Real Reductive Groups I

By Nolan R. Wallach

Real Reductive Groups I is an introduction to the representation theory of real reductive groups. It is based on courses that the author has given at Rutgers for the past 15 years. It also had its genesis in an attempt of the author to complete a manuscript of the lectures that he gave at the CBMS regional conference at The University of North Carolina at Chapel Hill in June of 1981.
This book comprises 10 chapters and begins with some background material as an introduction. The following chapters then discuss elementary representation theory; real reductive groups; the basic theory of (g, K)-modules; the asymptotic behavior of matrix coefficients; The Langlands Classification; a construction of the fundamental series; cusp forms on G; character theory; and unitary representations and (g, K)-cohomology.
This book will be of interest to mathematicians and statisticians.

• Language: English
• Publisher: Academic Press
• Release date: Mar 1, 1988
• ISBN: 9780080874517

Book preview

Real Reductive Groups I

Nolan R. Wallach

Department of Mathematics, Rutgers University, New Brunswick, New Jersey

ISSN  0079-8169

Volume 132 • Number (P1) • 1988

Table of Contents

Cover image

Title page

Inside Front Cover

Copyright page

Dedication

Preface

Introduction

Chapter 0: Background Material

Introduction

0.1 Invariant measures on homogeneous spaces

0.2 The structure of reductive Lie algebras

0.3 The structure of compact Lie groups

0.4 The universal enveloping algebra of a Lie algebra

0.5 Some basic representation theory

0.6 Modules over the universal enveloping algebra

Chapter 1: Elementary Representation Theory

Introduction

1.1 General properties of representations

1.2 Schur’s lemma

1.3 Square integrable representations

1.4 Basic representation theory of compact groups

1.5 A class of induced representations

1.6 C∞ vectors and analytic vectors

1.7 Representations of compact Lie groups

1.8 Further results and comments

Chapter 2: Real Reductive Groups

Introduction

2.1 The definition of a real reductive group

2.2 Parabolic pairs

2.3 Cartan subgroups

2.4 Integration formulas

2.5 The Weyl character formula

2.A Appendices to Chapter 2

Chapter 3: The Basic Theory of (g, K)-Modules

Introduction

3.1 The Chevalley restriction theorem

3.2 The Harish-Chandra isomorphism of the center of the universal enveloping algebra

3.3 (g, K)-modules

3.4 A basic theorem of Harish-Chandra

3.5 The subquotient theorem

3.6 The spherical principal series

3.7 A Lemma of Osborne

3.8 The subrepresentation theorem

3.9 Notes and further results

3.A Appendices to Chapter 3

Chapter 4: The Asymptotic Behavior of Matrix Coefficients

Introduction

4.1 The Jacquet module of an admissible (g, K)-module

4.2 Three applications of the Jacquet module

4.3 Asymptotic behavior of matrix coefficients

4.4 Asymptotic expansions of matrix coefficients

4.5 Harish-Chandra’s Ξ-function

4.6 Notes and further results

4.A Appendices to Chapter 4

Chapter 5: The Langlands Classification

Introduction

5.1 Tempered (g, K)-modules

5.2 The principal series

5.3 The intertwining integrals

5.4 The Langlands classification

5.5 Some applications of the classification

5.6 SL(2, R)

5.7 SL(2, C)

5.8 Notes and further results

5.A Appendices to Chapter 5

Chapter 6: A Construction of the Fundamental Series

Introduction

6.1 Relative Lie algebra cohomology

6.2 A construction of ()-modules

6.3 The Zuckerman functors

6.4 Some vanishing theorems

6.5 Blattner type formulas

6.6 Irreducibility

6.7 Unitarizability

6.8 Temperedness and square integrability

6.9 The case of disconnected G

6.10 Notes and further results

6.A Appendices to Chapter 6

Chapter 7: Cusp Forms on G

Introduction

7.1 Some Fréchet spaces of functions on G

7.2 The Harish-Chandra transform

7.3 Orbital integrals on a reductive Lie algebra

7.4 Orbital integrals on a reductive Lie group

7.5 The orbital integrals of cusp forms

7.6 Harmonic analysis on the space of cusp forms

7.7 Square integrable representations revisited

7.8 Notes and further results

7.A Appendices to Chapter 7

Chapter 8: Character Theory

Introduction

8.1 The Character of an admissible representation

8.2 The K-character of a (g, K)-module

8.3 Harish-Chandra’s regularity theorem on the Lie algebra

8.4 Harish-Chandra’s regularity theorem on the Lie group

8.5 Tempered invariant Z(g)-finite distributions on G

8.6 Harish-Chandra’s basic inequality

8.7 The completeness of the πτ

8.A Appendices to Chapter 8

Chapter 9: Unitary Representations and (g, K)-Cohomology

Introduction

9.1 Tensor products of finite dimensional representations

9.2 Spinors

9.2.5

9.3 The Dirac operator

9.4 (g, K)-cohomology

9.5 Some results of Kumaresan. Parthasarathy, Vogan, Zuckerman

9.6 u-cohomology

9.7 A theorem of Vogan-Zuckerman

9.8 Further results

9.A Appendices to Chapter 9

Bibliography

Index

Pure and Applied Mathematics

Inside Front Cover

This is Volume 132 in

PURE AND APPLIED MATHEMATICS

H. Bass, A. Borel, J. Moser, S.-T. Yau, editors

Paul A. Smith and Samuel Eilenberg, founding editors

A complete list of titles in this series appears at the end of this volume.

Copyright page

Copyright © 1988 by Academic Press, Inc.

All rights reserved.

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

ACADEMIC PRESS, INC.

1250 Sixth Avenue, San Diego, CA92101

United Kingdom Edition published by

ACADEMIC PRESS INC. (LONDON) LTD.

24-28 Oval Road, London NW1 7DX

Library of Congress Cataloging-in-Publication Data

Wallach, Nolan R.

Real reductive groups.

(Pure and applied mathematics; v. 132- )

Includes index.

1. Lie groups. 2. Representations of groups.

I. Title. II. Title: Reductive groups. III. Series: Pure and applied mathematics (Academic Press); 132, etc.

QA3.P8 vol. 132, etc. 510 s [512′.55] 86-32199

[QA387]

ISBN 0-12-732960-9 (v. 1: alk. paper)

88 89 90 91 9 8 7 6 5 4 3 2 1

Printed in the United States of America

Dedication

To my mother

Pauline Wallach

For as the sun is daily new and old, So is my love still telling what is told.

Preface

This book is intended as an introduction to the representation theory of real reductive groups. It is based on courses that the author has given at Rutgers for the past 15 years. It also had its genesis in an attempt of the author to complete a manuscript of the lectures that he gave at the CBMS regional conference at The University of North Carolina at Chapel Hill in June of 1981. When the manuscript for those lectures reached over 300 pages the author realized that the scope of the project involved much more than was expected for a CBMS volume. We apologize to the conference board for not having completed the volume that was expected. We, however, hope that this book will in part fulfill the obligation.

Initially, it was our intention to present the subject of representations of real reductive groups from the beginning to recent research, all in one volume. This has also been beyond the ability of the author. We have opted to present the material in two volumes in order to expand upon the original extremely terse exposition and to include recent developments in even the more classical aspects of the theory.

There are many people that have been helpful in the production of this volume. We thank our students (both former and present) for their patience over the years with the lectures on which this book is based. We especially thank Roberto Miatello for all of the errors that he has found in the various earlier versions of this material and for his many helpful comments. Hans Duistermaat pointed out a major blunder in our original exposition of Harish-Chandra’s regularity theorem. His explanation of the method of proof of this theorem that will appear in his forthcoming book with Kolk was very helpful. We also thank Kenneth Gross for having organized the above-mentioned CBMS regional conference so well. Finally, we take this opportunity to thank Armand Borel for his editorial help, encouragement and patience throughout the preparation of this opus.

We also take this opportunity to thank the National Science Foundation for the summer support during the preparation of this volume.

Introduction

You do not understand my philosophy. But that is the way science progresses each generation misunderstands the previous one.

—Harish-Chandra

The representation theory of real reductive groups is one of the most beautiful, demanding, useful and active parts of mathematics. Although there have been many important contributors to the field. Harish-Chandra, through his power and vision, almost single-handedly changed the field from a backwater of physics to what it is today. For better or for worse Harish-Chandra, in developing his awesome theory, also established the style of the field. Few disciplines in mathematics put as much emphasis on their technical details. This aspect of the subject makes it an extremely easy part of mathematics to read line by line and a very difficult part for those who would just like an over-all picture of the subject.

Although this book is a product of the Harish-Chandra legacy, we have attempted to allow the reader to get a feel of the subject without necessarily having understood every line. It is hoped that upon a first reading, the material will be studied by jumping from one part, that may seem interesting, to another. We have endeavored to do enough cross-referencing so that a reader could open the book in the middle and understand the material there by following the details backward. A careful reader will find mathematical gems in unlikely places. Kostant’s theorem on n-cohomology is in Chapter 9, Zuckerman’s translation principal is in an appendix to Chapter 6, radial component theory is in the appendices to Chapter 7, Kostant’s theorem on nilpotent orbits is in an appendix to Chapter 8.

As the title indicates, there is a forthcoming second volume which will contain, in particular, a proof of Harish-Chandra’s Plancherel theorem. Although both volumes emphasize the analytic aspects of the theory, the material in the volume at hand is more algebraic than the second volume. The reader who is predominantly interested in the algebraic aspects of the theory can read this volume without being too contaminated by analysis.

Let us now give a thumbnail tour of the present volume. Chapter 0 is a compendium of some of the basic results that usually appear in a first course in Lie groups and Lie algebras. It is included to establish notation and references. The purpose of Chapter 1 is to introduce the theory of infinite dimensional representations of Lie groups. The material presupposes no prior knowledge of the reader. Our account is tailored to the needs of the later chapters and since most of representation theory of general Lie groups is unnecessary to the case of real reductive groups, the reader should be aware that this chapter is just the tip of the iceberg. The chapter emphasizes representations on Hilbert spaces. Basic material on smooth, analytic and "K-finite" vectors is included. A novel aspect of this chapter is the development of the Peter-Weyl theory for compact Lie groups as a corollary to the theory of square integrable representations.

In Chapter 2, we introduce the class of Lie groups that will be studied throughout the remainder of the book. In particular we make the term real reductive group precise. The only prerequisites for this chapter are included in Chapter 0. We develop the theory of parabolic subgroups and Cartan subgroups. We take the more primitive notion to be that of parabolic subgroup and then show how the theory of Cartan subgroups is an outgrowth. Most of the classical groups are introduced in this chapter. We give the Iwasawa, Bruhat and Cartan decompositions for the groups. Integration formulas are given for these decompositions as are various versions of the Weyl integration formula. We also include a proof of the Weyl character formula (the standard one) since a similar proof will be used for the discrete series in Chapter 8.

The material of Chapter 3 is the heart of the algebraic approach to representation theory. It contains various forms of the Chevalley restriction theorem and the Harish-Chandra homomorphism. The formalism of (g, K)-modules is introduced. The critical notion of admissibility is developed. A proof is given of Harish-Chandra’s theorem that irreducible unitary representations are admissible. The chapter also includes the celebrated sub-quotient theorem of Harish-Chandra, Lepowsky, Rader and its corollary (in our development), the subrepresentation theorem of Casselman. The latter result is perhaps the most important single theorem to our development. It makes the theory of the real Jacquet module a viable approach to the representation theory of real reductive groups. Also our proof of this theorem contains ideas that will be critical to later developments in the book. The chapter also includes the basic theory of spherical functions. Most of the material in this chapter is algebraic or at least has algebraic statements. We have, however, given some analytic proofs of theorems that now have completely algebraic proofs. We indicate where the more algebraic approach can be found in the literature.

Chapter 4 is the core of our approach to the subject. It contains the theory of the real Jacquet module and its consequence (in our exposition) the asymptotic behavior of matrix coefficients. This chapter is strongly influenced by our joint work with Casselman (which was motivated by the pic [1]. The critical difference between our results and that of Harish-Chandra is that we give asymptotic expansions of smooth matrix coefficients rather than just "K-finite" ones.

The point of Chapter 5 is to give a proof of the Langlands quotient theorem (Langlands classification). This theorem reduces the classification of irreducible (g, K)-modules to the classification of tempered (g, K)-modules. The elementary aspects of tempered representations and their relationship with square integrable representations is also given. At this point in our development, the critical importance of the irreducible square integrable representations has become manifest. However, in this chapter these representations are described only in the case of SL(2, R).

Chapter 6 is devoted to a homologico-algebraic approach to constructing admissible (g, K)-modules that is equivalent to that of Zuckerman using derived functors of the "K-finite functor. Our approach follows the broad lines of our joint work with Enright. An approach that is closer to Zuckerman’s original ideas can be found in Vogan [2]. Using, what we call Zuckerman’s functors, we construct irreducible unitary representations. These representations had been conjectured to be unitary by Vogan (a generalization of a conjecture of Zuckerman). Vogan gave the first proof of this result, using Harish-Chandra’s theory of tempered representations. Our proof is elementary, and we use it as a basis for the theory of tempered representations. We single out the families constructed from so-called θ-stable Borel subalgebras and call them the discrete series. Using the theory of Jacquet module we prove that they are square integrable. In Chapter 8 it is shown that these representations exhaust the irreducible square integrable representations. The reader can go directly from this chapter to Chapter 9 which studies the twisted" (g, K)-cohomology with respect to unitary modules. A complete proof (mainly due to Vogan, Zuckerman and Kumaresan) of a conjecture of Zuckerman (that completely calculates this cohomology) is given there using the modules constructed in this chapter.

The next step is to prove that the discrete series exhausts the irreducible square integrable representations. In our approach, this is where the analysis begins in earnest. The next two chapters are very close to the spirit of Harish-Chandra’s original approach. In Chapter 7, the basics of Harish-Chandra’s theory of orbital integrals is given. Our approach differs in one important detail. We do not use the theory of the discrete series to prove that the orbital integrals define tempered distributions. Instead, we use a special case of Kostant’s convexity theorem (essentially due to Thompson [1]). The critical idea in this chapter is Harish-Chandra’s characterization of the matrix coefficients of the discrete series in terms of the vanishing of certain integral transforms. That is, these matrix coefficients span the space of cusp forms. We give Harish-Chandra’s formula for recovering a cusp form from its orbital integrals. This result implies Harish-Chandra’s basic theorem that says that irreducible square integrable representations can exist if and only if there is a compact Cartan subgroup. However, the completeness theorem must wait for the results in the next chapter.

At this point the reader should have noted a glaring omission in the contents of this book. The only mention of character theory has been in connection with the Weyl character formula. Chapter 8 is devoted to Harish-Chandra’s theory of characters of admissible representations. These characters are initially defined as distributions on the group (as traces of generalized convolution operators). The main theorem on characters is that they are given as integration against a locally integrable function (Harish-Chandra’s regularity theorem). Furthermore, on each Cartan subgroup this function has a form reminiscent of the Weyl character formula. With the "local L¹-theorem" in hand we prove that the Fourier coefficients of orbital integrals of cusp forms are multiples of characters of what we called the discrete series in Chapter 6. The completeness theorem is now immediate.

As we observed above, Chapter 9 could be read immediately after Chapter 6. This chapter contains a concise introduction to (g, K)-cohomology, vanishing theorems due to Kumaresan, Enright, Vogan-Zuckerman and the complete calculation of (g, K)-cohomology with respect to a tensor product of a finite dimensional and an irreducible unitary representation (due to Vogan and Zuckerman). The reader should consult Borel, Wallach [1] for an account of the general theory and its applications to discrete groups. We include tables of the vanishing theorems.

There are several books whose contents have significant overlaps with this one. Knapp’s recent book (ic. Other notable books on the subject are Warner [1], [2] and Varadarajan [1]. Both of these works follow Harish-Chandra’s original methods quite closely. Warner’s treatise in addition contains a very thorough introduction to representation theory (i.e., C∞-vectors, analytic vectors, induced representations). These books (and Helgason [1]) were valuable aids in the preparation of this work.

The literature in the field of reductive groups is vast. We have done our best to give adequate references. However, as is the case in any growing field, there are cases when a result has been proved (partially) by many authors. It would be a project beyond the scope of this book to give the precise history of the genesis of the theorems included in this book. However, in most cases the interested scholar should be able to determine a precise chronology by consulting the citations that we have included.

A reader who has mastered the basic graduate curriculum in mathematics should have all the mathematical background necessary to master the material in this volume. However, the serious student should approach this work with an ample supply of paper and pencils. Be patient and it will be yours.

0

Background Material

Introduction

The purpose of this chapter is to compile some of the background results, terminology and notation that will be used in this book. We recommend that the reader use this chapter basically for reference purposes. However, it might be worthwhile for the reader to skim through it on his first reading to become familiar with some of the notation and definitions. There are almost no proofs in this chapter. Everything covered can be found with adequate explanations in the references that we give, except for the material in Section 6. In Section 6 we give a noncommutative variant of the Artin-Rees Lemma of commutative algebra. There is a general Artin-Rees Lemma for nilpotent Lie algebras (see McConnell [1], Nouaze, Gabriel [1]). Lemma 0.6.4 appears for the first time in Stafford, Wallach [1].

0.1 Invariant measures on homogeneous spaces

0.1.1

Let G be a locally compact topological group. Then a left invariant measure on G is a positive measure, dg, on G such that

for all x ∈ G and all f in (say) Cc(G). If G is separable then it is well known (Haar’s theorem) that such a measure exists and that it is unique up to a multiplicative constant.

If G is a Lie group with a finite number of components then a left invariant measure on G can be identified with a left invariant n-form on G (here dim G = n). If μ is a non-zero left invariant n-form on G then the identification is implemented by integrating with respect to μ using the standard method of differential geometry.

If G is compact then we will (unless otherwise specified) use normalized left invariant measure. That is, the total measure is one.

If dg is a left invariant measure and if x ∈ G then we can define a new left invariant measure on G, μx, as follows:

The uniqueness of left invariant measure implies that

with δ a function of x which is usually called the modular function of G. If δ is identically equal to 1 then we say that G is unimodular. If G is unimodular then we will call a left invariant measure (which is then automatically right invariant) invariant. It is not hard to see that δ is a continuous homomorphism of G into the multiplicative group of positive real numbers. This implies that if G is compact then G is unimodular.

If G is a Lie group than the modular function of G is given by the following formula:

where Ad is the usual adjoint action of G on its Lie algebra.

0.1.2

Let M be a smooth manifold and let μ be a volume form on M. Let G be a Lie group acting on M. Then (g*μ)x = c(g, x)μx for each g ∈ G, x ∈ M. One checks that c satisfies the cocycle relation

(1)   

We will write ∫M f (x) dx for ∫M fμ. The usual change of variables formula implies that

for f (say) in Cc(G) and g ∈ G.

Let H be a closed subgroup of G. We take M to be G/H. We assume that G has a finite number of connected components. A G-invariant measure, dx, on M is a measure such that

(3)   

If dx comes from a volume form on M then (3) is the same as saying that |c(g, x)| = 1 for all g ∈ G, x ∈ M.

If M is a smooth manifold then it is well known that either M has a volume form or M has a double covering that admits a volume form. By lifting functions to the double covering (if necessary) one can integrate relative to a volume form on any manifold. Returning to the situation M = G/H, it is not hard to show that M admits a G-invariant measure if and only if the modular function of G restricted to H is equal to the modular function of H. Under this condition, a G-invariant measure on M is constructed as follows: let g be the Lie algebra of G and let h be the sub-algebra of g corresponding to H. Then we can identify the tangent space at 1H to M with g/h. The adjoint action of H on g induces an action Ad~ of H on g/h. The above condition says that |det Ad~(h)| = 1 for all h ∈ H. Thus if H⁰ is the identity component of H (as usual) and if μ is a non-zero element of Λm(g/h)* (m = dim G/H) one can translate μ to a G invariant volume form on G/H⁰.

Thus by lifting functions from M to G/H⁰ one has a left invariant measure on M. Now Fubini’s theorem says that we can normalize dg, dh and dx so that

(4)   

0.1.3

Let G be a Lie group with a finite number of connected components. Let H be a closed subgroup of G and let dh be a choice of left invariant measure on H. The following result is useful in the calculation of measures on homogeneous spaces.

Lemma

If f is a continuous compactly supported function on H\G (note the change to right cosets!) then there exists, g, a continuous compactly supported function on G such that

This result is usually proved using a partition of unity argument. For details see, for example, Wallach [1, Chapter 2].

0.1.4

Let G be a Lie group and let A and B be subgroups of G such that A ∩ B is compact and that G = AB. The following result is useful for studying induced representations.

Lemma

Assume that G is unimodular. If da is a left invariant measure on A and if db is a right invariant measure on B then we can choose an invariant measure, dg, on G such that

For a proof of this result see for example Bourbaki [1].

0.2 The structure of reductive Lie algebras

0.2.1

Let g be a Lie algebra over C. We use the notation z(g) for the center of g. Then g is said to be reductive if g = z(g) ⨁ [g, g] with [g, g] semisimple. We recall the basic properties of g that will be used in this book with appropriate references.

Recall that a subalgebra, h, of g is called a Cartan subalgebra if h is maximal subject to the conditions that h is abelian and if X ∈ h then ad X is semi-simple as an endomorphism of g. Here, if X, Y ∈ g then ad X(Y) = [X, Y] (as usual). Cartan subalgebras always exist and they are conjugate to one another under inner automorphisms (c.f. Jacobson [1. p.273]).

If X ∈ g then define the polynomials Dj on g by

here n = dim g. Let r be the smallest index such that Dr is not identically zero. Set D = Dr. X ∈ g is said to be regular if D(X) is nonzero.

Lemma

If X is regular then ad X is semi-simple. Futhermore, the centralizer in g of a regular element is a Cartan subalgebra of g (Jacobson [1, p.59]).

Fix, h, a Cartan subalgebra of g. If α ∈ h* then we set

If α and gα are non-zero then we call α a root of g with respect to h, and gα is called the root space corresponding to α. The set of all roots of g with respect to h will be denoted Φ(g, h)) and called the root system of g (with respect to h). We have

(2)   If α ∈ Φ(g, h)then dim(gα) = 1 (Jacobson [1, p.111]).

(3)   If α, β ∈ Φ(g, h))then [gα,gβ] = gα+β (Jacobson [1, p.116]).

(4)   If α ∈ Φ(g, h) then the only multiples of α in Φ(g, h) are α and – α (Jacobson [1, p. 116]).

0.2.2

Let g be as above. If B is a symmetric bilinear form on g then B is said to be invariant if

A non-degenerate invariant form on g always exists. On [g, g] one takes the Killing form Jacobson [1, p.69] and on z(g) one takes any non-degenerate symmetric form. The direct sum of the two forms is then a non-degenerate invariant form on g. Fix such a form, B. Fix a Cartan subalgebra, h, in g. It is clear that h is orthogonal, relative to B, to all of the root spaces. We therefore see that

(1)   B restricted to h is non-degenerate.

Thus, if μ ∈ h* then we can define Hμ ∈ h by

We can then define a non-degenerate symmetric bilinear form (, ) on h* by (μ, τ) = B(Hμ, Hτ) for μ, τ ∈ h*. One has

(2)   (α, α) is a positive real number for α ∈ Φ(g, h). (Jacobson [1, p. 110])

Let hR denote the real subspace of h spanned by the Hx for α ∈ Φ(g, h). Then one has

(3)   B restricted to hR is real valued and positive definite (Jacobson [1, p.118]).

0.2.3

We retain the notation of the previous number. If α ∈ Φ(g, h) we denote by sα the reflection about the hyperplane α = 0 in h. That is,

sx is called a Weyl reflection. The Weyl reflections have the following properties:

(1)   

We denote by W(g, h) the group generated by the Weyl reflections. W(g, h) is called the Weyl group of g with respect to h.

Let h′R denote the subset of all H ∈ hR such that α(H) is nonzero for all α ∈ Φ(g, h). Let C denote a connected component of h′R. Then C is called a Weyl chamber.

(3)   W(g, h) acts simply transitively on the Weyl chambers (Bourbaki [2, p.163]).

0.2.4

A subset P of Φ(g, h) is called a system of positive roots if Φ(g, h) is the disjoint union of P and – P (= {– α¦ α ∈ P}) and if whenever α, β ∈ P and α + β ∈ Φ(g, h) then α + β ∈ P. If C is a Weyl chamber then the set of all α ∈ Φ(g, h) that are positive on C is a system of positive roots. Conversely, if P is a system of positive roots then the subset of hR consisting of those H such that α(H) > 0 for all α ∈ P is a Weyl chamber. Thus specifying a Weyl chamber is the same as specifying a system of positive roots.

Fix a system of positive roots, P. Then α ∈ P is said to be simple if α cannot be written as a sum of two elements of P. The set of all simple roots of P is called a simple system for P or a basis for the root system Φ(g, h). Let π denote the simple system for P. Then π has the following properties (Jacobson [1, p.120]):

(1)   π is a basis for (hR)*.

(2)   If β ∈ P with nα ∈ N.

(3)   W(g, h) is generated by the sx for α ∈ π (Bourbaki [2, p. 155]).

0.3 The structure of compact Lie groups

0.3.1

Let G be a compact Lie group with Lie algebra g. Let gC denote the complexification of g. Then gC is a reductive Lie algebra over C. In fact, if (, ) is any positive non-degenerate symmetric bilinear form on g then we define a new form on g, 〈, 〉, as follows:

Here (as usual) dg denotes normalized invariant measure on G. The invariance of dg immediately implies that

By differentiating this formula one sees that 〈, 〉 is an invariant form on g. Thus, if u is an ideal of g then the orthogonal complement to u is also an ideal of g. Hence, dimension considerations imply that g is a direct sum of 1-dimensional and simple ideals. This clearly implies that g is reductive.

Recall that the Killing form of g, B, is defined by the following formula:

Since ad X is skew adjoint relative to 〈, 〉 for X ∈ g it is clear that B(X, X) ≤ 0 for X ∈ g. Also, B(X, X) = 0 if and only if ad X = 0. Thus, g is semisimple if and only if B is negative definite. The converse is also true.

Theorem

If g is a Lie algebra over R with negative definite Killing form then any connected Lie group with Lie algebra g is compact.

This theorem is known as Weyl’s theorem. For a proof see, for example, Helgason [1, Theorem 6.9, p. 133].

0.3.2

In this book a commutative compact, connected Lie group will be called a torus. Let T be a torus with Lie algebra t. If we look upon t as a Lie group under addition then exp is a covering homomorphism of t onto T. The kernel of exp is a lattice, L, in t. That is, L is a free Z module of rank equal to dim t.

Let T∧ denote the set of all continuous homomorphisms of T into the circle. If μ ∈ T∧ then the differential of μ (which we will also denote by μ) is a linear map of t into iR such that μ(L) ⊂ 2πiZ. If μ is a linear map of t into iR such that μ(L) ⊂ 2πiZ then μ is called integral. If μ is an integral linear form on t then we define for t = exp(X), tμ = exp(μ(X)). This sets up an identification of integral linear forms on t and characters of T.

0.3.3

Let G be a compact, connected Lie group. Then a maximal torus of G is (as the name implies) a torus contained in G but not properly contained in any sub-torus of G. Fix a maximal torus, T, of G. Then tC is a Cartan subalgebra of gC. The elements of Φ(gC, tC)are integral on t and thus define elements of T∧. Thus, we will look upon roots as characters of T. We now list some properties of maximal tori that will be used in this book.

(1)   A maximal torus of G is a maximal abelian subgroup of G (Helgason [1, p.287]).

(2)   If T and S are maximal tori of G then there exists an element g ∈ G such that S = gTg–1 (Helgason [1, p.248]).

(3)   Every element of G is contained in a maximal torus of G. That is, the exponential map of G is surjective. (Helgason [1, p.135].)

(4)   If T is a maximal torus of G then G/T is simply connected. (This follows from say Helgason [1, Cor.2.8, p.287].)

Let T be a maximal torus of G. Let N(T) denote the normalizer of T in G (the elements g of G such that gTg–1 = T). Let W(G, T) denote the group N(T)/T. Then W(G, T) is called the Weyl group of G with respect to T. If g ∈ s ∈ W(G, T) then we set sH = Ad(g)H for H ∈ t. This defines an action of W(G, T) on t.

(5)   Under this action W(G, T) = W(gC, tC) (Helgason [1, Cor.2.13, p.289]).

0.3.4

Let g be a semisimple Lie algebra over C. Then a real form of g, u, will be called a compact form if u has a negative definite Killing form. The following result is due to Weyl. Combined with Theorem 0.3.1 it is the basis of what he called the unitarian trick.

Theorem

If h is a Cartan subalgebra of g then there exists a compact form, u, of g such that u ∩ h is maximal abelian in u. (Jacobson, [1, p. 147].)

0.4 The universal enveloping algebra of a Lie algebra

0.4.1

Let g be a Lie algebra over a field F which we will think of as R or C. Then a universal enveloping algebra for g is a pair (A, j) of an associative algebra with unit, 1, over F, A, and a Lie algebra homomorphism, j, of g into A (here an associative algebra is looked upon as a Lie algebra using the usual commutator bracket, [X, Y] = XY – YX) with the following universal mapping property: If B is an associative algebra with unit and if σ is a Lie algebra homomorphism of g into B then there exists a unique associative algebra homomorphism σ~ of A into B such that σ(X) = σ~(j(X)).

It is easy to see that if (A, j) and (B, i) are universal enveloping algebras of g then there exists an isomorphism, T, of A onto B such that Tj = i. Thus, if a universal enveloping algebra exists then it is unique up to isomorphism.

The usual construction of a universal enveloping algebra of g is given as follows: Let T(g) denote the free associative algebra over F generated by the vector space g. That is, T(g) is the tensor algebra over the vector space g. Let I(g) denote the two sided ideal of T(g) generated by the elements XY – YX – [X, Y] for X, Y ∈ g. Set U(g) = T(g)/I(g). Let i denote the natural map of g into T(g). Let p denote the natural projection of T(g) into U(g). Set j = pi. Then it is easy to see that (U(g), j) is a universal enveloping algebra for g.

The basic result on universal enveloping algebras is the Poincare-Birkoff-Witt Theorem (P-B-W for short):

Theorem

Let X1, …, Xn be a basis of g. Then the monomials

form a basis of U(g) (Jacobson [1, p.159]).

0.4.2

In light of the uniqueness of universal enveloping algebras and P-B-W we will use the notation U(g) for the universal enveloping algebra of g and think of g as a Lie subalgebra of U(g). Thus, j will be looked upon as the canonical inclusion.

Let Um(g) denote the subspace of U(g) spanned by the products of m or less elements of g. Then Um(g) ⊂ Um+1(g) defines a filtration of U(g). This filtration is called the canonical filtration of U(g). With this filtration U(g) is a filtered algebra (that is, Up(g)Uq(g) ⊂ Up + q(g)). Let Gr U(g) denote the corresponding graded algebra. g generates U(g) and the elements XY – YX are in U¹ (g) for X, Y ∈ g. Hence Gr U(g) is a commutative algebra over F. Let S(g) denote the symmetric algebra generated by the vector space g. Then there is a natural homomorphism, μ, of S(g) onto Gr U(g). P-B-W implies that this homomorphism is an isomorphism. If X1, …, Xk are in g then set

the sum over all permutations σ of k letters. Then symm extends to a linear map of S(g) to U(g). Let q be the projection of Um(g) into Gr U(g). If x ∈ S(g) is homogeneous of degree k, then it is easily checked that q(symm(x)) = x. Hence symm defines a linear isomorphism of S(g) onto U(g). In particular, if X ∈ g then symm(Xm) = Xm (the multiplication on the left hand side is in S(g) on the right hand side it is in U(g)). symm defines a linear isomorphism of S(g) onto U(g) which is called the symmetrization mapping.

We note that if α the Lie algebra (0) then U(a) = Fbe the Lie algebra homomorphism of g onto a (Xextends to a homomorphism of U(g) onto F is called the augmentation homomorphism.

We denote by gopp the Lie algebra whose underlying vector space is g with bracket operation {X, Y} = [Y, X]. Then U(gopp) = U(g)opp (the opposite algebra). The correspondence X → – X defines a homomorphism of g onto gopp whose extension to U(g) will be denoted xT. We note that the linear map x → xT is defined by the following three properties:

0.4.3

Let b be a subalgebra of g. P-B-W implies that the canonical map of U(b) into U(g) is injective. We can thus identify U(b) with the associative subalgebra of U(g) generated by 1 and b. Let V be a subspace of g such that g = b ⨁ V. Then P-B-W implies that the linear map

Given by b v → b symm(v) for b ∈ U(b), v ∈ S(V), is a surjective linear isomorphism. Hence U(g) is the free module on the generators symm(S(V)) as a U(b) module under left multiplication. Similarly, U(g) is the free right U(b) module generated by symm(S(V)) under right multiplication by U(b).

0.5 Some basic representation theory

0.5.1

One of the most useful elementary results in representation theory is Schur’s Lemma. There is a Schur’s Lemma for most representation theoretic contexts (algebraic, unitary, Banach, etc.) In this book there will be several such Lemmas. We begin this section with a particularly useful one (usually called Dixmier’s Lemma). It is based on the following result:

Lemma

Let V be a countable dimensional vector space over C. If T is an endomorphism of V then there exists a scalar c such that T – cI is not invertible on V.

Suppose that T – cI is invertible for all scalars, c. Then P(T) is invertible on V for all non-zero polynomials P in one variable. Thus if R = P/Q is a rational function with P and Q polynomials then we can define R(T) to by the formula P(T)(Q(T)–1). This rule defines a linear map of the rational functions in one variable, C(x), into End(V). If v ∈ V is non-zero and if R ∈ C(x) is non-zero with R = P/Q as above then R(T)v = 0 only if P(T)v = 0. Thus the map of C(x) into V given by R → R(T)v is injective. Since C(x) is of uncountable dimension over C this is a contradiction.

0.5.2

We now come to Dixmier’s Lemma. Let V be a vector space over C. Let S be a subset of End(V). Then S is said to act irreducibly if whenever W is a subspace of V such that SW W then W = V or W = (0).

Lemma

Suppose that V is countable dimensional and that S ⊂ End(V) acts irreducibly. If T ∈ End(V) commutes with every element of S then T is a scalar multiple of the identity operator.

By 0.5.1 there exists c ∈ C such that T – cI is not invertible on V. Since the elements of S preserve Ker(T – cI) and Im(T – cI) and since at least one of the