The affine Grassmannian and the Springer resolution in positive characteristic

TL;DR

This paper establishes a positive-characteristic analogue of a key relationship between constructible sheaves on the affine Grassmannian and coherent sheaves on the Springer resolution, using mixed modular sheaves.

Contribution

It introduces a positive-characteristic version of the Arkhipov-Bezrukavnikov-Ginzburg result using new mixed modular sheaves framework.

Findings

Proves a positive-characteristic analogue of the classical result.

Links parity sheaves on the affine Grassmannian to exotic t-structures.

Develops the theory of mixed modular sheaves in this context.

Abstract

An important result of Arkhipov-Bezrukavnikov-Ginzburg relates constructible sheaves on the affine Grassmannian to coherent sheaves on the dual Springer resolution. In this paper, we prove a positive-characteristic analogue of this statement, using the framework of "mixed modular sheaves" recently developed by the first author and Riche. As an application, we deduce a relationship between parity sheaves on the affine Grassmannian and Bezrukavnikov's "exotic t-structure" on the Springer resolution.