Real Reductive Groups I

Nolan R. Wallach

• Publisher: Academic Press
• Year: 1988
• Print ISBN: 9780127329604
• eBook ISBN: 9780080874517
• Topic: Mathematics
• Subtopic: Group Theory
• Index: Mathematics

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.

1

Elementary Representation Theory

Introduction

In this chapter we develop most of the general representation theory that will be needed in this book. We have attempted to make the material as elementary as possible.

The infinite dimensional representation theory of Lie groups is a vast subject that has been studied by many authors in that last 40 years. Thus, a short chapter such as this one can only “scrape the surface” of the material. A much more encyclopedic account can be found in Chapters 4 and 5 of Warner [1]. The more general theory is not really necessary to our book, since we will be studying mainly reductive groups.

We now give a description of this chapter. The first section is canonical except for the introduction of the conjugate dual to a Hilbert representation. This notion is of great importance to the representation theory of reductive groups. In the second section we give a variant of Schur’s Lemma. As we indicated in Section 0.5 there are many variants of this Lemma. The one that we give for irreducible unitary representations is sufficient for our purposes. Section 3 is devoted to the most elementary properties of square integrable representations. As we will see in the later chapters, these representations are the basic ingredients in the harmonic analysis of real reductive groups. Section 4 contains the Peter-Weyl theory of representations of compact groups. It also contains the critical (for our purposes) notion of isotypic component. In Section 5 we study a very special class of induced representations. A good exposition of the general theory of induced representations can be found in Warner [1, Chap. 5]. Included in this section is Frobenius reciprocity for compact groups. In Section 6 we introduce just enough of the theory of smooth and analytic vectors to do the representation theory of the later chapters. Again, the serious reader can consult Warner [1, Chap. 4] for a much more comprehen...

Table of contents

• Cover image
• Title page
• Table of Contents
• Inside Front Cover
• Copyright page
• Dedication
• Preface
• Introduction
• Chapter 0: Background Material
• Chapter 1: Elementary Representation Theory
• Chapter 2: Real Reductive Groups
• Chapter 3: The Basic Theory of (g, K)-Modules
• Chapter 4: The Asymptotic Behavior of Matrix Coefficients
• Chapter 5: The Langlands Classification
• Chapter 6: A Construction of the Fundamental Series
• Chapter 7: Cusp Forms on G
• Chapter 8: Character Theory
• Chapter 9: Unitary Representations and (g, K)-Cohomology
• Bibliography
• Index
• Pure and Applied Mathematics