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 GL(n)(F) this implies that a smooth admissible representation p has a generalized Whittaker model WO(pi) corresponding to a nilpotent coadjoint orbit O if and only if O lies in the (closure of) the wave-front set WF(p). Previously this was only known to hold for F non-archimedean and O maximal in WF(p), see Moeglin andWaldspurger [Modeles de Whittaker degeneres pour des groupes p-adiques, Math. Z. 196 (1987), 427-452]. We also express WO(pi) as an iteration of a version of the Bernstein-Zelevinsky derivatives [Bernstein and Zelevinsky, Induced representations of reductive p-adic groups. I., Ann. Sci. Ec. Norm. Super. (4) 10 (1977), 441-472; Aizenbud et al., Derivatives for representations of GL(n, R) and GL(n, C), Israel J. Math. 206 (2015), 1-38]. This enables us to extend to GL(n)(R) and GL(n)(C) several further results by Moeglin and Waldspurger on the dimension of WO(pi) and on the exactness of the generalized Whittaker functor.