# Real Productive Groups I, Volume 132

## 1st Edition

**Authors:**Nolan Wallach

**Print ISBN:**9780127329604

**eBook ISBN:**9780080874517

**Imprint:**Academic Press

**Published Date:**28th February 1988

**Page Count:**412

## Description

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 exp

## Details

- No. of pages:
- 412

- Language:
- English

- Copyright:
- © Academic Press 1988

- Published:
- 28th February 1988

- Imprint:
- Academic Press

- eBook ISBN:
- 9780080874517

- Hardcover ISBN:
- 9780127329604

## About the Author

### Nolan Wallach

### Affiliations and Expertise

University of California

## Reviews

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.