any $M\in S$ and $g\in G$ there exists $N\in \S$ such that $M\circ g\cong N.$ Then every irreducible $V$-module $M$ produces naturally such a set by collecting all $M\circ g$ for $g\in G.$ Now we take a finite $G$-stable set $\S$ consisting of inequivalent irreducible $V$-modules. We construct a finite dimensional semisimple associative algebra $A_{\alpha}(G,\S)$ for a suitable 2-cocycle on $G$ (which is uniquely determined up to isomorphism by $V,$ $G$ and $\S$) such that $(A_{\alpha}(G,{\S}),V^G)$ forms a dual pair on ${\M}=\sum_{M\in \S}M$ in the precise sense of Howe \[H1\]-\[H2\]. That is, each simple $A_{\alpha}(G,\S)$ occurs in $\M$ and its multiplicity space is an irreducible $V^G$-module. Moreover, the different multiplicity spaces are inequivalent $V^G$-modules.

These duality results not only tell us the complete reducibility of every irreducible $V$-module as a $V^G$-module, but also provide an equivalence between the $A_{\alpha}(G,\S)$-module category and a subcategory of $V^G$-modules generated by the irreducible submodules of $\M$ by sending each simple $A_{\alpha}(G,\S)$-module to its multiplicity space. This kind of idea has appeared in \[DPR\] and \[DM5\] in the study of holomorphic orbifold conformal field theory.

Although the algebra $A_{\alpha}(G,\S)$ mentioned above appears naturally in the theory of vertex operator algebras, the construction itself is totally 