Frobenius semisimplicity for convolution morphisms

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.