This book aims to be a course in Lie groups that can be covered in one year with a group of good graduate students. I have attempted to address a problem that anyone teaching this subject must have, which is that the amount of essential material is too much to cover.

Preface

Part I: Compact Groups

 1 Haar Measure

 2 Schur Orthogonality

 3 Compact Operators

 4 The Peter-Weyl Theorem

Part II: Lie Group Fundamentals

 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 S1(2, C)

 13 The Universal Cover

 14 The Local Frobenius Theorem

 15 Tori

 16 Geodesics and Maximal Tori

 17 Topological Proof of Cartan's Theorem

 18 The Weyl Integration Formula

 19 The Root System

 20 Examples of Root Systems

 21 Abstract Weyl Groups

 22 The Fundamental Group

 23 Semisimple Compact Groups

 24 Highest-Weight Vectors

 25 The Weyl Character Formula

 26 Spin

 27 Complexification

 28 Coxeter Groups

 29 The Iwasawa Decomposition

 30 The Bruhat Decomposition

 31 Symmetric Spaces

 32 Relative Root Systems

 33 Embeddings of Lie Groups

Part III: Topics

 34 Mackey Theory

 35 Characters of GL(n,C)

 36 Duality between Sk and GL(n,C)

 37 The Jacobi-Trudi Identity

 38 Schur Polynomials and GL(n,C)

 39 Schur Polynomials and Sk

 40 Random Matrix Theory

 41 Minors of Toeplitz Matrices

 42 Branching Formulae and Tableaux...

 43 The Cauchy Identity

 44 Unitary Branching Rules

 45 The Involution Model for Sk

 46 Some Symmetric Algebras

 47 Gelfand Pairs

 48 Hecke Algebras

 49 The Philosophy of Cusp Forms

 50 Cohomology of Grassmannians

References

Index