Frobenius semisimplicity for convolution morphisms

This article concerns properties of mixed ℓ-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. As a crucial tool, we develop a new schematic theory of big cells for loop groups. With suitable reformulations, the main results are valid over any algebraically closed ground field.