  When the quotient of a symplectic vector space by the action of a finite
subgroup of symplectic automorphisms admits as a crepant projective resolution
of singularities the Hilbert scheme of regular orbits of Nakamura, then there
is a natural isomorphism between the Grothendieck group of this resolution and
the representation ring of the group, given by the Bridgeland-King-Reid map.
However, this isomorphism is not compatible with the ring structures. For the
Hilbert scheme of points on the affine plane, we study the multiplicative
behavior of this map.
