Lie Groups

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Springer Science & Business Media, Jun 17, 2004 - Mathematics - 451 pages
This book is intended for a one year graduate course on Lie groups and Lie algebras. The author proceeds beyond the representation theory of compact Lie groups (which is the basis of many texts) and provides a carefully chosen range of material to give the student the bigger picture. For compact Lie groups, the Peter-Weyl theorem, conjugacy of maximal tori (two proofs), Weyl character formula and more are covered. The book continues with the study of complex analytic groups, then general noncompact Lie groups, including the Coxeter presentation of the Weyl group, the Iwasawa and Bruhat decompositions, Cartan decomposition, symmetric spaces, Cayley transforms, relative root systems, Satake diagrams, extended Dynkin diagrams and a survey of the ways Lie groups may be embedded in one another. The book culminates in a "topics" section giving depth to the student's understanding of representation theory, taking the Frobenius-Schur duality between the representation theory of the symmetric group and the unitary groups as a unifying theme, with many applications in diverse areas such as random matrix theory, minors of Toeplitz matrices, symmetric algebra decompositions, Gelfand pairs, Hecke algebras, representations of finite general linear groups and the cohomology of Grassmannians and flag varieties. Daniel Bump is Professor of Mathematics at Stanford University. His research is in automorphic forms, representation theory and number theory. He is a co-author of GNU Go, a computer program that plays the game of Go. His previous books include Automorphic Forms and Representations (Cambridge University Press 1997) and Algebraic Geometry (World Scientific 1998).
 

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Contents

Haar Measure
3
Schur Orthogonality
6
Compact Operators
17
The PeterWeyl Theorem
21
Lie Group Fundamentals
27
Lie Subgroups of GLnC
29
Vector Fields
36
LeftInvariant Vector Fields
41
Coxeter Groups
189
The Iwasawa Decomposition
197
The Bruhat Decomposition
205
Symmetric Spaces
212
Relative Root Systems
236
Embeddings of Lie Groups
257
Topics
273
Mackey Theory
275

The Exponential Map
46
Tensors and Universal Properties
50
The Universal Enveloping Algebra
54
Extension of Scalars
58
Representations of sl2 C
62
The Universal Cover
69
The Local Frobenius Theorem
79
Tori
86
Geodesies and Maximal Tori
94
Topological Proof of Cartans Theorem
107
The Weyl Integration Formula
112
The Root System
117
Examples of Root Systems
127
Abstract Weyl Groups
136
The Fundamental Group
146
Semisimple Compact Groups
150
HighestWeight Vectors
157
The Weyl Character Formula
162
Spin
175
Complexification
182
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
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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