Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirkovic-Vilonen conjecture

Carl Mautner (1) , Simon Riche (2, 3)

1 UC Riverside - Department of Mathematics
2 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
3 LMBP - Laboratoire de Mathématiques Blaise Pascal

Let G be a connected reductive group over an algebraically closed field F of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of G, and Iwahori-constructible F-parity sheaves on the affine Grassmannian of the Langlands dual group. As applications we deduce in particular the missing piece for the proof of the Mirkovic-Vilonen conjecture in full generality (i.e. for good characteristic), a modular version of an equivalence of categories due to Arkhipov-Bezrukavnikov-Ginzburg, and an extension of this equivalence.

Domaines: Mathématiques [math]

Carl Mautner, Simon Riche. Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirkovic-Vilonen conjecture. Journal of the European Mathematical Society, 2018, 20, pp.2259-2332. ⟨hal-01110852v3⟩