Finite period vectors and Gauss sums

TL;DR

This paper explores the relationships between various gamma factors and sums associated with representations of general linear groups over finite fields, connecting them to local field gamma factors and the Langlands correspondence.

Contribution

It explicitly computes and relates gamma factors over finite fields to those over local fields, establishing new connections via Deligne--Kazhdan theory and the local Langlands correspondence.

Findings

Explicit formulas for gamma factors in terms of Gauss sums

Equivalence of gamma factors over finite and local fields for tamely ramified representations

Product formulas linking gamma factors to periods and vectors

Abstract

We study four sums including the Jacquet--Piatetski-Shapiro--Shalika, Flicker, Bump--Friedberg, and Jacquet--Shalika sums associated to irreducible cuspidal representations of general linear groups over finite fields. By computing explicitly, we relate Asai and Bump--Friedberg gamma factors over finite fields to those over nonarchimedean local fields through level zero supercuspidal representation. Via Deligne--Kazhdan close field theory, we prove that exterior square and Bump--Friedberg gamma factors agree with corresponding Artin gamma factors of their associated tamely ramified representations through local Langlands correspondence. We also deduce product formulae for Asai, Bump--Friedberg, and exterior square gamma factors in terms of Gauss sums. By combining these results, we examine Jacquet--Piatetski-Shapiro--Shalika, Flicker--Rallis, Jacquet--Shalika, and Friedberg--Jacquet periods and vectors and their connections to Rankin-Selberg, Asai, exterior square, and Bump-Friedberg gamma factors, respectively.