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?