Let $L$ be a lattice in a Lie group $G$. If $L = \mathbb{Z}^n$ and $G = \mathbb{R}^n$ then $L / G = (S^1)^n$, and there is a canonical isomorphism (Fourier transform) between $L^2(\mathbb{Z}^n)$ and $L^2((S^1)^n)$. Is there a general version of this? (These Hilbert spaces are isomorphic for dimension reasons, but I'm looking for a canonical isomorphism that one can efficiently write down and compute with.) (I'm aware that the Lie group structure on $S^1$ plays a big role here.)
The case I would be most interested to hear about is $G = PSL_2(\mathbb{R})$and $L = PSL_2(\mathbb{Z})$. (Or a comparison between $L^2(X(1))$ and $L^2(PSL_2(\mathbb{Z}))$... I'm curious about results that compare the geometry of Fuschian groups with the corresponding modular curves.)