submodules of $M?$ (3) What are the relation between irreducible $V^G$-submodules of $M$ and $V^G$-submodules of another irreducible $V$-module $N?$ These questions will be answered completely in this section and the next section.

In this section we deal with the case that $M$ is $G$-stable (see the definition below). The general case will be treated in the next section. The basis ideas in the proof of main theorems come from \[DLM1\].

Let $(M,Y_M)$ be an irreducible $V$-module and $g\in G$. Following \[DM1\] we set $M\circ g=M$ as vector spaces, and $Y_{M\circ g}(v,z)=Y_M(gv,z)$. Note that $M\circ g$ is also an irreducible $V$-module. We [*assume*]{} in this section that $M$ is $G$-[*stable*]{} in the sense that for any $g\in G,$ $M\circ g$ and $M$ are isomorphic. If $M=V$ this assumption is always true.

For $g\in G$ there is a linear isomorphism $\phi(g): M\to
M$ satisfying $$\label{3.4'}
\phi(g)Y_M(v,z)\phi(g)^{-1}=Y_{M\circ g}(v,z)=Y_M(gv,z)$$ for $v\in V.$ The simplicity of $M$ together with Schur’s lemma shows that $g\mapsto \phi(g)$ is a projective representation of $G$ on $M.$ Let $\a_M$ be the corresponding 2-cocycle in $C^2(G,\C^{\times}).$ Then $M$ is a module for $\C^{\a_M}[G]$ where $\C^{\a_M}[G]$ is the twisted group algebra. It is worth pointing out that if $M=V$ we 