A McKay correspondence for reflection groups

Abstract

We construct a noncommutative desingularization of the discriminant of a finite reflection group $G$ as a quotient of the skew group ring $A=S\ast G$. If $G$ is generated by order $2$ reflections, then this quotient identifies with the endomorphism ring of the reflection arrangement $\mathcal{A}\left(G\right)$ 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 $\mathcal{A}\left(G\right)$ 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.