Quantum differential surfaces of higher genera

We first construct a real family of $SL(2,\mathbb{R})$-invariant symbol composition product $\{{\mathrm{♯}}_{\theta }{\}}_{\theta \in ,\mathbb{R}}$ on the analogue of the Schwartz space $S(\mathbb{D})$ on the hyperbolic plane $\mathbb{D}:=SL(2,\mathbb{R})\mathrm{/}SO(2)$. The value $\theta =0$ consists in the pointwise commutative product of functions on $\mathbb{D}$. And admits an asymptotic expansion that deforms the pointwise product in the direction of the canonical $SL(2,\mathbb{R})$-invariant Kahler two form on $\mathbb{D}$. We then extend this construction to any (non-homogeneous) compact surface by considering the left action of an arithmetic Fuschian group $\Gamma\subset SL(2,$\mathbb{R}$)$on$$\mathbb{D}$$withassociatedRiemannsurface$\Sigma_\Gamma;:=;\Gamma\backslash$\mathbb{D}$$\mathrm{.}Moreprecisely,theproduct$\sharp_\theta$extendsfrom$S($\mathbb{D}$)$toasmooth$SL(2,$\mathbb{R}$)$- sub-module of $C^{\mathrm{\infty }}(\mathbb{D})$ that contains the $\mathrm{\Gamma }$-invariants $C^{\mathrm{\infty }}(\mathbb{D}{)}^{\mathrm{\Gamma }}\simeq C^{\mathrm{\infty }}({\mathrm{\Sigma }}_{\mathrm{\Gamma }})$ in $C^{\mathrm{\infty }}(\mathbb{D})$. In particular, ${\mathrm{♯}}_{\theta }$ defines a Fr'echet algebra structure on $C^{\mathrm{\infty }}({\mathrm{\Sigma }}_{\mathrm{\Gamma }})$. The resulting algebra is pre - $C^{\ast }$ and admits a continuous trace.