Affine braid group actions on derived categories of Springer resolutions

TL;DR

This paper constructs an affine braid group action on derived categories of Springer resolution-related varieties, providing a categorical perspective on affine Hecke algebra representations and aiding in proving Lusztig's conjectures.

Contribution

It introduces an explicit affine braid group action on derived categories, extending the categorical framework of affine Hecke algebras in geometric representation theory.

Findings

Explicit construction of the affine braid group action

Connection to Kazhdan--Lusztig--Ginzburg's affine Hecke algebra

Application to Lusztig's conjectures on equivariant K-theory

Abstract

In this paper we construct and study an action of the affine braid group associated to a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a "categorical version" of Kazhdan--Lusztig--Ginzburg's construction of the affine Hecke algebra, and is used in particular by the first author and Ivan Mirkovic in the course of the proof of Lusztig's conjectures on equivariant K-theory of Springer fibers.