GENERALIZED SPHERICAL FUNCTIONS ON REDUCTIVE p-ADIC GROUPS

Abstract. Generalized spherical functions on a reductive p-adic group G are eigenfunctions for the action of the Bernstein center, which satisfy a transformation property for the action of a maximal parabolic subgroup of G. In this paper we show that spaces of generalized spherical functions are finite dimensional. We compute dimensions of spaces of generalized spherical functions on a Zariski open dense set of infinitesimal characters. As a consequence, we get that on that Zariski open dense set of infinitesimal characters, the dimension of a space of generalized spherical functions is constant on each connected component of infinitesimal characters. 1