## Representation theory and automorphic functions |

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### Contents

Homogeneous Spaces with a Discrete Stability Group | 1 |

Additive and Multiplicative Characters of if 127 5 | 4 |

2 Representations of a Group G Induced by a Discrete | 17 |

Copyright | |

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algebra analytic function arbitrary assume automorphic forms compact horosphere compact set compact space complex numbers compute Consequently continuous function continuous series converges corresponding cosets decomposition defined denote disconnected field discrete series discrete subgroup eigenvalues elliptic elements equation equivalent everywhere dense expression finite function finite number following form following formula follows immediately Fourier transform function on Q functions p(g fundamental domain Gamma-function group G group of adeles Hence homogeneous space horospheres hyperbolic elements ideles integral irreducible representations irreducible unitary representations isomorphic kernel Lemma linear locally compact matrices of order Mellin transform multiplicative character obtain operators T(g orthogonal principal adeles proof rational numbers real numbers representation of G representation operator representation T(g satisfy the following Schwartz-Bruhat function sequence signr space H space of functions subset subspace supplementary series T+(g topological trace formula unimodular matrices unit element upper half-plane vector