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 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 $S^G/(\hat{I}”)$ 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 $S^G/(\hat{I}”)$. 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 $S^G/(\hat{I}”)$. We identify some of the corresponding matrix factorizations, namely the so-called logarithmic (co-)residues of the discriminant.

Major revision. Final version, to appear in Duke Math. Journal. 52pages