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Representation Theory of Lie Groups / Edition 1

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Brand new. We distribute directly for the publisher. 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. Read more Show Less

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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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Editorial Reviews

Booknews
Contains six series of lectures given at the Park City Mathematics Institute 1998 graduate summer school. The topics are representations of semisimple Lie groups, representations in Dolbeault cohomology, a geometric approach to unitary representations attached to elliptic orbits, the method of coadjoint orbits for real reductive groups, geometric methods in representation theory, and minimal representations and reductive dual pairs. The book can be used as a supplementary text for a course on introductory infinite-dimensional representation theory. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780821819418
  • Publisher: American Mathematical Society
  • Publication date: 1/25/2000
  • Series: IAS Park City Mathematics Series , #8
  • Edition number: 1
  • Pages: 340

Meet the Author

Jeffrey Adams
Jeffrey Adams is a historian of the Juniata region of Pennsylvania and author of Tyrone. The images in Harrisburg have been drawn from his own extensive collection of postcards.
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Table of Contents

Preface xi
Introduction 1
Representations of Semisimple Lie Groups 5
Introduction 7
Motivation 8
Lecture 1. Some Representations of SL(n, R) 15
Lecture 2. Semisimple Groups and Structure Theory 25
Lecture 3. Introduction to Representation Theory 33
Lecture 4. Cartan Subalgebras and Highest Weights 45
Lecture 5. Action by the Lie Algebra 53
Lecture 6. Cartan Subgroups and Global Characters 61
Lecture 7. Discrete Series and Asymptotics 71
Lecture 8. Langlands Classifications 81
Bibliography 87
Representations in Dolbeault Cohomology 89
Introduction 91
Lecture 1. Complex Flag Varieties and Orbits Under a Real Form 93
Lecture 2. Open G[subscript 0]-Orbits 103
Lecture 3. Examples, Homogeneous Bundles 109
Lecture 4. Dolbeault Cohomology, Bott-Borel-Weil Theorem 117
Lecture 5. Indefinite Harmonic Theory 123
Lecture 6. Intertwining Operators I 129
Lecture 7. Intertwining Operators II 135
Lecture 8. The Linear Cycle Space 141
Bibliography 145
Unitary Representations Attached to Elliptic Orbits. A Geometric Approach 147
Introduction 149
Lecture 1. Globalizations 151
Lecture 2. Dolbeault Cohomology and Maximal Globalization 157
Lecture 3. L[superscript 2]-Cohomology and Discrete Series Representations 163
Lecture 4. Indefinite Quantization 169
Bibliography 175
The Method of Coadjoint Orbits for Real Reductive Groups 177
Introduction 179
Lecture 1. Some Ideas from Mathematical Physics 181
Lecture 2. The Jordan Decomposition and Three Kinds of Quantization 187
Lecture 3. Complex Polarizations 197
Lecture 4. The Kostant-Sekiguchi Correspondence 203
Lecture 5. Quantizing the Action of K 207
Lecture 6. Associated Graded Modules 211
Lecture 7. A Good Basis for Associated Graded Modules 217
Lecture 8. Proving Unitarity 221
Exercises 229
Bibliography 237
Geometric Methods in Representation Theory 239
Introduction 241
Acknowledgments 241
Lecture 1. Overview 243
Lecture 2. Derived Categories of Constructible Sheaves 247
Lecture 3. Equivariant Derived Categories 253
Lecture 4. Functors to Representations 257
Lecture 5. Matsuki Correspondence for Sheaves 261
Lecture 6. Characteristic Cycles 265
Lecture 7. The Character Formula 271
Lecture 8. Microlocalization of Matsuki = Sekiguchi 275
Appendix. Homological Algebra 281
Bibliography 289
Minimal Representations and Reductive Dual Pairs 291
Lecture 1. Introduction 293
Lecture 2. The Oscillator Representation 297
Lecture 3. Models 301
Lecture 4. Duality 309
Lecture 5. Classification 315
Lecture 6. Unitarity 319
Lecture 7. Minimal Representations of Classical Groups 323
Lecture 8. Dual Pairs in Simple Groups 329
Bibliography 337
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