Frobenius semisimplicity for convolution morphisms

Abstract

Abstract. 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.

The test function conjecture for parahoric local models

“…G split connected reductive. The following corollary is an immediate consequence of Lemma 3.1, and is due to [LS97, Prop 4.6] for connected reductive groups (see also [Fal03,dHL]).…”

Smith-Treumann theory and the linkage principle

“…Then the arguments in [dCHL,Proofs of Lemma 3.6.3 and Proposition 3.6.4] show that we have a direct product decomposition…”

“…(Here, (2) follows from the fact that for any given w P W aff , for m " 0 we have L p´1q n Gpmq Ă ι n pwq ¨L´ń P an ¨ιn pwq ´1, see [dCHL]. For (1), we can use [dCHL,Remark 3.1.1] to reduce the claim to the case G " GL n pFq, which is clear from a matrix calculation.) Lemma 4.5.…”

On intersection cohomology with torus action of complexity one, II

“…Perspective. It was conjectured [CHL18] that the decomposition theorem exists, in a strong form, over finite fields, and confirmed for toric varieties [Cat15] and convolution morphisms of partial affine flag varieties [CHL18, Section 6]. Can this be verified in the setting of torus actions of complexity one?…”