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
$\text{GL}_{n}(\mathbb{F})$
this implies that a smooth admissible representation
$\unicode[STIX]{x1D70B}$
has a generalized Whittaker model
${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$
corresponding to a nilpotent coadjoint orbit
${\mathcal{O}}$
if and only if
${\mathcal{O}}$
lies in the (closure of) the wave-front set
$\operatorname{WF}(\unicode[STIX]{x1D70B})$
. Previously this was only known to hold for
$\mathbb{F}$
non-archimedean and
${\mathcal{O}}$
maximal in
$\operatorname{WF}(\unicode[STIX]{x1D70B})$
, see Moeglin and Waldspurger [Modeles de Whittaker degeneres pour des groupes p-adiques, Math. Z. 196 (1987), 427–452]. We also express
${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$
as an iteration of a version of the Bernstein–Zelevinsky derivatives [Bernstein and Zelevinsky, Induced representations of reductive p-adic groups. I., Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), 441–472; Aizenbud et al., Derivatives for representations of
$\text{GL}(n,\mathbb{R})$
and
$\text{GL}(n,\mathbb{C})$
, Israel J. Math. 206 (2015), 1–38]. This enables us to extend to
$\text{GL}_{n}(\mathbb{R})$
and
$\text{GL}_{n}(\mathbb{C})$
several further results by Moeglin and Waldspurger on the dimension of
${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$
and on the exactness of the generalized Whittaker functor.