Abstract: This is joint work with Ragnar-Olaf Buchweitz and Colin Ingalls.
The classical McKay correspondence relates the geometry of so-called
Kleinian surface singularities with the representation theory of finite
subgroups of SL(2,C). 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
subgroup 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 yelds a generalization of
Auslander's result, and moreover, a noncommutative resolution of the
discriminant of the reflection group G.
- Special Lecture