  This article concerns properties of mixed $\ell$-adic complexes on varieties
over finite fields, related to the action of the Frobenius automorphism. We
establish a fiberwise criterion for the semisimplicity and Frobenius
semisimplicity of the direct image complex under a proper morphism of varieties
over a finite field. We conjecture that the direct image of the intersection
complex on the domain is always semisimple and Frobenius semisimple; this
conjecture would imply that a strong form of the decomposition theorem of
Beilinson-Bernstein-Deligne-Gabber is valid over finite fields. We prove our
conjecture for (generalized) convolution morphisms associated with partial
affine flag varieties for split connected reductive groups over finite fields,
and we prove allied Frobenius semisimplicity results for the intersection
cohomology groups of twisted products of Schubert varieties. We offer two
proofs for these results: one is based on the paving by affine spaces of the
fibers of certain convolution morphisms, the other involves a new schematic
theory of "big cells" adapted to partial affine flag varieties, and combines
Deligne's theory of weights with a suitable contracting $\mathbb G_m$-action on
those big cells. Both proofs rely on our general result that the intersection
complex of the image of a proper map of varieties over a finite field is a
direct summand of the direct image of the intersection complex of the domain.
With suitable reformulations, the main results are valid over any algebraically
closed ground field.
