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.)