## Representations of Compact Lie GroupsThis book is based on several courses given by the authors since 1966. It introduces the reader to the representation theory of compact Lie groups. We have chosen a geometrical and analytical approach since we feel that this is the easiest way to motivate and establish the theory and to indicate relations to other branches of mathematics. Lie algebras, though mentioned occasionally, are not used in an essential way. The material as well as its presentation are classical; one might say that the foundations were known to Hermann Weyl at least 50 years ago. Prerequisites to the book are standard linear algebra and analysis, including Stokes' theorem for manifolds. The book can be read by German students in their third year, or by first-year graduate students in the United States. Generally speaking the book should be useful for mathematicians with geometric interests and, we hope, for physicists. At the end of each section the reader will find a set of exercises. These vary in character: Some ask the reader to verify statements used in the text, some contain additional information, and some present examples and counter examples. We advise the reader at least to read through the exercises. |

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### Contents

Lie Groups and Lie Algebras | 1 |

2 LeftInvariant Vector Fields and OneParameter Groups | 11 |

3 The Exponential Map | 22 |

4 Homogeneous Spaces and Quotient Groups | 30 |

5 Invariant Integration | 40 |

6 Clifford Algebras and Spinor Groups | 54 |

Elementary Representation Theory | 64 |

1 Representations | 65 |

8 The Complexification of Compact Lie Groups | 151 |

The Maximal Torus of a Compact Lie Group | 157 |

2 Consequences of the Conjugation Theorem | 164 |

3 The Maximal Tori and Weyl Groups of the Classical Groups | 169 |

4 Cartan Subgroups of Nonconnected Compact Groups | 176 |

Root Systems | 183 |

2 Roots and Weyl Chambers | 189 |

3 Root Systems | 197 |

2 Semisimple Modules | 72 |

3 Linear Algebra and Representations | 74 |

4 Characters and Orthogonality Relations | 77 |

5 Representations of SU2 SO3 U2 and O3 | 84 |

6 Real and Quaternionic Representations | 93 |

7 The Character Ring and the Representation Ring | 102 |

8 Representations of Abelian Groups | 107 |

9 Representations and Lie Algebras | 111 |

10 The Lie Algebra sl 2 C | 115 |

Representative Functions | 123 |

2 Some Analysis on Compact Groups | 129 |

3 The Theorem of Peter and Weyl | 133 |

4 Applications of the Theorem of Peter and Weyl | 136 |

5 Generalizations of the Theorem of Peter and Weyl | 138 |

6 Induced Representations | 143 |

7 TannakaKrem Duality | 146 |

4 Bases and Weyl Chambers | 202 |

5 Dynkin Diagrams | 209 |

6 The Roots of the Classical Groups | 216 |

7 The Fundamental Group the Center and the Stiefel Diagram | 223 |

8 The Structure of the Compact Groups | 232 |

Irreducible Characters and Weights | 239 |

2 The Dominant Weight and the Structure of the Representation Ring | 249 |

3 The Multiplicities of the Weights of an Irreducible Representation | 257 |

4 Representations of Real or Quaternionic Type | 261 |

5 Representations of the Classical Groups | 265 |

6 Representations of the Spinor Groups | 278 |

7 Representations of the Orthogonal Groups | 292 |

299 | |

305 | |

307 | |

### Common terms and phrases

adjoint representation automorphism basis bijection bundle called canonical Cartan subgroup closed subgroup compact connected Lie compact Lie group complex compute conjugation connected Lie group contained corresponding defined Definition denote diagram diffeomorphism differentiable direct sum dominant weight element exponential map finite finite-dimensional follows formula fundamental g e G G-invariant G-module given GL(n hence homomorphism induces injective inner product integral forms invariant inverse Irr(G irreducible characters irreducible modules irreducible representations isomorphism kernel left-invariant Lemma Let G Lie algebra Lie group G linear manifold matrix maximal torus morphism multiplication one-parameter group operation orthogonal polynomials positive roots PROOF Proposition quaternionic representation of G representation ring representative functions resH root system self-conjugate semisimple Show SO(n Sp(n Spin(2n structure SU(n submodules subspace summand surjective symmetric theorem theory topology trivial unitary vector space Weyl chamber Weyl group

### Popular passages

Page 303 - Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras. New York