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. Namely, in the first example we take X to be an arbitrary G-variety and we prove the above property for all $\oU$-equivariant sheaves on X where $\oU$ 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)). Analogous results hold also for D-modules when k is replaced by the field of complex numbers. 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.