Representations of Real Reductive Lie groups [1 ed.] 3764330376

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Progress in Mathematics Vol. 15 Edited by J. Coates and S. Helgason

Birkhauser Boston · Basel · Stuttgart

David A. Vogan, Jr.

Representations of Real Reductive Lie Groups

1981

Birkhauser Boston · Basel · Stuttgart

Author: David A. Vogan, Jr. Dept. of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139

-~)

)

JUN ?:2

1983

•"

,./

Library of Co~gress Cataloging in Publication Data Vogan, David A., 1954Representations of real reductive Lie groups. (Progress in mathematics ; vol .15) Bibliography: p. Includes index. 1. Lie groups. 2. Representations of groups. I. Title. II. Series: Progress in mathematics (Cambridge, ~·1ass.) ; 15. QA387.V63 512' .55 81-10099 ISBN 3-7643-3037-6 AACR2 CIP- Kurztitelaufnahme der Deutschen Bibliothek Vogan, DavidA.: Representations of real reductive Lie groups/ David A. Vogan. - Boston ; Basel ; Stuttgart Birkhguser, 1981. (Progress in mathematics ; Vol. 15) ISBN 3-7643-3037-6 NE: GT All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner. © Birkhguser Boston, 1981 ISBN 3-7643-3037-6 Printed in USA

To Jonathan, for eating the first draft.

TABLE OF CONTENTS Page

xi

?ref ace

xii

Introduction

Chapter 0.

Preliminaries

1

§1.

Assumptions on G

1

§ 2.

Roots and

4

§ 3.

Group representations and BarishChandra modules

11

Finite dimensional representations

26

§ 4.

Chapter 1.

SL (2, JR)

36

Group representations and Lie algebra representations

36

§2.

Structure of Barish-Chandra modules

44

§3.

The principal series

57

§1.

Chapter 2.

Z ('?)

+

§ 4.

SL- (2, IR)

68

§ 5.

An introduction to the R group

75

Geometry of the Kazhdan-Lusztig Conjecture

92

The Kazhdan-Lusztig conjecture for Verma modules

92

§2.

A special case of the classification

98

§3.

Geometry on the flag manifold

§1.

106

Chapter 3.

Chapter 4.

Kostant•s Borel-Weil Theorem

115

§1.

The Casselman-Osborne Theorem

115

§2.

Kostant's Theorem

122

Principal Series Representations and Quasisplit Groups §1.

The principal series

136

§2.

The Langlands quotient for principal series

149

§3.

Fine K-types and the R group

169

§4.

Principal series for quasisplit groups: reducibility and equivalences

191

An introduction to the translation principle

206

§5.

Chapter 5.

Cohomology of (t7,K) Modules

217

§1.

Representations of K

217

§2.

The Hochschild-Serre spectral sequence

226

Construction of parabolic subalgebras

234

Construction of cohomology

282

§3. §4.

Chapter 6.

136

Zuckerman's Construction and the Classification of Irreducible ( f/,K) Hodules

298

§1.

The category ~(~,B)

299

§2.

Zuckerman's functors

325

§3.

Parabolic induction

344

ix

Chapter 6.

(Continued) §4.

Principal series revisited

374

§5.

Construction of the standard representations

392

Other parametrizations of G, and the Langlands classification

409

Proof of Theorem 2.2.4

425

§6. §7.

Chapter 7.

Coherent Continuation of Characters

429

§1.

Complex groups and ideals in

§ 2.

Translation functors revisited

436

§ 3.

The translation principle, and an introduction to wall crossing

461

Translation functors and parabolic subalgebras

488

§ 4.

Chapter 8.

U(~)

429

Reducibility of the Standard Representations §1.

495

Cohomology of irreducible representations

495

Coherent continuation of the standard representations

527

Cayley transforms of regular characters

547

§4.

Non-complex root walls

568

§5.

More about Ua

588

§6.

Standard irreducible modules

602

§7.

Singular infinitesimal characters

619

§2. §3.

Chapter 9.

The Kazhdan-Lusztig Conjecture

624

§1.

Commutativity of certain diagrams

624

§ 2.

Blocks in ~(

§3.

Vanishing theorems for cohomology

670

§4.

Euler characteristics, character formulas, and Zuckerman's theorem

699

Vanishing theorems using Conjecture 7.3.25

712

§6.

The Kazhdan-Lusztig conjecture

725

§7.

Open problems

737

§5.

't/, K)

648

Bibliography

742

Index

748

Index of Notation

751

Preface Since this manuscript was completed in August, 1980, a great deal of progress has been made on some of the topics treated.

Working independently, Brylinski-Kashiwara and

Beilinson-Bernstein have established Conjecture 2.1.7 on the composition series of Verma modules.

Lusztig and I have used

the work of Beilinson and Bernstein to prove Conjectures 2.2.12 and 2.3.11 on Barish-Chandra modules, computing all composition series in the case of integral infinitesimal character.

[This settles Problems 9 and 10 of Section 9.7.)

The non-incegral case remains open.

The proofs rely on the

Weil conjectures and the Deligne-Goresky-MacPherson intersection homology theory.

It is not yet clear how these

developments will affect the material presented in this book -- they do not provide any obvious substantial simplifications, although in some cases they suggest significant improvements in the formulation of the theory. I would like to thank Janet Ellis for typing and proofreading a long and messy manuscript with great care and skill in the midst of many other responsibilities. I have the dubious satisfaction of knowing that no errors remain which are not of my own devising.

Cambridge, Massachusetts April, 1981

Introduction This book is a survey of some recent work on the (nonunitary) infinite dimensional representations of a real reductive Lie group G.

There are three major topics.

The

first is Langlands' theorem on the classification and realization of the irreducible representations of G (Theorem 6.5.12).

They are described in terms of certain "standard

representations"

(Definition 6.5.2), which generalize the

principal series and are sometimes reducible.

The second

topic is the reducibility of these standard representations. The main results

(Theorem 8.6.6 and Proposition 8.7.6) are

due to B. Speh and the author; they are not quite decisive. The third topic is a conjecture which describes explicitly the decomposition of standard representations into irreducible representations -- or, equivalently, the Harish-Chandra characters of the irreducible representations.

This

generalizes a conjecture of Kazhdan and Lusztig for Verma modules, and is described in Section 9.6. Since the theory of non-unitary representations of G was created by Harish-Chandra essentially as a means to study unitary representations, some apology might seem to be required for a book in which unitary representations play almost no part.

The first explanation for this omission is simply

a lack of space.

The first two topics at least are very

important for recent work on unitary representations.

For

example, the study of unitary representations with non-zero continuous cohomology (see [3]) has been advanced by the algebraic study of certain representations which are still

not known to be unitary.

(They are included in the conjec-

turally unitary representations of (6.5.17) below.)

The

theory of "complementary series" of unitary representations depends on (among other things) an understanding of the reducibility of standard representations.

Thus Theorem 8.6.6

provides a large number of unitary representations, and a proof of the conjecture of Section 9.6 would provide even more. 'rhe real explanation, however, is that non-unitary representation theory is interesting enough not to require any such justification.

Such a claim has to be supported in

the text and not in the introduction; but Chapter 2 attempts to describe the nature of the main results without too much technical clutter. In a little more detail, the book is organized as follows.

The reader is assumed to be quite familiar with

the structure and finite dimensional representation theory of complex reductive Lie algebras; this is really a prerequisite even for understanding the statements of most of the results.

Logically, the book also depends on Barish-

Chandra's basic theory relating group representations and Lie algebra representations ([12]) and on his subquotient theorem ([13]).

These topics are treated in [49] or [50],

and the results are summarized in Sections 0.3 and 4.1 of this book.

Since the ideas needed to prove them will not

be used here, the reader who is willing to take them on faith should have little difficulty.

The first three sections of

Chapter 1 summarize, with some proofs, the representation

theory of SL(2, ffi).

This is intended to provide examples as

guides to the rather abstract and technical treatment of the general case.

Some results are proved in general by

reduction to SL (2, ffi), and the necessary special cases of these are discussed in the rest of Chapter l. Chapter 2 contains a detailed statement of the Langlands classification of irreducible representations of G in a special case; and a geometric formulation of the conjecture of Section 9.6 on composition series of standard representations.

(The two formulations are not known to be equi-

valent.)

The entire chapter is meant as an extended intro-

duction to the rest of the book. The main technical tool used here to study representations is Lie algebra cohomology (of the nil radical of a parabolic subalgebra, with coefficients in a representation). Chapter 3 contains

t~m

fundamental theorems in that subject:

the Casselman-Osborne theorem relating cohomology and the center of the enveloping algebra, and Kostant's formulation of the Bott-Borel-Weil theorem. Chapter 4 discusses that part of the classification of irreducibles which can be obtained from ordinary principal series representations.

In addition to more standard inter-

twining operator techniques, it uses the Bernstein-GelfandGelfand theory of fine representations (see [2]); a detailed account of this theory is given in Section 4.3. Chapters 5 and 6 complete the classification of irreducible representations.

The method is discussed in some

detail in the introductions to those chapters.

Essentially

XV

it is a generalization of the highest weight theory of finite dimensional representations, with highest weight vectors replaced by a more general kind of cohomology classes for the representation.

The main problem is to find a construction

of representations, generalizing induction, which is nicely related to cohomology.

This was done (for reductive groups)

by G. Zuckerman (Definition 6.3.1).

This book uses only one

special case of this definition, in addition to ordinary induction.

It seems likely that one can do much more with

the idea. Chapter 7 is devoted to the Jantzen-Zuckerman "translation principle" and related matters.

This says that all

irreducible representations come in nice families

(like the

principal series, or finite dimensional representations). Hany results can therefore be proved by reducing to the case when the representation is in "general position" in some sense.

This is helpful technically, but the translation

principle plays an even more fundamental role.

Roughly

speaking, it provides a connection between the structure of irreducible representations, and the combinatorial structure of the Weyl group.

This is discussed more carefully in

Section 7.3; the key result is Theorem 7.3.16. In Chapter 8, the basic theorem on reducibility of standard representations is proved.

This is in principle

a trivial consequence of the results of Chapter 7, but requires some messy calculations.

(For example, we need the

Hecht-Schmid character identities for discrete series from [15] .)

Exactly the same ideas, with the judicious addition

xvi

of a technical conjecture (Conjecture 7.3.25), lead to the algorithm for computing composition series; this is the content of Chapter 9.' The book differs from the existing literature in several. ways.

Nost importantly, the standard representations are not

constructed by Langlands' method (that is, ordinary induction from discrete series) .

We use instead Zuckerman's "cohomo--

logical" induction from principal series. phic standard representations

This gives isomor ··

(Theorem 6.6.15), but that fact

is not proved in the text (or used) .

As far as the classifi-

cation itself goes, this choice is simply a matter of taste. However, the conjecture on composition series seems to be comprehensible only in the realization we have given. is an obvious

wa~

(There

to try to use ordinary induction, but then

any simple analogue of the critical Theorem 9.5.1 is false.) This is not to say that Zuckerman's realization is better; for analytic problems, it is Zuckerman's method which encounters obstacles. There are many less substantive changes.

What I had

perceived as the main theorem of [42], which relates cohomology and U ((j) K, does not appear here; the argument has been rearranged slightly to eliminate the need for it. nition of lowest K-type in [42]

The defi-

(Definition 5.4.18 below) has

been replaced by a much more technical one (Definition 5.4.1); they are equivalent for irreducibles but not for general (~,K)

modules.

The advantage of the new definition is that

a number of subsequent technical arguments become much simpler when it is used.

XVll

The proof of the Knapp-Stein reducibility theorem (Corollary 4.4.11) is new, avoiding both the rather delicate analysis used in [26] and the long case-by-case computation in the unpublished second part of [42]. The main result about translation "across a wall" is Theorem 7.3.16; it is (or was) the only non-formal part of the proof of the theorem on reducibility of standard representations.

About one-third of [39] is devoted to a proof

of it, which might charitably be described as sketchy.

A

second proof was given in [45], which was fairly short, but used Duflo's main theorem on primitive ideals (see [8]).

The

proof given here is entirely trivial; but the result is labelled as a theorem in memory of [39]. Each chapter begins with an introduction; these provide a more detailed guide to the main results.

Section 9.7 sum-

marizes some open problems. This book is based on lectures given at MIT during the 1979-80 academic year.

I would like to thank those who

attended for helpful comments and dogged perseverance. of Chapter 6 is unpublished work of G. Zuckerman.

Much

I thank

him for explaining it to me, and allowing it to appear here. J. Vargas provided a list of errors in the first draft, which was very helpful.

Many people have pointed out particular

errors and shortcomings; I apologize for those which undoubtedly remain. The author was supported in part by a grant from the National Science Foundation during the preparation of this book.

Chapter 0.

Preliminaries

General references for this chapter are [40] and [50]. §1.

Assumptions on G

Notation 0.1.1

Suppose H is a real Lie group.

H 0

identity component of H;

i0

Lie algebra of H;

Ad:

H

At=

(Jt )~, the complexification of

U(~)

End(~ ), the adjoint representation of H;

0

7

0

At;

universal enveloping algebra of

=

Write

)t_

~ is given the complex structure bar, defined by X + iY = X - iY

(X,Y

E

~Q)

This notation will be applied to groups denoted by other Roman letters in the same way without comment. By a real reductive linear group, we will mean a real Lie group G (not necessarily connected) , a maximal compact subgroup K of G, and an involution 8 of

~'

satisfying the

following conditions. a)

~O

b)

If g

is a real reductive Lie algebra; G, the automorphism Ad(g) of

E

Cf

is inner

(for the corresponding complex connected group); (0.1.2) c) d)

The fixed point set of 8 is ~ ;

0

Write ~O for the -1 eigenspace of 8; then the map K

x

~O

7

G,

(k,X)

7

k • exp(X) is a diffeo-

morphism; e)

G has a faithful finite dimensional representation;

Let -A..- ,0

~

0

be a Cartan subalgebra, and let H be

be the centralizer of ~O in G.

Then H is abelian.

(Notice that (d) =orces G to have only finitely many camponents.) Throughout this book, G will denote a real reductive linear group.

We will from time to time assume that G satis-

fies additional conditions, but these at least must always be met.

Conditions

(a) -

(d) define Barish-Chandra's category

of real reductive groups

(cf.

[40], §5) and require no fur-

ther justification here; we will use inductive arguments which lead from connected groups to disconnected ones.

As-

sumption (e) is in some sense only a convenience -- most of the results we obtain can be gotten without it, although some--times this requires more work and a less satisfactory formulation of the theorems.

However, one of our main goals is

the formulation of the Kazhdan-Lusztig conjectures discussed in the introduction; and this has not been carried out for non-linear groups.

(The problems do not seem to be very

deep, but they are quite messy.)

Assumption (f) is included

chiefly to make the Knapp-Stein "commutativity of intertwining operators" theorem (Corollary 6.5.14) hold. simplest case where it fails has XSL(2,IR);

(a) -

jG/G j = 4, G0 = SL(2, E)

0 (e) are satisfied, but (f) is not.)

Definition 0.1.3 G (k

(The

Make G an involution of G by setting

exp(X)) = k • exp(-X)

(X

We call C the Cart an involution of

t!o, k G or 7 0 .

K)

Example 0. 1. 4 a)

G

connected real linear semisinple group;

b)

G

GL(n, R), K

c)

G

real points of a reductive algebraic group

defined over d)

G~

If

=

O(n), 8(g)

=

tg-l

JR;

is a complex semisimple Lie algebra, and

is a real form of its Lie algebra G

=

=

normalizer in G--

u (k)

be the projection on the first factor in (c) . p =

Let

p(n,)

Regard U(~) as the algebra of

be as in (0.2.4).

polynomial functions on~*, and define

The

Harish-Chandra map from Z(~)

into U(~)

is by

definition

'r p

o

[

".

Theorem 0.2.8 (Harish-Chandra

see [22], p. 130).

In the setting of Definition 0"2.7, the map E, is an algebra isomorphism from~)

onto U(~)w, the algebra

of Weyl group-invariant elements of U(/v). only on -fv (and not on thro choice of Suppose ~ ~

Definition 0.2.9

?

It depends

J.- ) .

is a Cartan subalgebra

and E,

:

z (/ l _,_ u(.fi_,)

is the Harish-Chandra map.

If A

E

.J'v *,

let

be the composition of E, vlith ev; 0}; we call this the positive system defined by A. if /1,+ = f\,+(cp-,--ll

CA+

b)

= {A

Conversely,

is a fixed ;::lositive root system, set E

~* I 1'1

+

=

A is nonsingular, and ,+}

oA

--

this is the ~'ositive Weyl chamber defined by 1'1+. element A of the closure

7!6 +

An

of this positive Weyl

charooer is called dominant for /1,+; and /1,+ is called positive for A. Lemma 0. 2. 12 /1, +

~

f\, ( ' ,

.fl)

Let ~

'= if!

?e a Cartan subalgebra, and

a po::; '.ti ve root system.

Then the closed

11

Weyl chamber?';.+ '>

/1.

(Definition 0.2.11) is a fundamental

domain for the action of W on

"*

~

This result is most familiar for the real span of the roots in

~; but the general case is easily reduced to that. §3.

Group representations and Barish-Chandra modules.

Definition 0.3.1

A (continuous) representation of a Lie

group H is a pair ( n,

o/l ,

with ?-1 a complex Hilbert

space, and H ->- B ( ')/)

n

a continuous homomorphism of G into the semigroup B(/f) of bounded operators on

~

(endowed with the weak t0po-

logy); we assume that n(l) is the identity operator. An invariant subspace of (",Nl is a closed subspace ?; of TI

H

(H).

0

which is left invariant by all the operators in Such a subspace is called proper if it is not

equal to {O} or

7-/.

The representation (n,

P/l

is called

irreducible if ~:21- 7' 0, and there are no proper closed invariant subspaces.

It is called unitary if the opera-

tors in n(H) are all unitary. At some critical points in Chapter 4, we will need to deal with some representations on spaces of smooth functions, which are, of course, only Frechet spaces.

However, it seems

reasonable to deal then with some minor technical problems, rather than outline the theory in that generality.

12 Definition 0.3.2

Suppose (rr,}>/) and (rr

ous representations of a Lie group H. HomH ( rr , rr

1 )

HomH ( 71, r/ {L

:

1-1

+

1

1 ,

~ )

are continu-

Define

1 )

71' I

L

is

continuous, linear, and Tr

1

(g)L = Lrr(g) for all g

the space of intertwining operators between (rr, (rr

1

,1-/).

Hl

E

H},

and

'de say that the representations are boundedly

equivalent if there is an invertible intertwining operator between them. Definition 0.3.3

Suppose B is a direct product of a

compact group and an abelian group, so that every irreducible representation of H is finite dimensional. Define

H,

the dual object of H, to be the set of

bounded equivalence classes of irreducible representations of H.

If H is abelian, so that every irreducible

representation is one-dimensional, we make

Ha

group as

usual, and call it the group of characters of H. For infinite dimensional (non-unitary) representations, bounded equivalence fails to identify some representations which morally ought to be equivalent.

There is no completely

satisfactory way around this in general; but for reductive groups, the right notion of equivalence is provided by the work of Barish-Chandra in [12]. some preparation.

Its formulation requires

Suppose ('TT, ?{) is a continuous repre-

Definition 0.3.4 sentation of G.

A vector v

IV

E

is called K-finite if

dim for the linear span of all veetors of the form 'TT(k)v, fork c K. AfK = {v Fix (6,

v6 )

e

Put

~lv is K-finite}.

E

and define

n,

U

L (V6)

LcHomK ( 6, 'TI)

7/;

the 6 K-type or 6-primary subspace of

it is a

(possibly infinite) direct sum of copies of 6.

We say

that ('TT,~) is admissi0le if dim ~K(6)

< oo, all 6

E

In that case, the multiplicity of 6 in tion m ( 6, 'TI) = dim HomK (V , 6 Theorem 0.3.5

¥

an admissible representation of G. E

~O'

'TI

is by defini-

K) .

(Barish-Chandra [12]).

tion 0.3.4) and x

K.

Let ('TT, r;/) be

If v

E

~ K (DefiE~~

the limit

lim t('TT(exp(txl)v- v) t+O exists; call it 'TT(x)v.

Then 'TT(x)v

E

7/K,

and this defines a representation of the Lie algebra

1-o

on 1/K.

There is a lattice isomorphism

between

the closed invariant subspaces of fy!, and (arbitrary) K-

J 0 -invariant subspaces

invariant,

of ffK;

if

4

i.f :;_:

'=

closed and invariant, then

A nice exposition of this may be found in [49].

One does

no~

know whether an arbitrary irreducible representation is admissible; but there are various technical results which show that admissibility is a very weak condition on an irreducible representation (cf.

[12]).

As an illustration, we

mention Theorem 0.3.6

(Harish-Chandra [12]).

Any irreducible

unitary representation of G is admissible. Theorem 0.3.5 gives to tation of

(/6;

AtK the structure of a represen-

and since -.;,;.is complex, this structure can

be complexified to a At the same time

HK

(complex linear) representation of

~

is a representation of the group K.

These two representations satisfy the following (obvious) conditions: a)

!>~-~, is K-finite;

Every vector v

"

dim < b) (0. 3. 7)

oo

that is,

•

The differential of the representation of K is the restriction of the that is, if v

E

HK

.i(x)v

~

and x c

c

~

and k

E

..k0 ,

=lim~ (n(exp(tx))v- v). t-+0

If x

representation to

K, then

.

n(Ad(k)x) Definition 0.3.8

=

n{k)n{x)n(k)- 1 .

(Lepowsky) A (§1,K)

modul~

pair (n,X) with X a complex vector space and

is a TI

a

map n:

~

u K

+

End (X),

satisfying a)

1T

1,?"

is a com;:>lex linear Lie algebra representa-

tion, and niK is a group representation; and b)

the compatibility conditions 0.3.7 (a) -

hold, with X replacing

(c)

J/K.

The K-types of X and their multi?licities, and the notion of admissible, are defined exactly as for group representations (Definition 0.3.4).

/r[(at,K) for the category of c{(~,K)

We write

(~,K) modules, and

for the subcategory of admissible modules.

The ('1-,K) module (n,,;;¥K) attached. to an admissible group representation (n,~) by Theorem 0.3.5 is called the Harish-Chandra module of n. ~2

will often use module notation for

:·: · v instead of ~jove

TI

(x) v.

two

('1, K)

modules.

modules, writing

The category structure referred to

is the obvious one. Definition 0.3.9

(~,K)

Part of it is given by

Suppose (n,X) and (n' ,X') are Define

Hom

1

oj ,. ( 'il

0

, ·11

)

,J_\

Hom~ ,K(X,X

1

{L : X ~ X

I

1

)

linear, and the

({f

[ to

11

1

,K) -module maps

L is complex 1

TI

L = Lrr

1 },

(or intertwining operators) from

We call X and X 1 equivalent if there is an

•

invertible (cg-,K)-module map between them.

We write G

for the set of equivalence classes of irreducible (c;t,Kl modules. mally

Two representations of G are called infinitesi-

equiv~-~~~

if their Barish-Chandra modules are

equivalent.

The study of ( (/, K) modules instead of group represen·i:a·tions is justified by Theorems 0.3.5 and 0.3.6, and the next result. Theorem 0.3.10 [49]) •

(Harish-Chandra, Lepowsky, Rader -- see

Every irreducible ((/,K) module is the Harish-

Chandra module of an irreducible admissible representation of G.

Two

irreduci~!g

unitary representations of

G are boundedly equivalent if and only if they are infinitesimally equivalent. In particular, an irreducible

(~,K)

module is automatically

admissible. When G is connected, the group K plays a less serious role in the structure of a

( o;f• K) module:

for example, con-

dition (0.3.7) (c) is automatic, and (!",K)-module maps are just

~-module

maps.

Best of all is

17 'l'heorem 0.3.11 (Harish-Chandra [12]). nected. modules.

Suppose G is con-

Let (1r,X) and (n' ,X') be irreducible

(

(by Corol-

We have that information, so the results can

easily be checked.

Re A

n(~)

Definitio~.

0.

We leave the details to the reader. Q.E.= Fi~E

Let W = 0 if

E

~'

and

E =

±1; and suppose

= +1, and W = ±l if

E =

Then there is a unique (up to a scalar) non-zero

-1.

homomorphism of

(~,K)

modules

The image of A is isomorphic to X (A) (w); in particular, A is non-zero on the K-type W·

Proof.

Suppose first that A is not an integer of parity -E.

By Proposition 1.3.3(b) and Proposition 1.2.14(b),

Xc

(-A)

(wl -

So the result is true in this case; A must even be an isomorphism.

Next suppose A is an integer of parity -E; by

~1ypothesis

it is positive.

By Proposition l.3.3(c) and (d),

Xc(A) (w) is a quotient of Xc(E ®A) and a submodule of Xc(E ®-A); so there is a map A with the desired properties. let B be any other homomorphism between these principal

:~ow

series. ~sa

~s

By Proposition 1.3.3(c), Xc(E ®A) contains Xd(±A)

submodule; so Xc(E

a submodule.

0.

-A) contains

But the only irreducible submodule of

Xc(E ®-A) is Xc(-A) (wl Y±

0

(by Proposition l.3.3(d)); so

So B factors as B

Xc(E ®A) ~X (E ®-A) """Xc(A)

(w~/B

Since Xc(A) (w) is irreducible and occurs just once as a sub~odule

of Xc(E ®-A), the map

So B must be a multiple of A, so

is unique up to a scalar. B is a multiple of A.

Q.E.D.

We turn now to the analytic description of the principal series representations. Definition 1.3.5 G =

SL ( 2 , lR) :

M

~±[~ ~]}

centralizer of A inK=

H~ i]

N

P

If E

Define the following subgroups of

I

t

lR~

E

H~ ~-1]1

MAN

±1, and A

E

~,

E ®

0

I a

E

lR, t

E

lR~.

define a character

A

p +

a:

by (E

®A)[~ ~-1]

= (E) (sgn a) laiA·

Let ~®A denote the Hilbert space of complex valued functions f on G, satisfying a)

if g

E

G and p f(gp)

b)

=

E

[E

P, ®

(A+l)](p-l) f(g);

the restriction off to K lies in L 2 (K).

(We make

'fE®A a Hilbert space using the restriction

mapping into the Hilbert space L 2 (K) .)

The principal

series

(group) representation vli th parameters E: and A,

( rr, ~E:®A) , is defined by

:f A is purely imaginary, so that E:

0

A is a unitary character

::>f P, then

:.he usual induced representation in Hackey's sense [50]. is not purely imaginary, E: ~o

0

),

If

is not unitary; and there is

good general definition of non-unitary induced representa-

:.ions.

In the present setting, however, Definition 1.3.5 is

~ reasonable one, and we may speak of ~®A

as the induced

::epresentation. There are still several things to check:

that the opera-

:.ors rr(g) are bounded, that the representation is continuous, ~nd

that its Barish-Chandra module is Xc(E:®A).

All of this

:.s based on Lemma 1.3.6

(Iwasawa decomposition).

g =

Any element

[~ ~]

of SL (2, JR) can be written uniquely as a product g

(8

JR, y

E

> 0

I

X

E

IR)

Here k (8)

y

la 2

+

C

2 I cos 8

[ cos 8 -sin 8

sin cos

a /a2 + c2

~] sin 8

c /a2 + c2

Proof.

By matrix multiplication, k

(~

(8)

;-1]

(-~

=

cos 8

X

-1

cos 8 + y_l sin 8]

sin 8 -x sine + y

CC>S

3 •

I f this is equal to g, then clearly

a

2

+ c

2

y

2

2

cos 8 + y 2 sin 2 8

y

2

;

so since y is assumed to be positive,

(Since g is invertible, a and care not both zero.)

This, in

turn, forces a y

cos 8 So

c

sin 8

y

k(8) andy are uniquely determined; and xis fixed by k (8) )

-1

So the decomposition is unique. define

l(y0

g =

(

J

To prove that it exists,

k (8) by the formulas above. (k(8))-lg =

X

y-1

-1

Then

cy=i1 [ac bd)

il-ey ay_ 1

ay

J

X

(ad-bc)y y [0 2

y , and ad - be number.

Multiplying by

X

y

-1 ]

-1] =

1; here x is some real

k (8) proves the lemma.

Q.E.D.

Corollary 1. 3. 7

If

E

= ±1, let

~

denote the

Hilbert space of functions in L 2 (K) of parity E,

fiE=

E

{f

L

2

(K)

I

f(-x)

=

Ef(x)}.

Then restriction to K defines an isomorphism

?roof.

[-~ -~]

The element

of G lies in P

:n tile character E 0 A of P. :

').~...

E

1

S®t\

and

X

E

n K,

and acts by

E

So, by definition 1.3.5, if

K,

Ef (x).

:onversely, suppose f lies in ::m

G

Extend f to a function f

by f(g)

~hen

#s.

-A-1 = f(k)y

(g

f restricted to K is f, so f satisfies condition (b) of

Jefinition 1.3.5; and f was constructed to satisfy (a). Proposition 1.3.8

Q.E.D.

The principal series representation

(n, ~E®A) of Definition 1.3.5 is an admissible group representation (Definition 1.1.5). module

Its Barish-Chandra

(Definition 1.1.7) is isomorphic to

~(E

0

A) of

Definition 1.3.1. Proof.

Lemma 1.3.6 shows that the homogeneous space G/AN

(with A and N as in Definition 1.3.5) is naturally isomorphic to K, by the inclusion K

=

K/(K n AN)

+

G/AN .

Therefore, there is an action of G on K, by (1. 3. 9)

g

• k

k'

(gk

Define a function

a:

G

JRI

{y

-+

Y > o}

by (1.3.10)

a(g)

y

if g

=

k

•

(

~ ~-1 J

(k

e: K,

y

> 0)

Lemma 1. 3. 6 gives a formula for this function:

It also gives a formula for the action of G on K, which we will not write down; but we conclude by inspection of these formulas that the map G

X

K +

(K

X

IR)

(g,k)-+ (g-l · k, a(g-l k)! is continuous.

Suppose now that f lies in

Definition 1.3.5 and (1.3.9)

i¥E

By

c

and (1.3.10),

[1T (g) f] (k)

It is now clear that the operator 1T(g) is bounded (by the' m~ximum

of the Jacobian of the diffeomorphism of K given by

the action of g, times the maximum on K of the function a(g-l k )-A-l);

one simply performs a change of variable in

the integration over K which defines reasons 1T is continuous.

Recall that

I ITI(glfl I

For similar

~ is the space of

functions on K (the circle group) of parity s. series,

A?s

By Fourier

has an orthonormal basis given by functions

{f In e ~has parity c, and f n

n

(k(B))

= eine}.

Defining Pin as in Lemma 1.1. 3 (for "#_), we find

Therefore

(n[~ ~)fn)

(1)

'o lim 1t [ fn ( explo t+O

~~ n [10

lim t+O t

- fn (1)]

-t) 1 - fn (1)]

lim .!.[1 - 1] t 0;

-~))

0).

and similarly,

lim ! t t->-0

[fn [e0 -t e -t] 0

_ l fn {l)

)

lim !ret(A+ll - 1] t->-0 t l A + l.

Finally, the first formula of 1.3.11 gives

By

(l. 2 .1) ,

so the computations above and (1.3.12) show that

a

1

n

:zP +

1)

+

n

2

!(A + (n+l)). 2

Similarly, we find b

n

(n-1)).

These are exactly the structure constants for Xc(E

~

A)

(Definition 1.3.1), completing the proof of the proposition. Q.E.D.

For the remainder of this chapter, we return to our previous hypotneses on the level of experience of the reader; and in particular we provide no more detailed proofs.

In

69

:.-.is section, G wi 11 denote the group

{ [~ ~] Ia , b, c , d

+

SL-(2,IR)

:~viously

m,

E

ad - be

0

is SL (2,

ill).

Define sino)

K

0(2)

T

S0(2) = K 0

A

)[~~-l]la>O~

N

i[; ~]

M

j[±l ±l OH = 0

::.. 4 .1)

p

Lemma 1.4.2 tions of K, E

±1} .

G has two connected components, and the identity

:Jmponent G

n

=

Z':,

I

±coso) (

U~

MAN =

Em}

lx

. central~zer

±:-1]

I

of A in K.

0 ;i a EIR,

ill[.

XE

The set of irreducible unitary repre?enta-

K,

may be parametrized as follows:

if

let e

in8

be the indicated character; here k (8 )

= (

8)

c?s 8 sin 8 cos 8 ·

-s~n

If n is a positive integer, define

wn

=

K

IndK (x ) _

o

n

theJ1__jln i s a two-dimensional irreducible unitary repre~~nt~t:i,sl11_,

and

/U

Let ~~ denote the trivial representation, and ~~ the other one dimensional representation of K (which is trivial on K ); then 0

Then

The proof is left to the reader. Definition 1.4.3 of

I1

Identify the set

Mof

the characters

with pairs 0 or 1)

(cS.

cS

l

by

o·

ITt:. l . l

(For example, if 6 = {1, 0) , L1en

Mand

If 6 ~

v

E

~' define the principal series repre~

sentation with parameters

and v by G

Indp ( cS

v).

®

Here we regard v as a character of A as in Definition 1.3.5, and we make cS e v a character of P

= MAN by

making it trivial on N. More specifically,

~oev

=

{f: G ~ ~ [f(gp) all g c G and p

f(c:Jl

[(o

0

(v+l)) (p- 1 )] for ')

E

P; and fiK lies in L~(K)}

71

i;J: and v

cj:.

Lemma l. 4. 4

Fix

a)

rcl~

al IJ2 al IJ4 al

(0

(0, 0))

rll

al IJ3 al

ils al

(0

(l '0) or (0, l))

lPCi

al IJ2 al IJ4 Ell

(0

(l' l))

I

Nor&v K

b)

c5

E

E

The~

Let c be ti1e restriction of 6 to M n G-o.

here the representation on the right is the principal series for SL(2,

~)

Definition 1.4.5 principal series

defined in Definition 1.3.5.

o,

If

M and v

E

¢, define the

0;

0 fl.* I

is self-dual in the derived category;

fJ i \ 0~ --

C)

,l't'y

We regard ;;/( i of -c 0 . 1

~~i

< 0, and its support has dimension at most n -

for i B)

on

1

jJ:o' l

as a sheaf on all of

Conjecture 2.3.11

Suppose y,Jl

the zero sheaf for i odd, and

i i

t

0 0

tf3 by making it zero off

b.

--P/ i

Then d::>ly

is

113

M(v,y) =

(-l)£(y)-£(p)

N2

l: dim Hom(.J,

i

y

fl

il

Of course this would imply Conjecture 2.2.12(B).

08 fl

) .

I do not

know its relation to 2.2.12(A); this is a difficult, but probably

interesting, geometry problem.

Example 2. 3.12

continue with Example 2. 3. 9.

~1/e

Suppose first that y I sheaf on

~,

Then~~ is the constant

yp.

and .J~ is zero for i 0

the stalk of 7-f.y

i

0.

In particular,

at the point i is one dimensional:

f

so the conjecture says (-1)

• 1,

which agrees with what was computed in Example 2.2.7. If y = yp, 0. Gabber has computed that (extended by zero

to~), and~i =

~1

0 for i

is just

i 0.

x\

Thus

H i ('U,U)

The maps are

+

H i (U,V)

Z(~)

+

Hi ('",W) ·""

+

H i+l ("U,U)

and U(£) module maps.

+

•••

117 ?roof.

:ound in [5]. ~he

Z(~),

Except for the statement about

this may be

The maps which do not i:1crease the degree of

cohomology group are obtained by passage to the quotient

:rom the obvious maps between the various Homo: ( /1. -:;:>hat they commute with the

i

z(

u.,

*) •

o;)

action is obvious.

So we

:1eed only consider the "connecting homomorphism"

?ix a class w in the first

gro~p,

and choose a representative

The natural map i

Horn-

i

Horn 0.

Replacing p by pn changes nothing, so we may

assume z annihilates V. i

H (u,V).

By Theorem 3.1.5, y(z) annihilates

Since the infinitesimal character

)1

is assumed to

occur there, [ E,l',

o

y (z) l ( 11l

0.

Comparing this with (3.1.7) gives 0

[T_ P

!1.1-l

o

F, ( z ) ] ( 11 )

E, ( z) ( 11 + P (U) ) p

by the definition of z.

()1

+ p

('U))

This contradicts (3.1.8) (a), and

proves the result. §2.

Q.E.D.

Kostant's theorem.

In this section,

dt

will continue to denote a complex re-

ductive Lie algebra, and

the Levi decomposition of a parabolic subalgebra.

The results

will be applied mostly to the complexified Lie algebra

1

of

the maximal compact subgroup of G. again assume that

Gt

is the complexified Lie algebra of G.

-£

Fix a Cartan subalgebra

of

t , and let

W = W((/

~in

denote the Weyl group of positive root system

After (3.2.13), we will

t/(~

Definition 2.1.5; if w

OJ-

,/v) (Definition 0.2.5).

,)/.,) containing i'.(U,__f).

Fix a

We recall

W, set

c

i'.+(w) (3. 2 .1)

'1' ,

Using our fixed parabolic

w1 (3. 2. 2)

we define

hv

E

w I i'.+ (w) ~

{w

E

W

I

i'. (U.,.-l)}

whenever w

E

~* is dominant

for i'.+(Dj-,Ll, then ww is dominant for i'.+(t,Jz.,.)}. (The equivalence of the two definitions is easily checked.) Theorem 3.2.3

(Kostant - see [50], Theorem 2.5.2.1)

Suppose F is an irreducible finite dimensional representation of~' of highest weight A 0.4.3 (e)).

c

~* (Proposition

Then the irreducible finite dimensional

representation of £ of highest weight w occurs in H*(u,F) if and only if there is awE

w

w(A + p)

-

w1

such that

p

and in that case it occurs exactly once, in degree £(w).

124 The proof is based on several simple but important facts. (Some of the next results will not be needed until later, but are included now for simplicity of exposition.) Lemma 3.2.4 ([22] Whenever a

E

1

Corollary 10.2B)

(Notation (0.2.5)).

6+(~,~) is a simple positive root, we have

l.

Corollary 3. 2. 5 a)

Suppose a is a root of

./v in

b)

su12pose a is a root of

0.

=

L

Then

L

in

'U..-.

Then

> 0. Proof.

For (a), since the simple roots of ~in £ span all

the roots, we may assume a is simple. £is also simple with 6+.

in~·

since~

A simple root of

Jv in

was assumed to be compatible

So by Lemma 3.2.4,

v v + -

1 - 1

0. 6(~

For (b), notice that every root in

S +

L

y simple

is of the form

Y,

n

'f

with Sa simple root not in 6+(£), and ny a non-negative integer.

By (a), therefore, we may assume a is simple.

Write

y si;ple ay y; in 6+(£) then ay is a positive rational number.

If y is simple in 6+(£}

l.. .. ~ _)

then a I y

(since a is a root in U..) •

The inner product of

distinct simple roots is non-positive, so ::; 0.

Combining these gives

The first term is positive by Lemma 3.2.4, and the second is non-negative by the argument just given.

This proves

(b). Q.E.D.

From now on, many similar arguments will be left to the reader. Lemma 3.2.6

Let F be the irreducible finite dimensional

representation of ~

E

~(F)

1

1

of highest weight A.

(notation 0.2.3)

1

we cave

::; .

Equality holds if and only if Proof.

Then for all

~

A.

=

According to Proposition 0.4.3(b) and {g)

with na a non-negative integer.

1

::;

This equation and Lemma 3.2.4

imply

with equality if and only if ties leads to

)J

::; ,

A •

Adding the two inequali-

_I_LO

2

+

~

~

with equality if and only if

+ 2 + ,

= A.

This is exactly the asser-

tion of the lemma, Lemma 3.2.7

Q.E.D. The irreducible finite dimensional

representation F of highest weight A has infinitesimal character A+p Proof.

Obviously

Z(~)

acts on the highest weight space F(A)

by composition of s (Definition 0.2.7) and evaluation at A; for the positive root vectors annihilate F(A) by Proposition 0.4.3 (f).

Since sis TP

s (Definition 0.2.7) the lemma

o

follows.

Q.E.D.

-l*,

Lemma 3.2.8 a

E

ll+.

all w

E

Suppose A E and :;> 0 for all m Let {a.}i=l be any subset of ll+. Then for

w, (w(A+p)-p +l:ai,w(A+p)-p + l:ai>

:;>

Equality holds if and only if {ai} = ll+(w). Proof. We use the formula

P - wp

(3. 2. 9)

(see, for example,

l: SEt:.+(w)

[50], 2.5.2.4).

s

Notice also that

(3.2.10) this is immediate from (3.2.1) and Lemma 3.2.4. (*)

W(A+p)-p + l:a. l_

Set

WA -

Then

+ l:a l_..

A

/l,+(w)

n

{ai}

B

{ 131-13

E

/l,+(w) - A}

c

{ai} - A.

Thus {a.}= /l,+(w) if and only if B u Cis empty. l

of

Because

(3.2.10), if B e B, then < 13 ,wp> > 0.

Similarly, if y e C, then y is a positive root not in /l,+(w); so > 0. On the other hand,

(*) and (3.2.9) imply that

w {/..+p) -p +

w/.. +

L:ai

L: 13. 13eBuC

Since !.. and p are dominant, w/.. and wp lie in the same closed Weyl chamber; so by the inequalities above,


?

0 .

f~e::BuC

So expanding the left side gives

Q.E.D.

with equality if and only if B u C is empty. Corollary 3.2.11

In the setting of Theorem 3.2.3,

the representation of £ of highest weight w(!..+p)-p occurs in Proof.

Hom~(A

*u,F)

exactly once, in degree £(w).

Clearly the multiplicity of the

weight~

in

*

Hom~(A~,F)

is the number of expressions ~

with y e /I,(F) and {ai}

~

=

y - L: ai

/I,(U); here we count this expression

with the multiplicity of y.

Of course, the degree in A\t

corresponds to the cardinality of {ai},

In particular,

consider the multiplicity of the weight w(A+p)-p; fix an expression for it as above.

By Lemma 3.2.8

=

with equality only if {ai}

2

~

,

+ (w).

By Proposition 0.4.3(b),

the reverse inequality also holds; so we have equality, and {ai} =

~

+ (w).

Using (1.3.11), we get w(A+p)-p = y + wp-p;

so y is necessarily wA, which we know has multiplicity one (by Proposition 0.4.3

I~+ 0.

So L'l+(w) n 6+(~) is empty; sow

E

1

Now Corollary 3.2.11

1v .

implies that the representation of highest weight

*

in

Hom~(AU,F)

of

~

~(w).

exactly once, in degree

~

occurs

Since the action

commutes with the coboundary map d, this holds also for

the cohomology.

Q.E.D.

One more "folk theorem" about finite dimensional representations will be of some use later; it is the technique of proof which is most important, however. Proposition 3.2.12

Let F be the irreducible finite

dimensional representation of

~

of highest weight A,

and let V be another finite dj_mensional representation of

t?J·

ofF

0

Then the highest weight of any constituent Vis of the form

A+~,

with

~

E

li(V).

This result can also be extended to G in analogy with Carollary 3.2.16; we leave this to the reader, although it is the extended version we usually need. Proof.

Let~=~+

responding tot/.

of~

?V be the Borel subalgebra

cor-

We know from Proposition 1.3.2 that HO(?Z.-,F) '=

V y j certain conditions hold}.

The map of part (a) is defined by f

0

v

+

h, h(g)

=

f(g)

0

Clearly h is a map from G to V

y

(g-l • v) ®

F, and it is straightforward

to verify that h actually belongs to ~®F"

To see that this

correspondence is an isomorphism of Hilbert spaces, one can use the realization (4.1.13) and the corresponding fact about unitarily induced representations of compact groups.

That

the correspondence is an intertwining operator is formal. This proves (a).

Part (b) is an

i~nediate

consequence. Q.E.D.

Choose a Cartan subalgebra ~~ of ~O'

Notation 4.5.3 and set

centralizer of ~~ in M

Ts

Here s stands for split; this is a 9-stable maximally split Cartan subgroup of G (Definition 0.4.1). chapters, when abelian in

f 0,

~O

In later

will not necessarily be maximal

we will write As instead of A; so

108

we also fix a positive root system (1,

+ (1YI.-, ;ts ) '

and write p(Jrt}

(l

(/I,

+ (f?'I....,A......./..S )

) •

Thus if we put 6

+ (?Jt,A) ~s

u

Is

6(-rz,,-"ft,),

we get

Lemma 4.5.4

A principal series

strongly Z(?)-finite. V

E

A.

(1-,K)

module is

More precisely, suppose

cS

E

M,

Let .;.,J.'S

! lo e

denote the highest weight of 6 (Definition 0.4.3).

Let

A0 be the differential of A0 , and A = A

0

+

p ~)

(notation 4.5.3)

Then every infinitesimal character occurring in Tr(o

®

v)

F is of the form (notation 0.2.9):

®

Proof.

Fix a finite dimensional representation F of G.

We

:an choose a P-stable filtration of F so that the subquotients