On Bessel functions for representations of finite general linear groups

Abstract: In the 60s, Gel'fand defined a Bessel function associated with irreducible generic representations of GL(n, F_q). The Bessel function plays an important role in the representation theory of GL(n, F_q), and served as a key ingredient in the proof of the finite field analog of Jacquet's conjecture. However, the computation of explicit values of the Bessel function is difficult. In this talk, I'll explain a method based on gamma factors to compute values of the Bessel function. I will also explain a relation between some values of the Bessel function and Katz's generalized Kloosterman sheaves.