McKay correspondence and Hilbert schemes in dimension three

Let G be a nontrivial finite subgroup of SLn (C). Suppose that the quotient singularity Cn/G has a crepant resolution π: X→Cn/G (i.e. KX=OX). There is a slightly imprecise conjecture, called the McKay correspondence, stating that there is a relation between the Grothendieck group (or (co)homology group) of X and the representations (or conjugacy classes) of G with a “certain compatibility” between the intersection product and the tensor product (see e.g. [22]). The purpose of this paper is to give more precise formulation of the conjecture when X can be given as a certain variety associated with the Hilbert scheme of points in Cn. We give the proof of this new conjecture for an abelian subgroup G of SL3 (C).