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 $\mathbb{F}{q^{kn}}^{*}$. 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 $\pi{\theta \upharpoonright_{\mathbb{F}n^*}} \otimes \mathrm{St}^{k-1}$, 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.