Some results about geometric Whittaker model

Let G be an algebraic reductive group over a 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 χ:U→ G a commutes with Verdier duality. Namely, in the first example we take X to be an arbitrary G -variety and we prove the above property for all U -equivariant sheaves on X where U is the unipotent radical of an opposite parabolic subgroup; in the second example we take X = G and we prove the corresponding result for sheaves which are equivariant under the adjoint action (the latter result was conjectured by B. C. Ngo who proved it for G = GL ( n )). As an application of the proof of the first statement we reprove a theorem of N. Katz and G. Laumon about local acyclicity of the kernel of the Fourier–Deligne transform.