Quantum vertex representations via finite groups and the McKay correspondence

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.