# Real Reductive Groups I

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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.

## Table of Contents

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 9 groups

1.5. A class of induced representations

1.6. C" vectors and analytic vectors

1.7. Representations of compact Lie groups

1.8. Further results and comments

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

2.A.1. Some linear algebra

2.A.2. Norms on real reductive groups

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

3.A.1. Some associative algebra

3.A.2. A Lemma of Harish-Chandra

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 E-function

4.6. Notes and further results

4.A. Appendices to Chapter 4

4.A.1. Asymptotic expansions

4.A.2. Some inequalities

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

5.A.1. A Lemma of Langlands

5.A.2. An a priori estimate

5.A.3. Square integrability and the polar decomposition

Chapter 6. A Construction of the Fundamental Series

Introduction

6.1 Relative Lie algebra cohomology

6.2 A construction of (f, K)-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

6.A.1 Some homological algebra

6.A.2 Partition functions

6.A.3 Tensor products with finite dimensional representations

6.A.4 Infinitesimally unitary modules

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 integral 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

7.A.1 Some linear algebra

7.A.2 Radial components on the Lie algebra

7.A.3 Radial components on the Lie group

7.A.4 Some harmonic analysis on Tori

7.A.5 Fundamental solutions of certain differential operators

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 π

8.A. Appendices to Chapter 8

8.A.1 Trace class operators

8.A.2. Some operations on distributions

8.A.3. The radial component revisited

8.A.4. The orbit structure on a real reductive Lie algebra

8.A.5. Some technical results for Harish-Chandra’s regularity theorem

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

Introduction

9.1. Tensor products of finite dimensional representations

9.2. Spinors

9.3. The Dirac operator

9.4. (g, K)-cohomology

9.5. Some results of Kumaresan, Parthasarathy, Vogan, Zuckerman

9.6. μ-cohomology

9.7. A theorem of Vogan-Zuckerman

9.8. Further results

9.A. Appendices to Chapter 9

9.A.1. Weyl groups

9.A.2. Spectral sequences

Bibliography

Index

## Product details

- No. of pages: 412
- Language: English
- Copyright: © Academic Press 1988
- Published: February 28, 1988
- Imprint: Academic Press
- eBook ISBN: 9780080874517

## About the Author

### Nolan Wallach

#### Affiliations and Expertise

University of California