Lie Groups

Part I: Compact Topological Groups.- 1 Haar Measure.- 2 Schur Orthogonality.- 3 Compact Operators.- 4 The Peter-Weyl Theorem.- Part II: Compact Lie Groups.- 5 Lie Subgroups of GL( n , C ).- 6 Vector Fields.- 7 Left-Invariant Vector Fields.- 8 The Exponential Map.- 9 Tensors and Universal Properties.- 10 The Universal Enveloping Algebra.- 11 Extension of Scalars.- 12 Representations of sl(2, C ).- 13 The Universal Cover.- 14 The Local Frobenius Theorem.- 15 Tori.- 16 Geodesics and Maximal Tori.- 17 The Weyl Integration Formula.- 18 The Root System.- 19 Examples of Root Systems.- 20 Abstract Weyl Groups.- 21 Highest Weight Vectors.- 22 The Weyl Character Formula.- 23 The Fundamental Group.- Part III: Noncompact Lie Groups.- 24 Complexification.- 25 Coxeter Groups.- 26 The Borel Subgroup.- 27 The Bruhat Decomposition.- 28 Symmetric Spaces.- 29 Relative Root Systems.- 30 Embeddings of Lie Groups.- 31 Spin.- Part IV: Duality and Other Topics.- 32 Mackey Theory.- 33 Characters of GL( n , C ).- 34 Duality between Sk and GL( n , C ).- 35 The Jacobi-Trudi Identity.- 36 Schur Polynomials and GL( n , C ).- 37 Schur Polynomials and Sk .- 38 The Cauchy Identity.- 39 Random Matrix Theory.- 40 Symmetric Group Branching Rules and Tableaux.- 41 Unitary Branching Rules and Tableaux.- 42 Minors of Toeplitz Matrices.- 43 The Involution Model for Sk .- 44 Some Symmetric Alegras.- 45 Gelfand Pairs.- 46 Hecke Algebras.- 47 The Philosophy of Cusp Forms.- 48 Cohomology of Grassmannians.- Appendix: Sage.- References.- Index.