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- Title
Frobenius semisimplicity for convolution morphisms.
- Authors
de Cataldo, Mark Andrea; Haines, Thomas J.; Li, Li
- Abstract
This article concerns properties of mixed ℓ<inline-graphic></inline-graphic>-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.
- Publication
Mathematische Zeitschrift, 2018, Vol 289, Issue 1/2, p119
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-017-1946-4