Generalized and Degenerate Whittaker Models

We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized Whittaker model corresponding to a nilpotent orbit to any degenerate Whittaker model corresponding to the same orbit, and to certain degenerate Whittaker models corresponding to bigger orbits. We also give choice-free definitions of generalized and degenerate Whittaker models. Finally, we explain how our methods imply analogous results for Whittaker-Fourier coefficients of automorphic representations. For GLn(F) this implies that a smooth admissible representation π has a generalized Whittaker model WO(π) corresponding to a nilpotent coadjoint orbit O if and only if O lies in the (closure of) the wavefront set WF(π). Previously this was only known to hold for F nonarchimedean and O maximal in WF(π), see [MW87]. We also express WO(π) as an iteration of a version of the Bernstein-Zelevinsky derivatives [BZ77, AGS15a]. This enables us to extend to GLn(R) and GLn(C) several further results from [MW87] on the dimension of WO(π) and on the exactness of the generalized Whittaker functor.