Harmonic Analysis on Semi-Simple Lie Groups I

1 The Structure of Real Semi-Simple Lie Groups.- 1.1 Preliminaries.- 1.2 The Bruhat Decomposition-Parabolic Subgroups.- 1.3 Cartan Subalgebras.- 1.4 Cartan Subgroups.- 2 The Universal Enveloping Algebra of a Semi-Simple Lie Algebra.- 2.1 Invariant Theory I - Generalities.- 2.2 Invariant Theory II - Applications to Reductive Lie Algebras.- 2.3 On the Structure of the Universal Enveloping Algebra.- 2.4 Representations of a Reductive Lie Algebra.- 2.5 Representations on Cohomology Groups.- 3 Finite Dimensional Representations of a Semi-Simple Lie Group.- 3.1 Holomorphic Representations of a Complex Semi-Simple Lie Group.- 3.2 Unitary Representations of a Compact Semi-Simple Lie Group.- 3.3 Finite Dimensional Class One Representations of a Real Semi-Simple Lie Group.- 4 Infinite Dimensional Group Representation Theory.- 4.1 Representations on a Locally Convex Space.- 4.2 Representations on a Banach Space.- 4.3 Representations on a Hubert Space.- 4.3.1 Generalities.- 4.3.2 Examples.- 4.4 Differentiable Vectors, Analytic Vectors.- 4.5 Large Compact Subgroups.- 5 Induced Representations.- 5.1 Unitarily Induced Representations.- 5.2 Quasi-Invariant Distributions.- 5.3 Irreducibility of Unitarily Induced Representations.- 5.4 Systems of Imprimitivity.- 5.5 Applications to Semi-Simple Lie Groups.- 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.