Lie Groups

This book is intended for a one-year graduate course on Lie groups and Lie algebras. The book goes 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. The book is organized to allow different paths through the material depending on one's interests. This second edition has substantial new material, including improved discussions of underlying principles, streamlining of some proofs, and many results and topics that were not in the first edition.

For compact Lie groups, the book covers the Peter-Weyl theorem, Lie algebra, conjugacy of maximal tori, the Weyl group, roots and weights, Weyl character formula, the fundamental group and more. The book continues with the study of complex analytic groups and general noncompact Lie groups, covering the Bruhat decomposition, Coxeter groups, flag varieties, symmetric spaces, Satake diagrams, embeddings of Lie groups and spin. Other topics that are treated are symmetric function theory, the representation theory of the symmetric group, Frobenius-Schur duality and GL(n) × GL(m) duality with many applications including some in random matrix theory, branching rules, Toeplitz determinants, combinatorics of tableaux, Gelfand pairs, Hecke algebras, the "philosophy of cusp forms" and the cohomology of Grassmannians. An appendix introduces the reader to the use of Sage mathematical software for Lie group computations.

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.