Abstract: Let $G$ be a finite subgroup of $GL(n,\mathbb{C})$. Then $G$ acts linearly on the polynomial ring $S$ in $n$ variables over $\mathbb{C}$. When $G$ is generated by reflections, then the discriminant $D$ of the group action of $G$ on $S$ is a hypersurface with singular locus of codimension 1. The classical McKay correspondence relates the geometry of the resolutions of singularities of so-called Kleinian surfaces with the representation theory of finite subgroups of $SL(2,\mathbb{C})$. In particular, there is an algebraic version of this correspondence, due to M. Auslander.

In this talk we present a version of the McKay correspondence when $G$ is a finite group generated by reflections: We give a natural construction of a so-called noncommutative resolution of the coordinate ring of $D$ as a quotient of the skew group ring $A=S*G$. We will explain this construction, which allows to extend Auslander's theorem to reflection groups. This is joint work with R.-O. Buchweitz and C. Ingalls.

Speaker's web page: http://www-personal.umich.edu/~emfaber/index.html