$s\in \S$ and simple $D({\O}_s)$-modules are precisely $\Ind_{S(s)}^{D(s)}M$ where $M$ ranges over the simple $\C^{\a_{s}}[G_{s}]$-modules.

ii\) $A_{\a}(G,{\S})$ is semisimple and simple $A_{\a}(G,{\S})$-modules are precisely $\Ind_{S(s_j)}^{D(s_j)}M$ where $M$ ranges over the simple $\C^{\a_{s_j}}[G_{s_j}]$-modules and $j\in J.$

i\) follows from Theorem \[t3.3\] and that fact that $\C^{\a _s}[G_s]$-Mod is a semisimple category. ii) follows from Lemma \[l3.2\] and i).

[If we only wanted to know the semisimplicity of $A_{\alpha}(G,\S),$ we could find it in the literature of Hopf algebra when we regard $A_{\alpha}(G,\S)$ as a crossed product (see Remark \[ra\]). But in the later sections we need to know the explicit structure of simple $A_{\alpha}(G,\S)$-modules given in Theorem \[t3.3\]. The semisimplicity of $A_{\alpha}(G,\S)$ is a trivial corollary of Theorem \[t3.3\].]{}

Next we determine the center $Z(D({\O}_s))$ of $D({\O}_s)$. However, we are not going to use it in this paper but still include it for general interest. We certainly expect that the result on center will be used in our future study on general orbifold conformal field theory.

Since any cocycle in $Z^2(G_s,\C^*)$ is cohomologous to a normal cocycle (cf. Lemma \[a2.1\]). Then we may [*assume $\a_s$ is a normal cocycle*]{} for all $s\in S.$ We also assume that $\a_{sg_i}(g_j^{-1}hg_i,g_i^{-1}ag_i)=\a_{sg_i}(g_j^{-1}hah^{-1}g_j,g_j^{-1}hg_i)$ for all $1\leq 