Starting with Bargmanns 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.
作者簡介
暫缺《SL2（R）》作者簡介
圖書目錄
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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