Lie Groups

Front Cover
Springer Science & Business Media, Apr 17, 2013 - Mathematics - 454 pages
This book aims to be a course in Lie groups that can be covered in one year with a group of good graduate students. I have attempted to address a problem that anyone teaching this subject must have, which is that the amount of essential material is too much to cover. One approach to this problem is to emphasize the beautiful representation theory of compact groups, and indeed this book can be used for a course of this type if after Chapter 25 one skips ahead to Part III. But I did not want to omit important topics such as the Bruhat decomposition and the theory of symmetric spaces. For these subjects, compact groups are not sufficient. Part I covers standard general properties of representations of compact groups (including Lie groups and other compact groups, such as finite or p adic ones). These include Schur orthogonality, properties of matrix coefficients and the Peter-Weyl Theorem.
 

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Contents

10
33
The Local Frobenius Theorem
79
Tori
86
Geodesics and Maximal Tori
94
Topological Proof of Cartans Theorem
107
The Weyl Integration Formula
112
The Root System
117
Examples of Root Systems
127
Topics
273
Mackey Theory
275
Characters of GLn C
284
Duality between Sk and GLn C
289
The JacobiTrudi Identity
297
Schur Polynomials and GLn C
308
Schur Polynomials and Sk
315
Random Matrix Theory
321

Abstract Weyl Groups
136
The Fundamental Group
146
Semisimple Compact Groups
150
HighestWeight Vectors
157
The Weyl Character Formula
162
Spin
175
Complexification
182
Coxeter Groups
189
The Iwasawa Decomposition
197
The Bruhat Decomposition
205
Symmetric Spaces
212
Relative Root Systems
236
Embeddings of Lie Groups
257
Minors of Toeplitz Matrices
331
Branching Formulae and Tableaux
339
The Cauchy Identity
347
Unitary Branching Rules
357
The Involution Model for Sk
361
Some Symmetric Algebras
370
Gelfand Pairs
375
Hecke Algebras
384
The Philosophy of Cusp Forms
397
Cohomology of Grassmannians
428
References
438
Index
446
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