The decomposition of $L^{2}\left(S^{2}\right)$ under $SO\left(3,\mathbb{R}\right)$ is well-known.


Focus now on the hyperbolic plane $H$ presented as the quotient $SL\left(2,\mathbb{R}\right)/SO\left(2,\mathbb{R}\right)$. It is non-compact, therefore my understanding is that infinite-dimensional 
representations of $SL\left(2,\mathbb{R}\right)$ will appear in the decomposition of $L^{2}\left(H\right)$. 


(a) Is there an algebraic part of the spectrum and does it have a description 
similar to the one in $L^{2}\left(S^{2}\right)$? 


(b) How to classify the $SL\left(2,\mathbb{R}\right)$ representations and what is the whole spectrum? 


(c) Consider $X\_{0}\left(1\right):=SL\left(2,\mathbb{Z}\right)\setminus H$. How does $L^{2}\left(X\_{0}\left(1\right)\right)$ decompose? 


(d) The same for $X\_{0}\left(N\right):=\Gamma\_{0}\left(N\right)/H$. How does $L^{2}\left(X\_{0}\left(N\right)\right)$ decompose? 



Functions on hyperbolic space and modular curves
a) Weyl's unitary trick implies there are no nontrivial irreducible finite dimensional unitary representations of $SL\left(2,\mathbb{R}\right)$. This is basically the opposite of $SO\left(3\right)$.


b) [Wikipedia](http://en.wikipedia.org/wiki/Representation_theory_of_SL2(R)) has a classification of all unitary irreps. An irreducible representation given as a space of functions on H can be viewed as a massive particle state in relativistic QM on $R^{\left(1,2\right)}$.


c) I think you get real-analytic Eisenstein series and discrete series. Eisenstein series form a continuous spectrum, while discrete series give modular forms. You can find more in Gelbart's book "Automorphic forms on adele groups"


d) Same thing, except the Eisenstein series involve a summation over a smaller range of cosets of translation, and the modular forms are invariant under a smaller group. I am told that the Maass forms and holomorphic forms for congruence groups that I mentioned only give a countable collection of unitary representations, while the principal series has a continuous parameter.


