We know that for every $l\in\N$ the polynomials $p=\Re(x+iy)^l$ and $q=\Im(x+iy)^l$ form a basis of ${\cal H}_l$. $\G(U)f\colon=f\circ U^{-1}$ is a representation of $\OO(2)$ in ${\cal H}_l$. Turning to complex notation we get for $z\colon=x+iy$ $$ \G(C_m)z^l =(e^{-2\pi i/m}z)^l =e^{-2\pi il/m}z^l =\cos(2\pi l/m)p+\sin(2\pi l/m)q+i(-\sin(2\pi l/m)p+\cos(2\pi l/m)q) $$ and $$ \G(\s)z^l =(\bar z)^l =\cl{z^l} =p-iq $$ Therefore we obtain the matrices: $$ \G(C_m)=\left(\begin{array}{cc} \cos(2\pi l/m)&\sin(2\pi l/m)\\ -\sin(2\pi l/m)&\cos(2\pi l/m) \end{array}\right) \quad\mbox{and}\quad \G(\s)=\left(\begin{array}{cc} 1&0\\ 0&-1 \end{array}\right) $$ Checking traces we see that this representation is equivalent to the representation $\Psi_l$ in subsection.