內容簡介

Starting with Bargmann’’s paPer on the tnhmte dimenslonal representattons olSL2（R），the theory of representations of semisimple Lie groups has evolved toa rather extensive production.Some of the main contributors have been：Gelfand-Naimark and Harish-Chandra.who considered the Lorentz group inthe late forties；Gelfand-Nalmark.who dealt with the classical complexgroups.while Harish•Chandra worked out the general reaI case。especiallythrough the derived representation of the Lie algebra.establishing thePlancherel formula （Gclfand-Graev also contributed to the teal case）；Cat-tan。Gelfand-Naimark.Godement.Harish.Chandra。who developed thetheory of spherical functions （Godement gave several Bourbaki seminarreports giving proofs for a number of spectral results not accessible other-wise）；Selberg，who took the group modulo a discrete subgroup and obtainedthe trace formula；Gelfand.Fomin，Piateckii.Shapiro，and Harish.Chandra，who established connections with automorphic forms；Jacquet-Lanlands，who pushed through the connection with L-series and Hecke.
 

目錄

Notation
Chapter Ⅰ General Results
　1 The representation on Co(G)
　2 A criterion for complete reducibility
　3 L2 kernels and operators
　4 Plancherel measures
Chapter Compact Groups
　l Decomposition over K for SL2(R)
　2 Compact groups in general
Chapter Ⅲ Induced Representations
　1 Integration on coset spaces
　2 Induced representations
　3 Associated spherical functions
　4 The kernel defining the induced representation
Chapter Ⅳ Spherical Functions
　1 Bi-invariance
　2 Irreducibility
　3 The spherical property
　4 Connection with unitary representations
　5 Positive definite functions
Chapter Ⅴ The Spherical Transform
　2 The Harish transform
　3 The Mellin transform
　4 The spherical transform
　5 Explicit formulas and asymptotic expansions
Chapter Ⅵ The Derived Representation on the Lie Algebra
　1 The derived representation
　2 The derived representation decomposed over K
　3 Unitarization of a representation
　4 The Lie derivatives on G
　5 Irreducible components of the induced representations
　6 Classification of all unitary irreducible representations
　7 Separation by the trace
Chapter Ⅶ TracesⅡ
　2 Integral formulas
　3 The trace in the induced representation
　4 The trace in the discrete series
　5 Relation between the Harish transforms on A and K
　Appendix. General facts about traces
Shapter Ⅷ The Planeherel Formula
　1 Calculus lemma
　2 The Harish transforms discontinuities
　3 Some lemmas
　4 The Plancherel formula
Chapter Ⅸ Discrete Series
　1 Discrete series in L2(G)
　2 Representation in the upper half plane
　3 Representation on the disc
　4 The lifting of weight m
　5 The holomorpbic property
Chapter Ⅹ Partial Differential Operators
　1 The universal enveloping algebra
　2 Analytic vectors
　3 Eigenfunctions of (f)
……
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