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The McKay correspondence and noncommutative resolutions of discriminants of reflection groups 2 Faber, Eleonore


Joint work with Ragnar-Olaf Buchweitz and Colin Ingalls. The classical McKay correspondence relates irreducible representations of a finite subgroup $G$ of $SL(2,\mathbb{C})$ to exceptional curves on the minimal resolution of the quotient singularity $\mathbb{C}^2/G$. Maurice 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. Auslander's isomorphism holds more generally for small finite subgroups $G$ of $GL(n,K)$, that is, $G$ does not contain any (pseudo-)reflections. The goal of this work is to establish a similar 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 noncommutative resolution of the discriminant of the reflection group $G$. In particular, we obtain a correspondence between the nontrivial irreducible representations of $G$ and certain maximal Cohen--Macaulay modules over the discriminant, and we can identify some of these modules, namely the so-called logarithmic (co-)residues of the discriminant.

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