On the operator content of nilpotent orbifold models

Let $V$ be a simple vertex operator algebra and $G$ be a finite nilpotent group of automorphisms of $V.$ We prove the following in this paper: (1) There is a Galois correspondence between subgroups of $G$ and the vertex operator subalgebras of $V$ which contain $V^G$ given by the map $H\mapsto V^H.$ (2) Assume that for every G\in G$there is unique simple$g$-twisted$V$-module$M(g).$Then there exists a Hochschild 3-cocycle$\alpha$on the integral group$Z[G]$such that there is an equivalence of categories between$V^G$-module category (whose objects are$V^G$-submodules of direct sums of copies of$\oplus_{g\in G}M(g),$and whose morphisms are$V^G$-module homomorphisms) and the module category for the twisted quantum double$D_{\alpha}(G)$associated to$\alpha.$