## Lie GroupsThis book is intended for a one year graduate course on Lie groups and Lie algebras. The author proceeds 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. For compact Lie groups, the Peter-Weyl theorem, conjugacy of maximal tori (two proofs), Weyl character formula and more are covered. The book continues with the study of complex analytic groups, then general noncompact Lie groups, including the Coxeter presentation of the Weyl group, the Iwasawa and Bruhat decompositions, Cartan decomposition, symmetric spaces, Cayley transforms, relative root systems, Satake diagrams, extended Dynkin diagrams and a survey of the ways Lie groups may be embedded in one another. The book culminates in a "topics" section giving depth to the student's understanding of representation theory, taking the Frobenius-Schur duality between the representation theory of the symmetric group and the unitary groups as a unifying theme, with many applications in diverse areas such as random matrix theory, minors of Toeplitz matrices, symmetric algebra decompositions, Gelfand pairs, Hecke algebras, representations of finite general linear groups and the cohomology of Grassmannians and flag varieties. Daniel Bump is Professor of Mathematics at Stanford University. His research is in automorphic forms, representation theory and number theory. He is a co-author of GNU Go, a computer program that plays the game of Go. His previous books include Automorphic Forms and Representations (Cambridge University Press 1997) and Algebraic Geometry (World Scientific 1998). |

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### Contents

Haar Measure | 3 |

Schur Orthogonality | 6 |

Compact Operators | 17 |

The PeterWeyl Theorem | 21 |

Lie Group Fundamentals | 27 |

Lie Subgroups of GLnC | 29 |

Vector Fields | 36 |

LeftInvariant Vector Fields | 41 |

Coxeter Groups | 189 |

The Iwasawa Decomposition | 197 |

The Bruhat Decomposition | 205 |

Symmetric Spaces | 212 |

Relative Root Systems | 236 |

Embeddings of Lie Groups | 257 |

Topics | 273 |

Mackey Theory | 275 |

The Exponential Map | 46 |

Tensors and Universal Properties | 50 |

The Universal Enveloping Algebra | 54 |

Extension of Scalars | 58 |

Representations of sl2 C | 62 |

The Universal Cover | 69 |

The Local Frobenius Theorem | 79 |

Tori | 86 |

Geodesies and Maximal Tori | 94 |

Topological Proof of Cartans Theorem | 107 |

The Weyl Integration Formula | 112 |

The Root System | 117 |

Examples of Root Systems | 127 |

Abstract Weyl Groups | 136 |

The Fundamental Group | 146 |

Semisimple Compact Groups | 150 |

HighestWeight Vectors | 157 |

The Weyl Character Formula | 162 |

Spin | 175 |

Complexification | 182 |

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 |

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

analytic assume automorphism bilinear Chapter commutative compact connected Lie compact group complex representation complexification conjugacy classes conjugate connected Lie group contained corresponding cuspidal representations decomposition defined denote diagonal dimension double coset Dynkin diagram eigenvalues element embedding equal example exists finite finite-dimensional formula geodesic GL(fc GL(n group G Haar measure Hermitian homomorphism identity induced inner product integral intertwining operator invariant involution irreducible character irreducible representation isomorphic kernel Lemma Let G Lie algebra Lie subgroup Lie(G linear manifold matrix coefficient maximal torus multiplication neighborhood nonzero orthogonal parabolic subgroup parametrized partition path permutation polynomial positive definite positive roots Proof prove R-module representation of G restriction ring root system Satake diagram semisimple simple reflections simple roots span SU(n subgroup of G subset subspace symmetric spaces tangent Theorem topological trivial unipotent unique upper triangular vector field vector space Weyl group zero