  We establish a $q$-analog of our recent work on vertex representations and
the McKay correspondence. For each finite group $\Gamma$ we construct a Fock
space and associated vertex operators in terms of wreath products of
$\Gamma\times \mathbb C^{\times}$ and the symmetric groups.
  An important special case is obtained when $\Gamma$ is a finite subgroup of
$SU_2$, where our construction yields a group theoretic realization of the
representations of the quantum affine and quantum toroidal algebras of $ADE$
type.
