On gamma factors for representations of finite general linear groups

TL;DR

This paper defines and studies the Shahidi gamma factor for irreducible generic representations of finite general linear groups, establishing its properties, relations, and applications to converse theorems and explicit formulas.

Contribution

It introduces the Shahidi gamma factor for finite groups, proves its multiplicativity, relates it to existing gamma factors, and applies it to converse theorems and explicit Bessel function formulas.

Findings

Shahidi gamma factor is multiplicative.

Relation established between Shahidi and Jacquet--Piatetski-Shapiro--Shalika gamma factors.

Explicit formulas derived for Bessel function values.

Abstract

We use the Langlands--Shahidi method in order to define the Shahidi gamma factor for a pair of irreducible generic representations of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$ and $\operatorname{GL}_m\left(\mathbb{F}_q\right)$. We prove that the Shahidi gamma factor is multiplicative and show that it is related to the Jacquet--Piatetski-Shapiro--Shalika gamma factor. As an application, we prove a converse theorem based on the absolute value of the Shahidi gamma factor, and improve the converse theorem of Nien. As another application, we give explicit formulas for special values of the Bessel function of an irreducible generic representation of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$.