The classical McKay correspondence relates irreducible representations of a finite subgroup G of SL(2,C) to exceptional curves on the minimal resolution of the quotient singularity C^2/G. M.Auslander observed an algebraic version of this correspondence: let G be a finite subgroup of SL(2,K) for a field K whose characteristic does not divide the order of G. The group acts linearly on the polynomial ring S=K[x,y] and then the so-called skew group algebra A=G*S can be seen as an incarnation of the correspondence. In particular, A is isomorphic to the endomorphism ring of S over the corresponding Kleinian surface singularity.
Our goal is to establish an analogous result when G in GL(n,K) is a finite group generated by reflections, assuming that the characteristic of K does not divide the order of the group. Therefore we will consider a quotient of the skew group ring A=S*G, where S is the polynomial ring in n variables. We show that our construction yields a generalization of Auslander's result, and moreover, a so-called noncommutative resolution of the discriminant of the reflection group G. This is joint work with Ragnar-Olaf Buchweitz and Colin Ingalls.