Ragnar-Olaf Buchweitz: A McKay Correspondence for Reflection Groups

This is joint work with Eleonore Faber and Colin Ingalls. Let G be a finite subgroup of GL(n,K) for a field K whose characteristic does not divide the order of G. The group G then acts linearly on the polynomial ring S in n variables over K and one may form the corresponding twisted or skew group algebra A = S*G. With e in A the idempotent corresponding to the trivial representation, consider the algebra A/AeA. If G is a finite subgroup of SL(2,K), then it is known that A is Morita-equivalent to the preprojective algebra of an extended Dynkin diagram and A/AeA to the preprojective algebra of the Dynkin diagram itself. This can be seen as a formulation of the McKay correspondence for the Kleinian singularities.

We want to establish an analogous result when G is a group generated by reflections. With D the coordinate ring of the discriminant of the group action on S, we show that A/AeA is maximal Cohen-Macaulay as a module over D and that it is of finite global dimension as a ring.

In all cases we can verify the ring A/AeA is the endomorphism ring of a maximal Cohen-Macaulay module over the ring of the discriminant, namely of the direct image of the coordinate ring of the associated hyperplane arrangement.

In this way one obtains a noncommutative resolution of singularities of that discriminant, a hypersurface that is a free divisor, thus, singular in codimension one.