On the Godement-Jacquet functional equation for finite matrix monoids

Abstract

In previous work with Harman and Snowden, we found explicit formulas for units of rank ideals of the matrix monoid algebra $\mathbb{K}\left[\operatorname{Mat}_n\left(\mathbb{F}_q\right)\right]$, where $\mathbb{K}$ is an algebraically closed field with characteristic not dividing $q$. In this work, we use these explicit formulas to establish a regularized Godement--Jacquet functional equation valid for irreducible representations of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$ and of the algebra $\mathbb{K}\left[\operatorname{Mat}_n\left(\mathbb{F}_q\right)\right]$. Using this functional equation we give an explicit expression for primitive central idempotents corresponding to irreducible representations of $\mathbb{K}\left[\operatorname{Mat}_n\left(\mathbb{F}_q\right)\right]$, where the characteristic of $\mathbb{K}$ does not divide $\left|\operatorname{GL}_n\left(\mathbb{F}_q\right)\right|$. Interestingly, the coefficients in this formula are closely related to degenerate non-abelian Gauss sums.