The McKay correspondence as an equivalence of derived categories

Abstract

The classical McKay correspondence relates representations of a finite subgroup G ⊂ SL(2,C) to the cohomology of the well-known minimal resolution of the Kleinian singularity C2/G. Gonzalez-Sprinberg and Verdier [10] interpreted the McKay correspondence as an isomorphism on K theory, observing that the representation ring of G is equal to the G-equivariant K theory of C2. More precisely, they identify a basis of the K theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of G