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

TL;DR

This paper establishes deep connections between tilting sheaves, parity sheaves, and the Mirkovic-Vilonen conjecture, advancing understanding in geometric representation theory and confirming conjectural equivalences in good characteristic.

Contribution

It relates exotic tilting sheaves to parity sheaves on affine Grassmannians, proving the Mirkovic-Vilonen conjecture in full generality for good characteristic.

Findings

Proof of the Mirkovic-Vilonen conjecture in good characteristic.

Establishment of a modular equivalence of categories.

Extension of known categorical equivalences.

Abstract

Let $\mathbf{G}$ be a connected reductive group over an algebraically closed field $\mathbb{F}$ 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 $\mathbf{G}$, and Iwahori-constructible $\mathbb{F}$-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.