Representation Theory and Automorphic Functions

algebra arbitrary automorphic forms compact set compact space complex numbers compute Consequently continuous function converges corresponding cosets decomposition defined denote disconnected field discrete series discrete subgroup duality theorem eigenvalues elliptic elements equation everywhere dense expression finite number following form following formula follows immediately Fourier transform functions g fundamental domain G₁ group G h+(p H₁ Hence homogeneous space horospheres hyperbolic elements ideles integral invariant irreducible representations irreducible unitary representations isomorphic kernel L₂(X lemma linear locally compact matrices of order Mellin transform obtain operators T(g p-adic P(Yo principal adeles proof rational numbers real numbers representation of G representation operator representation T(g satisfy the following Schwartz-Bruhat function space H space of functions subset supplementary series topological trace formula unimodular matrices unit element upper half-plane vector zero πρ