The representation theory of locally compact groups has been vig­ orously developed in the past twenty-five years or so; of the various branches of this theory, one of the most attractive (and formidable) is the representation theory of semi-simple Lie groups which, to a great extent, is the creation of a single man: Harish-Chandra. The chief objective of the present volume and its immediate successor is to provide a reasonably self-contained introduction to Harish-Chandra's theory. Granting cer­ tain basic prerequisites (cf. infra), we have made an effort to give full details and complete proofs of the theorems on which the theory rests. The structure of this volume and its successor is as follows. Each book is divided into chapters; each chapter is divided into sections; each section into numbers. We then use the decimal system of reference; for example, 1. 3. 2 refers to the second number in the third section of the first chapter. Theorems, Propositions, Lemmas, and Corollaries are listed consecutively throughout any given number. Numbers which are set in fine print may be omitted at a first reading. There are a variety of Exam­ ples scattered throughout the text; the reader, if he is so inclined, can view them as exercises ad libitum. The Appendices to the text collect certain ancillary results which will be used on and off in the systematic exposi­ tion; a reference of the form A2.

Inhalt

1 The Structure of Real Semi-Simple Lie Groups.- 1.1 Preliminaries.- 1.1.1 The Structure of Complex Semi-Simple Lie Algebras.- 1.1.2 Root Systems I - Basic Properties.- 1.1.3 Root Systems II -?-Systems.- 1.1.4 Structure of the Nilpotent Constituent in an Iwasawa Decomposition.- 1.1.5 Reductive Lie Algebras and Groups.- 1.2 The Bruhat Decomposition-Parabolic Subgroups.- 1.2.1 Tits Systems.- 1.2.2 The Complex Case.- 1.2.3 Boundary Subgroups and Parabolic Subgroups of a Real Semi-Simple Lie Group.- 1.2.4 Levi Subgroups of a Parabolic Subgroup. The Langlands Decomposition.- 1.3 Cartan Subalgebras.- 1.3.1 Conjugacy of Cartan Subalgebras in a Real Reductive Lie Algebra.- 1.3.2 Classification of Roots.- 1.3.3 Fundamental Cartan Subalgebras.- 1.3.4 Regular and Semi-Regular Elements in a Reductive Lie Algebra.- 1.3.5 Semi-Simple and Nilpotent Elements in a Reductive Lie Algebra.- 1.4 Cartan Subgroups.- 1.4.1 Structure Theorems.- 1.4.2 The Groups W(G,J) and W(G,J0).- 1.4.3 Semi-Simple and Unipotent Elements in a Reductive Lie Group.- 2 The Universal Enveloping Algebra of a Semi-Simple Lie Algebra.- 2.1 Invariant Theory I - Generalities.- 2.1.1 Modules.- 2.1.2 The Fundamental Theorem of Invariant Theory.- 2.1.3 Invariants of Finite Groups Generated by Reflections.- 2.1.4 Symmetric Algebras and Formal Power Series.- 2.1.5 Weyl Group Invariants.- 2.2 Invariant Theory II - Applications to Reductive Lie Algebras.- 2.2.1 A Theorem of Harish-Chandra.- 2.2.2 Theorems of Finitude.- 2.3 On the Structure of the Universal Enveloping Algebra.- 2.3.1 Generalities.- 2.3.2 Existence of Sufficiently Many Finite Dimensional Representations.- 2.3.3 The Reductive Case.- 2.4 Representations of a Reductive Lie Algebra.- 2.4.1 Simple Modules - The Theorem of Highest Weight.- 2.4.2 The Formula of H. Weyl and B. Kostant.- 2.4.3 The Characters of a Reductive Lie Algebra.- 2.5 Representations on Cohomology Groups.- 2.5.1 The Riemann-Roch Theorem for Lie Algebras.- 2.5.2 Theorems of Bott and Kostant.- 3 Finite Dimensional Representations of a Semi-Simple Lie Group.- 3.1 Holomorphic Representations of a Complex Semi-Simple Lie Group.- 3.1.1 Generalities.- 3.1.2 The Borel-Weil Theorem.- 3.2 Unitary Representations of a Compact Semi-Simple Lie Group.- 3.2.1 The Invariant Integral on a Compact Semi-Simple Lie Algebra.- 3.2.2 The Planche rel Theorem for a Compact Connected Semi-Simple Lie Group.- 3.3 Finite Dimensional Class One Representations of a Real Semi-Simple Lie Group.- 3.3.1 The Theorem of É. Cartan and S. Helgason.- 3.3.2 Inequalities.- 4 Infinite Dimensional Group Representation Theory.- 4.1 Representations on a Locally Convex Space.- 4.1.1 Basic Concepts.- 4.1.2 Operations on Representations.- 4.1.3 Intertwining Forms and Operators.- 4.2 Representations on a Banach Space.- 4.2.1 Banach Representations of Associative Algebras.- 4.2.2 Banach Representations of Groups.- 4.3 Representations on a Hubert Space.- 4.3.1 Generalities.- 4.3.2 Examples.- 4.4 Differentiable Vectors, Analytic Vectors.- 4.4.1 Passage to U?.- 4.4.2 Absolute Convergence of the Fourier Series.- 4.4.3 A Density Theorem. Fourier Series in Function Spaces.- 4.4.4 Elliptic Elements in the Enveloping Algebra.- 4.4.5 Density of Analytic Vectors - The Theorem of Nelson.- 4.4.6 Analytic Domination - Applications to Representation Theory.- 4.4.7 The Paley-Wiener Space.- 4.5 Large Compact Subgroups.- 4.5.1 The Algebras Cc,?(G), Ic,?,(G).- 4.5.2 Groups with Large Compact Subgroups.- 4.5.3 Properties of Largeness.- 4.5.4 Naimark Equivalence.- 4.5.5 Infinitesimal Equivalence.- 4.5.6 Jordan-Hölder Series - Multiplicities.- 4.5.7 Theorems of Finitude.- 4.5.8 Characters.- 4.5.9 Square Integrable Representations.- 5 Induced Representations.- 5.1 Unitarily Induced Representations.- 5.1.1 The Definition.- 5.1.2 Unitarily Induced Representations and Measures of Positive Type.- 5.1.3 Elementary Properties of Unitarily Induced Representations.- 5.2 Quasi-Invariant Distributions.- 5.2.1 The Global Situation.- 5.2.2 The Local Situation.- 5.2.3 A Fundamental Estimate.- 5.2.4 The Case of Countably Many Orbits.- 5.3 Irreducibility of Unitarily Induced Representations.- 5.3.1 On the Notion of Induced Representation.- 5.3.2 Estimation of the Intertwining Number.- 5.3.3 Reciprocity Theorems.- 5.3.4 Decomposition Theorems.- 5.4 Systems of Imprimitivity.- 5.4.1 Mackey's Orbital Analysis.- 5.4.2 Examples.- 5.5 Applications to Semi-Simple Lie Groups.- 5.5.1 The Subquotient Theorem.- 5.5.2 Irreducibility of the Principal P-Series - P Minimal.- 5.5.3 The Characters of the Principal P-Series - P Minimal.- 5.5.4 The Riemann-Lebesgue Lemma for the Principal P-Series - P Minimal.- Appendices.- 1 Quasi-Invariant Measures.- 2 Distributions on a Manifold.- 2.1 Differential Operators and Function Spaces.- 2.2 Tensor Products of Topological Vector Spaces.- 2.3 Vector Distributions.- 2.4 Distributions on a Lie Group.- General Notational Conventions.- List of Notations.- Guide to the Literature.