The projective McKay correspondence

TL;DR

This paper extends the McKay correspondence to the cotangent bundle of P^1, establishing derived equivalences between equivariant sheaves and modules over preprojective algebras, linked by spherical twists.

Contribution

It develops an analogous McKay correspondence for T*P^1, connecting equivariant sheaves with preprojective algebra modules via spherical twists.

Findings

Derived equivalences between sheaves on T*P^1 and preprojective algebra modules.

Spherical twists relate different equivalences, replacing reflection functors.

Extension of McKay correspondence to cotangent bundles.

Abstract

Kirillov has described a McKay correspondence for finite subgroups of PSL_{2}(C) that associates to each `height' function an affine Dynkin quiver together with a derived equivalence between equivariant sheaves on the projective line P^1 and representations of this quiver. The equivalences for different height functions are then related by reflection functors for quiver representations. The main goal of this paper is to develop an analogous story for the cotangent bundle of P^1. We show that each height function gives rise to a derived equivalence between equivariant sheaves on the cotangent bundle T*P^1 and modules over the preprojective algebra of an affine Dynkin quiver. These different equivalences are related by spherical twists, which take the place of the reflection functors for P^1.