Exterior square gamma factors for cuspidal representations of $\mathrm{GL}_n$: finite field analogs and level zero representations

TL;DR

This paper defines and analyzes exterior square gamma factors for cuspidal representations of finite general linear groups, connecting finite field analogs with local field representations via level zero methods.

Contribution

It introduces a new finite field analog of exterior square gamma factors for cuspidal representations and relates them to local field gamma factors through level zero representations.

Findings

Gamma factors expressed via Bessel functions and regular characters

Finite field gamma factors linked to local field gamma factors

Extension of Jacquet-Shalika and Matringe's frameworks

Abstract

We follow Jacquet-Shalika, Matringe and Cogdell-Matringe to define exterior square gamma factors for irreducible cuspidal representations of $\mathrm{GL}_n(\mathbb{F}_q)$. These exterior square gamma factors are expressed in terms of Bessel functions, or in terms of the regular characters associated with the cuspidal representations. We also relate our exterior square gamma factors over finite fields to those over local fields through level zero representations.