On Ginzburg-Kaplan gamma factors and Bessel-Speh functions for finite general linear groups

TL;DR

This paper introduces a new finite field construction of gamma factors for pairs of irreducible representations of general linear groups, linking Bessel functions and Kloosterman sums, and proving their multiplicativity.

Contribution

It provides a finite field analog of doubling constructions for gamma factors, connecting Bessel functions of Speh representations with Kloosterman sums, and proves their multiplicativity.

Findings

New construction of tensor product gamma factors for finite general linear groups.

Established a relation between Bessel functions and Kloosterman sums.

Proved the multiplicativity of twisted matrix Kloosterman sums.

Abstract

We give a new construction of tensor product gamma factors for a pair of irreducible representations of $\operatorname{GL}_c\left(\mathbb{F}_q\right)$ and $\operatorname{GL}_k\left(\mathbb{F}_q\right)$. This construction is a finite field analog of a construction of doubling type due to Kaplan in the local field case and due to Ginzburg in the global case, and it only assumes that one of the representations in question is generic. We use this construction to establish a relation between special values of Bessel functions attached to Speh representations of generic principal series representations and twisted matrix Kloosterman sums. Using this relation, we establish the multiplicativity identity of twisted matrix Kloosterman sums.