Twisted vertex representations via spin groups and the McKay correspondence

Abstract

We establish a twisted analog of our recent work on vertex representations and the McKay correspondence. For each finite group $\Gamma$ and a virtual character of $\Gamma$, we construct twisted vertex operators on the Fock space spanned by the super spin characters of the spin wreath products $\Gamma\wr \tilde {S}_n$ of $\Gamma$ and a double cover of the symmetric group $S_n$ for all $n$. When $\Gamma$ is a subgroup of ${\rm SL}_2(\mathbb {C})$ with the McKay virtual character, our construction gives a group-theoretic realization of the basic representations of the twisted affine and twisted toroidal Lie algebras. When $\Gamma$ is an arbitrary finite group and the virtual character is trivial, our vertex operator construction yields the spin character tables for $\Gamma\wr \tilde {S}_n$.