On the spherical automorphic spectrum of a split reductive group

Date(s) : 03/11/2015
14 h 00 min - 15 h 00 min

Let G be connected reductive group over a number field F and let A denote the ring of adèles of F.
Langlands gave a decomposition of the space of square-integrable functions on the quotient G(F)\G(A) (modulo the center) as a direct sum in terms of equivalence classes of cuspidal data. Assume G is split and let K be a maximal compact subgroup of G(A) with factors in each finite place equal to a hyperspecial maximal compact subgroup. The summand corresponding to the cuspidal datum (T,1), where T is a maximal split torus and 1 the trivial character of T(F)\T(A), contains a non-empty subspace consisting of K-fixed vectors. It was described by several authors in many special cases.
In a joint work with E. Opdam and V. Heiermann, we provide a uniform description of this space of K-fixed vectors which holds for every split reductive group over F.

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