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

Abstract

We find a recursive expression for the Bessel function of S. I. Gelfand for irreducible generic representations of GL n (F q ). 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.

On gamma factors for representations of finite general linear groups

“…Remark 4.3. In [27], a vast generalization of the method in the proof of Theorem 4.5 is used in order to find formulas for…”

“…However, [27] relies on the results of [26], which in turn rely on the local Langlands correspondence. The proof given here does not rely on such results.…”

“…If one naively extends the definition γ(π × σ, ψ) using the approach suggested above, it is not clear that the gamma factor would be multiplicative. Both of these difficulties need to be resolved for applications as in [27].…”