This book is intended for a one year graduate course on Lie groups. Rather than providing a comprehensive treatment, the author emphasizes the beautiful representation theory of compact groups. However, this book also discusses important topics such as the Bruhat decomposition and the theory of symmetric spaces.
Author(s): Daniel Bump
Series: Graduate texts in mathematics 225
Edition: 1
Publisher: Springer
Year: 2004
Language: English
Pages: 467
City: Berlin
Contents......Page 9
Preface......Page 5
Part I: Compact Groups......Page 13
1 Haar Measure......Page 15
2 Schur Orthogonality......Page 18
3 Compact Operators......Page 29
4 The Peter-Weyl Theorem......Page 33
Part II: Lie Group Fundamentals......Page 39
5 Lie Subgroups of GL(n, C)......Page 41
6 Vector Fields......Page 48
7 Left-Invariant Vector Fields......Page 53
8 The Exponential Map......Page 58
9 Tensors and Universal Properties......Page 62
10 The Universal Enveloping Algebra......Page 66
11 Extension of Scalars......Page 70
12 Representations of sl(2, C)......Page 74
13 The Universal Cover......Page 81
14 The Local Frobenius Theorem......Page 91
15 Tori......Page 98
16 Geodesics and Maximal Tori......Page 106
17 Topological Proof of Cartan's Theorem......Page 119
18 The Weyl Integration Formula......Page 124
19 The Root System......Page 129
20 Examples of Root Systems......Page 139
21 Abstract Weyl Groups......Page 148
22 The Fundamental Group......Page 158
23 Semisimple Compact Groups......Page 162
24 Highest-Weight Vectors......Page 169
25 The Weyl Character Formula......Page 174
26 Spin......Page 187
27 Complexification......Page 194
28 Coxeter Groups......Page 201
29 The Iwasawa Decomposition......Page 209
30 The Bruhat Decomposition......Page 217
31 Symmetric Spaces......Page 224
32 Relative Root Systems......Page 248
33 Embeddings of Lie Groups......Page 269
Part III: Topics......Page 285
34 Mackey Theory......Page 287
35 Characters of GL(n, C)......Page 296
36 Duality between S_k and GL(n, C)......Page 301
37 The Jacobi-Trudi Identity......Page 309
38 Schur Polynomials and GL(n, C)......Page 320
39 Schur Polynomials and S_k......Page 327
40 Random Matrix Theory......Page 333
41 Minors of Toeplitz Matrices......Page 343
42 Branching Formulae and Tableaux......Page 351
43 The Cauchy Identity......Page 359
44 Unitary Branching Rules......Page 369
45 The Involution Model for S_k......Page 373
46 Some Symmetric Algebras......Page 382
47 Gelfand Pairs......Page 387
48 Hecke Algebras......Page 396
49 The Philosophy of Cusp Forms......Page 409
50 Cohomology of Grassmannians......Page 440
References......Page 450
Index......Page 458
