In Bump, Automorphic forms and representations, Thm. 2.3.3 (p. 174), there is a theorem to the effect that the right regular representation of $G$ on the Hilbert space
$L^2(\Gamma \backslash G, \chi)$ decomposes into a Hilbert space direct sum of irreducible subrepresentations. Here $G$ is, say, a connected Lie group with $\Gamma$ a cocompact discrete subgroup, and $L^2(\Gamma \backslash G, \chi)$ is the Hilbert space of square-integrable (on a fundamental domain for $\Gamma \backslash G$) functions on $G$ with the property that $f(\gamma g) = \chi(\gamma)f(g)$ for $\gamma \in \Gamma$. In Bump we have $G = PGL_2^+(\mathbf{R})$ but I don't think this matters in the context of this question. The method of proof is by applying the spectral theorem for compact operators to the operator $\rho(\phi)$ obtained by integrating the right regular representation $\rho$ against an appropriate compactly-supported smooth function $\phi$ on $G$. I think the exact same proof can be applied to prove part II of the Peter-Weyl theorem on Wikipedia: any unitary Hilbert space representation of a compact group $G$ can be decomposed as a Hilbert space direct sum of irreducible subrepresentations.
Is there a way to use part II of the Peter-Weyl theorem to directly deduce the theorem in Bump? $L^2(\Gamma \backslash G, \chi)$ is isomorphic as a vector space to $L^2(\Gamma \backslash G)$, but I don't know if this isomorphism can be chosen to be $G$-intertwining.
Also, can one also show in this case that the irreducible components of $L^2(\Gamma \backslash G, \chi)$ are finite-dimensional?
Best Answer
$\chi$ has finite image ? $H=\ker(\chi)$.
Given an irreducible representation of $G$ (acting on the right) on $L^2(H\setminus G)$, as $H\setminus \Gamma$ is abelian ie. $H \gamma\gamma'=H \gamma'\gamma$, the map $$\Pi_\chi f(x) = \sum_{\gamma\in H\setminus \Gamma} \chi(\gamma)f(H\gamma x)$$ sends it either to $0$ or to an irreducible representation of $G$ on $$L^2(\Gamma\setminus G,\chi)=\{ f\in L^2(H\setminus G), \forall \gamma\in \Gamma,x\in G, f(H\gamma x)=\chi(H\gamma)f(Hx)\}$$