On Jacobi sums arising from the classical doubling method

TL;DR

This paper introduces non-abelian Jacobi sums linked to representations of classical and general linear groups over finite fields, expressing them via Gauss sums and analyzing their properties within the framework of the doubling method.

Contribution

It defines non-abelian Jacobi sums in the context of the doubling method and provides explicit formulas relating them to Gauss sums for various groups and characters.

Findings

Expressed non-abelian Jacobi sums in terms of Kondo's Gauss sums for GL groups.

Derived explicit formulas for classical groups involving Deligne--Lusztig data.

Proved that these Jacobi sums are constant on geometric Lusztig series.

Abstract

We define the notion of a non-abelian Jacobi sum $\mathcal{J}^{\mathrm{dbl}}\left(\pi, \chi\right)$ attached to an irreducible representation $\pi$ of a general linear group or a classical group over a finite field and a character $\chi$ of the multiplicative group of the finite field or its quadratic extension. These sums emerge in the study of the doubling method of Piatetski-Shapiro--Rallis and Lapid--Rallis. For general linear groups, we express these non-abelian Jacobi sums in terms of Kondo's non-abelian Gauss sums. For classical groups and for characters that are not conjugate-dual, we give an explicit formula for these non-abelian Jacobi sums in terms of Gauss sums attached to the Deligne--Lusztig data of the representation, and we prove that these Jacobi sums are constant on geometric Lusztig series. Our results rely on a multiplicativity result of non-abelian Jacobi sums obtained by Girsch--Zelingher.