A McKay correspondence for reflection groups

TL;DR

This paper develops a noncommutative desingularization of the discriminant of finite reflection groups, establishing a correspondence between group representations and Cohen–Macaulay modules over the discriminant's coordinate ring.

Contribution

It constructs a new noncommutative desingularization framework for reflection group discriminants and links irreducible representations to Cohen–Macaulay modules via matrix factorizations.

Findings

Constructs a noncommutative desingularization as a quotient of the skew group ring.

Establishes a correspondence between irreducible representations and Cohen–Macaulay modules.

Identifies specific matrix factorizations related to the discriminant.

Abstract

We construct a noncommutative desingularization of the discriminant of a finite reflection group $G$ as a quotient of the skew group ring $A=S*G$. If $G$ is generated by order two reflections, then this quotient identifies with the endomorphism ring of the reflection arrangement $\mathcal{A}(G)$ viewed as a module over the coordinate ring $S^G/(\Delta)$ of the discriminant of $G$. This yields, in particular, a correspondence between the nontrivial irreducible representations of $G$ to certain maximal Cohen--Macaulay modules over the coordinate ring $S^G/(\Delta)$. These maximal Cohen--Macaulay modules are precisely the nonisomorphic direct summands of the coordinate ring of the reflection arrangement $\mathcal{A} (G)$ viewed as a module over $S^G/(\Delta)$. We identify some of the corresponding matrix factorizations, namely the so-called logarithmic (co-)residues of the discriminant.