On quantum Galois theory

For a simple vertex operator algebra $V$ and a finite automorphism group $G$ of $V$ then $V$ is a direct sum of $V^{\chi}$ where $\chi$ are irreducible character of $G$ and $V^{\chi}$ is the subspace of $V$ which $G$ acts according to the character $\chi.$ We prove the following: 1. Each $V^{\chi}$ is nonzero. 2. $V^{\chi}$ is a tensor product $M_{\chi}\otimes V_{\chi}$ where $M_{\chi}$ is an irreducible $G$-module affording $\chi$ and $V_{\chi}$ is a $V^G$-module. If $G$ is solvable, $V_{\chi}$ is a simple $V^G$-module and $M_{\chi}\mapsto$V_{\chi}$is a bijection from the set of irreducible$G$-modules to the set of (inequivalent) simple$V^G$-modules which are contained in$V.$