## Lie GroupsThis 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. One approach to this problem is to emphasize the beautiful representation theory of compact groups, and indeed this book can be used for a course of this type if after Chapter 25 one skips ahead to Part III. But I did not want to omit important topics such as the Bruhat decomposition and the theory of symmetric spaces. For these subjects, compact groups are not sufficient. Part I covers standard general properties of representations of compact groups (including Lie groups and other compact groups, such as finite or p adic ones). These include Schur orthogonality, properties of matrix coefficients and the Peter-Weyl Theorem. |

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### Contents

10 | 33 |

The Local Frobenius Theorem | 79 |

Tori | 86 |

Geodesics and Maximal Tori | 94 |

Topological Proof of Cartans Theorem | 107 |

The Weyl Integration Formula | 112 |

The Root System | 117 |

Examples of Root Systems | 127 |

Topics | 273 |

Mackey Theory | 275 |

Characters of GLn C | 284 |

Duality between Sk and GLn C | 289 |

The JacobiTrudi Identity | 297 |

Schur Polynomials and GLn C | 308 |

Schur Polynomials and Sk | 315 |

Random Matrix Theory | 321 |

Abstract Weyl Groups | 136 |

The Fundamental Group | 146 |

Semisimple Compact Groups | 150 |

HighestWeight Vectors | 157 |

The Weyl Character Formula | 162 |

Spin | 175 |

Complexification | 182 |

Coxeter Groups | 189 |

The Iwasawa Decomposition | 197 |

The Bruhat Decomposition | 205 |

Symmetric Spaces | 212 |

Relative Root Systems | 236 |

Embeddings of Lie Groups | 257 |

Minors of Toeplitz Matrices | 331 |

Branching Formulae and Tableaux | 339 |

The Cauchy Identity | 347 |

Unitary Branching Rules | 357 |

The Involution Model for Sk | 361 |

Some Symmetric Algebras | 370 |

Gelfand Pairs | 375 |

Hecke Algebras | 384 |

The Philosophy of Cusp Forms | 397 |

Cohomology of Grassmannians | 428 |

438 | |

446 | |

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### Common terms and phrases

Abelian Aroot assume automorphism bilinear Chapter commutative compact connected Lie compact group complexification conjugacy classes conjugate connected Lie group contained corresponding cuspidal cuspidal representations decomposition defined denote diagonal dimension dēt double coset Dynkin diagram eigenvalues element embedding Exercise finite finite-dimensional geodesic GL(k GL(n group G Haar measure Hermitian homomorphism identity induced inner product integral invariant involution irreducible character irreducible representation isomorphic Lemma Let G Lie algebra Lie subgroup linear manifold Matn matrix coefficients maximal torus multiplication neighborhood nonzero orthogonal parabolic subgroup parametrized partition path polynomial positive definite positive roots Proof prove R-module representation of G restriction ring root system Satake diagram semisimple SO(n span subgroup of G subset subspace symmetric spaces tangent Theorem trivial unipotent unique vector field vector space Weyl group zero