2-cocycle $\a_M\in Z^2(G,\C^*)$ such that $(\C^{\alpha_M}[G],V^G)$ forms a dual pair on $M$ where $\C^{\alpha_M}[G]$ is the twisted group algebra. Section 6 is a continuation of Section 5 without assuming that $M$ is $G$-stable. The algebra $A_{\alpha}(G,S)$ based on a $G$-stable set $\S$ of $V$-modules enters the picture naturally. The two main theorems of this paper are proved in this section.

We would like to thank A. Wassermann for pointing out a gap in the proof of Theorem 2.4 (i) of \[DLM1\]. This gap has been filled in this paper (see Remark \[rcorect\]).

Twisted group algebras
======================

In this elementary section we review some results on twisted group algebras which will be used in Section 3.

Throughout this paper, $F^*$ denotes the multiplicative group of a field $F$ and $G$ a finite group which acts trivially on $F^*$. Let $\a$ be any element in $Z^2(G,F^*)$ . Thus $\a$ is a map $$\a:G\times G \rightarrow F^*$$ which satisfies the following properties: $$\begin{aligned}
& &\a(x,1)=\a(1,x)=1\ \ {\mbox for\ \ all}\ \ x\in G\\
& &\a(x,y)\a(xy,z)=\a(y,z)\a(x,yz)\ \ {\mbox for\ \ all}\ \ x,y,z\in G.\end{aligned}$$

The [*twisted group algebra*]{} $F^{\a}[G]$ of $G$ over $F$ is defined to be the vector space over $F$ with basis 