McKay's correspondence for cocompact discrete subgroups of SU(1,1)

TL;DR

This paper explores a generalization of McKay's correspondence from finite subgroups of SU(2) to cocompact discrete subgroups of the universal cover of SU(1,1), linking representations to vector bundles on algebraic surfaces.

Contribution

It extends McKay's correspondence to cocompact discrete subgroups of the universal cover of SU(1,1), connecting unitary representations with vector bundles on algebraic surfaces.

Findings

Established a correspondence between certain unitary representations and vector bundles.

Linked the representation theory of cocompact subgroups to algebraic geometry of surfaces.

Generalized classical McKay correspondence to a new geometric setting.

Abstract

The classical McKay correspondence establishes an explicit link from the representation theory of a finite subgroup G of SU(2) and the geometry of the minimal resolution of the quotient of the affine plane by G. In this paper we discuss a possible generalization of the McKay correspondence to the case when G is replaced with a cocompact discrete subgroup of the universal cover of SU(1,1) such that its image in PSU(1,1) is a cocompact fuchsian group with quotient of genus 0. We establish a correspondence between a certain class of finite-dimensional unitary representations of G and vector bundles on an open algebraic surface with the trivial canonical class canonically associated to G.