## Introduction to the Representation Theory of Compact and Locally Compact GroupsBecause of their significance in physics and chemistry, representation of Lie groups has been an area of intensive study by physicists and chemists, as well as mathematicians. This introduction is designed for graduate students who have some knowledge of finite groups and general topology, but is otherwise self-contained. The author gives direct and concise proofs of all results yet avoids the heavy machinery of functional analysis. Moreover, representative examples are treated in some detail. |

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### Contents

Compact groups and Haar measures p | 3 |

Representations general constructions | 13 |

A geometrical application | 21 |

Finitedimensional representations of compact groups | 29 |

Decomposition of the regular representation | 40 |

Convolution Plancherel formula Fourier inversion | 53 |

Characters and group algebras | 63 |

Induced representations and FrobeniusWeil reciprocity | 78 |

l2 Invariant measures on locally compact groups | 117 |

l3 Continuity properties of representations | 128 |

l4 Representations of G and of LG | 144 |

l6 Discrete series of locally compact groups | 151 |

l7 The discrete series of Sl2K | 164 |

l8 The principal series of Sl2R | 172 |

l9 Decomposition along a commutative subgroup | 179 |

Type I groups | 187 |

Tannaka duality | 90 |

Groups with few finitedimensional representations lll | 111 |

2l Getting near an abstract Plancherel formula | 194 |

### Common terms and phrases

Banach space canonical Cc(G character circle group closed subgroup coefficient commutative compact subgroup conjugate continuous functions convergence convolution Corollary countable decomposition defined definition denote dimension discrete series dm(x dual eigenvalue End(H End(V equivalent factor representation finite dimensional representation follows formula function f G-morphism group G Haar measure hence hermitian Hilbert space Hilbert sum Hilbert-Schmidt operator homomorphism implies integral invariant measure invariant subspaces irreducible representation isomorphism isotypical component kernel left regular representation left translations Let G Lie groups locally compact group matrices multiplication neighbourhood Neumann algebra neutral element non-zero orthonormal basis particular Plancherel polynomials Proof Proposition prove regular representation representation of G right regular representation right translates rotation s e G scalar operator scalar product Schur's lemma self-adjoint space H subset tc(s theorem topology unimodular unitary irreducible representation unitary representation vector space von Neumann algebra