We construct a noncommutative desingularization of the discriminant of a finite reflection group as a quotient of the skew group ring . If is generated by order reflections, then this quotient identifies with the endomorphism ring of the reflection arrangement viewed as a module over the coordinate ring of the discriminant of . This yields, in particular, a correspondence between the nontrivial irreducible representations of to certain maximal Cohen–Macaulay modules over the coordinate ring . These maximal Cohen–Macaulay modules are precisely the nonisomorphic direct summands of the coordinate ring of the reflection arrangement viewed as a module over . We identify some of the corresponding matrix factorizations, namely, the so-called logarithmic (co-)residues of the discriminant.
Ragnar-Olaf Buchweitz. Eleonore Faber. Colin Ingalls. "A McKay correspondence for reflection groups." Duke Math. J. 169 (4) 599 - 669, 15 March 2020. https://doi.org/10.1215/00127094-2019-0069