On Certain Degenerate Whittaker Models for Cuspidal Representations of ${\mathrm{GL}}_{k\cdot n}\left({\mathbb{F}}_q\right)$

Abstract

Let $\pi$ be an irreducible cuspidal representation of ${\mathrm{GL}}_{kn}\left({\mathbb{F}}_q\right)$. Assume that $\pi ={\pi }_{\theta }$, corresponds to a regular character $\theta$ of ${\mathbb{F}}_{q^{kn}}^{\ast }$. We consider the twisted Jacquet module of $\pi$ with respect to a non-degenerate character of the unipotent radical corresponding to the partition $(n^k)$ of $kn$. We show that, as a ${\mathrm{GL}}_n\left({\mathbb{F}}_q\right)$-representation, this Jacquet module is isomorphic to ${\pi }_{\theta {\upharpoonright }_{{\mathbb{F}}_n^{\ast }}}\otimes {\mathrm{St}}^{k-1}$, where $\mathrm{St}$ is the Steinberg representation of ${\mathrm{GL}}_n\left({\mathbb{F}}_q\right)$. This generalizes a theorem of D. Prasad, who considered the case $k=2$. We prove and rely heavily on a formidable identity involving $q$-hypergeometric series and linear algebra.

27 pages. Comments are welcome