## decomposition into irreducible unitary representations: references for explicit formulas?

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!

-
The questions here cover a huge amount of territory, so it's important to make clear what your own background in the literature is: what have you already read? – Jim Humphreys Nov 7 2010 at 22:23
I've started with two books of Iwaniec on modular forms, mainly the chapters on spectral decomposition. I haven't plunged into the volumes of Helgason yet, and I don't know if I should find detailed presentations of the locally symmetric manifolds there. The second example I'm considering is a non-reductive group: $\mathbb{R}^2\rtimes GL_2(\mathbb{R})$. This group can produce univrsal elliptic curves, also called elliptic surfaces. I want to now check the spectral decomposition for the universal elliptic curve. I'm quite lost for literatures in this topic, or maybe I'd try some Jacobi forms. – turtle Nov 8 2010 at 14:23