The decomposition of
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
SL(2; R) will appear in the decomposition of
(a) Is there an algebraic part of the spectrum and does it have a description
similar to the one in
(b) How to classify the
SL(2; R) representations and what is the whole spectrum?
X_0(1) := SL(2; Z)\H. How does
(d) The same for
X_0(N) := \Gamma_0(N)/H. How does