# 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.

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Are you interested only in those cases, or also in the higher rank cases? Because then you will need to know something about langlands' theory of Eisenstein series (the spectral resolution works roughly the same, by looking at the resolvent, but you might have several Casimirs to consider). – Asaf May 11 '14 at 18:58
A good reference for the theory of $SL_{2}(\mathbb{C})$ would be the Sarnak-Cohen notes, which are freely available in Peter's IAS site. – Asaf May 11 '14 at 18:59
@Asaf, also in higher rank (yes, I'm interested in Langlands' theory of Eisenstein series too), even though the main cases I care about now are the two I mentioned. and Thanks for the reference to the notes! Didn't know about it. – user31415 May 11 '14 at 20:18

## 3 Answers

A very nice discussion of this can be found in chapter 2 of the book "Automorphic forms on adele groups" by Stephen Gelbart.

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Thanks! That's a really good heads-up - I looked into Bump a little and don't think I have a satisfactory answer there, and I didn't look into Gelbart afterwards. I am still interested in the more general situation though. :) – user31415 May 10 '14 at 20:59
I would look at Borel's SL2 book, which is roughly in the same spirit as Gelbart's book. – Asaf May 11 '14 at 19:04

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.

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Thanks! Is there a survey about the situation in cuspidal part? For example, it is expected that the spectrum of Laplacian for $SL_2(\mathbb{Z}) \backslash SL_2(\mathbb{R})$ is simple - what is the expectation for general semisimple groups and the center of universal enveloping algebra? The only place I know of for the surface case is Sarnak's 2003 survey in Bulletin, and unfortunately I don't even know where I can look for even the expectations in the general situation. – user31415 May 13 '14 at 0:56
Multiplicity one is known not to hold for SL(n), so no simplicity of the eigenvalues. It is a reasonable conjecture to assume that given a lattice, the multiplicity is bounded by a constant depending on the lattice and the weights. – Marc Palm May 13 '14 at 5:02
1. 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.

2. 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 $GL(2,\mathbb A_\mathbb Q)$.

3. 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.

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