The infinite-dimensional unitary representations of SL2(R)$SL_{2}\left(\mathbb{R}\right)$ appearing in the right-regular representation on L2(H)$L^{2}\left(H\right)$ are precisely the unitary representations of SL2(R)$SL_{2}\left(\mathbb{R}\right)$ possessing a SO2(R)$SO_{2}\left(\mathbb{R}\right)$-fixed vector. These are parametrized by R \union [0,1]$\mathbb{R}\cup\left[0,1\right]$, where R$\mathbb{R}$ parametrizes unitary principal series representations and [0,1]$\left[0,1\right]$ parametrizes the "complimentary"complementary series" representations. This is implicit in Knapp's chapter in the Corvallis volume; see also Iwaniec's book on the spectral theory of automorphic forms for a classical treatment of this case.
Anyway, the point of this is that L2(H)$L^{2}\left(H\right)$ has a "direct integral" decomposition into irreducible representations, so the proper analogy in this situation is not L2(S^2)$L^{2}\left(S^{2}\right)$ but rather L2(R)$L^{2}\left(\mathbb{R}\right)$. By contrast, the cofinite quotients X0(N)$X_{0}\left(N\right)$ have a "mixed" spectral decomposition, that is L2(X0(N))$L^{2}\left(X_{0}\left(N\right)\right)$ breaks into a continuous part (Eisenstein series, parametrized by R$\mathbb{R}$) and a discrete part, the so-called cusp forms. This theory is due to Selberg and is by no means straight-forward. Again, see Iwaniec's book for a nice classical treatment.
