Generalized Whittaker models and the Bernstein center

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.