Spectral decomposition of $\Gamma\backslash X$

Question

Let $X$ be a reasonable manifold of non-positive curvature (could be $\mathbb{H}^n$, symmetric or locally symmetric space, homogeneous Hadamard manifold etc.), and let $\Gamma$ be a reasonable group of isometries (with or without elliptic elements). Consider the quotient $\Gamma \backslash X$ (which has the structure of an orbifold in general). In many situations of interest, we have the decomposition $$ L^2(\Gamma \backslash X) = L^2_\text{pp}(\Gamma \backslash X) + L^2_\text{ac}(\Gamma \backslash X), $$ where $\text{pp}$ denotes pure point spectrum and $\text{ac}$ denotes the absolutely continuous spectrum of the Laplace operator. This includes for example, when $X = \mathbb{H}^2$, $\Gamma$ is a cocompact lattice or $\operatorname{SL}_2(\mathbb{Z})$ or some congruence subgroup. One can also take $X$ to be the Siegel modular space, and $\Gamma = \operatorname{Sp}_4(\mathbb{Z})$.

I am trying to get a sense of the literature on this decomposition. The main question is, how general can one go with $X$ and $\Gamma$ so that the above decomposition is true? In particular, I am interested in cases where $\Gamma$ has some arithmetic origin. I would also appreciate information on what is not known.

Answer 1

I am only aware of the case, where $X$ is a symmetric space of non-compact type. If $\Gamma$ is a cocompact lattice, you always get your desired decomposition (in this case there only is pure point spectrum). When the lattice $\Gamma$ is cofinite (i.e. the quotient $X/\Gamma$ has finite volume), this decomposition should still be true. Here, the absolutely continuous spectrum is described by Eisenstein series (explaining the references provided by MBN) and the discrete spectrum consist of cusp forms and some additional part coming from residues of Eisenstein series. While the absolutely continuous spectrum can be explicitly written down, the cusp forms remain mysterious. Some pieces of the Langlands program could be understood as giving a description of these cusp forms for congruence lattices.

The reason that one gets your decomposition in these cases is that there exist a representation-theoretic description of the Laplace spectrum for semisimple Lie groups. I do not know what happens in more general settings.