really gives the projection operator on noncommutative orbifold $T^{2}/Z_{N}$. The form of (\[eq:4\]) is a generalization of $GHS$ construction.[^5]. From the above result, we have $$M(k,q_{0})_{nn^{\prime }}=\frac{<k,q_{0}+\frac{ln}{A}|\phi
_{1}><\phi
_{2}|k,q_{0}+\frac{ln^{\prime }}{A}>}{\sum_{n^{"}}<k,q_{0}+\frac{ln"}{A}%
|\phi _{1}><\phi _{2}|k,q_{0}+\frac{ln"}{A}>}.$$Noticing that this equation satisfies (\[eq:1\]), we have $$\begin{aligned}
  \label{eq:30}
\psi _{st}(k,q_{0})
&=&\frac{1}{A}\sum_{n=0}^{A-1}M(k,q_{0})_{n+s,n}e^{2\pi
i(q_{0}/l+n/A)t}  \nonumber \\
&=&\frac{\frac{1}{A}\sum_{n=0}^{A-1}<k,q_{0}+\frac{l(n+s)}{A}|\phi
_{1}><\phi
_{2}|k,q_{0}+\frac{ln}{A}>e^{2\pi i(q_{0}/l+n/A)t}}{%
\sum_{n}<k,q_{0}+\frac{ln}{A}|\phi _{1}><\phi
_{2}|k,q_{0}+\frac{ln}{A}>}
\nonumber \\
&=&\frac{F_{st}(k,q_{0})}{AF_{00}(k,q_{0})},\end{aligned}$$where $$\label{eq:42}
F_{st}(k,q_{0})\equiv
\sum_{n=0}^{A-1}<k,q_{0}+\frac{l(n+s)}{A}|\phi _{1}><\phi
_{2}|k,q_{0}+\frac{ln}{A}>e^{2\pi i(q_{0}/l+n/A)t},$$with $$\begin{aligned}
F_{st}(k,q_{0}) &=&F_{st}(k,q_{0}+l/A)=F_{st}(k+2\pi /l,q_{0}), \\
F_{st}(k,q_{0}) &=&F_{s+A,t}(k,q_{0})e^{-ilk} \\
&=&F_{s,t+A}(k,q_{0})e^{-2\pi iq_{0}A/l}.\end{aligned}$$So the function $F_{st}$ is the function of independent variables $%
X=e^{-ilk} $ and $Y=e^{-2\pi iq_{0}A/l}$, namely $F_{st}(k,q_{0})=\Phi _{st}(X,Y)$. Similarly $$\psi _{st}(k,q_{0})=\Psi _{st}(X,Y)=\frac{\Phi _{st}(X,Y)}{A\Phi
_{00}(X,Y)}. \label{eq:12}$$If we change the variable $X$ and $Y$ into $u_{1}^{A}$ and $u_{2}^{A}$ respectively, the standard form (\[eq:6\]) of the projection operator can be easily obtained. So the key question is to find out $F_{st}(k,q_{0})$.

 Coherent State Representation
==============================

Introduce coherent states $$|z>=e^{-\frac{1}{2}z\bar{z}}e^{a^{+}z}|0>,$$where $z=x+iy,\bar{z}=x-iy$, which satisfies $$\frac{1}{\pi }\int_{-\infty }^{\infty }d^{2}z|z><z|\equiv \frac{1}{\pi }%
\int_{-\infty }^{\infty }dxdy|z><z|=identity,$$ $$R_{N}|z>=|\omega _{N}z>.$$

We can show [@9] $$\label{eq:27}
<k,q|z>=\frac{1}{\sqrt{l}\pi ^{1/4}}\theta (\frac{q+\frac{\tau }{\tau _{2}}%
k-i\sqrt{2}z}{l},\frac{\tau }{A})e^{-\frac{\tau }{2i\tau _{2}}k^{2}+ikq+%
\sqrt{2}kz-(z^{2}+z\bar{z})/2},$$where

$$\theta (z,\tau )\equiv \theta \left[
\begin{array}{c}
0 \\
0%
\end{array}%
\right] (z,\tau )$$

and$$\label{eq:34}
\theta \left[
\begin{array}{c}
a \\
b%
\end{array}%
\right] (z,\tau )=\sum_{m}e^{\pi i\tau (m+a)^{2}}e^{2\pi
i(m+a)(z+b)}.$$Thus we can expand the 