Harmonic Analysis on Semi-Simple Lie Groups I

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.