On Jacobi sums arising from the classical doubling method

We define the notion of a non-abelian Jacobi sum ${\mathcal{J}}^{\mathrm{d}\mathrm{b}\mathrm{l}}(\pi ,\chi )$ 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.