Let $V$ be a vertex algebra of countable dimension, $G$ a subgroup of ${ \rm Aut} V$ of finite order, $V^{ G}$ the fixed point subalgebra of $V$ under the action of $G$, and ${ \mathscr S}$ a finite $G$-stable set of inequivalent irreducible twisted weak $V$-modules associated with possibly different automorphisms in $G$. We show a Schur--Weyl type duality for the actions of ${ \mathscr A}_{ \alpha}(G,{ \mathscr S})$ and $V^G$ on the direct sum of twisted weak $V$-modules in ${ \mathscr S}$ where ${ \mathscr A}_{ \alpha}(G,{ \mathscr S})$ is a finite dimensional semisimple associative algebra associated with $G,{ \mathscr S}$, and a $2$-cocycle $\alpha$ naturally determined by the $G$-action on ${ \mathscr S}$. It follows as a natural consequence of the result that for any $g\in G$ every irreducible $g$-twisted weak $V$-module is a completely reducible weak $V^G$-module.
Categories: math.QA