## Real Productive Groups IReal Reductive Groups I is an introduction to the representation theory of real reductive groups. It is based on courses that the author has given at Rutgers for the past 15 years. It also had its genesis in an attempt of the author to complete a manuscript of the lectures that he gave at the CBMS regional conference at The University of North Carolina at Chapel Hill in June of 1981. This book comprises 10 chapters and begins with some background material as an introduction. The following chapters then discuss elementary representation theory; real reductive groups; the basic theory of (g, K)-modules; the asymptotic behavior of matrix coefficients; The Langlands Classification; a construction of the fundamental series; cusp forms on G; character theory; and unitary representations and (g, K)-cohomology. This book will be of interest to mathematicians and statisticians. |

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### Contents

1 | |

17 | |

Chapter 2 Real Reductive Groups | 41 |

Chapter 3 The Basic Theory of g KModules | 73 |

Chapter 4 The Asymptotic Behavior of Matrix Coefficients | 107 |

Chapter 5 The Langlands Classification | 137 |

Chapter 6 A Construction of the Fundamental Series | 173 |

Chapter 7 Cusp Forms on G | 225 |

Chapter 8 Character Theory | 289 |

Chapter 9 Unitary Representations and g KCohomology | 353 |

403 | |

411 | |

413 | |

### Common terms and phrases

6-stable a e Cl(A admissible assertion assume that G Cartan involution Cartan subalgebra Cartan subgroup chapter compact completes the proof contained corresponding define denote diffeomorphism direct sum eigenvalues elements equal exact sequence exists formula function functors g e G G-invariant g-module Harish-Chandra Hence highest weight Hilbert representation homomorphism induced infinitesimal character invariant measure irreducible g isomorphic K-finite K)-module Langlands Let F Let G Let H Let q Lie algebra Lie group module nilpotent non-zero notation orthonormal basis parabolic subalgebra parabolic subgroup polynomial positive constant positive roots prove real reductive group representation of G resp result now follows semi-norm semi-simple square integrable square integrable representations subalgebra of g subgroup of G submodule subspace system of positive Theorem theory U(gc unitary representation vector Weyl Weyl character formula