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?