On values of the Bessel function for generic representations of finite general linear groups

TL;DR

This paper derives recursive formulas for Bessel functions of generic representations of finite general linear groups, linking their special values to $L$-functions, sheaves, and polynomial roots, with applications in number theory.

Contribution

It introduces a recursive expression for Bessel functions of irreducible generic representations of finite GL(n), connecting their values to $L$-functions and sheaf theory, and explores their applications.

Findings

Special values of Bessel functions relate to $L$-functions and sheaves.

Certain polynomials with Bessel function coefficients have roots on the unit circle.

Bessel functions of base change representations relate via Dickson polynomials.

Abstract

We find a recursive expression for the Bessel function of S. I. Gelfand for irreducible generic representations of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$. We show that special values of the Bessel function can be realized as the coefficients of $L$-functions associated with exotic Kloosterman sums, and as traces of exterior powers of Katz's exotic Kloosterman sheaves. As an application, we show that certain polynomials, having special values of the Bessel function as their coefficients, have all of their roots lying on the unit circle. As another application, we show that special values of the Bessel function of the Shintani base change of an irreducible generic representation are related to special values of the Bessel function of the representation through Dickson polynomials.