Some results about the geometric Whittaker model

TL;DR

This paper explores conditions under which averaging functors on l-adic sheaves commute with Verdier duality in algebraic groups, providing new examples and applications in geometric representation theory.

Contribution

It introduces two new cases where averaging functors commute with Verdier duality, extending known results and confirming a conjecture for specific groups.

Findings

Averaging functors commute with Verdier duality under certain conditions.

Established new cases for sheaves on G-varieties with nondegeneracy conditions.

Reproved a theorem on local acyclicity of Fourier-Deligne kernels.

Abstract

Let G be an algebraic reductive group over a an algebraically closed field of positive characteristic. Choose a parabolic subgroup $P$ in $G$ and denote by $U$ its unipotent radical. Let $X$ be a $G$-variety. The purpose of this paper is to give two examples of a situation in which the functor of averaging of l-adic sheaves on $X$ with respect to a generic character of $U$ commutes with Verdier duality. In the first example we take $\chi$ to be an arbitrary $G$-variety and we prove the above property for all $\overline{P}$-equivariant sheaves on $X$ where $\overline{P}$ is an opposite parabolic subgroup assuming $\chi$ satisfies a strong nondegeneracy condition (such a $\chi$ exists for some but not all choices of $P$). In the case when $P$ is a Borel subgroup it is enough to require that the sheaf in question is $\overline{U}$ equivariant where $\overline{U}$ is the unipotent radical of $\overline{P}$. In the second example we take $X = G$ where $G$ acts by left translations and we prove the corresponding result when $P$ is a Borel subgroup for sheaves equivariant under the adjoint action of $G$ (the latter result was conjectured by B. C. Ngo who proved it for $G = GL(n)$). As an application we reprove a theorem of N. Katz and G. Laumon about local acyclicity of the kernel of the Fourier-Deligne transform.