  Let $G$ be a nontrivial finite subgroup of $\SL_n(\C)$. Suppose that the
quotient singularity $\C^n/G$ has a crepant resolution $\pi\colon X\to \C^n/G$
(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
$\SL_3(\C)$.
