Reductive groups, the loop Grassmannian, and the Springer resolution

TL;DR

This paper establishes categorical equivalences linking modular representations of reductive groups over fields of characteristic p to equivariant coherent sheaves on the Springer resolution, extending geometric representation theory in positive characteristic.

Contribution

It proves new equivalences of derived categories connecting modular representation theory with geometric objects like the Springer resolution, including a graded version of a conjecture by Finkelberg-Mirkovi\'c.

Findings

Derived category equivalences for blocks of reductive group representations.

A modular analogue of Lusztig's quantum group constructions.

A graded version of the Finkelberg-Mirkovi\'c conjecture.

Abstract

In this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic $p$ bigger than the Coxeter number and a derived category of equivariant coherent sheaves on the Springer resolution (or a parabolic counterpart). In the case of the principal block, combined with previous results, this provides a modular version of celebrated constructions due to Arkhipov-Bezrukavnikov-Ginzburg for Lusztig's quantum groups at a root of unity. As an application, we prove a "graded version" of a conjecture of Finkelberg-Mirkovi\'c describing the principal block in terms of mixed perverse sheaves on the dual affine Grassmannian.