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?