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Part Ⅰ I: Compact Groups 3

1 Haar Measure 3

2 Schur Orthogonality 6

3 Compact Operators 17

4 The Peter-Weyl Theorem 21

Part Ⅱ: Lie Group Fundamentals 29

5 Lie Subgroups of GL(n, C) 29

6 Vector Fields 36

7 Left-Invariant Vector Fields 41

8 The Exponential Map 46

9 Tensors and Universal Properties 50

10 The Universal Enveloping Algebra 54

11 Extension of Scalars 58

12 Representations of sl(2, C) 62

13 The Universal Cover 69

14 The Local Frobenius Theorem 79

15 Tori 86

16 Geodesies and Maximal Tori 94

17 Topological Proof of Cartan's Theorem 107

18 The Weyl Integration Formula 112

19 The Root System 117

20 Examples of Root Systems 127

21 Abstract Weyl Groups 136

22 The Fundamental Group 146

23 Semisimple Compact Groups 150

24 Highest-Weight Vectors 157

25 The Weyl Character Formula 162

26 Spin 175

27 Complexification 182

28 Coxeter Groups 189

29 The Iwasawa Decomposition 197

30 The Bruhat Decomposition 205

31 Symmetric Spaces 212

32 Relative Root Systems 236

33 Embeddings of Lie Groups 257

Part Ⅲ: Topics 275

34 Mackey Theory 275

35 Characters of GL(n, C) 284

36 Duality between Sk and GL(n, C) 289

37 The Jacobi-Trudi Identity 297

38 Schur Polynomials and GL(n, C) 308

39 Schur Polynomials and Sk 315

40 Random Matrix Theory 321

41 Minors of Toeplitz Matrices 331

42 Branching Formulae and Tableaux 339

43 The Cauchy Identity 347

44 Unitary Branching Rules 357

45 The Involution Model for Sk 361

46 Some Symmetric Algebras 370

47 Gelfand Pairs 375

48 Hecke Algebras 384

49 The Philosophy of Cusp Forms 397

50 Cohomology of Grassmannians 428

References 438

Index 446