Generalized and degenerate Whittaker models

TL;DR

This paper develops a unified framework for generalized and degenerate Whittaker models over local fields, establishing new epimorphisms, definitions, and connections to automorphic forms, extending classical results to archimedean cases.

Contribution

It constructs epimorphisms between Whittaker models, provides choice-free definitions, and extends known results to archimedean fields for GL(n).

Findings

Constructed epimorphisms between Whittaker models.

Extended dimension and exactness results to archimedean fields.

Connected Whittaker models to wave-front sets and automorphic coefficients.

Abstract

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.