Godement-Jacquet gamma factors of distinguished representations of $\mathrm{GL}_n(\mathbb{F}_q)$

Abstract

Let $k$ be a finite field of characteristic $p$. In the 1960s, Kondo attached non-abelian Gauss sums to irreducible $\mathbb{C}$-representations of $\mathrm{GL}_n(k)$, and computed them in terms of Green parameters. On the other hand, the Godement-Jacquet functional equation in which they occur was established by Macdonald in the 1980s. We first revisit Macdonald's and Kondo's results with a different perspective, in the process of generalizing their constructions to representations with coefficients in $\mathbb{Z}[\sqrt{p}^{-1},\mu_p]$-algebras. Then, when $p$ is odd and $R$ is an algebraically closed field of characteristic different to $p$, our main result shows that the Godement-Jacquet gamma factor of a cuspidal irreducible $R$-representation, which is distinguished with respect to the subgroup fixed by a Galois or an inner involution, coincides with the sign of the associated period under the normalizer of this subgroup. Finally, we compute the gamma factors of these distinguished representations in terms of Green's and James' parametrizations of irreducible cuspidal $R$-representations.