I'm looking for references of the decomposition of $L^2(\Gamma\backslash G)$, where $G$ is a connected Lie group, and $\Gamma\subset G$ a discrete lattice; for simplicity one may assume that $G$ is the real point associated to a linear algebraic group defined over $\mathbb{Q}$, without characters defined over $\mathbb{Q}$, and $\Gamma$ is an arithmetic/congruence lattice in $G$. Write $\Omega=\Gamma\backslash G$. Then $\Omega$ has a canonical probability measure induced by the left Haar measure of $G$, and the right translation gives a unitary representation on $L^2(\Omega)$. When I consider the decomposition of $L^2(\Omega)$ into irreducible unitary representaions, I heard about the notion of continuous spectra and discrete spectra, but why are they called spectra and where may I find explicit descriptions for the classical groups?
Also what if one considers the double quotient $M=\Gamma\backslash G/K$, where $K$ is a maximal compact subgroup of $G$? Is it also described via representations of $G$? I don't see an explicit action of $G$ on it. Also if I conjugate $K$ to a second maximal compact subgroup, is there any invariant description of the decomposition of $L^2(M)$, namely independent of the choice of $K$? I heard about the notion of Shimura varieties, so maybe here I should restrict to the case where $M$ is a locally symmetric hermitian manifold.
Thanks a lot!
