The decomposition of `L^2(S^2)`

under `SO(3; R)`

is well-known.

Focus now on the hyperbolic plane H presented as the quotient `SL(2; R)/SO(2; R)`

. It is non-compact, therefore my understanding is that infinite-dimensional
representations of `SL(2; R)`

will appear in the decomposition of `L^2(H)`

.

(a) Is there an algebraic part of the spectrum and does it have a description
similar to the one in `L^2(S^2)`

?

(b) How to classify the `SL(2; R)`

representations and what is the whole spectrum?

(c) Consider `X_0(1) := SL(2; Z)\H`

. How does `L^2(X_0(1))`

decompose?

(d) The same for `X_0(N) := \Gamma_0(N)/H`

. How does `L^2(X_0(N))`

decompose?