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

TL;DR

This paper investigates certain degenerate Whittaker models for cuspidal representations of general linear groups over finite fields, generalizing previous results and revealing explicit structures of twisted Jacquet modules.

Contribution

It generalizes Prasad's theorem for $k=2$ to arbitrary $k$, describing the structure of twisted Jacquet modules for cuspidal representations of $ ext{GL}_{kn}( ext{F}_q)$.

Findings

Jacquet module isomorphic to a tensor product involving a restricted character and Steinberg representation

Generalization of Prasad's theorem from $k=2$ to arbitrary $k$

Key identity involving $q$-hypergeometric series and linear algebra

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.