i,j\leq k,$ $h\in 
G_s$ and $\a_s$-regular $a\in G_s$. In the orbifold theory, these conditions are satisfied by chosing $\alpha_s$ carefully.

Let $\{l_1,...,l_r\}$ be a full set of representatives for the $\a_s$-regular conjugacy classes of $G_s$ and for each $t$ in $\{1,...,r\}$, let $L_t$ be $\a_s$-regular conjugacy class of $G_s$ containing $l_t,1\leq t\leq r.$ Set $$Z(L_t)=\sum_{a\in L_t}\sum_{i=1}^k g_i^{-1}ag_i\o e(sg_i).$$

$Z(L_t)$ is an center element of $D(\O_s).$

Let $b\o e(sg_j)\in D({\O}_s)$. Then there exists $1\leq i'\leq k$ and $h\in G_s$ such that $b=g_{i'}^{-1}hg_j.$ We have $$\begin{aligned}
& &\ \ \ \ b\o e(sg_j)\cdot Z(L_t)\\
& &=b\o e(sg_j)\cdot \sum_{a\in L_t}\sum_{i=1}^k g_i^{-1}ag_i\o e(sg_i)\\
& &=\sum_{a\in L_t}\sum_{i=1}^k\a_{sg_i}(b,g_i^{-1}ag_i)bg_i^{-1}ag_i\o e(sg_jg_i^{-1}ag_i)e(sg_i)\\
& &=\sum_{a\in L_t}\a_{sg_j}(b,g_j^{-1}ag_j)bg_j^{-1}ag_j\o e(sg_j)\\
& &=\sum_{a\in L_t}\a_{sg_j}(g_{i'}^{-1}hg_j,g_j^{-1}ag_j)g_{i'}^{-1}hag_j\o e(sg_j)\end{aligned}$$ and $$\begin{aligned}
& &Z(L_t)\cdot b\o e(sg_j)\\
& &=\sum_{a\in L_t}\sum_{i=1}^k g_i^{-1}ag_i\o e(sg_i)\cdot b\o e(sg_j)\\
& &=\sum_{a\in L_t}\sum_{i=1}^k\a_{sg_j}(g_i^{-1}ag_i,b)g_i^{-1}ag_ib\o e(sg_ib)e(sg_j)\\
& &=\sum_{a\in L_t}\a_{sg_j}(g_{i'}^{-1}ag_{i'}, g_{i'}^{-1}hg_j)g_{i'}^{-1}ahg_j\o e(sg_j).\end{aligned}$$ Now the assertion follows from $$\begin{aligned}
& &\ \ \ \ \sum_{a\in L_t}\a_{sg_j}(g_{i}^{-1}hg_j,g_j^{-1}ag_j)hah^{-1}\\
& &=\sum_{a\in L_t}\a_{sg_j}(g_{i}^{-1}hah^{-1}g_{i},g_{i}^{-1}hg_j)hah^{-1}\\
& &=\sum_{a\in L_t}\a_{sg_j}(g_{i}^{-1}ag_{i},g_{i}^{-1}hg_j)a.\end{aligned}$$

Let $\{l_1,..., l_r\}$ be a full set of representatives for the $\a_s$-regular conjugacy classes of $G_s$ and for each $t\in \{1,...,r\}$, let $L_t$ be a $\a_s$-regular conjugacy class of $G_s$ containing $l_t, 1\leq t\leq r$. Then the elements $Z(L_t)$ constitute a $\C$-basis of $Z(D({\O}_s))$.

By Theorem \[t3.3\] the 