McKay correspondence and Hilbert schemes in dimension three

Let $G$ be a nontrivial finite subgroup of ${\text{SL}}_n(\text{C})$. Suppose that the quotient singularity ${\text{C}}^n\mathrm{/}G$ has a crepant resolution $\pi \text{\hspace{0.17em}⁣}:X\to {\text{C}}^n\mathrm{/}G$ (i.e. $K_X={\text{shfO}}_X$). 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 ${\text{C}}^n$. We give the proof of this new conjecture for an abelian subgroup $G$ of ${\text{SL}}_3(\text{C})$.