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 $mathrm{GL}_n(F)$ this implies that a smooth admissible representation $pi$ has a generalized Whittaker model $mathcal{W}_{mathcal{O}}(pi)$ corresponding to a nilpotent coadjoint orbit $mathcal{O}$ if and only if $mathcal{O}$ lies in the (closure of) the wave-front set $mathrm{WF}(pi)$. Previously this was only known to hold for $F$ non-archimedean and $mathcal{O}$ maximal in $mathrm{WF}(pi)$, see [MW87]. We also express $mathcal{W}_{mathcal{O}}(pi)$ as an iteration of a version of the Bernstein-Zelevinsky derivatives [BZ77,AGS15a]. This enables us to extend to $mathrm{GL_n}(mathbb{R})$ and $mathrm{GL_n}(mathbb{C})$ several further results from [MW87] on the dimension of $mathcal{W}_{mathcal{O}}(pi)$ and on the exactness of the generalized Whittaker functor.