Harmonic Analysis and Representations of Semisimple Lie Groups : Lectures Given at the NATO Advanced Study Institute on Representations of Lie Groups and Harmonic Analysis, Held at Liège, Belgium, September 5–17, 1977

This book presents the text of the lectures which were given at the NATO Advanced Study Institute on Representations of Lie groups and Harmonic Analysis which was held in Liege from September 5 to September 17, 1977. The general aim of this Summer School was to give a coordinated intro­ duction to the theory of representations of semisimple Lie groups and to non-commutative harmonic analysis on these groups, together with some glance at physical applications and at the related subject of random walks. As will appear to the reader, the order of the papers - which follows relatively closely the order of the lectures which were actually give- follows a logical pattern. The two first papers are introductory: the one by R. Blattner describes in a very progressive way a path going from standard Fourier analysis on IR" to non-commutative harmonic analysis on a locally compact group; the paper by J. Wolf describes the structure of semisimple Lie groups, the finite-dimensional representations of these groups and introduces basic facts about infinite-dimensional unitary representations. Two of the editors want to thank particularly these two lecturers who were very careful to pave the way for the later lectures. Both these chapters give also very useful guidelines to the relevant literature. Front Matter....Pages i-viii General Background....Pages 1-67 Front Matter....Pages 69-70 Structure of Semisimple Lie Groups....Pages 71-93 Finite-Dimensional Representation Theory....Pages 94-108 Infinite-Dimensional Representations....Pages 109-130 Infinitesimal Theory of Representations of Semisimple Lie Groups....Pages 131-255 The Role of Differential Equations in the Plancherel Theorem....Pages 257-315 A Geometric Construction of the Discrete Series for Semisimple Lie Groups....Pages 317-378 Erratum to the Paper: A Geometric Construction of the Discrete Series for Semisimple Lie Groups....Pages 379-383 Front Matter....Pages 385-386 1-Differentiable Deformations of the Poisson Bracket Lie Algebra....Pages 387-396 Differentiable Deformations of the Poisson Bracket Lie Algebra....Pages 397-407 Physical Applications Related to 1-Differentiable Deformations of Poisson Brackets....Pages 408-418 Physical Applications Related to Differentiable Deformations of Poisson Brackets: Quantum Mechanics....Pages 419-428 Separate and Joint Analyticity of Vectors in Group Representations and an Application to Field Theory....Pages 429-438 Nonlinear Representations of Lie Groups and Applications....Pages 439-448 Introduction to the 1-Cohomology of Lie Groups....Pages 449-465 Random Walks on Lie Groups....Pages 467-489 Back Matter....Pages 491-496 Lectures given at the NATO Advanced Study Institute on Representations of Lie Groups and Harmonic Analysis, held In Liege, Belgium, September 5-7, 1977 Lectures given at the NATO Advanced Study Institute on Representations of Lie Groups and Harmonic Analysis, held In Liège, Belgium, September 5-7, 1977