  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.
