Generalized Whittaker models and the Bernstein center

Abstract

Let F be a non-Archimedean local field and G the group of F -points of a connected reductive algebraic group defined over F . Let B be a minimal F -parabolic subgroup of G with Levi component T and unipotent radical U . We let ξ be a smooth character of U, nondegenerate in a certain sense. We consider the theory of smooth ξ-Whittaker functions on G via the structure of the representation c -Ind ξ of G which is compactly induced by ξ. We prove a finiteness result for these generalized Whittaker models, corresponding to the uniqueness property for classical Whittaker models on quasi-split groups. In many cases, including the one where G is quasi-split, we describe the ring of G -endomorphisms of c -Ind ξ in terms of the Bernstein Center. This has consequences for the structure of the category of smooth representations of G . It gives an algebraic technique for finding improved (and more general) versions of results on classical Whittaker functions originally obtained by more analytic methods.