Decomposition of $L^2(\Gamma \backslash G)$ Let $G$ be a semisimple Lie group, and $\Gamma$ be an lattice (arithmetic) - typical examples I am thinking about would be $(SL_2(\mathbb{R}), SL_2(\mathbb{Z})$, or $(SL_2(\mathbb{C}), PGL_2(O_F))$ (here $F$ is say, an imaginary quadratic field of class number 1, and we do a central twist to make the determinant 1). I would like to a reference for the basic facts of  decomposition of $L^2(\Gamma \backslash G)$ into unitary irreps of $G$ (or irreducible admissible $(g,K)$-modules). 
I am aware of this very similar question (Decomposition of Regular Representation of Non-compact Lie group), and also tried to look at two books by Knapp (Lie groups beyond an introduction, and representation theory of semisimple groups). Maybe I overlooked things when I was skimming them, but I can't quite pin down the sections where this is done. (Especially where the results are spelled out). Can anyone point me to surveys/books that would state the facts/proofs of this decomposition, in particular which unitary irrep would show up, and with what multiplicity? Thanks!
Edit: The answer below answers the case of $SL_2(\mathbb{R}), SL_2(\mathbb{Z})$, but I am still interested in the higher rank case.
 A: A very nice discussion of this can be found in chapter 2 of the book "Automorphic forms on adele groups" by Stephen Gelbart.
A: The continuous part(=Eisenstein series) is understood by Moeglin-Waldspurgers book on Eisenstein series. 
The cuspidal part(=discrete part) is not well understood. Many things are still open. I am referring here to Maass forms. The multiplicity will be finite. I think this is due to Harish-Chandra's LNM "Automorphic forms on Lie groups".
I suggest for SL(2, o_F), you can look at Iwaniec-Spectral theory of automorphic forms for $\mathbb{Q}$ or J. Elstrodt, F. Grunewald, and J. Mennicke,
Groups acting on hyperbolic space for imaginary quadratic fields as well. But this addresses SO(2) respective SU(2) invariant vectors only.
A: *

*I highly recommend A. Borel's Automorphic Forms on $SL(2,\mathbb R)$ as an introduction to the general theory. While done over $\mathbb R$, Borel exposition is very clear and sets up the machinery such that it readily extends to more general settings. The book in fact culminates with the spectral decomposition. I would say once you've read Borel's book you'll be in a good place to tackle the cases you are interested in.

*Regarding Gelbart's Automporhic Forms on Adele Groups, he treats $GL(2,\mathbb A_\mathbb Q)$, but is much more sketchy as his aim is to give an overview of Jacquet-Langlands' book. As a guide to reading this I suggest Knapp's article Theoretical Aspects of Trace Formula, modeled after a certain Gelbart-Jacquet paper is a very readable account of the transition from $SL(2,\mathbb R)$ to $GL(2,\mathbb A_\mathbb Q)$.

*For higher rank cases Moeglin-Waldspurger provides a key to Langlands' theory of Eisenstein Series. You might find the notation burdensome, but in the least case you should read the introduction where an outline of the induction is given.
