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.