McKay correspondence and Hilbert schemes in dimension three

Abstract

Let $G$ be a nontrivial finite subgroup of $\SL_n(\C)$. Suppose that the quotient singularity $\C^n/G$ has a crepant resolution $π\colon X\to \C^n/G$ (i.e. $K_X = \shfO_X$). 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. \cite{Maizuru}). 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 $\SL_3(\C)$.