McKay correspondence and Hilbert schemes in dimension three

Let G be a nontrivial finite subgroup of SLn (C). Suppose that the quotient singularity C-n/G has a crepant resolution pi: X --> C-n/G (i.e. K-X = O-X). 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 C-n. We give the proof of this new conjecture for an abelian subgroup G of SL3 (C). (C) 2000 Elsevier Science Ltd. All rights reserved.