A Schur-Weyl type duality for twisted weak modules over a vertex algebra

Let $V$ be a vertex algebra of countable dimension, $G$ a subgroup of $\mathrm{A}\mathrm{u}\mathrm{t}V$ of finite order, $V^G$ the fixed point subalgebra of $V$ under the action of $G$, and $\mathcal{S}$ 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 ${\mathcal{A}}_{\alpha }(G,\mathcal{S})$ and $V^G$ on the direct sum of twisted weak $V$-modules in $\mathcal{S}$ where ${\mathcal{A}}_{\alpha }(G,\mathcal{S})$ is a finite dimensional semisimple associative algebra associated with $G,\mathcal{S}$, and a $2$-cocycle $\alpha$ naturally determined by the $G$-action on $\mathcal{S}$. 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.