The main feature in the study of orbifold theory is the appearance of dual pairs. It is proved in \[DLM1\] and \[DM2\] that $G$ and $V^G$ form a dual pair in the sense of Howe \[H1\]-\[H2\]. More precisely, it is shown in \[DLM1\] that all the irreducible $G$-modules occur in $V$ and the space of multiplicity of each irreducible $G$-module in $V$ is an irreducible $V^G$-module. Moreover, inequivalent irreducible $G$-modules produce inequivalent $V^G$-modules in this way.

In this paper we extend the duality result in \[DLM1\] to any irreducible $V$-module and obtain again several duality theorems of Schur-Weyl type. In the process we realize that it is better to consider a finite set of inequivalent irreducible modules which is $G$-stable instead of one single module. The general setting is more natural and the results are more beautiful.

More explicitly, for an irreducible $V$-module $M=(M,Y_M)$ (see Section 4 for the details of the definition of a module) and $g\in G$ we define a new irreducible $V$-module $M\circ g=(M\circ g,Y_{M\circ g}).$ Here $M\circ g$ is equal to $M$ as a vector space and $Y_{M\circ g}(v,z)=
Y_M(gv,z)$ for $v\in V$ following \[DM1\]. A set $\S$ of irreducible $V$-modules is called $G$-stable if for 