Quantum vertex representations via finite groups and the McKay correspondence

Abstract

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 $S{U}_2$, where our construction yields a group theoretic realization of the representations of the quantum affine and quantum toroidal algebras of $ADE$ type.

34 pages, amslatex, to appear in Comm. Math. Phys

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