Dimension formula for the twisted Jacquet module of a cuspidal representation of $\GL(2n,\mathbb{F}_q)$

TL;DR

This paper computes the dimension of a specific twisted Jacquet module for cuspidal representations of the general linear group over a finite field, providing explicit formulas based on matrix rank.

Contribution

It introduces a formula for the dimension of twisted Jacquet modules of cuspidal representations of GL(2n, F) based on matrix rank.

Findings

Derived explicit dimension formulas for twisted Jacquet modules

Connected module dimensions to matrix rank and representation properties

Enhanced understanding of cuspidal representation structures

Abstract

Let $F$ be a finite field and $G=\GL(2n,F)$. In this paper, we calculate the dimension of the twisted Jacquet module $\pi_{N,\psi_{A}}$ where $A\in \M(n,F)$ is a rank $k$ matrix and $\pi$ is an irreducible cuspidal representation of $G$.