On Jacobi sums arising from the classical doubling method

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.