Example of infinite automorphic multiplicity Let $G$ be a locally compact group and $\Gamma$ a lattice in $G$. For an irreducible unitary representation $\pi$ of $G$ let 
$$
m_\Gamma(\pi)=\dim\mathrm{Hom}_G(\pi,L^2(\Gamma\backslash G))
$$
be its automorphic multiplicity.
In [1], Chapter 3 one finds that $m_\Gamma(\pi)$ is finite if $G$ is a reductive Lie group and $\Gamma$ is arithmetic. The authors seem to indicate that there are examples when $m_\Gamma(\pi)=\infty$, but they don't give one.
Where can I find such an example? Are there examples with $G$ being a Lie group?
[1] Osborne, M. Scott; Warner, Garth:
The theory of Eisenstein systems. 
 Academic Press, New York-London, 1981.
 A: Assuming cocompactness, I think that this is finite in much greater generality than what you are mentioning. (no need to deal with Lie groups, no reference to any sort of arithmeticity)
Theorem: Let $\Gamma$ be a cocompact discrete subgroup of a locally compact group $G$. Then $L^{2}(\Gamma \backslash G)$ is the Hilbertian direct sum of countably many irreducible representations of $G$, each occuring with finite multiplicity.
Some comments. Here $G$ acts by right translations on $L^{2}(\Gamma \backslash G)$, this defines the unitary representation. The theorem is valid for "twisted coefficient" or "induced representations" (depending on one's preferred vocabulary); i.e we can start with a finite dimensional unitary representation of $\Gamma$, and induce it. The same statement will apply to the corresponding representation of $G$.
(This theorem is stated on page 23 in the book "Representation theory and automorphic functions" by Gelfand, Graev, Pyatetskii-Shapiro. The reference to that book is pointed out in Borel and Wallach's book "Continuous cohomology, discrete subgroups and representations of reductive groups". The above theorem is fundamental to start discussing Matsushima's formula in the context of reductive or semisimple Lie groups, as done by Borel and Wallach.)
The non-cocompact case if more involved (and I don't know much about it so cannot say more).
(see also When does a unitary Hilbert space rep of a reductive Lie group decompose into a direct sum of irreps with finite multiplicities? )
