# 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!

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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 '10 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 '10 at 14:23

The discrete spectrum is a bit hard to come by, this is one of the central problems in the theory of automorphic forms. The continuous spectrum is given by Eisenstein-series, as is due to Langlands. A good survey on Langlands work is in Arthur's article in the Corvallis proceedings, Symosia in Pure Maths No 33. As for the K-invariants, well they are just that, K-invariant vectors in the representation. The group doesn't act on this space, but a certain subalgebra of the universal envolope of the Lie-algebra does, providing a host of interesting differential operators, see Helgason's books about this.

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The following works in the generality of a locally compact unimodular group $G$ and $\Gamma$ being a closed unimodular subgroup and $K$ being compact.

$C_c^\infty(G)$ acts on $L^2(\Gamma \backslash G)$ by convolution operators: $$\phi \in C_c^\infty(G): f \in L^2 \longmapsto \left( x \mapsto \int\limits_{G} \phi(g) f(xg) d g\right).$$ Call this algebra representation $\pi$.

Restrict this algebra representation to $C_c^\infty(K\backslash G/K)$ and call it $\sigma$. It acts non-trivial only on the subspace $L^2(\Gamma \backslash G /K)$ by the Schur orthogonality relations.

Let $\hat{G}$ denote the unitary dual of $G$. Note that both algebra representations have a direct integral decomposition into irreducible representations $$\pi = \int\limits_{\hat{G}} \tau \; d \mu_\pi(\tau), \qquad \pi = \int\limits_{\hat{G}} \tau \; d \mu_\sigma(\tau).$$

The support of $\mu_\sigma$ is then contained in the set of the spherical representations of $G$, i.e., those with a $K$-invariant vector. On its support, $\mu_\sigma$ coincides with $\mu_\pi$, whose support may be larger.

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