The GL(2, C) McKay correspondence

In this paper we show that for any affine complete rational surface singularity the quiver of the reconstruction algebra can be determined combinatorially from the dual graph of the minimal resolution. As a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph. Also, for any finite subgroup G of GL(2,C)GL(2,C), it means that the endomorphism ring of the special CM CC [[x, y]]G-modules can be used to build the dual graph of the minimal resolution of C2/GC2/G, extending McKay’s observation (McKay, Proc Symp Pure Math, 37:183–186, 1980) for finite subgroups of SL(2,C)SL(2,C) to all finite subgroups of GL(2,C)GL(2,C)