Representation Theory of Lie Groups / Edition 1

Representation Theory of Lie Groups / Edition 1

by Jeffrey Adams
     
 

ISBN-10: 0821819410

ISBN-13: 9780821819418

Pub. Date: 01/25/2000

Publisher: American Mathematical Society

This book contains written versions of the lectures given at the PCMI Graduate Summer School on the representation theory of Lie groups. The volume begins with lectures by A. Knapp and P. Trapa outlining the state of the subject around the year 1975, specifically, the fundamental results of Harish-Chandra on the general structure of infinite-dimensional

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Overview

This book contains written versions of the lectures given at the PCMI Graduate Summer School on the representation theory of Lie groups. The volume begins with lectures by A. Knapp and P. Trapa outlining the state of the subject around the year 1975, specifically, the fundamental results of Harish-Chandra on the general structure of infinite-dimensional representations and the Langlands classification. Additional contributions outline developments in four of the most active areas of research over the past 20 years. The clearly written articles present results to date, as follows: R. Zierau and L. Barchini discuss the construction of representations on Dolbeault cohomology spaces. D. Vogan describes the status of the Kirillov-Kostant ''philosophy of coadjoint orbits'' for unitary representations. K. Vilonen presents recent advances in the Beilinson-Bernstein theory of ''localization''. And Jian-Shu Li covers Howe's theory of ''dual reductive pairs''. Each contributor to the volume presents the topics in a unique, comprehensive, and accessible manner geared toward advanced graduate students and researchers. Students should have completed the standard introductory graduate courses for full comprehension of the work. The book would also serve well as a supplementary text for a course on introductory infinite-dimensional representation theory.

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Product Details

ISBN-13:
9780821819418
Publisher:
American Mathematical Society
Publication date:
01/25/2000
Series:
IAS Park City Mathematics Series, #8
Pages:
340

Related Subjects

Table of Contents

Prefacexi
Introduction1
Representations of Semisimple Lie Groups5
Introduction7
Motivation8
Lecture 1.Some Representations of SL(n, R)15
Lecture 2.Semisimple Groups and Structure Theory25
Lecture 3.Introduction to Representation Theory33
Lecture 4.Cartan Subalgebras and Highest Weights45
Lecture 5.Action by the Lie Algebra53
Lecture 6.Cartan Subgroups and Global Characters61
Lecture 7.Discrete Series and Asymptotics71
Lecture 8.Langlands Classifications81
Bibliography87
Representations in Dolbeault Cohomology89
Introduction91
Lecture 1.Complex Flag Varieties and Orbits Under a Real Form93
Lecture 2.Open G[subscript 0]-Orbits103
Lecture 3.Examples, Homogeneous Bundles109
Lecture 4.Dolbeault Cohomology, Bott-Borel-Weil Theorem117
Lecture 5.Indefinite Harmonic Theory123
Lecture 6.Intertwining Operators I129
Lecture 7.Intertwining Operators II135
Lecture 8.The Linear Cycle Space141
Bibliography145
Unitary Representations Attached to Elliptic Orbits. A Geometric Approach147
Introduction149
Lecture 1.Globalizations151
Lecture 2.Dolbeault Cohomology and Maximal Globalization157
Lecture 3.L[superscript 2]-Cohomology and Discrete Series Representations163
Lecture 4.Indefinite Quantization169
Bibliography175
The Method of Coadjoint Orbits for Real Reductive Groups177
Introduction179
Lecture 1.Some Ideas from Mathematical Physics181
Lecture 2.The Jordan Decomposition and Three Kinds of Quantization187
Lecture 3.Complex Polarizations197
Lecture 4.The Kostant-Sekiguchi Correspondence203
Lecture 5.Quantizing the Action of K207
Lecture 6.Associated Graded Modules211
Lecture 7.A Good Basis for Associated Graded Modules217
Lecture 8.Proving Unitarity221
Exercises229
Bibliography237
Geometric Methods in Representation Theory239
Introduction241
Acknowledgments241
Lecture 1.Overview243
Lecture 2.Derived Categories of Constructible Sheaves247
Lecture 3.Equivariant Derived Categories253
Lecture 4.Functors to Representations257
Lecture 5.Matsuki Correspondence for Sheaves261
Lecture 6.Characteristic Cycles265
Lecture 7.The Character Formula271
Lecture 8.Microlocalization of Matsuki = Sekiguchi275
Appendix.Homological Algebra281
Bibliography289
Minimal Representations and Reductive Dual Pairs291
Lecture 1.Introduction293
Lecture 2.The Oscillator Representation297
Lecture 3.Models301
Lecture 4.Duality309
Lecture 5.Classification315
Lecture 6.Unitarity319
Lecture 7.Minimal Representations of Classical Groups323
Lecture 8.Dual Pairs in Simple Groups329
Bibliography337

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