dimension of $Z(D({\O}_s))$ equals to the number of inequivalent irreducible $\C^{\a_s}[G_s]$-modules which is $r$ (see Theorem \[a2.2\]).

The following corollary is immediate.

Let $\{l_1^{j},...,l_{r_j}^{j}\}$ be a full set of representatives for the $\a_{s_j}$-regular conjugacy classes of $G_{s_j}$ and for each $t_j\in \{1,...,r_j\}$, let $L_{t_j}$ be $\a_{s_j}$-regular conjugacy class of $G_{s_j}$ containing $l_{t_j}, 1\leq t_j\leq r_j.$ Set $$Z(L_{t_j})=\sum_{a\in L_{t_j}}\sum_{i_j=1}^{k_j}g_{i_j}^{-1}ag_{i_j}\o e(sg_{i_j}).$$ Then for all $j\in J$, for all $t_j\in \{ 1,..., r_j\}$, $Z(L_{t_j})$ constitutes a $\C$-basis for $Z(A_{\a}(G,{\S})).$

Modules for vertex operator algebras and related results
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In this section we turn our attention to the theory of vertex operator algebras. In particular we shall defines various notion of modules for a vertex operator algebra $V$ following \[FLM\], \[DLM2\] and \[Z\]. We also recall from \[DLM4\] the associative algebras $A_n(V)$ for any nonnegative integer $n$ and relevant results. These results will be used extensively in Sections 5 and 6 to study dual pairs arising from an action of a finite group $G$ on $V.$

Let $V=(V,Y,{\bf 1},\omega)$ be a vertex operator algebra (see \[B\] and \[FLM\]). For a vector space $W$, let $W\{z\}$ be the space of $W$-valued formal series in arbitrary complex powers of $z$. We present three different notion of 