) = f(gi2gz)

for gi E GL(k, c) and gz E GL(n, @) .

This representation is multiplicity free and decomposes as

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

337

with the sum over all partitions p with a t most min{k,n} parts (see [16, Sec. 5.2.41 or the article by Benson-Ratcliff in this volume). Let H k C GL(k,C) be the maximal torus of diagonal matrices, and embed 6 k c GL(k, C)as the permutation matrices. If we only consider the together action of the subgroup Norm ( H k ) 2 Gk K H k of GL(k, C)on with the right action of GL(n,C), then

x

Y

k summands

Hence

P(X)r S(C")8 . .. @ S(Cn) k factors

as a representation of Norm ( H k ) x GL(n, C).Here 6 k acts by permuting the tensor factors, while h = diag[zl,. . , ,xk] E Hk acts by multiplication by xj on the j t h factor. The weight space decomposition of P ( X ) relative to the H k action is thus

P ( X ) ( p )% Sml(C") @ . . . @ Smk(C") for p = [ m l ,. . . , mk] .

(22)

Here 6 k acts by permuting the factors in this decomposition while GL(nC) acts as usual on each copy of C".In particular, the weight detk = [l,1,. . . ,1]is fixed by 6 k and the corresponding weight space is P(X)(detk) % sl(@") €9.. * @ sl(@")= (Cn)@'" with the usual commuting actions of 6 k and GL(n, C). On the other hand, if we calculate this weight space using (21), we see that

(Cn ) @k 2

@

f'i\,)(detd @ F $ )

XEPar(k,n)

as a module for Gk x GL(n, C).Invoking Theorem 3.8 we conclude: For all

X E Par(k,n),

as a $k

module, with the action of

6 k

coming from its embedding into

GL(k, C). Examples (1) Take X = [ l , .. . , I] E Par(k). Then the representation Pi\,, of GL(k,C)

is Ak C k , on which GL(k, C)acts by g H det(g). This shows once again that EXis the sgn representation of 6 k .

R. Goodman

338

(2) Now take X = [k].Then the representation F h ) of GL(lc, C ) is Sk(Ck)Z Pk((Ck)*).The detk weight space is one-dimensional and spanned by the monomial X I . . ' x k , which is fixed by 6 k . Again we see that E['] is the trivial representation of 6 k . 6. Polynomial invariants and FFT

6.1. Invariant polynomials

Let G be a reductive linear algebraic group. Recall from Section 1that given a regular representation (T,V )of G, we have a locally regular representation p of G as automorphisms of the commutative algebra P ( V ) of complexvalued polynomial functions on V :

p ( g )f(v)

= f(g-'v)

for f E P ( V ) and g

E

G

(here we write g v for 7r(g)v when the action 7r is clear from the context). Since G acts by automorphisms of P ( V ) , the space J' = P ( V ) Gof Ginvariant polynomials is a subalgebra of P ( V ) .Thus we can consider P ( V ) as a module for 3 under the action of pointwise multiplication, which commutes with the G action. Then in the isotypic decomposition

P ( V ) = @P(V)(A)

x€a each summand is invariant under J . By Corollary 1.2 there is a projection f H f b from P ( V ) onto J , with degfb 5 d e g f . If f E P ( V ) and cp E 3' then

(PfIb= cpfb

+

(Decompose f = f b . . . into isotypic components; then q f = c p f b is the isotypic decomposition of cpf .)

(23)

+

Theorem 6.1 (Hilbert-Hurwitz): J' is finitely generated as an algebra over @.

Proof: Let J+ = {f E 3 : f(0) = 0) and write R = P ( V ) . Since R is a polynomial ring in dim V generators, the Hilbert basis theorem implies that the ideal R3+ is finitely generated as an R module: there exist q j E J'+ such that n

339

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

Furthermore, since ,7+ is invariant under the C x action on R (f(v) H f (Cv) for E Cx ), we may take each 'pj to be homogeneous of some degree d j 2 1. We claim that {'pj} generate 3 as an algebra over C. Let f E ,7 be of degree d and assume inductively that all polynomials in 3 of degree less than d are polynomials in 91,. . . ,9,.We can find fj E R so that f = C jfjcpj. Now project onto ,7 and use (23):

<

j

j

fj

Since degcpj 2 1, we have deg 5 deg f j < deg f . Hence by induction each f jb is a polynomial in ~ 1 ,. .. vn,so the same is true of f . 0 We shall say that {PI,. . . , cp},

c J' is a basic set of G invariants if

. . ,cp,} generates 3 as an algebra over C (i) (91,. (ii) each cpi is homogeneous (of some degree d i ) and n is as small as possible, subject to (i) and (ii). By Theorem 6.1 there always exists a basic set of invariants (the polynomials 'pi are not unique but the set { d l , . . . , d,} of degrees is uniquely determined). Example: Let G = 6, and V = Cn,with G acting as permutations of the coordinates. Then p ( s ) f ( x l l . .. , x n ) = f ( X s ( l ) l . . . ? x s ( n ) ) for

f

E @ [ X I , . . . rxnj and

E

en

and ,7 is the algebra of symmetric functions in n variables. By the fundamental theorem of symmetric functions one has ,7 = C[al1. . . ,a,], where op(x1,. . . , x n ) =

c

(pth elementary symmetric function)

Xj1. .'XjP

1ij1<...<jp
. . . ,a,} are algebraically independent, so Furthermore, the functions {a~, they give a basic set of invariants with degrees d p = p . (The function ap is the restriction to the diagonal matrices of the character of Apenas a representation of GL(n, C ).) 6.2. Invariants of vectors and covectors

-

Take G as a classical group (GL(n, C ) ,O ( P , B ) ,or Sp(Cn,0) with n even) and V = C n the defining representation of G. Let

v m = -V' @ * . . @ V m column vectors

v*k =

v*

@

. . . @ v*

k row vectors

R. Goodman

340

with the natural G action on each summand. Then P ( V * k@ Vm)Gis the algebra of G-invariant polynomial functions of k covectors and m vectors. The First Fundamental Theorem (FFT) of invariant theory for the group G gives an explicit description of sets of basic invariants (for all values of k and m). There is an alternate picture that reveals the hidden symmetries in this situation and gives an obvious algebra of G-invariant polynomials together with a set of quadratic generators. Namely, we have the G-isomorphisms

v*kG Mkxn

right G action (matrix multiplication)

V m G M,,,

left G action (matrix multiplication)

where Mkxn is the vector space of k x n complex matrices. In this picture we see that the reductive group L = GL(k, C) x GL(m, C) acts on M k x n@ Mnxm by

(a,b)(z@y)=az@yb-'

f o r a E G L ( k , C ) and b E G L ( m , C ) .

This action obviously commutes with the G action. The induced action on functions makes P ( M k x n@ M n x m ) Ginto an L module. Note that the maximal torus of diagonal matrices in L acts in the original picture V * k@ V" by scalar multiplication in each vector summand, while the Weyl group 6 k x of L acts by permutation of positions of the summands. Define the multiplication map

em

p : Mkxn @ M a x , + M k x m

2 @

y

H zy

(matrix multiplication).

Obviously p(sg @ g-'y) = p(z @ y) for all g E GL(n, C ) . Hence we have a n algebra homomorphism

In particular, if we take f = xij (the (i,j ) matrix entry function on M k x m ) , then

~ * ( z i j ) ( v ; ,. . . > v i , 211,'

>urn)=

( ~ 5vj) ,

(the contraction of the ith covector with the j t h vector). There is a natural action of L on Mkxm with GL(k,C) acting by left multiplication and GL(m, C) acting by right multiplication, Hence L acts . map p intertwines the two L actions. on P ( M k x m ) GThe

Multiplicity-Ree Spaces and Schur- Weyl-Howe Duality

341

6.3. Polynomial F F T for GL(n)

The FFT for G L ( n , C ) is the assertion that the method just indicated to construct invariants furnishes the complete algebra of polynomial invariants. Theorem 6.2: Let G = GL(n,C). T h e n the m a p p* is surjective. Hence the km quadratic polynomials q5ij = p*(xij) with 1 5 i 5 k and 1 5 j 5 m give a set of basic invariants for P ( M k x n @ Mnxm)G . After discussing tensor invariants in the next section we shall show there how this theorem is an immediate consequence of Proposition 3.1. At this point we observe that the image of p consists of all k x m matrices x with rank(x) 5 min(k, m,n). This gives rise to the following dichotomy: (1) If n 2 min(k,m), then p is surjective. Hence p* is injective and

P(Mkxn @ hfnxm)GL(n'@) P(Mkxm) is a polynomial algebra with km generators. One says that h f k x m is the algebraic quotient of V*'"@ V" by GL(n,C). The representation of L on P ( M k x m ) is multiplicity-free (see [16, Theorem 5.2.71 or the article by Benson-Ratcliff in this volume). (2) If n < min(k,m) then Ker(p*) # 0. The group L acts on Ker(p*), and from the multiplicity-free decomposition of P ( M k x m ) under L one finds that Ker(p*) is a determinantal ideal generated by ( n4- 1) x ( n 1) minors. Thus P ( M k x n CB Mnxm)GL(niC) is the algebra of regular functions on the associated determinantal variety. This is the Second Fundamental Theorem (SFT) for GL(n,C) invariants (see [16, Theorem 5.2.151 for the complete statement).

+

6.4. Polynomial F F T for the orthogonal group

We next consider the full orthogonal group relative to the bilinear form B ( z ,y) = xty on V = Cn: G = O ( n , C ) = ( 9 E G L ( n , C ) : gtg =I}. Since V 2 V * as a G-module, via the form B , it suffices to consider the invariants of k vector arguments P ( V k ) G= P ( M n x k ) G ,where G acts on Mmxk by left multiplication. Define a map 7

: MnXk 4

SMk

(k x k symmetric matrices),

~ ( x =) x t x

R. Goodman

342

For g E G we have r ( g x ) = xtgtgx = r ( x ) . Hence T*

:P(sMk)+ P(Vk)G

as in the case of GL(n, C). In particular, if we take f = xij (the ( i ,j ) matrix entry function on S M k ) , then T*(Xij)(VI,.

.. ,Uk)

= VZVj t

(the inner product of the ith and j t h vectors). The map r intertwines the right action of the hidden symmetry group L = GL(k,C) on Mnxk. Here the action of L on SMk is given by x H bxbt (for b E L ) . Theorem 6.3: Let G = O(n,C). Then the map T * is surjective. Hence the k(k 1)/2 quadratic polynomials 8ij = r * ( x i j ) with 1 5 i 5 j 5 k give a set of basic invariants f o r P(Mnxk)G.

+

Proof for the case n 2 k: There is a natural G-equivariant embedding Mnxk C M n x n ; just add n - k columns of zeros on the right to make x E Mnxk into an n x n matrix. Hence we may assume that k = n . Now 0 see [16, Proposition 4.2.61 for the proof.e We shall complete the proof for the general case n < k after discussing tensor invariants in the next section. Here we observe that the image of r consists of all k x k symmetric matrices x with rank(x) 5 min(k,n). This gives rise to the following dichotomy:

(1) If n 2 k, then 7 is surjective. Hence T * is injective and P ( M n x k ) G P ( s M k ) is a polynomial algebra with k(k + 1)/2 generators. One says that SMk is the algebraic quotient of Adnx,+by O ( n ,C).The representation of L on P(hfn,k)G is multiplicity-free (see [16, Theorem 5.2.91 or the article by Benson-Ratcliff in this volume).

(2) If n < k then Ker(r*) # 0. From the multiplicity-free decomposition of P ( s M k ) under L one finds that Ker(.r*) is a determinantal ideal generated by ( n 1)x (n 1) minors. Thus P ( M n x k ) Gis the algebra of functions on the associated symmetric determinantal variety. This is the Second Fundamental Theorem (SFT) for O(n, C)invariants (see [16, Theorem 5.2.171 for

+

+

the complete statement). eThe proof is by induction on n and can be viewed a s an algebraic group version of the Q R factorization for M , and the Cholesky Decomposition for SM,. This result is associated with a particular partial compactzfication of the symmetric space GL(n, @ ) / O ( nC). ,

Multiplicity-he Spaces and Schur- Weyl-Howe Duality

343

6.5. Polynomial FFT for the symplectic group Now consider the symplectic group G = Sp(Cn,R), where n = 2p is even and

Here I p is the p x p identity matrix. Thus G is the subgroup of GL(n,@) defined by g t J g = J . Since (Cn)* 2 @" via the form 0, it suffices to consider the invariants of k vector arguments P ( V k ) ) G= ? ( M n x k ) . Define a map : Mnxk

-+

AMk

(k x k skew-symmetric matrices), y ( x ) = x t J x .

For g E G we have y ( g x ) = xtgt J g x = y(z). Hence

y* : P ( A M k ) -+ p ( V k ) ) G as in the case of O(n,C). In particular, if we take f entry function on AMk), then

= xi?

(the ( i ,j ) matrix

y*(xij)(vl,"' ,vk) = fl(vi,vj) (contraction of the ith and j t h vectors by 0). The map r intertwines the right action of the hidden symmetry group L = G L ( k , C ) on Mnxk with the action of L on AMk given by x H bxbt (for b E L ) .

Theorem 6.4: Let G = Sp(Cn,0). Then the map y* is surjective. Hence the k(k - 1 ) / 2 quadratic polynomials wij = y * ( x i j ) with 1 5 i < j 5 k give a set of basic invariants f o r P(Mnxk)G. Proof for the case n

2 k: The

same citation as for the orthogonal

0

case.

We shall complete the proof for the general case n < k after discussing tensor invariants in the next section. Here we observe that the image of y consists of all k x k skew-symmetric matrices x with rank(x) 5 min(k,n). This gives rise to the following dichotomy:

2 k, then y is surjective. Hence y* is injective

and P(Mn,k)G E ?(Ahfk) is a polynomial algebra with k(k - 1 ) / 2 generators. One says that AMk is the algebraic quotient of Mnxk by Sp(C", 0).The representation of L on P ( h & x k ) Gis multiplicity-free (see [16,Theorem 5.2.111 or the article by Benson-Ratcliff in this volume).

( 1 ) If n

R. Goodman

344

(2) If n < k then Ker(y*) # 0. From the multiplicity-free decomposition of P ( A M k ) under L one finds that Ker(y*) is generated by a set of Pfafian polynomials of degree n/2 1. Thus P ( I I ~ , , ~is)the ~ algebra of functions on the associated skew-symmetric Pfafian variety. This is the Second Fundamental Theorem (SFT) for O ( n ,C ) invariants (see [16, Theorem 5.2.181 for the complete statement).

+

Summary: For a classical group G (general linear, orthogonal, symplectic), the G-invariant polynomial functions of vectors and covectors are generated by all the possible G-invariant contractions of vectors and covectors. 7. Tensor invariants and proof of FFT 7.1. Tensor invariants f o r GL(V)

We turn now from consideration of invariant polynomials to the general case of invariant tensors. Let V = C" and consider the mixed tensor space V@m8 V*Bk as a GL(V) module. For E (Cx the element
<

V@""8 V*@'" End(@)

(24)

as a GL(V) module, and hence

By Schur duality (Corollary 1.6 and Proposition 3.1) we know that EndGL(V)(vBk)is spanned by the transformations ~ ( s ) s, E 6 k . Let { e l , . . . , e n } be the standard basis for @" and let {e:, . . . , e z } be the dual basis. For an index I = [ i l l . .. , i k ] with 1 5 i, 5 n we set 111 = k and

Recall from Section 3 that the action of s E e,.I. Define

6 k

on k-tensors is ok(s)eI =

Then C, corresponds to (Tk(s) under the isomorphism (24). Thus we obtain the First Fundamental Theorem of Tensor Invariants for GL(V): Theorem 7.1: For Ic 2 1 one has (Vmk8 V*@k)GL(V) = Span{C, : s E 6k}.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

345

The vector space V@'@IV*@~ is self-dual as a GL(V) module, and hence each of the mixed tensors C, can also be viewed as a linear functional. This gives the alternate version of the Tensor FFT for GL(V) in terms of total contractions of vectors with covectors:

Corollary 7.2: The space of GL(V) -invariant linear forms on VBk@ V*@"" is spanned by the contractions k 211

8 . .. @ V k @ u; @ . . . @vv;

H n(V;(j),

Vj)

j=1

for s E Gk. 7.2. Proof of polynomial FFT f o r GL(V)

The polynomial form of the FFT for GL(V) is a consequence of Corollary 7.2. To prove this, we need to view GL(V)-invariant polynomials as tensors with additional symmetries, as we did in Section 5 . 3 . Let ~ k , , = ( c " ) x~(ex), and write t E Tic," as t = [XI,. . . , Xk,y1,. . . ,ym]. Denote the regular characters of T k , m as k

for [p,q] E zk x Vk @ V*" by

z".Let ~

k

m

act, on ~z = [ul ,...,V k ] CB [v;, . . . ,v;]

E

t . 7. = [ X l u l , . . . , X k v k ] @ [ylv;, . . . , y,u;1. This action commutes with the GL(V)-action on V k @ V*", so GL(V) leaves invariant the weight spaces of Tk,, in P ( V k @ V*m). These weight spaces are described by the degrees of homogeneity of f E P ( V k @ V*,) in ui and u; as follows. For p E N k and q E N" set

P [ p q v k@ V*")

=

{f E P(Vk @ v*m): f ( t . z )

=t[P+(z)}.

Then P(Vk @ v*m) =

@ @

P[p>qI(vk @

pENk qENm

and this decomposition is GL(V)-invariant. Thus

v*m),

R. Goodman

346

We now give another realization of these weight spaces in terms of tensors. Given p E Nk and q E Nm we set

v*@p V@q= V*@pl8 . .. @

@

V*@pk@ V@q' 8 . .. @ V@q-

This space is isomorphic to V*@IPI8 V*@lq1and is a GL(V) module with the usual action. Let Bp = BPI x . . . x B,,, with each factor acting as a group of permutations of the corresponding tensor factor in V@P. This gives a representation of 6, x Bq on V*@p@ VBq that commutes with the action of GL(V), Lemma 7.3: Let p E Nk and q E Nm. There is a linear isomorphism p [ P > q l ( v k@ v * m ) G L ( V ) N -

[(V*@IPI@ v@lql)GL(")

1

EPX%

.

(26)

Proof: We have the isomorphisms

--

P(Vk 63 v*m) S((V*)k @ Vm) S(V*) @ ' * @ S(V*) @ S ( V ) @ * ' . @ S ( V ) m factors

k factors

as GL(V) modules. Hence

p[p,ql(Vk @ v * m ) 2 S[PI(V*)@ S q V ) ,

(27)

where S[P](V*)= SPl(V*) 8 . . . @ SPk(V*) and S[ql(V) = Sq'(V) 8 . .. @ SQm (V). We also have a GL(V)-module isomorphism

sr(v)2 [ v @ yc v@r with 6, acting by permuting the tensor positions as usual. Combining this with (27) we obtain the linear isomorphisms p[PAl(p@

V*m)GL(V)g [S[PI(V*) @ Srql(V)]G W f ) N

-

[(V*@IPI V@hl B p x G @

)

I GL(V) .

This implies (26) since the actions of GL(V) and 6, x 6, mutually commute . 0 We now prove the polynomial version of the First Fundamental Theorem for GL(V). This theorem asserts that for each p E Nk and q E N", the space

p[P>sl(Vk@ v * m ) G L ( V )

Multiplicity-&ee Spaces and Schur- Weyl-Howe Duality

347

is spanned by monomials of the form

for suitable choices of k , m and rij. The subgroup GL(V) acts on PIP)ql(Vk@ V * m )by the character assume that IpI = lq(= T , say. By Lemma 7.3,

:

C E C " } of

H C1qI-lP1,

so we may

T = {Clv

C

From Theorem 7.1 we know that the space (V*@r @ V @ r ) G L (is V) spanned by the complete contractions Csfor s E er.Hence the right side of (29) is spanned by the tensors

c

4 ( g ) €4 G(h)CS

(g,h)E6.,x%

for s E 6,. Under the isomorphism (29), the action of G P x 6 , disappears and these tensors correspond to the polynomials

Fs(vl,.. . , ' U k , ' U ; , . . . , v L ) = c,(Uy1@ ." =

(.,""I

@..

@ U p k @U ' ;@"

€4

" '

@u&@q")

. @ U P L , w;€4 . . . @ w;)

where each wi is uj for some j and each wi+ is v;, for some j' (depending 0 on s). Obviously Fs is of the form (28). 7.3. Tensor invariants f o r orthogonal and symplectic QrozlpS

Consider now a nondegenerate bilinear form w on V , which we assume to be either symmetric or skew-symmetric. Let G be the subgroup of GL(V) that preserves w (so G is either the full orthogonal group or the symplectic group). Any mixed tensor that is invariant under GL(V) is also invariant under G, of course. To find additional tensor invariants, we can use the Gmodule isomorphism V V* furnished by w to restrict attention to V * @ k . Furthermore, (V*@k)G = 0 if k is odd, since -I E G. Hence we need only find a linear basis for (V*@zk)G. The given form w E (V*@2)G by definition. Since G preserves tensor multiplication, it follows that e k = W@'

E

(V*@zk)G.

R. Goodman

348

The representation 0 2 k of 6 2 k on V*@2kcommutes with the action of G, of course, so the tensors 0 2 k ( s ) & are also G invariant, for every s E 62k.

Theorem 7.4: For all integers k s E 62k}.

2 1 one has (V*@")'

= Span{gzk(s)Bk :

Because of the symmetries of the tensor e k under the action of 6 2 k , there are redundancies in the spanning set of Theorem 7.4. A labeling that factors out these symmetries is the following, which we will also use in Section 13. Define a two-partition of the set (1,. . . ,2k} to be any set of k pairs E = {{il,jl}, . . . , { i k , j k } } such that { i l , j l , . . . , i k , j k } consists of the integers 1,.. . ,2k.Denote the set of all 2-partitions of k by Zk. For E El, define the complete contraction

c

n k

A((U1 @ " ' @ l ? 2 k )

=

w('uip,v j p ) .

p= 1

(We label the pairs in so that i, < j 'p ; then this definition has no sign ambiguity, even when w is skew-symmetric.) The invariant tensors in Theorem 7.4 are just these contractions.

Corollary 7.5: For all integers k

2 1 one has (V*@")'

=

Span{& :

( E zk}.

Proof of Theorem 7.4:Following Attiyah-Bott-Patodi, we shift the action of G from V*@2kto EndV by a tensor algebra version of the classical polarization operatorsf This transforms the space of G-invariant tensors into a different space of GL(V)-invariant tensors built from G-invariant polynomials on End(V). The results of Section 6 allow us to express these G-invariant polynomials as covariant tensors with no further G-invariance condition. By this means each G-invariant tensor gives rise to a unique mixed GL(V)-invariant tensor. But we know that all such tensors are linear combinations of complete vector-covector contractions. Finally, specializing the polarization variables, we find that the original G-invariant tensor is in the span of the complete contractions relative to the form w. In more detail, given A E V*Bm we define @ A E Pk(EndV) @ V*@mby

@ . x ( Xw) , = (A, X @ m w ) for X E EndV and Since (A, w) = ( a ~ ( l , w ) we , see that the map X

H @A

20

E VBm.

is injective.

fThis can be viewed as unseparation of variables, and is another instance of a hidden symmetry.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

349

Let G C GL(V) be any subgroup, for the moment. Let G act on Pm(EndV) @ V*@mby left multiplication on End(V) only (no action on V*@m).Let GL(V) act by right multiplication on the EndV factor and in the usual way on V*@m.Since these actions of G and GL(V) mutually commute, we obtain a representation of the product group G x GL(V). In particular, (9, h) . @'X(X,w)= (A, (g-lxh)@'mh-l

'

w)= @g.x(x, w)

for g E G , h E GL(V). Hence @ A is automatically invariant under GL(V) for any A, while if X is G-invariant, then so is @ A . Conversely, if @ E Pm(EndV) @ V*@mis invariant under G x GL(V), then @ is determined by the linear functional X : w H @ ( I , w )since @ ( h , w )= @ ( I h, . w) for h E GL(V) and GL(V) is dense in EndV. Furthermore, for g E G we have

@ ( I w) , = @(gI,w) = @(Ig, w)= @ ( I g, . w) (here we have used the inclusion G c GL(V) to pass from the left action of G on EndV to the right action of GL(V) on EndV). Hence @ = @ A with X E (V*@m)G. The map X H @A thus gives a linear isomorphism (V*@m)G2 [pm(EndV)L(G)€9 V*@m]GL(V)

(30)

where L(G) denotes the left-multiplication action of G on End(V). Let X E [V*@2k]G. Then by (30) with m = 2k, we have E [P2k(EndV)G€9 V*@2k] GVV)

. Theorems 6.3 and 6.4 (which we have proved in the case and X = @ x ( I )By = dimV), there is a polynomial FA on S M , x V@2kwhen G = O(V) or on AM, x V @ 2 kwhen G = Sp(V), SO that for all X E Mn and w E V@",

k

@.,(X,w) =

G { F x ( X t X , w)w) when when G Fx(XtJnX,

= O(V), =

Sp(V).

We view FA as an element of Pk(SMn)€9V*@2k (resp.of Pk(AMn)@V*@2k). Note that (A, w)= @A(In,w)=

c

Fx(In, w) when G = O(V) , Fx(J,, w) when G = Sp(V) .

The next step is to translate the GL(V)-invariance of priate invariance property of FA. The map

0 : M,

4

v*€9 v*,

0 ( 2 )=

c

2 i j e;

i,j

@A

€9 e;

into an appro-

R. Goodman

350

furnishes GL(V)-module isomorphisms AM, Hence

Pk(AM,)

Sk(A2V),

E

A2V* and S M ,

Pk(SM,)

Thus there is a GL(V)-invariant tensor C E

E

S2V*.

E Sk(S2V).

I8 V*@2kso that

Fx(A,w)= ( A B k18W , C) for w E V@2kand A in either S2V* or A2 V*. By the tensor FFT for GL(V) (tensor form) we may assume that C is a complete contruction: lil=2k

for some s E B 2 k . When G = O(V) we take A = I , to recover the original G-invariant tensor A as

(A, w)= Fx(I,, w)= (@(In)@k 8 w,C )

( e l , @ ( & d @ k ) ( we:.I) , =(

=

~ ( s ) Q ( Lw). )@~,

JII=2k

When G=Sp(V), we likewise take A = J , to get (A, w)= ( 0 2 k ( s ) O ( J , ) @ ' ~w). , Since when G = O(V) ol = @(J,)@k when G = S p ( V ) ,

{

0

we conclude that A = 0 2 k ( S ) 8 ; . 7.4. Proof of polynomial FFT for orthogonal and

symplectic groups We finally complete the proof of the FFT for the action of G = O(V) or G = Sp(V) on P ( V ) , using an argument similar to the case of GL(V) to deduce the polynomial version of the FFT from the tensor version. Let p E Wm. Since -1 E G and acts by (-1)IPI on PIPl(Vm), we may assume that IpI = 2 k . We now show that the space PIP1(Vm)G is spanned by monomials

n m

cp(v1,. . . , v,)

=

W(Ui,

v

p

i,j=l

of weight p. This will prove the FFT (polynomial version) for G. By Lemma 7.3, p[PI(Vm)G gg [(V*@2k)G]BP,

(31)

M u l t i p l i c i t y - h e Spaces and Schur- Weyl-Howe Duality

351

The space (V*@zk)G is spanned by the tensors u&(s)OE for s E 6 z k (Theorem 7.4). Hence the right side of (32) is spanned by the tensors

tE6,

for s E 6 2 k . Under the isomorphism (32), the action of 6,disappears and these tensors correspond to the polynomials k

F,(v~, .. .

=

c;k(s)e;(vyl

8 . .. 8 ll$p-)

= ~ w ( u , , u ~ ,+ ~ ) a= 1

where each ui is v j for some j (depending on s). Thus Fs is of the form (31). 0 8. Weyl algebra and Howe duality

8.1. Daality in the Weyl algebra

We shall now apply the general duality theorem from Section 1 to the following situation. Let V be an n-dimensional vector space over C and let 5 1 , . . . , z, be coordinates on V relative to a basis { e l , . . . ,e n } . Let E l , . . . , En be the coordinates for V' relative to the dual basis { e ; , . , . , e k } . We denote by P D ( V ) the algebra of polynomial coeficient diferential operators on V . This is the subalgebra of End(P(V)) generated (as an associative algebra) by the operators

d D a. -- ,

8x2

Mi = multiplication by z i (i = 1,.. . , n ).

+

Since (a/dxi)(zjf) = (dxj/dxi)f zj(af/axi) for f E ators satisfy the Heisenberg commutation relations

P(V), these oper-

[Oil M j ] = d i j I for i , j = I , . . . , n

(33)

(the algebra P D ( V ) is often called the Weyl algebra). Define P D o ( V ) = @I and for k 2 1 let P D k ( V ) be the linear span of all products of k or fewer operators from the generating set (01,. . . , D,, M I , . . . , Mn}. This defines an increasing filtration of the algebra PD(V): PDo(V) c . . . c P D k ( V ) c PDk+1(V) c . . . with

u

PDk(V)= 'PD(V)

k20

pDk(v) ' P D m ( v )

c

PDk+m(V).

R. Goodman

352

ekto

Let Gr(PD(V)) = Grk(PD(V)) be the associated graded algebra. If T E PV(V) then we say T has filtration degree k if T E PDk(V) but T $ PVk-l(V), and we write deg(T) = k. We write Gr(T) = T

+ PDk-l(V)

-

E Grk(PD(V))

when deg(T) = k . The map T Gr(T) is a linear isomorphism (but not an algebra homomorphism) from P D ( V ) to Gr(PD(V)). From (33) it is easily verified that deg(MaDP) = la1

+ 1/31

and the set of operators { M a D P : a,/3 E P D ( V ) , where we write ~a

.

= MY'.. Mzn,

for a,/3 E Wn

W"} is a (vector-space) basis for

DP =

. . . D,P,.

Let p be the representation of GL(V) on P ( V ) with P(S)f(Z) = f ( g - W

for

f

EW ) .

We view P D ( V ) as a GL(V)-module relative to the action g T = p(g)Tp(g-l) For g E GL(V) with matrix calculate that

for T E P D ( V ) , g E GL(V)

[gij]

relative to the basis { e l , . . . ,en}, we n

n

p(g)ojp(g-l) = C g i j D i > i=l

p(g)Mip(g-') = C g i j M j . j=1

(34)

+

The set {Gr(M"DP) : la1 IpI = k } is a basis for Grk(PD(V)).Thus the nonzero operators of filtration degree k are those of the form

+

with cap # 0 for some pair a,@with JaI 1/31 = k (note that the filtration degree of T is generally larger than the order of T as a differential operator). If T in (35) has filtration degree k then we define the symbol of T to be the polynomial o(T)E P k ( V@ V') given by

Lemma 8.1: The symbol map gives a linear isomorphism P V ( V ) P ( V @ V') as GL(V)-modules.

Z

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

353

Proof: Using (33), one shows by induction on k that any monomial of degree Ic in the operators D1,. . . ,D,, M I , . . . , Mn is congruent (modulo P D I , - ~ ( V ) to ) a unique ordered monomial MOD0 with Icy1 = Ic. Hence o ( T )= o(S)if Gr(T) = Gr(S). Thus (T gives a linear isomorphism

+

Grk(PD(V))

P k ( V @ V*)

Since p(g)PDk(V)p(g-') = PDk(V), there is a representation of GL(V) on Grk(PD(V)) for each k. From (34) we see that Di transforms as the vector ei under conjugation by GL(V), while Mi transforms as the dual vector e t . Since GL(V) acts by algebra automorphisms on Gr(PD(V)) and on P ( V @V*), this implies that (36) is an isomorphism of GL(V) modules. Now compose these maps with the canonical quotient maps P'Dk(v) -+ Grk( P D ~ ( v ) ) . 0 We can now obtain the general Weyl algebra duality theorem:

Theorem 8.2: Let G be a reductive algebraic group acting regularly on V. Then there is a multiplicity-free decomposition P(V)E

@

EX@@,

(37)

X€C(V)

as a module under the joint actions o ~ P D ( Vand ) ~ @[GI.Here C ( V ) c 6?, FAis an irreducible regular G-module of type A, and E X is an irreducible module for P V ( V ) G that uniquely determines A.

Proof: We apply Theorem 1.4, with L = P ( V ) and R = P D ( V ) . Note that L is the direct sum of the finite-dimensional G-invariant subspaces Lk = P k ( V )of homogeneous polynomials of degree k . The action of G on each LI, is regular, so L is a locally regular G module. We shall show that R satisfies conditions (i) and (ii) of Theorem 1.4. Let 0 # f E P ( V ) be of degree d. Then there is some Q E W" with ( Q (= d such that 0 # D"f E C. Given any g E P ( V ) ,let M g E P D ( V ) be the operator of multiplication by g . Then g E @MgDaf. This proves that R acts irreducibly on P ( V ) (condition (i)). The algebra R is the union of the finite-dimensional G-invariant subalgebras PDk (V), and the action of G on FDI,(V) is regular by Lemma 8.1. Hence R also satisfies condition (ii). 0 To use Theorem 8.2 effectively for a particular G-module V we need a more explicit description of the algebra PD(V)G. The following result is a first step in that direction.

R. Goodman

354

Theorem 8.3: Let {$I,. . . , $J,} generate the algebra P(VCBV*)~. Suppose Tj E PD(V)G are such that a(Tj) = $j for j = 1,.. . , T . Then {TI,.. . , T,} generates the algebra PD(V)G.

Proof: Let 3 C PD(V)G be the subalgebra generated by T I ,. . . ,T,.. Then P D o ( V )=~ CI c 3. Let S E PD,(V)G have filtration degree k. We may assume by induction on k that P D ~ - I ( Vc) 3. ~ Since a(S) E P k ( V ~ V * ) G by Lemma 8.1, we can write

where ~ j ~ . .E. ~CC., Set

Although R is not unique (it depends on the enumeration of the Tj), we have a ( R ) = a ( S ) since 0 is an algebra homomorphism. Hence R - S E PDk-I(V) by Lemma 8.1. By the induction hypothesis, R. - S E 3,so we have S E 3. 0

Corollary 8.4: (Notation as in Theorem 8.3) Suppose T I , .. . ,T, can be chosen so that

g’ = Span(T1,. . . ,T,.} is a Lie subalgebra of PD(V)‘.

Then in the canonical decomposition (37) the spaces E X are irreducible modules for the Lie algebra g‘, and X is uniquely determined b y the equivalence class of E X as a 0‘-module. Hence there is a bijection (duality correspondence)

W )* W)

1

where A(V) is the set of irreducible representations ofg‘ that occur in P ( V ) . Proof: The representations of the Lie algebra g’ are the same as the representations of the universal enveloping algebra U(g’). Let p’ : U(g’) -+ End(P(V)) be the representation associated to the action of g’ as differential operators. The assumption on T I ,. . . , T, implies that p’(U(g’)) = PD(V)G. Hence the irreducible PD(V)G-modulesare the same as irreducible 8’-modules. 0 Remark: Theorem 8.2 is also valid when V is any smooth connected affine G-variety. Here we take R = D(V) to be the ring of algebraic diferen-

355

Multiplicity-Fkee Spaces and Schur- Weyl-Howe Duality

tial operators on V and use Theorem 1.4.9 The algebra D(V)G in this case (G connected, reductive), has been studied by Knop [25]. He proves that its center 3G(V) is a polynomial ring in rankG(V) generators, where rankG(V) = dim Bx - dim N z for a generic point z E V (here B is a Bore1 subgroup of G with nilradical N ) . Furthermore, ItD(V)G is a free module over 3G(V) (this is a generalization of results of Kostant [26] for the case V = G, with G acting by left multiplication). The representation theory of D(V)G seems to be unknown, in general, although special cases have been studied by I. Agricola, F. Knop, T. Levasseur, G. Schwarz, J. Stafford, and others. 8.2. Howe duality f o r orthogonal/symplectic groups We now determine the structure of PD(V)G when G is an orthogonal or symplectic group and V is the sum of n copies of the fundamental representation of G. Using the First Fundamental Theorem of classical invariant theory, we will show that the assumptions of Corollary 8.4 are satisfied. This will give the Howe duality between the (finite-dimensional) regular representations of G occurring in P ( V ) and a set of irreducible representations of the dual Lie algebra 8'. Let w be a nondegenerate bilinear form on C kthat is either symmetric or skew symmetric, and let G c GL(lc,C) be the isometry group of w .Thus G is the (complex) orthogonal group when w is symmetric, and G is the (complex) symplectic group when w is skew (and k even). Let V = (C'))". Then

--

P ( V @ V*) = P( C k63 * * . @ ck@

(P)* @ . . . @ (Ck)*) .

n copies

n copies

Hence if T E P D ( V ) then the symbol of T is a polynomial function

f(x1,.. . ,xn,J1,. . . ,&)

,

where xi E Ck,

&

E (Ck)*.

From Section 6 we know that the algebra of G-invariant polynomials P ( V @ V*)G is generated by three types of quadratic polynomials: evaluation of w on two vectors: evaluation of W * on two covectors: contraction of vector-covector:

. . , xn,tl,. . . ,&) . . ,En)

pij(x1,.. . ,xn,&,.

= w(xi,xj) = w*(&, [ j )

cij(x1,.. . ,x,,&,

= (xi,&)

rij(x1,.

. . . ,&)

gSee [l];the smoothness assumption on V is essential here, since the action of D(V) on Aff(V) can fail to be irreducible when V is not smooth.

R. Goodman

356

where 1 5 i , j 5 n and w* is the form on (Ck)* dual to w . There is a canonical GL(V)-module isomorphism d from P(V*) E S(V) to the algebra of constant-coefficient differential operators on V. The linear span of the quadratic invariant polynomials above furnish symbols for the following Lie algebras of G-invariant differential operators:

p- = Span(multip1ication by rij : 1 5 i , j 5 n } p+ = Span(differentiati0n by Aij = a ( p i j ) : 1 5 i , j 5 n } t = Span{Eij + $ ~ i :j 1 5 i , j 5 n> Here it is convenient to identify V with M n x k , with G acting by right multiplication. If xi denotes the ith row of x E Mnxkrthen k

Eij = xi . VXj=

d 1 xir -. r=l axji"

The operators Eij , which correspond to vector-covector contractions, commute with the right action of all of GL(lc,C) (Eij is the classical polarization operator). Obviously [p-,p-] = 0 and [p+,p+] = 0. An easy calculation shows that lelP,I

The choice of shift tation relation.

f&j

= P,

9

[P-,P+I = I!.

for the operators in t arises from the last commu-

p- + t + p + . Then 5' is a Lie algebra and it generates the associative algebra P v ( h f n x k ) G . Furthermore, Theorem 8.5: Set 5' 5,

~

=

{ sp(n,@) 50(2n,@)

The subalgebra t representation

S

when w is symmetric when w is skew

5K(n,@) acts o n

p(j?fnxk)

b y the differential of the

of K = GL(n,@) (replace K by its two-fold cover if Ic is odd).

Proof: The first statement follows from Corollary 8.4. The other parts are 0 easy calculations (see [16, Sec. 4.51 for details). We call g' the Howe dual of g = Lie(G) associated to the representation of 5 on V. Notice that the correspondence between 5 and g' interchanges orthogonal and symplectic Lie algebras. There is an asymmetry between g and g', however. The action of 5 on P ( V ) is by vector fields (corresponding

Multiplicity-Free Spaces and schur- Weyl-Howe Duality

357

to the representation of G on V ) ,whereas the subalgebras pk of g' act by second-order differential operators and multiplication by quadratic polynomials, which do not come from a geometric action on V . We will show in Section 11 how to exponentiate the action of a real form of g' on P ( V ) to a unitary representation of an associated real Lie group on a Hilbert-space completion of P ( V ) . 8.3. Howe duality for GL(k)

--

Now consider G = GL(k,@) acting on

v = ck@ . . . €9 ck€9 (C"*@ ' . . €9 (C"* p copies

q copies

In this case the symbol of T E P D ( V ) is a polynomial function

f(x1,.. . , x p , v 1 , .. . , 7 7 q , F l , . . . , E P , Y l , .. . , Y q ) , where [XI,. . . , xP,q1,.. . , vq] E V and [(I,. . . , Q , Y I , . . . ,yql E V * (xi,y j are vectors in C k and &, vj are covectors in ( C k ) * ) Theorem . 6.2 asserts that the algebra of G-invariant polynomials on V @ V* is generated by contractions of a vector with a covector. Now there are four possibilities for contractions: (1) vector and covector in V : (xi,q j ) for 1 5 i 5 p and 15j.59

(2)

vector and covector in

V*:

(yj,

ti)

for 1 5 i

< p and

I l j < q

(3) vector from V , covector from V * : (xi,&) (4) covector from V , vector from V * : (yi,733)

We can identify V with Ad(,+,)

k

for 1 5 i , j 5 p for 15 i , j < q

if we make g E G act on the right by

Here xi is the ith row of x and v3 the j t h row of v. Contractions of type (1) and ( 2 ) furnish symbols for the G-invariant operators

p- = Span{ multiplication by rij = (xi,vj)) p+ = Span{ differentiation by Aij} , where

Aij

= V,,

. V,

=

8 C" T=l

dxiT

%jT

for 1 < i < p a n d 1 < j 5 q .

R. Goodman

358

The linear span of contractions of type (3) and (4) furnishes symbols for the G-invariant operators t = Span{E$)

+ ~ d i :j 1 5 i,j L p)

+ $6ij

Span{,$)

:

1I i,j

I q) ,

where E$) is the polarization operator for the x variables and Ei:) for the 77 variables. By the same argument as in Theorem 8.5 we conclude that

P V ( V ) Gis generated by

+ e + p+ .

g’ = p-

These subalgebras have the commutation relations

P, P*I

= P*

[P-,P+l c e

I

*

+

In this case g’ is isomorphic to gI(p q , C ) , with t 3 gI(p, C) @ g ( ( q , C). The action of e on P(M,,k @Mpxk)is the differentialof the representation

p(g,h)f(x,7 ) = (det g det h)-”’f(g-’x,.h-’7) for (9,h ) E K = GL(p, C ) x GL(q, C). (We must replace K by the two-fold covers of each factor when n is odd). 9. Harmonic duality

9.1. Harmonic polynomials

Let G be O ( C k , w ) ,Sp(Ck,w),or GL(lc,C) acting on V = M n x k on the right. In the case of GL(Ck) the first p rows of 2 E Mnxk transform as vectors, whereas the remaining q rows transform as covectors. From Section 8 the Howe dual to G is, respectively, g’

sp(n, C), so(2n,C), or gI(p

+ q1C )

with p

+q =n.

We will assume that p > 0 and q > 0 in the third case.h With G fixed, the spectrum C ( V ) of G on P ( V ) only depends on n (or the pair p,q in the third case); we can thus denote it by E(n) (or C ( p ,4)). From the dual point of view if we fix g’, then the set A(V) of irreducible representations of g‘ that occur in P ( V ) only depends on k ; we can thus denote it as A ( k ) . The general duality theorem gives a bijection E ( n ) H R(lc). We now show how to express this bijection in terms of harmonic duality. hIf p = 0 or q = 0 then e = g’ and the modules E X8 F A that occur in the decomposition of P(V)are finite-dimensional, This is the well-known GL(n)-GL(k) duality (see the lectures of Benson-Ratcliff in this volume).

Multiplicity-FI-ee Spaces and Schur- Weyl-Howe Duality

359

In all cases there is a triangular decomposition g1 = p- @ e e 3 p+

.

Here C is the Lie algebra of the reductive group K (a two-fold cover of GL(n, C ) or GL(p,C ) x GL(q, C ) in general). The representation of K on P ( V ) is the natural representation associated with the left multiplication action of GL(n, C ) on V tensored with the one-dimensional representation g

H

(det g)-'/'

or

(g, h ) H (det g det h)-'/'.

Let 6 denote this character, viewed as a weight of the maximal torus of K . The subalgebra p- acts by multiplication by G-invariant quadratic polynomials, whereas p+ acts by G-invariant constant-coefficient Laplace operators {Aij}. We define the G-harmonic polynomials to be

ni,j

'FI = P ( v ) ~ =+

Ker(Aij)

Since Ad(K)p+ = p + , the space 'FI is invariant under the reductive group K x G. In this section we will show that 'FI gives a multiplicity-free duality pairing between irreducible representations of K and G; furthermore, the decomposition of Fl generates the decomposition of P ( V ) under g1 and G. Let U ( k ) c GL(k,C) denote the unitary group. Then KO= K n U ( n ) is a compact real form of K . We assume that the bilinear form w is chosen so that Go = G n U ( k ) is a compact real form of G and w is real on R k . Define an inner product on Mnxk by

(x I y) = tr(y*x) for x , y E M n x k (y* = g t ) . This inner product is invariant under U ( n )x U ( k ) ,acting by left and right multiplication, hence it is invariant under KOx UO.We set 11z11' = (z I x). Let f(x) = C, caxa be in P ( V ) , where LY is a multi-index and xa = as usual (xij are the matrix entry functions on i k f n x k ) . Define the constant-coefficient differential operator

a(f)

is an algebra isomorphism from P ( V ) to the constantThen f H coefficient differential operators on V that is equivariant relative to the action of U ( k )x U ( n ) .Set g*(z) = s(z)for g E P ( V ) .If g ( x ) = C , daze, then

(a(f)g*)(o)=

c

a! c a d , .

a

(38)

R. Goodman

360

We define (f I g ) = ( a ( f ) g * ) ( O ) .From (38) we see that this is a positive definite Hermitian inner product on P ( V ) ,called the Fischer inner product. We note that

( f g I h) = (f I d(g*)h)

(39)

for all f , g , h E P ( V ) . The Fischer inner product has the following analytic definition. Denote Lebesgue measure on V by d X ( z ) , where we identify V with RZnkvia the real and imaginary parts of the matrix coordinates.

Lemma 9.1: For f , g E P ( V ) one has

( d = dim@V = nk).

Proof: See the lectures of Benson-Ratcliff in this volume.

0

9.2. Main theorem

We now apply the Weyl algebra theorem from Section 8 to obtain multiplicity-free decompositions of the harmonic polynomials and the entire space P (V).

Theorem 9.2 (Harmonic Duality): (1) The space 'H of G-harmonic polynomials on V decomposes under K x G into mutually orthogonal subspaces (relative to the Fischer inner product) as 'H = @

&'(")+6@3"

oEC(V)

Here C ( V ) c is the spectrum of P ( V ) as a G module, 3' c H ' is an irreducible G-module of type 0, and c 'H is an irreducible finite-dimensional K-module with highest weight r(c)+ 6. In particular, every irreducible representation of G in P ( V ) is realized in the harmonic polynomials. (2) Set E 7 ( u ) f 6= P ( V ) G.&'(')f6. ule and

Then E'(")+' is an irreducible g' mod-

P ( V ) = @ E'(")+6@3" UEC(V)

is an orthogonal decomposition of P ( V ) (relative to the Fischer inner product) under the mutually commuting actions of g' and G. (3) The map c H ~ ( 0from ) C(V) free as a K x G module.

---f

k is injective. Thus 'H is multiplicity-

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

361

Proof: Since 3-t is an invariant subspace for the reductive group K x GI x and multiplicity function m : -+ { 1 , 2 , . . .} there is a subset I? c such that

2

3-t g @

@"(PP)

@

&P+b

@

30

(40)

(p,+r

with K x G acting trivially on the multiplicity spaces @ " ( P i u ) . Indeed, we first consider 3-t as a locally-finite K x G-module relative to the natural left-right action on V = M n x k (omitting the determinant twist from the K representation) and use Proposition 1.1and Example 1 in Section 2.3. Then tensor with the character det-"' of K to shift the highest weights from p, to p, b. To prove that r = { ( T ( o ) , o ) : D E C ( V ) } and ~ ( T ( o ) , D = ) 1, we need to examine the action of 0' on P ( V ) in more detail. Let J' = P ( V ) G be the G-invariant polynomials, and let 3. be the homogeneous polynomials of degree j in 3. Then J'j = 0 for j odd, and ( p - ) j acts by multiplication by 3 2 . on P ( V ) . Since the bilinear form w is real on R k , we have J * = J and 3-t* = 3-t. Let J+ = {f E J' : f ( 0 ) = 0). We claim that

+

7-P = J+.P ( V )

(41)

(orthogonal complement relative to the Fischer inner product). Indeed, if

f E ,7+ . P ( V ) and h E 3-t then a(f)h = 0 by definition of 'HI and thus f Ih. Conversely, if h IJ+ . P ( V ) then for all f E P ( V ) and g E J+, 0 = (fg I h ) = (f I a ( g * ) h ) . Hence d ( g ) h = 0 for all g E ,7+,so we have h E 3-t. We can now determine the general structure of the irreducible 0'modules in P ( V ) . The commutation relations in 0' can be expressed as

P+P-

c P-P+ + e

I

eP-

c P-(e

+ 1)

in the universal enveloping algebra U(0'). Hence by induction, one has

P+(P-)"

C

(P-)"P++(P-)"-l(e+l)

for all integers m 2 1. Thus if 2 then (42) implies that

p+(p-)".

2 c (p-)"-'

e(P-)"

C

(P-)"(t+l)

(42)

c 3-t is any P-invariant linear subspace,

.Z and t ( p - ) " . 2 c (p-)" .Z

(43)

for all m 2 1. (a) L e t & c 3-t be a n y t-irreducible subspace. S e t E = J .&. T h e n E is a n irreducible 0'-module and & = E n 3-t.

R. Goodman

362

Indeed, (43) implies that E is invariant under g’. Also every f E E is of the form m

f=xgjhj

whereOfgjEJ2jandhjEE.

(44)

j=O

Suppose F c E is a nonzero g‘-invariant subspace. Take f E F so that the integer rn in (44)is minimal. Then (43) implies that p + f = 0. Hence f E 3-1. Thus m j=1

Since the left side is in J+.P(V), it must be zero by (41). Hence we conclude that f E &. But t acts irreducibly on &, so U(0)f = & and thus F = E . The same argument shows that E n 3-1 = &, completing the proof of (a). (b) Let E c P ( V ) be an irreducible g’module. Set & = E n 3-1. Then & is an irreducible t-module and E = J .&. Note that the action of p+ on P ( V ) lowers the degree of polynomials, so & # 0. If 0 # 3 c & were a proper t-submodule, then J .F c E would be a proper irreducible g’-submodule by (a). Hence & must be irreducible as a t-module and E = J . &, proving (b). (c) Let & and 3 be t-invariant subspaces of 3-1. Assume that & I3 (relative to the Fischer inner product). Set E = J ‘ & and F = J .3. Then E I F . By (41) we have the orthogonal decompositions

E

=&

@J+.&

F = 3@ J+ .3.

Thus E I 3 and F I&, so we only need to verify that Now

J+ . & I J+ .3.

(J+ I Jt . 3)= (E I 8(J+)J+. 3 ) . ’

But 8(J+)Jt . 3 proving (c).

c F since 3 is &invariant. Hence & I 8(J+)Jt . 3,

We now complete the proof of the theorem. It is clear from the integral formula for the Fischer inner product (Lemma 9.1) that Go and KO act by unitary operators on P ( V ) , hence the decomposition (40) of X is orthogonal relative to the Fischer inner product because Go and KOhave the same finite-dimensional invariant subspaces in P ( V ) as G and K , respectively (see [16, Sec. 2.4.41). Also, since K is connected, a finite-dimensional subspace of P ( V ) is invariant under K if and only if it is invariant under 0.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

363

Let ( p , a) E I’ occur in (40). By (a), (b) and Theorem 3.4 we know that the irreducible 8’-module E’”+6= ,7 . €’”+6 uniquely determines p. On the other hand, Theorem 8.2 asserts that P ( V ) is semisimple as a 8‘-module, with the 8’ multiplicity spaces being irreducible regular G-modules corresponding bijectively to the associated 8’-modules. Hence J? is determined If we call this projection C and write the elements by its projection onto of I’ as (~(a),cr), then the map a H .(a) is injective. The multiplicities m ( ~ ( c ) ,= c )1 for all a E C, since otherwise (a) and (c) would imply that 3“ is paired with more than one copy of an irreducible 8‘ module, contradicting Theorem 8.2. Finally, (b) implies that C = C ( V ) ,since P ( V ) is semisimple as a g’-module 0

e.

Remarks: (1) Theorem 9.2 was obtained by Howe in his influential paper [21] (which circulated as a preprint for more than a decade); his proof used an argument based on a filtration by finite-dimensional subspaces and the classical double commutant theorem, instead of Theorem 8.2. Knowing that the decomposition of the harmonics is multiplicity free simplifies the task of finding harmonic highest weight vectors, as we will see in Section 10. (2) The shift by S in the highest weights for K in the harmonic decomposition would appear to be a minor nuisance. In fact, it plays an important analytic role. For T E P D ( V ) let T * denote the adjoint of T relative to the Fischer inner product:

If we write T in polarized form as T = Cj f j a ( g j ) with fj, g j E P ( V ) ,then we see from (39) that T* = Cjg,*a(f,’). Hence ( p * ) * = pr and t* = t. It follows that := {T E 8’ :

T* = -T}

is a real form of 8’.We will show in Section 11 that the irreducible representation of gb on the space E X ,with X = .(a) 6, can be integrated to a

+

unitary representation .TT’ of a (noncompact) real group Gb with Lie algebra gb. The shift by 6 controls the rate of decay a t infinity on Gb of the matrix entries of d. We will show in Section 12 that for lc large enough (relative to n),the representations .TT’ are square-integrable (recall that 6 = lc60, where So is a fixed weight of 8’).

(3) The injective map a ++ is called the theta-correspondence (more precisely, the local theta-correspondence over R) because of the connection between the oscillator representation and theta-functions (see [5]), There

364 h

R. Goodman

are many recent papers devoted to the problem of understanding the thetacorrespondence from a geometric orbit perspective (see [19, Chap. 121 for a survey). 10. Decomposition of harmonic polynomials

We now turn to the explicit determination of the harmonic duality from Section 9 when G is the orthogonal group and g' the symplectic Lie algebra (for the other two cases, when G is the symplectic or general linear group, see [22]and [S]).It is convenient to take G as the orthogonal group O ( C k w) , for the symmetric form w ( x ,y ) = X t C k y on C k , where

when

when

Here Il denotes the 1 x 1 identity matrix. This choice of w ensures that the diagonal matrices in G give a maximal torus. Also G is a self-adjoint matrix group (invariant under g H g * ) , so the subgroup Go = G n U ( k ) is a compact real form of G, and w is real on the real matrices, as we assumed in Section 8. In accordance with the block decomposition of c k , we write elements z E Mnxk as

z = [ x y ] whenk=21,

z=[x y

t ] whenk=21+1,

(45)

where x , y E Mnxl and t E @". Define the map

P : Mnxk

-+

SMn,

P(Z)

= zCkzt.

From Theorem 6.3 we know that the algebra of G-invariant polynomials on Mnxk (relative to right G-multiplication) is generated by the matrix entries of 0:

P(Z)P,

=

EL1

(YPS

xqs

+ XPS Yqs)

c",=, (yPsxqs + xPsy q s ) + tptq a(Ppq)

when Ic = 21, when k

= 21

+ 1.

(46)

We denote by Apq = the corresponding constant-coefficient differential operators, as in Section 9.1. The space of G-harmonic polynomials is

Multiplicity-he Spaces and Schur- Weyl-Howe Duality

365

Denote by 'Ft(j) the G-harmonic polynomials that are homogeneous of degree j . The space IH is invariant under GL(n, C)x G with the action ~ ( hg ), f ( z )= f(h-l.9)

for h E GL(n, C ) and g E G .

(Note that we have omitted the factor (deth)-k/2 that occurs in Theorem 8.5, so IT is single-valued on GL(n, C)even when k is odd). From Theorem 9.2 we know that 'H decomposes under the representation IT as a multiplicity-free direct sum' 'Ft

=

@ E'(")

@ FU.

UTEC

We will now determine C and the duality correspondence c H ~ ( 0 )The . key point is to find generators for the algebra W N n x Nof harmonic highest weight vectors, relative to a Borel subgroup B, x B c GL(n,C) x G. Here B, = D,N, is the upper-triangular subgroup of GL(n, C ) (D,the diagonal matrices, N , the unipotent upper-triangular matrices), and B = H N is a Borel subgroup of G. The fact that 'Ft is multiplicity-free under GL(n, C )x G will play a crucial role.

Notation: We denote by ~j the character diag[al,. . . , a,] ++ aj of D,. We write N$+ for the integer ptuples X = [ml, . . . ,m,] with ml 2 m2 2 . . 2 mp 2 0. Set 1x1 = m l + . . . + m , and define the depth of X to be the smallest integer i such that mi > 0 (if X = 0, set depth(0) = 0). +

10.1. O ( k ) Harmonics (k odd) Assume that k = 21+1 is odd. Then G = G ox {H}, where Go = SO(C',w) is the identity component of G. We fix the Borel subgroup B = H N c Go as follows. The maximal torus H consists of the diagonal matrices h = diag[zl,. . . ,zi,zF1, . . . ,zcl,11, xi E C x .

The unipotent radical N has Lie algebra n consisting of the matrices with block decomposition

[: O

b -at -Ct

O

,

a E Mixi strictly upper-triangular, b = -bt € M i x i ,C E C ~ .

'The irreducible GL(n, C)-module

&r(u)

I$::',

in the notation of Section 3.

(47)

R. Goodman

366

The weights of H are parameterized by Z1. For h E H and X = [ml,. . . , ml] E Z'we set hX = xy' . . . xy' for the corresponding character of H . The irreducible representations of G remain irreducible on restriction to Go and (? is parameterized as {d>'where }, X E.N'++- is the highest weight ~ l . G = G I U G-1, where for G o ,E = f l , and 7 r X l ' ( - I ) = ~ ( - 1 ) lThus h

21 = {(A,

h

1) : X E

,

N:+}

G-1 = {(A, -1) : X

E

PI;+},

(see 116, Sec. 5.2.21).

+

Theorem 10.1: (G = O(Ck,u), k = 21 1) Let C be the spectrum of G on the G-harmonic polynomials 'Ft C P ( M n x k ) . (a) Assume k 5 n. Then C

=

2 and

hence C does not depend on n

(G-stable range). (b) Assume 1

< n < k. Then (?I c C and C n (?-I

= {(A, -1) :

k - n 5 depth(X) 5 l }

(unstable range: C depends on k and n ) . (c) Assume n I 1. Then C n

2-1= 8 and

C n (?I = {(A, 1) : depth(X) 5 n} .

Thus C does not depend on k (GL(n)-stablerange). The duality correspondence is given as follows: Let X = [ml,. . . , md, 0 , . . . , O] E Pi$+ have depth d with 0 5 d 5 min{l,n}. Then '

-

[o,. . . ,o, -md,. . . , -ml]

for u = (A, 1) E

c n El,

n-d

.(a)

=

[O,. . . , O , -1,.

. . -1, - m d , . . . , -ml]

-A n-k+d

k-2d

for u = (A, -1) E

cnC-l.

Remark: The parameter E for the representation diE is determined by the corresponding GL(n)highest weight .(A, E ) , since left multiplication by -In on Mnxk is the same as right multiplication by -Ik. Hence 'Ft is also multiplicity-free as a module for GL(n,C)x Go (this property will be used in the proof). The first step in the proof of Theorem 10.1 is to find a set of generators for the joint eigenfunctions of B, x B in 'H. Just as in the case of GL(n) x

Multiplicity-free Spaces and Schur- Weyl-Howe Duality

367

GL(k) duality (see the article by Benson-Ratcliff), the general strategy is to take appropriate minor determinants. By (46) the operators Apq are given in coordinates as

The minors of z = [x y t] are linear functions of each column of the matrix components x, y, t. If the minors are chosen to depend only on x or to be linear in t , then they will obviously be harmonic. If they depend on both x and y, then interchanging a n x column for a y column will change the sign of the minor but not change the action of the operators Apq= Aqp, so once again the minor will be harmonic. We now proceed to carry out this program. Let p 5 n and q 5 1. For u E Mnxl define p x q submatrices Lp,q(u)and Rp,q(u) of u by

For t E @" and j 5 n define

qj)=

[tnI:"]

EP - j

(the bottom j entries o f t ) . Let z = [ x y t ] E M n x k as in (45). Define f j ( z ) = detLj,j(x) for 15j 5 min(1,n).

If n

2 1 + 1 then we also define

Sj(Z)

=

i

+

I

for j = 1 1 , det [ L j , l ( 4 t ( j ) det [Lj,l(x) Rj,j-~-~(y) tcj)] for 1 2 I j I min{n, Ic}.

+

Lemma 10.2: (a) Let 1 5 j 5 min{l,n}. Then B, x B eigenfunction of weight ( p ,v), where p=

-&,-j+l

- . . . - E~

and

fj E

> 1 and let 1 + 1 5 j 5 min{n, Ic}. Then g j a B, x B eigenfunction of weight (ply), where

(here y = 0 zfj = I c ) ,

- . . . - E,

fj

is a

v = E ~ + . . . + E ~ .

(b) Assume n

p = -&n-j+l

' H ( j ) and

and y = E I +

9

.

.

E 'H(j)

+&k-j

and g j is

R. Goodman

368

Proof: Assume 1 i j 5 min{l, n}. It is clear from (48) that Apqf j ( z ) = 0 for 1 5 p , q 5 n, since f j ( z ) only depends on x. The diagonal matrices in Bn and B act on f E P(hfnxk)by with a E Dn, b E D1 .

f(x,y , t ) H f (a-'zb, a-lyb-l, a-'t),

(49)

+

Since f j involves columns 1 , .. . ,j and rows n - j 1,.. . ,n of x, it has weight ( p , v ) as stated (the columns of x transform as vectors under H , whereas the rows of x transform as covectors under D,).To verify that f j is fixed under the left action of N,,note that u E N, acts by x H ux.Since u is unipotent upper triangular, this action transforms the ith row of x by adding multiples of rows below the ith row, so it fixes f j . To verify that f j is fixed under the right action of the unipotent radical N = exp n of B , we observe from (47) that N is generated by the subgroups N A , N B , NC consisting of the matrices

[:

0 (a;-1

:j [:0' :] [: 11

0

I

I b O

'

'

1 -2cc

YCt

t

1

(50)

respectively, where a is upper-triangular unipotent, bt = -b, and c E C1. The elements of the subgroup N A act by [x y t] H [xu y t ] ,while the subgroups N B and Nc fix x. Hence the action of N transforms the ith column of x by adding multiples of columns to the left of the ith column, so f j is invariant under N . This proves part (a) of the lemma.

+

Now assume n > 1 and 1 1 5 j 5 min{k,n}. Then g j ( z ) is a linear function of t and does not depend on the variables yrs for s 5 k - j . Hence by (48) we have A p q g j ( z ) = 0 if min{p, q } 5 k - j . If min{p, q } 2 Ic - j 1, then

+

+

Fix s with Ic - j 1 5 s 5 1. If the column #s of x and column #s of y are interchanged in the determinant defining g j ( z ) , then g j ( z ) changes sign. Hence the function

likewise changes sign since the differential operator is symmetric in the variables x and y. But h ( z ) is of degree zero in the variables xps,xqs,yps,

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

369

yqs since g j ( z ) depends linearly on each variable. Hence h ( z ) = 0. This proves that g j ( z ) is G-harmonic. Since gj involves columns 1,. . . ,1 of z and columns k - j + 1,.. . ,1 of y , we see from (49) that gj transforms under H by the weight

7 = (El

+ + ' ' '

El)

- (&k-j+1 + ' * ' + E l )

= &1

+ + &k-j ' ' '

.

+

Since gj involves rows n - j 1,.. . , n of 2 , it transforms under D, by the same weight p as does fj. It is clear that g j is fixed under the left action of N,. To verify that gj is fixed under the right action of N , it suffices to check the action of the matrices in (50). These give the transformations

respectively. The determinant defining g j involves all the columns of z and t . Since the columns of zcct, zc, and tct are linear combinations of the columns of z and t , it is clear that these transformations fix g j . 0

Corollary 10.3: Let m = [ m l ,. . . ,m,] E NI'++, where r = min(1,n). Assume that m has depth d and set X = [m,0,.. . ,O] E N :+. Define q m = flml-mz . . . md-1-md fT" (when m = 0 set cpo(z) = 1). fd-1 (a) pm is a G-harmonic polynomial, homogeneous of degree Iml. Thus cpm(-z) = (-l)lmlcpm(t) for t E M n x k . Furthermore, cpm is a B, x B eigenfunction of weight ( a ,A), where

-

a = [ 0,. .. ,O,

-md,..

. ,- m l ]

n-d

(when m = 0 take a = 0).

+

(b) Suppose n > 1 and n - k d 2 0.For m # 0 , define $m = v m g k - d l fd (when m = 0 set $0 = g k ) . Then $m is a G-harmonic polynomial, homogeneous of degree JmJ+ Ic - 2d. Thus ?lm(-,z) = - ( - l ) ~ m ~ $ m ( zfor ) z E M n x k . Furthermore, $ J is~ a B, x B eigenfunction of weight @,A), where /? = [ 0 , . . . ,0, -1,. . . , -1, -md,. . . , -ml]

-

vn-k+d

k-2d

(When m = 0 take ,6 = [ 0 , . . . , 0,-1 n-k

k

R. Goodman

370

Proof: Since p m ( z ) is a function of IC alone (where z = [z y above), it is clear that pm is G-harmonic. For the same reason, A p q $m = ( p m / f d ) A p q g k - d

t] as

=0.

Thus we see that $m is also G-harmonic. The other assertions are immediate consequences of Lemma 10.2 0

Now we turn to the proof of Theorem 10.1. A B, x B joint eigenfunction generates an irreducible subspace under the action of GL(n) x Go by Theorem 3.5.Since the space of harmonic polynomials on Mnxk is a multiplicity-free GL(k,cC) x Go module, it follows that a Bk x B eigenfunction is uniquely determined (up to a scalar multiple) by its weight and parity. If m E PI$+ has depth d 5 n, then from Corollary 10.3 we see that the right translates of pm under G span an irreducible space of type (m, l), while the right translates of $m under G span an irreducible space of type (m, -1). When k 5 n,then the conditions n > 1 and n - k d 2 0 in part (b) of Corollary 10.3 are automatic. Thus every irreducible representation of G occurs in ‘H in this case, as asserted in part (a) of the theorem. To prove parts (b) and (c) of the theorem, assume that Ic > n. Let f E ‘H be a B, x B eigenfunction. Define a polynomial f on MkXk by

+

-

We claim that f is G-harmonic. Indeed, if min{p, q } 5 k - n then Apqf = 0 since does not depend on the variables zpq for p 5 k - n. On the other hand, if min{p, q } > k - n then

f

A P q f ( z )= Ap!qif(z’’) =0

+

+

(where p’ = p - k n and q‘ = j - k n),since f is G-harmonic. To see that Tis a Bk x B eigenfunction, write b E Bk as

where Then ?(b-lzb’) = f(S-lz”b’) for b’ E B . Since f is B, x B eigenfunction, it follows that is a Bk x B-eigenfunction. Furthermore this shows that Bk weight p of ?is of the form

7

-

p = [ 0 , . . . , O , a,, k-n

. . . ,all

with a, 2

. . . 2 a1 .

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

371

Thus by part (a) we know that ?is a multiple of either vrn or qrnfor some m E N!++of depth d 5 n, since f is homogeneous. If 1 < n < k,then vm is defined for all m E N\+,but $rn is only defined when the depth d of m satisfies k - n 5 d 5 1. This implies part (b) of the theorem. If n 5 1, then (pm is defined for all m of depth d 5 n, but in this case $rn is never defined. This implies part (c) of the theorem. The formula for the map 7 follows from the formulas for Q and p in Corollary 10.3. 0 10.2. O ( k ) Harmonics (k e v e n )

We now assume that k = 21 is even. We take the Bore1 subgroup B c G whose Lie algebra consists of the matrices with block decomposition (block sizes 1 x 1) upper-triangular,

(If k = 2 then b = 0 and B 2 C " ) . Let N c B be the unipotent radical (the matrices as above with a upper-triangular unipotent). Recall that O(C',,W)= Go x ( 1 , s )

where Go = SO(C',u) is the identity component and s E G is the reflection interchanging the basis vectors el and e21 and fixing all other basis vectors ei. Since s normalizes B it acts on the characters of B. Let X = [rnl,. . . ,ml] E Nk+.If rnl # 0, then s . X # X (since s changes rnl to -ml). In this case there is a unique irreducible G representation 7rITx>Osuch that

(where p ) . If

7rp

rnl

denotes the irreducible Go representation with highest weight

= 0, then there are two irreducible representations

of G whose restriction to Go is &E

7rA.

7rA+ ( E

= fl)

They are related by

= det @7rAi-'

and labeled so that 7rA+(s)acts by E on the Go highest weight vector (see [16, Sec. 5.2.21). Thus can be written as a disjoint union = G-1 U u el,where

G*I

z

e

A

h

=

{7rTTX+ :

depth(X) < 1, E = fl} , Go = {7rAi' : depth(X) = 1 , =~ 0 ) .

R. G o o d m a n

372

Theorem 10.4: (G = 0 ( C k l w ) ,k = 21)

the G-harmonic polynomials 7-1 C (a) Assume k 5 n . Then C = range).

e and thus C does not depend on n (G-stable

(b) Assume 1 < n < Ic. Then

C ne - 1

Let C be the spectrum of G on

P(kfn,k).

c C, whereas

U

{(A, -1) : Ic - n 5 depth(A) < 1 )

=

(unstable range: C depends on k and n ) . (c) Assume n = 1. Then C (d) Assume n

=

61U eo.

< 1. Then C = {(A, 1) : depth(X) 5 n} C 6:.

Thus C does not depend on k when n 5 1 (GL(n)-stablerange). The duality : N correspondence is given as follows: Let A = [ml,. . . ,m d , 0 , . . . , 01 E + have depth d with 1 5 d 5 min{ 1 , n}. Then

r(o)=

I

--

[O, . . . , O ,

-md,.

. ., - m ~ ]

for o = (A, E ) E En (El uEo),

n-d

[O, . . . , 0 , -1,. . . , -1, n-kfd

-md,.

. . ,-ml]

for

(T

= (A, -1) E

c nE d 1 .

k-2d

To prove the theorem, we will find a set of generators for the joint eigenfunctions of B, x B in 7-1. For u E M n x l , p 5 n , and q 5 1, define ) Rp,q(u) as in the proof of Theorem 10.1. In this case matrices L p , q ( uand we write z = [ x y ] as in (45) and we define fj(z)

If n 2 1

= detLj,j(x)

for 15 j 5 min{l,n}.

+ 1 then we also define

g j ( z ) = det [ L j , l ( x ) Rj,j-l(y)

]

for 1

+ 15 j

Lemma 10.5: (a) Let 1 5 j 5 min{l,n}. Then B, x B eigenfunction of weight ( p , v ) , where - . . . - E,

p = -&n-j+l

Furthermore fj(zs)

=

and

fj(z) if j < 1, where s

+

fj

5 min{k,n}. E

7-1(j)

and

is a

v = ~ l + . . . + ~ j .

E

G is the reflection el

+

(b) Suppose n 2 1 1 and take 1 1 5 j 5 min{n, k } . Then g j E g j is a B, x B eigenfunction of weight ( p l y ) , where p = -Enpj+1

fj

- . . . - cn and y = E I + . ' .

+

H

e21.

'Fl(j)

and

&k-j

(here y = 0 if j = k). Furthermore g j ( z s ) = - g j ( z ) , with s E G as in (a).

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

373

Proof: Essentially the same as the proof of Lemma 10.2. Note that in this case the unipotent radical N of B is generated by the transformations

z

+-+

[ z a y] , z

H

[z y + z b ] ,

0

with a upper-triangular unipotent and bt = -b.

Corollary 10.6: Let m = [ m l ,. . . ,m,] E PI:+, where r = min(1,n). Assume m has depth d and set X = [m,O,. .. ,0]E N\+. Define (Pm = f;"'-"". . f dm- dl - l - m d f r d (when m = 0 set cpo = 1).

-

(a) cpm is a G-harmonic polynomial, homogeneous of degree \mi. Furthermore, vrn is a B, x B eigenfunction of weight (a,A), where

a = [ 0 , . . .,0 , -md,. . . ,-m1]

,

n-d

(when m = 0 take a d < 1.

=

0). Set

cpk(z)= cpm(zs). Then cp&

=

vm

when

+

(b) Let n > 1 . If d < 1 and n - k d 2 0 define Gm = (Pm g k - d l f d (when m = 0 set $0 = g k ) . Then qbrn is a G-harmonic polynomial, homogeneous of degree Im(+ Ic - 2d. Furthermore qm is a B, x B eigenfunction of weight (D, 4, where

--

p = [ 0 , . . . ,o, -1,. . . , -1, n-k+d

(when m = 0 take

. . , -ml]

k-2d

p = [ O , . . . , 0 , ,1, .:. , -y). Set $&(z) n-k

$"m = -$lrn.

-md,.

= $m(zs).

Then

k

Proof: This follows from Lemma 10.5 by the same arguments as in the proof of Corollary 10.3. 0 To prove Theorem 10.4, assume first that n 2 k. By Corollary 10.6 the functions cpm are defined for all m E N$+.If m has depth 1 then the right G-translates of ym span an irreducible subspace of type (m,O). If m has depth less than 1 then $m is also defined. In this case the right G-translates of (Pm span a G-irreducible subspace of type (m, I), whereas the right Gtranslates of $m span an irreducible subspace of type (m,-l). Thus we get all irreducible representations of G in X ,as asserted in part (a) of the theorem. The argument when n < Ic proceeds as in the proof of Theorem 10.1 by lifting harmonic B, x B eigenfunctions from M n x k to M k x k . Note that

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374

prnis defined for all m of depth d i min{n, l } , whereas $, when n > 1 and k - n I d < 1. We omit the details.

is only defined 0

10.3. Examples of harmonic decompositions

+

(1) Assume n 5 1 and k = 21 1 or 21, so that we are in case (c) of Theorem 10.1 or cases (c) and (d) of Theorem 10.4. The restrictions to S O ( k ) of the representations in C are the class n representations of SO(k) - those that have a vector fixed under the subgroup S O ( k - n). This follows from the branching law (see [16, Sec. 8.11). In this case the harmonic polynomials on M n x k decompose under GL(n) x S O ( k ) as

Here = [-mn,. . . , - m l ] and U X is the irreducible S O ( k ) module with highest weight X (when n = 1, k = 21 is even and ml # 0, then ItXis the sum of the irreducible representations with highest weights X and s A). For n = 1, (51) is the classical spherical harmonic decomposition and gives the decomposition of polynomials restricted to the sphere S O ( k ) / S O ( k - 1).For n > 1 (51) gives the decomposition of polynomials restricted to the Stiefel manifold S O ( k ) / S O ( k- n). This decomposition was obtained by Gelbart [12] and Ton-That [32] before Kashiwara and Vergne [22] worked out the general case that we have presented here. ( 2 ) Now assume 1 < n < k , so that we are in case (b) of Theorems 10.1 and 10.4. The decomposition of the harmonics in this case was obtained by Strichartz [29]. For example, let n = 2 and k = 3. Then we have the decomposition

of the harmonic polynomials on M 2 x 3 . Here Vim] denotes the irreducible SO(3) representation with highest weight m ~ 1The . B2 x B harmonic eigenfunction p(m) ( 2 ) = x y generates the summand & [ o ~ - m @] The B2 x B harmonic eigenfunction $ ( m , ( ~ = ) z F - ' ( x l t 2 - 2 2 t 1 ) generates the summand & [ - ' ~ - ~ l @ Vlrnl. Here we write

=

[;:

Y1

tl

y2

t2]

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

375

(3) Let TI = 3 and Ic = 3 so that we are in case (a) of Theorem 10.1. Then we have the decomposition

of the harmonic polynomials on M3x3 as a module for GL(3) x SO(3). The B3 x B harmonic eigenfunction ~ ( ~ ) = ( xz a) generates the summand The B3 x B harmonic eigenfunction (2) = xT-'(zzt3 z&) generates the summand E[o*-ly-m] @ l/""]. Here we write &[0103-ml@V[m1.

[; ;

z1 y1 tl

z=

b]

For m = 0 the function ~ ( o ) ( z=) det z generates the one-dimensional summand &[-11-1,-11 @ ),J"31. Let C ( m ) denote the one-dimension representation g H (detg)m of GL(3). Let wl = [1,0,0] and w2 = [1,1,0].Then the GL(3) representations occurring in 'FI are c(-'),

&[o,o>-"l

c(-")

@ &""2

for all m 2 0, and

for all m 2 1. 11. Symplectic group and oscillator representation

We now turn to the functional-analytic aspects of the harmonic duality decomposition in Theorem 9.2 (recall Example 4 in Section 2.3). If we replace the complex group G by its compact real form Go = GnU(V) then the finite-dimensional representations 3"remain irreducible under Go and the action of Go is unitary relative to the Fischer inner product. We would like to have a similar picture for the dual representations E X (where X = T ( ( T )+ 6). At the Lie algebra level it is clear that to obtain a unitary representation, we should take the real form gb of g' that acts by skew-hermitian operators relative to the Fischer inner product. The analytic problem is to construct a unitary representation of an associated real Lie

R. Goodman

376

group Gb on the completion of P ( V ) , and to describe its action on the Hilbert space completions of the infinite-dimensional spaces EX. We will construct Gb as a subgroup of the rnetaplectic group Mp(nk, W) (the two-sheeted cover of the real symplectic group Sp(nk,W)). The associated unitary representation will be the restriction to Gb of the oscillator representation of the metaplectic group. This representation already appears in the harmonic decomposition as a Lie algebra representation by elements of degree 2 in the Weyl algebra. However, when we try to exponentiate it to a unitary group representation, we encounter the conflict between the particle and the wave description of quantum mechanics; the representation has a simple description (the holomorphic model) relative to the maximal compact subgroup KO U(n) of Sp(n,W), and another simple description (the real-wave model) relative to the maximal parabolic subgroup P E GL(n,IW) K SM,(R) of Sp(n,R). In both descriptions KOn P 2 O ( n ) acts geometrically, but some of the remaining group elements act in a more subtle way. Thus it will be necessary to consider two matrix forms of the real symplectic group and the intertwining operator (the Bargmann-Segal transform) that relates the two versions of the oscillator representation. 11.1. Real symplectic group

Let Sp(n, C) be the subgroup of GL(2n, C) that preserves the skew-form n

n(5,Y) = C ( x i ~ n +-i zn+iYi) i=l

on C2,. Thus g E Sp(n,C) if and only if gtJng matrix transpose and J , is the matrix

=

J,, where gt denotes

We can also describe Sp(n,C) as the fixed-point group of the involution 7 : g H J n ( g t ) - l J l l on GL(2n,C). The Lie algebra sp(n,C) of Sp(n, C) consists of all X E M2n such that J,X + X t J , = 0. These matrices have block form

X =

[” ] C -At

withAEM,andB,CESM,

Here we use the notation Mn for the n x n complex matrices and S M , for the n x n symmetric complex matrices.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

377

The real symplectic group Sp(n,R) = Sp(n,C) n GL(2nlR). Its Lie algebra 5p(n,W) consists of all the real matrices in 5p(n,C).

Maximal Compact Subgroup. A fundamental technique for studying a unitary representation of a real reductive group such as Sp(n,R) is to restrict the representation to a maximal compact subgroup, under which the representation space decomposes as the (Hilbert-space) direct sum of multiples of irreducible (finite-dimensional) subspaces. The real orthogonal group O ( k ) c U(k) is the subgroup of real unitary matrices. Since Sp(n, C)and Sp(n, R) are invariant under the map g H g * , the groups Sp(n) = Sp(n, C)n U(2n) and Sp(n, R)n U(2n) = Sp(n, R)n O(2n) are maximal compact subgroups of Sp(n,C) and Sp(n, R), respectively (see [24, Proposition 1.21). The subgroup of diagonal matrices in Sp(n) is a maximal torus in Sp(n). However, the subgroup of diagonal matrices in Sp(n,R)n O(2n) is finite and is not a maximal torus in Sp(n,R). Hence it is convenient to replace Sp(n,R) by an isomorphic real form Go so that the diagonal matrices in Go n U(2n) comprise a maximal (compact) torus in Go. Define

and let cr be the conjugation (conjugate-holomorphic involution) cr(g) = In,n(g*)-lln,non GL(2n,C). The fixed-point set of cr is the real form U(n, n)of GL(2n, C). Set

Then J;'In,, = Kn,so it follows that (TT = T C J .Hence cr leaves Sp(n,C) invariant and its restriction to Sp(n,C)defines a conjugation of Sp(n, C) which we continue to denote as cr. If g E Sp(n,C) then o ( g ) = m ( g ) = KngK,. In terms of the n x n block decomposition, cr acts by

CJ[t;]=

D C [B A ] '

Define Go = { g E Sp(n, C) : a(g)= g } . Then Go is a real form of Sp(n,C). Its Lie algebra go = Lie(G0) consists of all matrices X E 5p(n,C)such that a ( X ) = X . In terms of the block decomposition, go consists of the matrices

A* = -A, B

= Bt

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378

Lemma 11.1: The subgroup KO= Go n U ( 2 n ) is a maximal compact subgroup of Go and consists of all matrices

[t 11,

withA e U ( n ) .

Hence KO U ( n ) and the subgroup of diagonal matrices in KOis a mmimal compact torus of Go. Proof Since a(g*)= a(g)* for g E G L ( 2 n , C ) , the group Go is invariant under g H g*. Hence KO is a maximal compact subgroup of GO. Write g E G L ( 2 n , C ) as

["C

'=

"1

D

with A,B,C,D E M,.

Then g E U ( n , n ) if and only if g*I,,,g written as

A*A - C*C = I,,

= In,,. This condition can be

B*B - D*D = - I n ,

A*B

- C*D = 0 .

(53)

On the other hand, g E U ( 2 n ) if and only if g*g = Iz,. This condition can be written as

A*A+C*C=I,,

B*B+D*D=I,,

A*B+C*D=O.

(54)

If g E U ( 2 n )n U ( n , n ) then from (53) and (54) we have C*C = 0 and B*B = 0. Hence B = 0 and C = 0, so A*A = I, and D*D = I,. Thus U ( 2 n )n U ( n , n ) = U ( n ) x U ( n ) . But if g E S p ( n , C ) is in block-diagonal form, then

0

Hence g E KOif and only if A E U ( n ) . Define an involution 6 on Sp(n, C)by

O(g) = In,ngIn,n (note that 6 is an inner automorphism of Sp(n, C)). If g E Go then (St)-' = J n g J i l and 3 = K,gK,. Hence (g*)-' = J n S J i ' = JnKngKnJi'. Since J,K, = I,,,, it follows that B ( g ) = (g*)-l

for g E Go.

Thus the maximal compact subgroup KO is the fixed-point set of 6' in Go. Its complexification is

K = ( 9 E Sp(n,C) : O(g) = 9 ) .

Multiplicity-he Spaces anu Schur- Weyl-Howe Duality

379

Note that if g E GL(2n, C), then O(g) = g if and only if

[; i] ,

g=

a,dEGL(n,C).

If g E K , then in this block decomposition d = ( u t ) ) - ' . Hence K via the homomorphism aH

[;

GL(n, C)

.

(a;-']

The complexification of the Lie algebra t o of KO is the Lie algebra t Z gI(n,C) of K . The involution O gives a decomposition of 5p(n,C). The f l eigenspace of 8 on 5p(n,C) is t, whereas the -1 eigenspace is

We have 5p(n,C)

=t@p

with commutation relations

[t,t] c t

1

[t,PI

c P , [P, PI c t

'

The center of t is spanned by I,,, and t = CI,,, @ [t,t], with the derived algebra [t,t] 2 4 7 2 , C). The f l eigenspaces of adI,,, on p are

P + = {O [ ~B o] :BESM,},

P - = { [ ~0

0] 0 :CESMn}.

These subspaces are invariant under t and have the commutation relations

[P+,P+l

=0,

[P-,P-I

=0,

[P+,P-I c t .

Thus there is a triangular decomposition

5p(n,C) = p- @ t @ p + (as we already noted in Section 8). The conjugation u interchanges p+ and

p - , since g([:

:]

:I)=[;

for B E SMn. We can describe these decompositions in terms of root spaces as follows (see [16, Sec. 2.3.11).The complexification T of TOis a maximal (algebraic) torus in Sp(n, C) and has Lie algebra

t ={X

= diag[zl,. . . , z,

-21,.

. . , -x,]

: xj E C } .

R. Goodman

380

The set of roots @ = @(g,t) of t on g is f ~ fi~j for 1 5 i,j 5 n, where E ~ ( X=) xi for X E t as above. We have @ = Q C u an,where Qc={+(&i-&j)

:l
is the set of compact roots (the roots o f t on P) and @ , = { f ( & i + & j ) :l

< i l j l n }

.

is the set of noncompact roots (the roots o f t on p). Take the set of positive roots @+ to be ~i f ~j for 1 5 i 5 j 5 n, and let (a: (respectively Q:) be the positive compact (respectively noncompact) roots. Then

e=t+

c

c

P*=

Bcr,

g*:P.

P€@L

a€@,

The simple roots in Q+ are a l , . . . , a,, where for i = 1,.. . , n - 1 and a,

a( = ~i - &,+I

= 2~,.

The unique simple noncompact root is the long root a,, and the highest root is y = 2 ~ (it 1 is noncompact). Let p be one-half the sum of the positive roots. Then p = nE1

+ ( n- 1 ) E Z + . + * '

(55)

En.

Cuyley Trunsform. We now show that the group Go = Sp(n, C)nU ( n ,n ) is conjugate to Sp(n,R) within Sp(n,C). To understand this in terms of the adjoint representation of 5p(n,C),consider first the case n = 1 (recall that Sp(1,C) = SL(2,C)).Set

a.

where i is a fixed choice of Then [k,x]= 2x,[k, y] = -2y, [x,y] = k, so {x,y, k} is a TDS (three-dimensional simple) triple. Furthermore, the one-parameter subgroup

t H exp(itk) =

cos t sin t

- sin t

cos t

is a maximal compact torus in SL(2,EX). Let 1

h=

[o

0 -1]

e=

[:;]

0 0

f = [1 0 ]

M u l t i p l i c i t y - h e Spaces and Schur- Weyl-Howe Duality

381

be the standard TDS in d ( 2 , C). We can conjugate {x,y, k} to { e ,f , h} as follows: Since x y = h, we have

+

(ad(y - x))(k) = 2h,

(ad(y - x))(h) = -2k,

and so fort E C .

etad(Y-x)k=(cos2t)h+(sin2t)k, Setting t = ~ 1 4we , obtain e(n/4)ad(Y-X)k= h . Since (y - x ) = ~ - I , we have exp[t(y - x)] = (cost)I

+ (sin t)(y - x),

for t E C

Define 7r

c = exp[-(y - x)] = -

4

Then c(ik)c-' = ih. Thus c conjugates the compact torus in SL(2,R) generated by i k to the compact torus in Go generated by ih:

[

cost

c sint

[

-sint] c-1 = eit cost

O e-it

]

7

tEW*

The automorphism g H cgc-' is called the Cayley transform. A similar construction works in Sp(n,C) (and for other real semisimple Lie groups). Set

Then c E Sp(n) and c-l = E . Lemma 11.2: Let Go = U ( n ,n) n S p ( n , C). Then c-'Goc = Sp(n,W). Proof: Since Go is closed under g H g* it has a polar decomposition Go = Koexp(p0). We already have shown that KO E U(n), hence KO is connected. Thus Go is also connected. So it will suffice to show that Ad(c)-'go = sp(n,R) . Let X E go be given by (52). Then one calculates that

+

(A A ) + i ( B - B) i(A - A ) ( B B)

+ +

i(A - A) + ( B

+ B)

1

( A + A) - i ( B - B)

'

R. Goodman

382

Set u we have

Also ; ( A

+ A ) - $ ( B- B)= -ut. c-'xc =

;[

Hence

-4

E 5p(n,R)

Since go and sp(n,R) are real forms of sp(n,C),they have the same real dimension. Hence the map X H c - ' X c is a real Lie algebra isomorphism from go to 5p(n,R). 0

Maximal Parabolic Subgroup. Let P be the subgroup of Sp(n,R) consisting of the matrices

where SMn(R) denotes the real n x n symmetric matrices. The group P is a maximal parabolic subgroup of Sp(n,R). It has the structure of a semidirect product M N , where M consists of the block-diagonal matrices

and N consists of the matrices

Thus as Lie groups M E GL(n,R) and N 2 SM,(R) (an abelian group). The group KOn P consists of all matrices

where O ( n ) = (9 E GL(n,R) : gtg = I,} is the usual real orthogonal group. Define N - = N t . Thus N - is the group of matrices

Multiplicity-fie Spaces and Schur- Weyl-Howe Duality

383

Then P- = M N - is the opposite parabolic subgroup to P and P n P - = M . Note that

and

Thus P- = JnPJ;' Let

is conjugate to P in Sp(n,R).

be in Sp(n,R). If detA

#

0, then

D

= (At)-'

+ C B (this follows from

gtJng = Jn). Hence we can factor g as

Thus the subset N - M N is open and dense in Sp(n,R). This shows that a continuous representation K of Sp(n,R) is uniquely determined by its restriction to the subgroup P together with the single operator n ( J n ) . 11.2. Holomorphic (coherent-state) model for oscillator

representation We write W z ( C n )= W2(Cn,e - ~ ~ " ~ ~ z d Xfor ( z the ) ) Hilbert-space completion of P(C") relative to the Fischer inner product introduced in Section 9. The elements of this space are naturally identified with holomorphic functions f on Cn such that

L

I f ( z ) l ze-IIzI12dX(z) < 0 0 .

For each w E C" the function K,(z) = e(zlw)is in W2(Cn)and plays the role of reproducing kernel for this space:

f(w)= (f I GJ) for

f E W2(Cn)

(see [7, Proposition XI.1.11). Define the annihilation operators Aj and creation operators Af by

a

~ j f ( z= ) Gf(z)

~ J t f ( 2= ) z

j . 1 ~ ~ 1for

f E p(cn).

R. Goodman

384

These operators satisfy the commutation relations [Ai,A;] = are mutually adjoint relative to the Fischer inner product:

6ij I .

They

(Ajcp I %4= (P I A;$)

(56)

for cp, 7c, E P(C"). The operators { A l ,. . . , A,, A!, . . . , AL} generate the Weyl algebra PD(Cn). Define a representation a of sp(n, C ) on P(Cn) as follows. For b E S M , let Qb(z) = ztbz be the quadratic form on Cn defined by b. Let

where b, c E SM,. Define operators a ( X ) and a ( Y )on P(Cn) by 1

1

a ( x ) f ( z )= ,a(Qb)f(z)

7

a(Y)f(z= ) ,Qc(z)f(z)

J--r

where i = (recall the map Q H a ( Q ) from P(Cn) to constantcoefficient differential operators on P(C")that was introduced in the proof of Theorem 9.2). We calculate that

[ a ( X )a , ( Y ) ]= - x ( b C ) k j zjj,k

Let H =

a dzk

-

1-tr(bc) 2

[! -it]

Since [ X , Y ]=

E t, where h E M,. Define the operator a ( H ) by

[

-:b],

we see that [ a ( X ) a, ( Y ) ]= a ( [ X , Y ] ) One .

calculates that

(note that H H a ( H ) is the standard representation of t 2 gI(n,C) on P ( P ) tensored with the one-dimensional representation H ++ - $tr(h)). Thus a is a representation of sp(12, C) on P(Cn) (this is the representation of 5' = sp(n,C) in Theorem 8.5 for the case G = O(1,C)).

Multiplicity-Fbee Spaces and Schur- Weyl-Howe Duality

385

We can write the representation w in terms of the annihilation and creation operators as

for X , Y , H as above. Now take Y = o ( X ) , where (T is the conjugation defining the real form go of sp(n,@).Then C j k = bjk, so from (56) we see that ( w ( X ) c pI $) = -(P I w(@))$)

(57)

for all X E p+ and cp,$ E F ' ( P ) . Since ( ~ ( p & )= pr and p + , p - generate sp(n,C ) , it follows that relation (57) holds for all X E sp(n,C). Since go is the fixed-point set of

we conclude that

(T,

( w ( Z ) c p I $) = -(cp

IW(o/J)

for

E go .

One says that w is a unitarizable representation of go. By the Cartan decomposition Go = Koexp(p0) the group KO is a topological retract of Go. Since t o has a one-dimensional center, it follows that Go has an m-sheeted covering group for every integer m. Let y : Mp_(n,R) + Go be the two-sheeted covering and let = y-'(Ko). Then KO is a two-sheeted covering of U(n). If ie and z E @" we set IC. z = uz, where

KO

-

y(i) =

The function

x:

H

KO

[::]

with u E U(n) .

det(u)-1/2 is a (single-valued) character of

KO.

Theorem 11.3: There is a unitary representation w of Mp(n,R) o n then W2(C") whose differential is the representation w of go. If E

KO

w(K)f(z) = x(K)f(i-' . z) f o r f E

w'(c").

Proof: Let

H o = :z[

0 -iIn

]

Et0.

Take the basis for w ( s p ( n , @ )to ) be the operators AiAj and AIA) for 1 5 i 5 j 5 n and +(AfAj AjA!) for 1 5 i,j 5 n.Define the seminorms p k on P(C") in terms of this basis and the norm JIcpJIas in Section 11.5.

+

386

R. Goodman

Lemma 11.4: T h e inequality

holds f o r all cp E P(@"). Proof of Lemma 11.4: Let cp = la1 we have

+ 2,

Cac,za.

Since a(iH0) acts on zQ by

We calculate that

Likewise,

Finally,

The estimate (58) now follows from these formulas and the inequality 2ab 5 ( a b)'. 0

+

Proof of Theorem 11.3: From Lemma 11.4 and Corollary 11.9 we obtain a unitary representation a of the universal covering group of Sp(n,R) whose differential is the representation a. Now HO spans the center of to and exp(27rHo) = I in Sp(n,C). Since the one-parameter unitary group a(exp(tH0)) acts by eit(k+n/2)on P k ( C n ) ,we have a(exp(47rHo)) = I . Hence the representation a descends to a single-valued representation of Mp(n, R). 0

We will call a the oscillator representation of Sp(n,R). We write ~ if the dependence on n is not evident from the context. An explicit formula for a ( g ) as an integral operator was obtained by Bargmann (see [8, Theorem 4.371).

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

387

11.3. Bargmann -5'egal transform

To obtain a realization of the oscillator representation in which the action of the maximal parabolic subgroup P of Sp(n,R) is easily described, we use another representation of the creation and annihilation operators. Let S(Rn)denote the Schwartz space of rapidly-decreasing smooth functions on R".Define creation and annihilation operators

on S(Rn) for 1 5 j 5 n. These operators satisfy the commutation relations [ai,as] = & I and they are mutually adjoint relative to the L2(R", dX(x)) inner product. They leave invariant the space P ( R n ) of (complex-valued) polynomial functions on Rn and act irreducibly on this space. Following [2] and [28], we construct a unitary operator

B : L ~ ( R " ,d~(x))

---f

H~(c", e-IIZ112dX(zt))

that intertwines aj with Aj and a: with AS. Since the space L2(Rn)(respectively W 2 ( P ) ) is the n-fold Hilbert-space tensor product of the space L2(R) (respectively W2(C)),it suffices to do the calculation for the case n = 1. Because W2 has a reproducing kernel, any such operator B will be given as

[-B(z1 .)f(x) 00

Bf(z) =

dx.

To intertwine the two pairs of creation-annihilation operators, the kernel B ( z ,x) must be a holomorphic function of z and smooth function of z that satisfies

These equations imply that

dB

-=

dz

dB ( h x - z ) B and - = ( h z - x ) B . dX

The solution is easily found to be

{

1 B ( x ,z ) = Cexp h x z - 2(x2

I

+ 2)

R. Goodman

388

with C a constant. It remains to verify that the operator defined by this kernel is unitary (with appropriate choice of C). For this, take p E P(R) and the normalized Gaussian (ground state) 1 cpO(x) = -exp

J;;

(-ix2)

(note that alp0 = 0). Then

"Ja

B(Pcpo)(z)= -

J;;

{

p(x)exp -x2

-02

+ fizz

- -z2

2l

l

dx.

Completing the square in the exponential and using the translationinvariance of the measure dx, we obtain

The right side of this equation is obviously a polynomial of the same degree asp, so it follows that B is a bijection from P(R)cpo onto P ( C ) .Furthermore B q o ( z ) = C. Since cpo has L2-norm 1 and the constant function 1 has W2norm 1,we conclude that C = 1. To complete the proof that B is a unitary operator, we observe that P(B)cpois the cyclic space generated by cpo under the action of the operators a1 and a!. Likewise P ( C ) is the cyclic space generated by the constant function 1 under the action of the operators A1 and A!. The creationannihilation operators act irreducibly on these spaces. Since B* intertwines the pair {al,af} with {Al,A!}, it follows that B*B commutes with a1 and a! on P(R)(po, while BB* commutes with A1 and Af on P(C). Thus B*B and BB* are both multiples of the identity by Lemma 2.1. Since we have normalized the kernel B ( x , z ) so that B carries the unit vector cpo to the unit vector 1, it follows that B is unitary in the case n = 1. This implies that B is unitary for any n as remarked above. For any integer n 2 1 and polynomial p E P(Rn), define

for z E C",where cpo(z) = ~ - " / ~ e x(-$ztx) p for x E J P.Notice that if g E O ( n ) then cpo(gx) = cpo(x). Thus if we set fg(x) = f(g-lz), then B ( f g )= B ( f ) , for f = cpop and g E O ( n ) .We have proved the following.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

389

Theorem 11.5 (Bargmann-Segal Transform): The operator B maps the space P(Rn)(p,onto P(C") bzjectively. It extends to a unitary operator from L2(Rn;dx) onto W2(Cn).Iff E S ( R n ) then

Furthermore, B intertwines the representations f ++ f, of O ( n ) on L2(Rn;dx) and on W2(Cn). Also BajB-l = Aj and BaJB-' = Atj on

P(C"). Remark: Since B is unitary, B-' = B' is an integral operator on W2(Cn) with kernel B ( z ,x) = B(Z,x). 11.4. Real (oscillatory-wave) model for oscillator

representation We now use the Bargmann-Segal transform to obtain the real-wave (Schrodinger) model of the oscillator representation. Let y : Mp(n,R) 4 Sp(n,R) be the covering homomorphism and let c E Sp(n) be the Cayley transform. We define a unitary representation n of Mp(n,R) on L2(Rn) by r(g) = B-lw(cgc-')B

(60)

(here cgc-' E Mp(a,R) is the element such that y(cgc-') = cr(g)c-'). For f E S(R") let Ff be the Fourier transform

Theorem 11.6: The action of n(g) on f E S(Rn) is as follows: (1)

If y(g)

(det

=

(A:)-1]

with A

E

GL(n,W then n(g)f(z) =

f (Ap1.).

(2) I f y(g) =

[ y ] with

(3) If Y(S) = e(i/2)ztbz

[t

b E SM,(R) then r(g)f(s) = e--(i/2)ztbz f(XI.

[i !]

with b

E

Ff (XI.

These formulas uniquely determine r*

SM,(IR) then T(r(g)f)(s)

=

R. Goodman

390

Proof: For X E sp(n,R) write ~ T ( X = ) B a ( A d ( c ) X ) B - l for the Lie algebra representation corresponding to T . Here the operator d n ( X ) acts on S(Rn).

]

(1) Let X = [ h with h = diag[hl, . . . ,h,] and h j E R. Then 0 -h 0 -ih Hence the formulas of Section 11.2 give Ad(c)X = ih

[

1.

a(Ad(c)X) = =-

5l

1

n

{

h j (Aj)2 -

j=1 n

c

hj(Aj

+ A:)(Aj

- A:)

-

j=1

1 -tr(h) . 2

Now B - l A j B = aj and B - l A i B = a;. Also we have (aj

+ af) f (x)= &xj f (x),

(aj

af

- a;) f (x)= &-(x)

(61)

dXj

for f E S(Rn).Thus we obtain

c n

~ T ( X= ) -

hjxj-

j=1

a

1

axj - -tr(h) 2

.

This shows that (1) is true for A = exp h (in this case det A > 0 and there is no need to pass to the metaplectic group). If A E O ( n ) then (1)holds for g, since B intertwines the action of O ( n ) on L 2 ( R n )and W2((Cn) (note that y(g) is in the maximal compact subgroup KOof Go, so a ( g ) is described in Theorem 11.3). By the polar decomposition, GL(n, R) = O(n)(exp a)O(n) , where a is the subspace of real diagonal matrices. Hence (1) holds for all A E GL(n,R).

(2) Let X =

[i :]

with b E SMn(R). Then Ad(c)X

Hence the formulas of Section 11.2 give

=

[ib b

-ib

1.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

391

Now applying the Bargmann-Segal transform and using (61), we obtain l

d T ( X ) f ( Z )= 2i

{

n bjkZjZk}f(Z)

*

j,k=1

This proves (2). (3) Let X =

[:i]

with b E SM,(R). Then Ad(c)X =

Thus

-ib

[b

b ib].

Applying the Bargmann-Segal transform and using (61) again, we obtain

for f E: s ( R n ) .Since F ( d 2 / a z j d Z k ) F - ' is the operator of multiplication by - X j Z k , this proves ( 3 ) . Formulas (l),( 2 ) , and ( 3 ) uniquely determine x since N - M N is dense 0 in Sp(n,R).

11.5. Analytic vectors Here we present refinements of some results of Nelson [27] concerning exponentiation of Lie algebra representations, following the approach in [14]. Suppose go is a real finite-dimensional Lie algebra, represented as skewHermitian (unbounded) operators on a complex inner product space V (not assumed complete). Let g be the complexification of go. Then X H X* (the Hermitian adjoint of X relative to the inner product on V ) is a conjugatelinear anti-automorphism of g such that X * = -X for X E go. Fix a basis { X I , .. . , X d } for g. Define seminorms Pn on V by setting Po(.) = 1141 and

for n = 1,2,.. . . Here 21,. . . ,in run over 1,2,.. . , d and llull denotes the norm of u E V . Let V be the Hilbert-space completion of V relative to

R. Goodman

392

the norm llzlll and let V" be the completion of V relative to the family of seminorms { p n } . Then the representation of g on V extends continuously to a representation on V". The seminorm p n ( w ) is also defined for v E V". One says that v E V" is an analytic vector for g if there is an r > 0 such that

v,W c V" be the subspace for which (62) holds. Ur,o v,W of analytic vectors for g is invariant under g. Let

The space V" =

Theorem 11.7 (Nelson): Suppose there exists an r > 0 so that V,W is dense in V . Then the representation of 00 integrates to a strongly continuous unitary representation on V of the simply-connected Lie group Go with Lie algebra 00. Proof: (Sketch) Define a norm on g by then

( 1 c,"=, ciXi(I = c,"=,

Icil. If X E g

llXn41 5 IlxlInPn(v). , to V by the exponential series Hence the operator e x is defined from VW provided llXll < r. The map e x p X H e x defines a local representation of the complex Lie group germ corresponding to g (the rearrangement of the exponential series needed for the Campbell-Hausdorff formula is justified by convergence of (62); see [3] or [18]).The operator e x is unitary for X E go, since X is skew-Hermitian, and the local representation extends to a strongly continuous unitary representation of Go on W . 0

-

Suppose there is an element HO E g such that pl(v) 5

11~1+ 1 llHovll

for all v E V .

(63)

Let A = m a x l l i g II[Ho,Xi](I(the norm of adHo on 8). Note that A = 0 if and only if HO is in the center of 6 , and this case is of no interest here. So we assume A > 0.

Theorem 11.8: Every analytic vector for HO is an analytic vector for g. More precisely, if

then w E

v,W

for all r < min{A-l,A-'(l

- e-A5)}.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

393

Remark: If s can be arbitrarily large in (64) (one says that w is an entire vector for Ho in this case), then v E V,W for all r < A-l. Note that this upper bound for r is controlled by the non-commutativity of g and it is finite if A # 0. In general v is not an entire vector for g (see [15] for more precise results along this line). Proof: Let y3 be any of the basis elements X i . Then the a priori estimate (63) implies that

I IYm+1 Ym .

' *

y14

I L I lYm . . Yl vI 1 + I p o y m . . . Yl v )I '

for all t~ E V". Now m

HOYm"*Yl = Ym"'Y1HO

+ CY,"'Yk_1[HO,Yk]Yk+l

'**Y1,

k=l and by definition of p m and the constant A we have IlY,... [Ho,Yk]...Y1wlI 5 Ap,(w). Hence

+

IIYm+1...YlvII L I l y m * . . y ~ H o ~ I I(l+mA)p,(~)

I pm(H0v) + (1+ mA)pm(v). Since this holds for any choice of Yl , . . . , Ym+l,it implies pm+i(v) 5 pm(&w)

+ (1+ mA)p,(w)

for all w E V .

(65)

Now fix w E V" and set am,n = p,(H;w). Replacing w by Hgw in (65), we see that the sequence {am,n} satisfies the recursive inequalities

+

am+l,n 5 am,n+l+ (1 mA)am,n. To estimate the rate of growth of am,n, we introduce the majorant sequence bm,n defined by b ~= ,aO,n ~ for all n and b , + ~ , ~= bm,n+i

+ (1+ mA)b,,,

for all m 2 0, n

Clearly am,n 5 bm,n for all m,n. Consider the generating function

The recursion for bm,n implies that (as a formal series)

20 .

R. Goodman

394

We assume that the series f ( y ) = cp(0,y) converges for y = s. The Cauchy problem for this analytic first-order p.d.e. is easily solved by the method of characteristics, and one obtains cp(z, y) = (1 -

f (y - A-' log(1- A x ) ) .

(the analytic solution must agree with the formal solution since the line z = 0 is non-characteristic). Setting y = 0, we see that the series for p(z,0)

converges absolutely for 1x1 5 r provided

r < min{A-',A-'(l Since a,o

- e-As)}

5 bm,O this proves the theorem.

0

Corollary 11.9: Suppose Ho E 0 and V has an (algebraic) basis consisting of eigenvectors for H o . Then V c VW , for all T < A-' and hence VW , is dense in V for all T < A-l. Thus the representation of 00 integrates to a strongly continuous unitary representation of Go on V .

-

Proof: If HOV = Xu, then the left side of (64) is e3IxI, and hence is finite for all s > 0. By the remark after Theorem 11.8 this implies that (62) holds 0 for all r < A-l. Now apply Theorem 11.7.

12. Dual pair Sp(n, R)-O(k) The oscillator representation has many applications to analysis and physics (see [8]and [20], for example). Here we apply it in the context of unitary representation theory and highest weight representations (see [6] for more on this point). To determine which of the representations that occur in the decomposition of the oscillator representation are square-integrable, we apply Harish-Chandra's criterion to the explicit formula for the &correspondence that we calculated in Theorems 10.1 and 10.4. In particular, we show that all the square-integrable highest-weight representations of Sp(n,IR) occur in the duality correspondence with O(2n) (this was first proved by Gelbart [ll]). 12.1. Decomposition of H 2 ( M n x k ) under Mp(n, IR) x O ( k ) Let G = O(lc,C) = { g E GL(lc,C) : ggt = I } and let G' = Sp(n,C) c GL(2n, C ) be the symplectic group relative to the skew-form with matrix Jn as in Section 11.1. Define a skew form 0 on Mznxk by

n ( w , Z) = tr(wtJnz) for W, z E

M2nxk.

M u l t i p l i c i t y - h e Spaces and Schur- Weyl-Howe Duality

395

Then R is nondegenerate. We embed G' x G into SP(MZnxk,R) as follows. Let g E G and h E G'. Then R(hwg, hzg) = tr(wt(htJnh)zggt) = R(w, z ) since ggt = I and htJnh = J,. Hence we have an injective regular homomorphism L x R : G' x G + sp(M2nxk, 0) given by R(g)z = zg-',

L(h)z = h~ for g E G, h E G', z E M2nxk.

We identify M2nxk with Cznk by the map

z = [ Z I , . . . , Zk]

H

5=

[

EC=2nk zk

where z j E Czn is the j t h column of z . It is easy to check that

R(z, W ) = z t J n k ' & , so Sp(MZnxk,52) becomes Sp(nk,C)under this identification. Thus we will view z either as a 2n x k matrix or a vector in Cznk,whichever is more convenient for the calculation at hand. Define a hermitian form on Mznxk by

( z ,W ) = tr(w*In,nZ) 7

where In,n is the matrix in Section 11.1. w e have (z,w) = '&*Ink,nkZ,so when z E M z n x k is identified with Z E CZnk,the form ( z ,w) becomes the one used in Section 11 to define the group U ( n k ,n k ) . Thus we will denote the isometry group of this form as U ( n k ,n k ) . If g E U(n, n)then (gZ99W) = tr(W*g*In,ngz)= (z, W) since g*ln,,g = Thus the left multiplication homomorphism L : GL(2n,C) -+ GL(Mznxk) carries U(n,n) into U ( n k , n k ) .If h E U(k) then (zh, wh) = tr(w*ln,nZhh*)= ( z , W) . Furthermore,

[;I

h=

[f]

.

Hence the right multiplication homomorphism R : GL(k, C ) GL(Mznxk) carries U(k) into the maximal compact subgroup U(nk) x U(nk) of U ( n k ,n k ) . -+

R. Goodman

396

Let Go = G n U(k) = O ( k ) be the compact real form of G, and let Gb = G'nU(n,n ) 2 Sp(n,R) be the real form of Sp(n, C)as in Section 11.1. Let KO = Gb n U(2n) 2 U(n) be the maximal compact subgroup of Gb. Then the embedding L x R : Sp(n,@) x O ( k , C ) -+ Sp(nk,(C) gives an embedding of the real forms Gb x Go

-

Sp(nk,CC) n U ( n k ,n k ) 2 Sp(nk,R)

and carries the maximal compact subgroup K O x Go into the maximal compact subgroup Sp(nk, C)r l ( U ( n k ) x U ( n k ) )of Sp(nk, C)n U ( n k ,n k ) .

If u E U(n) and ko

=

ti

[o

GI pair (k0,g) E KOx Go acts on

is the corresponding element of K O ,then the M2nxk

by

We now calculate the restriction of the oscillator representation w(nk) to L(K0) x R(Go) in the holomorphic model on P ( V ) ,where V = Mnxk. Let (k0,g) E KO x Go. F'rom (66) and Theorem 11.3 we see that cdnk)(L(ko)R(g))f(s)= (det u)-"'(det

g)"/2f(u-1sg)

for f E P ( V ) (67)

(note that the determinant of the map s H u-lzg is (det u)-'((det g),). If k and n are both even, formula (67) defines a representation of KOx Go. For the general case, let c Mp(n, R) be the two-sheeted cover of KOand let be the lift of R(G0) to Mp(nk,R). Then (67) gives a single-valued unitary representation of x In the following we shall simply drop the factor (det g ) n / 2 from (67) to make the representation single-valued on Go. is essential for extending the representation However, the factor (det from K O to Mp(n, R). Let 6 denote the differential of this character of Let gb be the Lie algebra of Gb, The complexification of gb is g' =

KO

eo

G.

&.

sp(n,C). We now calculate the action of t ~ ( ~ ~ ) ( L (Let g ' )X) .= Y=

[::]

with b, c E SM,. Then

[::]

and

Multiplicity-Free Spaces

and Schur- Weyl-Howe Duality

397

(where L ( b ) x = bx for x E M n x k ) . The quadratic form on V associated with L ( b ) is

Hence

) the operator of multiplication by -;Qh(.,. Since p* and c d n k ) ( L ( Y )is generate g', these operators determine a ( n k ) ( g ' ) . F'rom Theorem 8.5 we conclude that the algebra P D ( V ) Gis generated by z d n k ) ( g ' ) . We recall some notation that was introduced earlier. Let 'FI denote the space of G-harmonic polynomials on M n x k and let C c be the spectrum of G on 7-f. Let the map T :

C + A c Z+ :

be as in Theorems 10.1 and 10.4. Let T' c 7-f be an irreducible G-module in the class (T.Let &'(")f6 c IH be the irreducible finite-dimensional kmodule with highest weight T ( U ) 6, as in Theorem 9.2. Let V = M n x k . Consider the unitary representation of Mp(n,W) x O ( k ) on W2(V), where Mp(n,W) acts by the restriction of the oscillator representation m(nk)and O ( k ) acts geometrically by right multiplication on V . For D E C let IET(u)f6 be the closure in W2(V) of the 8'-irreducible subspace P ( V ) .

+

Theorem 12.1: The spaces Er(u)+6,f o r r~ E C, are irreducible and mutually inequivalent unitary representations of Mp(n,W). Furthermore, W2(V) decomposes as a multiplicity-free Halbert space orthogonal s u m

under the action of Mp(n,R) x O ( k ) .

Remark: When k is even the character u H (det u)-'12 occuring in the oscillator representation is well-defined on U ( n ) and ET(u)+6gives an irreducible unitary representation of Sp(n, W). Proof: The key point is the following density result:

R. Goodman

398

(*) Suppose E c W2(V) is a closed subspace that is invariant under Mp(n,IR). Set EO = E n P ( V ) . Then EO is dense in E and is invariant under 8’. To prove this, let f(x) = C , c , P be in W2(V) and set

fq(x)=

C c,xa

for q = 0 , 1 , 2 , . . . .

la154

Then f, E P ( V ) and it is clear from (38) that Now take

/If

-

fqll

-+

0 as q

-+

m.

as in the proof of Theorem 11.3. Since

we see from (67) that d n k ) ( H o )acts by -i(j define

+ 3) on P j ( V ) .Hence if we

then Pq is a bounded operator on W2(V) and Pqf = fq for q = 0 , 1 , 2 , . . . . Since E is closed and invariant under Mp(n, R), we know that E is invariant under the one-parameter unitary group t H a ( n k ) ( e x p t H o )Hence . PqE C E for q = 0 , 1 , 2 , . . . . This shows that Eo = P ( V )n E is dense in E. To prove that EO is invariant under g’, take cp E EO and E E l . If X E gb, then

+

(dnk)(exptX)cp1 +) = o for all t E

IR ,

(70)

since E is invariant under Mp(n,IR). But since cp E P ( V ) ,the left side of (70) is an analytic function o f t for It( near zero by Corollary 11.9. Taking the derivative in t and setting t = 0, we conclude that

(dnk) (x>cp 1 $) = o

for all

+ E E‘- .

Hence d n k ) ( X ) c pE E , completing the proof of (*) To prove the theorem, first observe that if E c ET(u)f6is a proper closed subspace that is invariant under Mp(n, R), then by (*) the space Eo is invariant under g’. Hence EO= 0 by the irreducibility of & T ( “ ) + 6 . But EO is dense in E , so E = 0, showing that lF,T(a)f6 is irreducible.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

399

If o,o' E C and T is a bounded Mp(n, R) intertwining operator from IET(u)+6 to IEr('")+61 then T commutes with the projection operators Pq in (69). Hence T maps & T ( u ) + 6 to & T ( u ' ) + 6 . Since the functions in P ( V ) are analytic vectors for Mp(n,R), we conclude (as in the proof of (70)) that T intertwines the g' actions. Hence o = (T' by Theorem 9.2. The orthogonality of the decomposition also follows from Theorem 9.2. 0 12.2. Square-integrable representations of Sp(n, R)

The irreducible unitary representations of Go = Mp(n,R) that occur in Theorem 12.1 are called highest-weight representations. Some of them also appear as discrete summands in the decomposition of the left regular representation of on L2(Go) (these representations are called squareintegrable). We now apply Harish-Chandra's criterion [17] to determine which of the representations E'(")f6 are square-integrable. It is convenient to give separate statements of the result depending on the parity of k.

Theorem 12.2: (notation of Theorem 10.1) Let k = 21 uE

c.

(a) i f n > I

+ 1 then

+ 1 be odd. Let

is newer square-integrable.

I E T ( ~ ) + ~

(b) ifn = L+1 then E'(')+6 is square-integrable if and only if u = (A) -1) E G-1 and depth()\) = 1 . h

(c) If n _< 1 then IET(u)+6 is square-integrable for all o E C. Proof: The general condition on the highest weight A for squareintegrability is ( A + P , y-)< 0 ,

(71) where p is one-half the sum of the positive roots and y- is the co-root to the highest noncompact root y. For q ( n ,R) we have p = [n, n - 1,.. . , 2,1] and y = 2 ~ 1 so , y- = ~1 (see Section 11.1).We must check this condition when A = ~ ( o )b, with 6 = [ - k / 2 , . , . , -k/2]. ). (A+p, y-)= T ( o ) ~Let T ( D ) ~denote the first coordinate of ~ ( u Then k / 2 n.Since k = 21 1, the Harish-Chandra condition (71) is

+

+

+

7(0)1

< l + 1- n.

(72)

Case (a): n > Z+1. The formulas for T ( O ) in Theorem 10.1 show that ~ ( 6 is either 0 or -1 in this case. But 1 - n 1 5 -1, so (72) is never satisfied.

+

+

Case (b): n = 1 1. Now the right side of (72) is zero. The formulas for T ( O ) show that T ( O ) I < 0 if and only if o = (A, -1) E with d = 1.

c-1

)

R. Goodman

400

Case (c): n 5 1. Now the right side of ( 7 2 ) is positive. The formulas for T ( ( T ) show that T ( c ) ~I 0 for all (T E C, so ( 7 2 ) is always satisfied. 0

Theorem 12.3: (notation of Theorem 10.4) Let k = 21 be even. Let (T E C. (a)

If n > I then TE'(')+6

is never square-integrable.

(b) If n = 1 then lET(u)+6 is square-integrable if and only i f u = (A, 0 ) E and depth()\) = 1. (c) I f n

< 1 then E T ( u ) f 6is square-integrable for all

(T

Go

E C.

Proof: When k = 21 the Harish-Chandra condition ( 7 1 ) becomes

<1-n.

T((T)~

(73)

Case (a): n > 1. The formulas for T ( ( T )in Theorem 10.4 show that T ( ( T ) is~ either 0 or -1 in this case. But 1 - n 5 -1, so ( 7 3 ) is never satisfied. Case (b): n = 1. Now the right side of ( 7 3 ) is zero. The formulas for show that T ( ( T ) ~< 0 if and only if (T = (X,O) E GOand depth()\) = 1.

T((T)

Case (c): n < 1. Now the right side of ( 7 3 ) is positive. The formulas for T ( ( T ) show that T ( ( T ) 5 ~ 0 for all (T E C, so ( 7 3 ) is always satisfied. 0

Examples (1) Assume k is even. Then the oscillator representation a ( n kis) single2n then we see from the formula for the valued on Sp(n,R). If k &correspondence that every GL(n, C)-highest weight )\ that satisfies the Harish-Chandra inequality is of the form T ( ( T ) 6, for some (T E C. Thus every highest-weight discrete-series representation of Sp(n, R)occurs in the reduction of a ( n kin) this case.

>

+

( 2 ) Let (T be the trivial representation of O(Ic) (denoted by 7r(',l) in Sections 10.1 and 10.2). Then (T E C for all n 2 1 and T ( ( T ) = 0, by Theorems 10.1 and 10.4. The representation, call it 7 r + , of Mp(n,R) that corresponds to (T is square integrable if and only if 2n < k . It occurs with multiplicity one in W 2 ( M n x k ) , and has highest weight 6 = [ - k / 2 , . . . I - k / 2 ] . This weight parameterizes the one-dimensional representation g H det(g)-'/' of the maximal compact subgroup of Mp(n, R).Since consists of the constant functions, the space lHI: := lE6 of 7r+ is the completion (in the Fischer norm) of the space P ( M n x k ) G ,where G = O ( k , C ) . ( 3 ) Let (T be the representation g H det(g) of O ( k ) (denoted by 7r('>-') in Sections 10.1 and 10.2). Then (T E C if and only if n 2 Ic. In this case

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality T(0) = [ 0,.

401

. . ,o, -1,, . , , -11

n-k

k

by Theorems 10.1 and 10.4. The representation, call it T - , of Mp(n, R)that corresponds to cr is never square integrable. It occurs with multiplicity one in H 2 ( k f n x k ) when n 2 k , and it has highest weight

x = [ - l c / 2 , . . .,-Ic/2] n

+ [ O , . . . ,o, -1,. * .,-11. n-k

k

This is the highest weight of the representation (det)-k/2 @ Ak(Cn)*of the maximal compact subgroup of Mp(n, R). In this case T" = C g k , where g k ( Z ) is the determinant of the bottom k x Ic block of z E M n x k when we take the orthogonal group G = 0 ( C k , w ) as in Section 10. Thus the space MI? := Ex of T - is the completion (in the Fischer norm) of the space ( P ( h ' l n x k ) G g k ) @ A'((@.")*. Note that for fixed n , one obtains a distinguished set of n irreducible unitary highest-weight representations of Mp(n, R) this way by taking k = 1,.. . , n.

(4) By the harmonic duality theorem, (W:,n*) are the only irreducible Mp(n,R) modules that occur with multiplicity one in w 2 ( h f n x k ) . More details and other models for the representations lE7(")+6 can be found in

161. Final Remarks: In Schur-Weyl duality we took tensor powers of the representation of GL(n, C)on Cn (the representation of smallest dimension) to obtain all the irreducible finite-dimensional polynomial representations of GL(n, C). The two irreducible components r* of the oscillator representation on MI2 (C")are the smallest unitary highest-weight representations of Mp(n,R) in the sense of Gelfund-Kirillov dimension (see [31]). As we already noted in Section 11.3the representation on w 2 ( M n x k ) is the Ic-fold tensor product of this representation: k

w2(Mn,k)

=@

w2((Cn) (Hilbert-space tensor product).

(74)

Thus the action of the group O ( k ) on the right-side of (74) is another instance of a hidden symmetry.j jThe unitary representations that occur in the decomposition of this tensor product are the mathematical analog of the elementary particles, some familiar and some exotic, that physicists create by high-energy collisions of the basic particles.

R. Goodman

402

13. Brauer algebra and tensor harmonics In this final section we use duality to decompose the space of Ic-tensors under the action of the orthogonal or symplectic group G. This was first done by Brauer [4], who determined the generators and relations of the Gcentralizer algebra. The complication here is that this algebra is not a group algebra (as was the case when G = GL(n,C)). However, just as in the case of Howe duality, there is an analog of the harmonic duality of Section 9 in this situation. The centralizer algebra contains C [ 6 k ]as a subalgebra, and there is a subspace of harmonic tensors (in Weyl’s terminology completely traceless) which decomposes in a multiplicity-free way under the jointly commuting actions of G and 6:k.The full space of Ic-tensors then decomposes as the sum of spaces of partially harmonic tensors (see [16, Sec. 10.31, [9] and [lo] for details). 13.1. Centralizer algebra and Brauer diagrams Let G be the full isometry group of a nondegenerate bilinear form w on a finite-dimensional complex vector space V . We assume w to be either symmetric or skew-symmetric. For f E V * define f b E V by w(fb,

v) = ( f ,v> for all v

E

v.

The map f H f b is then a G-isomorphism between V * and V. Define a G-module isomorphism T : V*@2k ---f End(VBk) by T ( f 1€3

‘ . . €3 f2k)U

= W ( f ; €3

fi €4

”

’

€3 f k k ,

u)fi” @3 fi €4

’

’ . €4 f k k - 1

(75)

for fi E V * and u E V B k . Here we have extended w to a bilinear form on V@‘“by k

w(u1 CZJ

. . . CZJ uk,v1 CZJ . . ’ CZJ V k )

w(ui, vi) for ui,vi E

=

v.

k l

Theorem 13.1: Let Zk be the set of two-partitions of (1,.. . ,2k}. For E E sk let A, E (V*B2k)Gbe the corresponding complete contraction. Then EndG(VBk)= Span{T(Xc) :

E

E

%} .

Proof: Since T is a G-module isomorphism this is a immediate consequence of Corollary 7.5. 0 Theorem 13.1 only gives a spanning set for the centralizer algebra EndG(Vwk)as a vector space. To describe the multiplicative structure of this algebra it is convenient to introduce a graphic presentation of the set

Multiplicity-Ree Spaces and Schur- Weyl-Howe Duality

403

of two-partitions. We display the set { 1 , 2 , . . . , 2 k } as an array of two rows of lc labeled dots, with the dots in the top row labeled 1 , 3 , . . . , 2k - 1 from left to right, and the dots in the bottom row labeled 2 , 4 , . . . , 2 k . Consider the set XI, of all (unoriented) graphs whose vertices are the two rows of dots, and such that each dot is connected with exactly one other dot by an edge. (A dot in the top row can be connected either with another dot in the top row or with a dot in the bottom row.) An example with lc = 5 is shown in Figure 1. We call an element of XI, a Brauer Thus we can identify the set 3 k of two-partitions with x k ; if E E k corresponds to the Brauer diagram x E xk, we shall write A, for the complete contraction

<

Fig. 1. A Brauer diagram.

The group 6 j 2 k acts transitively on XI, by permuting the dots according to their labels. If x E XI, and s E 6 2 k then s . x is the graph obtained by permuting the dots by s and maintaining the edge connections (dot s ( i ) is connected to dot s ( j ) in s . x if and only if dot i is connected to dot j in x). Clearly O&(S)x,

= As.,

for

S

E 6 2 k and X E X I , .

(76)

Here C k denotes the representation of 6 k on P k a,s in Section 3, and UII, is the contragredient representation on V * @ 2 kLet . xo be the graph with each dot in the top row connected with the dot below it (see Figure 2 for 1

3

5

xo=lI I 2

Fig. 2.

4

6

7

8

9

II 1

0

Basic diagram with labeled dots.

kKerov [23] uses the term chip because of the analogy with an integrated circuit chip, where the top row of dots are the input ports and the bottom row of dots the output ports. For a development of Kerov's approach, see 191 and [lo].

R. Goodman

404

the case k = 5). Then the Brauer diagram z1 in Figure 1 is s . zo where s E 610is the cyclic permutation (2594). Let T : 6 k + 6 2 k be defined by ~(s)(2j- 1) = 2s(j) - 1 and 7-(s)(2j) = 2j for j = 1 , . . . , k . If s E 6 k , then T ( S ) acts on a Brauer diagram by permuting the top row of dots according to s while leaving each dot in the bottom row fixed. Clearly T is an injective homomorphism, and from (75) and (76) we see that

Q(S)T(X,)= T(&(s).z) for

SE

6jk

and

5

Ex k

.

(77)

For the basic diagram zo we have r ( s ) . zo = xo if and only if s is the identity, so the permutations in 6 k correspond to the diagrams in the orbit ~ ( 6 k‘20 ) (these diagrams are just the two-line notation for a permutation). Hence by (77) the operators in EndG(VBk) associated with the orbit of zo come from the natural action of 6 k on V B k In . particular, the basic diagram xo corresponding to the G-invariant tensor w B k gives the identity operator on V B k . The complete set of T ( 6 k ) orbits on x k can be described as follows. For x E x k let r be the number of edges in the diagram of z that connect a dot in the top row with another dot in the top row (call such an edge a top bar). The bottom row of z also must have r such edges (call them bottom bars), and we call x an r-bar diagram. All diagrams in the T ( 6 k ) orbit of 2 also have r top bars, and there is a unique z in this orbit with all its edges either horizontal or vertical (that is, if z is considered as a twopartition of 2k, then every odd-even pair {2i - 1,2j} that occurs in z has i = j ) . We will call such a Brauer diagram (or two-partition) normalized. The normalized diagrams give a set of representatives for the T ( 6 k ) orbits on x k . For example, when k = 3 and r = 1 then there are three orbits of l-bar diagrams, with normalized representatives indicated in Figure 3.

Fig. 3.

Normalized l-bar Brauer diagrams ( k = 3).

Theese orbits coresspond to the two-partitions

Multzplicity-&ee Spaces and Schur- Weyl-Howe Duality

405

If z is a normalized Brauer diagram, then for every top bar in z joining the dots numbered 2i - 1 and 2 j - 1 there is a corresponding bottom bar joining the dots numbered 2i and 2 j . We will say that z contains an (i,j)-bur in this case (with the convention that i < j). For example, the normalized diagram in the orbit ~ ( 6 5 ) z(with l z1 from Figure 1) is shown in Figure 4; it contains a (2,5)-bar.

Fig. 4.

Normalized Brauer diagram with (2,5)-bar

13.2. Generators f o r the centralizer algebra

A normalized Brauer diagram determines an element of the algebra EndG(V@'")by Theorem 13.1. For example, the diagram shown in Figure 5

-I I I " '

Fig. 5 .

Brauer diagram for

712

= DlzClz.

contains a single (1,2)-bar corresponding to the tensor 0,*~(23)w@'", where (23) is the transposition 2 ts 3. Since 0 , * ~ ( 2 3 ) w = @ (~a , * ( 2 3 ) ~ @ ' ) @ w @ ( we have T(~&(23)Ld@~) ui @ 212 @ u =

i

c w ( v 1 , f p Z ) W ( 7 J z ,f P 2 ) PZ

= w(v1, v z p C3 'u.

for

V and u E V@("'). Here = bP4.1 and

w1,v2 E

4 f p , f9)

{fp}

and

{fp}

I

fp1

@ f p l @ 'u.

Pl

are bases for V with

P

is the tensor dual to w. Thus this I-bar diagram gives an operator which is the composition

T(OZk(23)ek)

3V@.(k-Z)% v@k,

v@k

T ~ = Z

R. Goodman

406

with C12(v18 vz 8 u ) = W ( V ~ , V ~ ) Ua contraction operator (contract the first and second tensor positions by w) and Dlz(u) = 13 8 u an expansion operator (multiply on the left by 0). These operators obviously intertwine the actions of G on V@'"and V @ ( k - 2 ) . In general, for any pair 1 5 i < j I k we define the ij-contraction operator Cij : V B k+ V @ ( k - 2by )

Cij(v1 @ * ' ' 8 v k ) = w(vi, vj) vl 8 ' ' ' 8 v', 8 ' ' ' 8 6 8

"

'

8 ?,Jk

(omit vi and vj in tensor product) and the ij-expansion operator Dij + V@kby

:

v@(k-z)

n

These operators intertwine the action of G and are mutually adjoint, relative to the invariant form w on V B k :

~ ( C i j uW) , = W ( U , DijW) Set

rij = Dijcij

for

E V B k ,w E

V@("') .

E EndG(VBk).If u = 211 8 * . . 8 v k with

Vi

E

(78)

v,then

n

The contraction and expansion operators satisfy the symmetry properties

D a.j. E Dj a. .I Ca.j . - EC.. j a 7

(80)

since C pf p 8 f P = E C pf P 8 fp. Hence rij = rji, so the operator rij only depends on the set { i , j } . Let z k , T C XI, be the set of normalized r-bar Brauer diagrams, and set

T=o

Lemma 13.2: Suppose that z E z k l r as a normalized r-bar Brauer diagram with bars {il,jl},.. . , { i r , j T } . Then T. w

..j-. p w q

=.j-'..7-.. aqjq

wp forp # 4 .

(81)

Thus the operator rz = rilj, . . .ri,.j,. only depends on z and not on the enumeration of the bars in z . Furthermore, rt = T(X,). Proof: The commutativity relation (81) is clear since rij only operates on the ith and j t h tensor positions. 0

407

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

Proposition 13.3: The algebra EndG(V@'") is spanned by the set of operators U k ( S ) T z with s E 6 k and z E zk.Furthermore, if dim V 2 2k then this set of operators is linearly independent. Proof: Since zk gives a cross-section for the r ( 6 k ) orbits on X k , the first statement follows from Theorem 13.1, Lemma 13.2, and the intertwining relation (77). The proof of linear independence when dim V 2 2k is straightforward (see [16, Corollary 10.1.41). 13.3. Relations in the centralizer algebra

We next determine the algebraic relations among the operators in Proposition 13.3.

Lemma 13.4: Let n = dimV and set E = 1 if w is symmetric and E = -1 if w is skew. The operators rij (where 1 5 i , j 5 k and i # j ) satisfy the following relations, where ( i l ) denotes the transposition of i and 1: = r j i and 72. $3 = o r$3

(1)

rij

(2)

rijqrn= r l m r i j

(3)

~ i j ~= j la k ( i 1 ) ~ j fl o r

(4)

ak(s)Tijok(s)-l

(5)

Ok(ij)Tij

f o r distinct i l j l l l m distinct i l j l l

= ~ ~ ( i ) , ~f o( rj )all s E 6 k

= &Tij

Proof: The contraction and expansion operators satisfy CijDij = nI , (82) which follows from C&,w ( f,, f p ) = n. This implies property (1).Property (2) was already checked in Lemma 13.2. To verify (3), note that T i j T j l ( u 1 @ ' . . @ u k )= w ( . j , ~ ) C ~ ( u i , f p ) u p q , P,q

where upq= w1 @ . . . @

fq

me..@

v

ith

jth

c W ( u i ,f p ) u p q = & u l @ " ' @ P

f q

v fq

v ith

@...@

@"'@

f P @ . . . @ u k . But

v lth

f q

v jth

@'"@

ui

v

@*'*@uk,

lth

which gives (3). Relations (4) and ( 5 ) are simple calculations from the definition of rij. 0 Define the Braver Algebra B k ( & , n ) with parameters JC,&,nto be the associative algebra generated by 6 k and elements {rij : 1 5 i < j 5 k}

R. Goodman

408

subject to the relations (1)-(5) in Lemma 13.4; here n can be any complex number and E = 4 1 . From these relations it is clear that B k ( E , n) is finitedimensional. If 7 is the subalgebra generated by { q j } , then 7 is an ideal and we have the decomposition ak(&,

n ) = e[ek]@ 7 .

(83)

f i o m Proposition 13.3 and Lemma 13.4 we see that there is a surjective algebra homomorphism Bk(&,n)

-

(n= dimV)

EndG(V@'")

with E = k1 determined as in Lemma 13.4 (5). The two algebras are isomorphic if n 2 2k. In any case, the centralizer algebra EndG(V@'")is the quotient of the associated Brauer algebra by a two-sided ideal, so the representations of the centralizer algebra can be viewed as representations of the Brauer algebra.'

I

...

I

Fig. 6.

x r

r f l

I

Brauer diagram for

".

I

5,.

We can describe the multiplication in B k ( ~ , n and ) the relations in Lemma 13.4 in terms of concatenation of Brauer diagrams. Let s, E (?jk be the transposition r t) r 1. It corresponds to the Brauer diagram shown in Figure 6. Let z , = T,,,+I be the operator corresponding to the normalized Brauer diagram with a single ( r ,r 1) bar, as in Figure 7. Since 6 k is generated by s 1 , . . . , S k - 1 , we see from Proposition 13.3 and property (3) in Lemma 13.4 that the algebra B k ( ~ n is ) generated by the operators ~ 1 ,. .. , s k - 1 and z 1 , . . . , z k - 1 . If x, y are Brauer diagrams, then their product xy in the Brauer algebra is obtained by placing the x above y and joining the lower row of dots in x to the upper row of dots in y. When x, y correspond to elements of 61,(no bars) this procedure obviously gives the multiplication in 6 k . When x or y have bars, we remove the closed loops from the concatenated graph using relation (1) in Lemma 13.4.

+

+

'See [9]and [lo] for recent work on the representation theory of the Brauer algebra and citations of earlier work.

Multiplicity-Free Spaces and Schur- Weyl-Howe Duality

r Fig. 7.

409

r+l

Brauer diagram for .'2

The general recipe for transforming the concatenated Brauer diagrams of z and y into a scalar multiple of the Brauer diagram for zy is as follows: (1) Delete each closed loop in the concatenated diagram and multiply by a scalar factor of 'n if there are T such loops. (2) Multiply by a factor of E for every path in the concatenated diagram that begins and ends on the top row of x or on the bottom row of y.

For example, if z = 4236)735746 and y = (~(46)712734756,then xy is obtained as shown in Figure 8 (see [16, Sec. 10.1.2 and Exercises 10.1.31 for further details and examples).

Fig. 8.úThe relation

13.4. Harmonic tensors

Let k 2 2. A tensor u E V@'"is called w-harmonicm if it is annihilated by all the contraction operators Cij. Denote by

the space of all w-harmonic k-tensors. We will simply call these tensors harmonic and write Fl(V@',w ) = ',Jf(V@') when w is clear from the context. Example: Assume w is symmetric and let v E V. Then Cijv@'" = w ( ~ , v ) v @ ( " ~Thus ). the symmetric tensor v@' is harmonic if and only mWeyl uses the term traceless; we prefer the term harmonic because when w is symmetric the contraction operators Cij act as Laplacians on the symmetric tensors.

410

R. Goodman

-

if v is an isotropic vector for w . This is the same as the polynomial function 5 (5,x ) on ~ V* being harmonic relative to the Laplace operator defined by w . On the other hand, every skew-symmetric tensor is harmonic when w is symmetric. Theorem 13.5 (Harmonic tensor duality): T h e space IFt(V@'k) is invariant under 6 k x G and decomposes as %(V@'k) %

@ E X @ ux.

(84)

XEA

Here A c Par(k), E X is the irreducible 6 k - m o d u l e corresponding t o the partition X by Schur-Weyl duality, and U x is a n irreducible G-module. Furthermore, the modules U X are all distinct.

Proof: Since Ci3rij = CijDijCij = nCij, we have Ker(Cij) = Ker(rij). Hence u is harmonic if and only if riju = 0 for 1 5 i < j 5 k . Since rij commutes with pk(G), we see that IFt(V@'") is invariant under G. Proposition 13.3 and Lemma 13.4 imply that 'FI(V@"") is invariant under B k ( & , n ) and the action of B k ( & ,n) on IFt(V@'") reduces to the action of 6.k. Now apply Theorem 1.4. 0 13.5. Decomposition of harmonic tensors for Sp(V)

We now determine the set A of partitions of k occurring in Theorem 13.5 and the corresponding irreducible representations U X when G is the symplectic group." We take V = C" with n = 21 and the bilinear form

(so the standard basis vectors ei are w-isotropic and el is paired with en, e2 is paired with en-l, and so forth). Let G = Sp(V,w) and let H be the diagonal matrices in G. Then H is a maximal torus whose elements are of the form

h = diag[zl,. . . , z i , x ; 1 , . . . ,z;'],

zi E C x .

(85)

Following the notation in Section 10, we let D, be the diagonal matrices, B, the upper-triangular matrices, and N , the upper-triangular unipotent matrices in GL(V). With our choice of w the group B = G n B, is a Bore1 "Similar methods work for the orthogonal groups but the details are considerably more intricate - see [16, Sec. 10.21 for a full treatment.

Multiplicity- Free Spaces and Schur- Weyl-Howe Duality

41 1

subgroup of G and N = G n Nn is its unipotent radical. The weight lattice of H is identified with Z 1 , where X = [ m l ,. . . , ml] gives the character h H xyl . . . x r l

when h is given by (85). The Weyl group W of G acts by all permutations and sign changes of the coordinations of A. The set of B-dominant weights is thus identified with I+?! (see [16, Sec. 2.51). For X E Nk+ let (7rX,UX) be the irreducible representation of G with highest weight A. If IXI = Ic then we view X as a partition of Ic with a t most 1 parts. Let E X = (V@k)Nn(X) be the corresponding irreducible representation of 6 k on the space of GL(n, C ) highest weight vectors of weight A, as in Theorem 3.8. Theorem 13.6: Let X E Par(k,n). Then E X c X ( V B k ) if and only ZfX has at most 1 parts. Furthermore, the space of w-harmonic k-tensors has isotypic decomposition 'H(VBk)E

@

EX@lJX

(86)

X EPar( k , l )

under 6 k x Sp(V, w ) . Thus all the irreducible representations of sp(V,w) occur in the decomposition of the harmonic k-tensors, f o r k = 1 , 2 , . . . . The general form of decomposition (86) follows from Theorem 13.5. To determine the spectrum A of Sp(V,w) on the harmonic tensors, we will compare the spaces of B eigenvectors in V@""with the spaces E X . If p. = [ml,. . . ,m,] E Z" is a weight of D,, then we denote by ji the restriction of p to H . From (85) G , = [ml - m,, m2 - m,-l,..

. , ml - ml+l].

(87)

Hence if p. E PIT+ is a B,-dominant weight, then ji is a B-dominant weight. We introduce the notation

w'((x)= ( v @ . " ) ~ ( xfor) ,x E I+?;+. Since N

c N,, we have EP c Wk(,G).

Proposition 13.7: There are the following dichotomies:

(1) Assume X E

PI$+. Then either Wk((x)n'FI(VBk)= 0 or else W k ( X )c

'FI(V@k).

(2) Assume p. E Par(k,n). Then either EP Wk

(a.

n 'FI(VBk) = 0 or else EP

=

R. Goodman

412

Proof: (1) By Theorems 3.7 and 13.1 we know that Wk((x) is an irreducible module for B ~ ( E n). , Since Wk((x)n?i(V@k) is a Bk(&,n)-invariantsubspace of Wk((x), it must be 0 or W k ( X ) . (2) Assume EP n'FI(VQk)# 0. Since EP c Wk((cL), it follows by (1) that W k ( J ic ) 7-l(Vmk). Furthermore, Wk((cL) is irreducible under 6:k.Indeed, it is irreducible under B ~ ( En), by (l),and on the harmonic tensors the action of B ~ ( E n), is the same as the action of 6 k . By Theorem 3.8 it follows that W k ( ( c L )= E'1. 0

Proposition 13.8: Let p E Par(k,n). T h e n E l p has at most 1 parts.

c ?l(Vmk)if and only if

Proof: Let p = [ml,. . . , m,]. Define up = u'f"' 8 . . . 8n'@ as in Theorem 3.8, where up = el A. . A e p and c j = m3-mj+l (with mntl = 0). Then up E E P and the depth of p is the largest integer d such that c d # 0. We first verify that up is harmonic if and only if p 5 1. To see this, take any pair i , j with 1 . 5 i < j 5 p . If p 5 1, then Cijup = 0 since w ( e i , e j ) = 0. Conversely, if p > 1 then

-

cl,l+lup=

el A . . . A el-1 A e1+2 A . . . A e p # 0 ,

since w(e1, el+l) = 1. So up is not harmonic in this case. Thus to finish the proof of the proposition, we may assume that p has at most 1 parts. Let 1 5 p 5 q 5 1. We now show that

Cij(up8 u q )= 0

for all 1 5 i 5 p

<j 5 p +q .

Set u = up 8 uq. In terms of the basis { e I } , w is obtained by a double alternation:

The contraction operator Cij removes es(i) and et(j-p) from each term of the sum and multiplies the resulting ( p q - 2)-tensor by w(es(i),et(j-p)). But s ( i ) t ( j - p ) 5 p q I n,while w(e,, eb) = 0 unless a b = n 1. Hence Ci,j(w)= 0. 0

+

+

+

+

+

We now complete the proof of Theorem 13.6. By Propositions 13.7 and 13.8, it will suffice to prove the following:

(*)

IfX E

N$+ is such that 0 # Wk((x)c ?l(Vmk),then 1x1 = k .

Multiplicity-flee Spaces and Schur- Weyl-Howe Duality

413

To establish (*), take a nonzero tensor u E Wk(X)and decompose u under the action of D, as u=

c

u p , where p E Par(k, n) and up

P

Fix some p = [ml,. . . ,mn] such that u p # 0. Then G, = X and from (87) we see that n

1

1x1 = C(mi - m,+i-i)

= k - 2r,

where r

>, mi. n

=

i=l+l

i=l

Thus X E Par(k - 2r,l) so from Theorem 3.8 we know that 0 # E X c V@(k-2r). Suppose for the sake of contradiction that r > 0. Since the expansion operator D12 is injective and intertwines the action of G on tensors, we have

0 # (D12)rEx c W k ( ( x ) . Since Cl2Dl2 = n l , this implies that

But we have assumed that Wk((x)in contained in the harmonic tensors, a 0 contradiction. Thus r = 0 and (*) is proved.

Examples (1) Assume k 5 1 and take X = [l,. . . ,1] E Par(lc). Then E Xis the sgn representation of 6 k and U x is the kth fundamental representation of sp(v,w ) . From Theorem 13.6 we know that uXis the sgn-isotypic component for 6 k in 3-I(VBk). Hence

is the space of harmonic skew k-tensors. (2) Let k = 2 and assume 1 2 2. Then the two partitions of 2 give the trivial and sgn representations of 6 k , respectively. Because the form w is skewsymmetric, every symmetric tensor is harmonic. Hence by Theorem 13.6 we have

h!(VB2) = s2(v)@%skew(v). The summands are the irreducible representations of G with highest weights 2 ~ and 1 ~1 ~ 2 .

+

414

R. Goodman

References 1. I. Agricola, Invariante Differentialoperatoren und die Frobenius-Zerlegung einer G-Varietat, J . Lie Th. 11 (2001) 81-109. 2. V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Part I, Comm. Pure Appl. Math. 14 (1961) 187-214.

3. N. Bourbaki, Elkments de mathkmatique, Fascicule X X X V I I , Groupes et algbbres de Lie, Ch. 2 & 3, Hermann, Paris 1972. 4. R. Brauer, On algebras which are connected with semisimple continuous groups, Ann. of Math. 38 (1937) 857-872. 5. P. Cartier, Quantum mechanical commutation relations and theta functions, in Algebraic Groups and Discontinuous Subgroups (Proc. Symp. in Pure Mathematics IX), Amer. Math. SOC., 1966, 361-386. 6. M. Davidson, T. Enright and R. Stanke, Differential Operators and Highest Weight Representations (Memoirs of the AMS # 455), Amer. Math. SOC., 1991. 7. J. Faraut and A. KorBnyi, Analysis on Symmetric Cones, Oxford University Press, 1994. 8. G. B. Folland, Harmonic Analysis in Phase Space (Annals of Math. Study # 122), Princeton University Press, 1989. 9. F. Gavarini and P. Papi, Representations of the Brauer algebra and Littlewood’s restriction rules, J . Algebra 194 (1997) 275-298. 10. F. Gavarini, A Brauer algebra theoretic proof of Littlewood’s restriction rules, J . Algebra 212 (1999) 24C-271. 11. S. Gelbart, Holomorphic discrete series for the real symplectic group, Invent. Math. 19 (1973) 49-58. 12. S. Gelbart, A theory of Stiefel harmonics, Trans. Amer. Math. SOC.192 (1974) 29-50. 13. S. Gelbart, Examples of dual reductive pairs, in Automorphic Forms, Representations and L-Functions (Proc. of Symp. in Pure Math. XXXIII, vol. l ) , Amer. Math. SOC.,1979, 287-296. 14. R. Goodman, Analytic domination by fractional powers of a positive operator, J. Funct. Anal. 3 (1969) 246-264. 15. R. Goodman, Analytic and entire vectors for representations of Lie groups, Trans. Amer. Math. SOC.143 (1969) 55-76. 16. R. Goodman and N.R. Wallach, Representations and Invariants of the Classical Groups, Cambridge University Press, 1998. 17, Harish-Chandra, Representations of semisimple Lie groups V, Amer. J . Math. 78 (1956) 1-41. 18. G. Hochschild, The Structure of Lie Groups, Holden-Day, 1965. 19. J.-S. Huang, Lectures on Representation Theory, World Scientific, 1999. 20. R. Howe, Dual pairs in physics: harmonic oscillators, photons, electrons, and singletons, in Applications of Groups Theory in Physics and Mathematical Physics, Lectures in Appl. Math. 21, Amer. Math. SOC.,1985, 179-206. 21. R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. SOC. 313 (1989) 539-570.

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22. M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978) 1-47. 23. S. V. Kerov, Realizations of representations of the Brauer semigroup, J . Soviet Math. 47 (1989) 2503-2507. 24. A. W. Knapp, Representation Theory of Semisimple Groups, Princeton University Press, 1986. 25. F. Knop, A Harish-Chandra homomorphism for reductive group actions, Ann. of Math. 140 (1994) 253-288. 26. B. Kostant, Lie group representations on polynomial rings, Amer. J . Math. 85 (1963) 327-404. 27. E. Nelson, Analytic Vectors, A n n . of Math. 70 (1959) 572-615. 28. I. E. Segal, Tensor algebras over Hilbert spaces I, Trans. Amer. Math. SOC. 81 (1956) 106-134. 29. R. Strichartz, The explicit Fourier decomposition of L 2 ( S O ( n ) / S O ( n- m ) ) , Canad. J. Math. 27 (1975) 294-310. 30. S. Sternberg, Group Theory and Physics, Cambridge University Press, 1994. 31. E.-C. Tan and C.-B. Zhu, Poincarb series of holomorphic representations, Indag. Mathem., N.S., 7 (1996) 111-126. 32. T. Ton-That, Lie group representations and harmonic polynomials of a matrix variable, R a n s . Amer. Math. SOC.216 (1976) 1-46.