# Compactness of integral operators on $L^2_0(\Gamma \backslash G, \chi)$

Let $$G = PGL(2, \mathbb{R})$$, $$\Gamma \subset SL(2, \mathbb{Z})$$ a congruence subgroup, $$\chi: \Gamma \to S^1$$ a character, and $$(\pi, L^2_0(\Gamma \backslash G, \chi))$$ be the usual Hilbert space right regular representation of $$G$$ consisting of $$L^2$$ functions on $$G$$ satisfying $$f(\gamma g) = \chi(\gamma)f(x)$$ and the condition (via Fourier expansion at the cusps) of being cuspidal.

I learned how to deduce the fact that this $$L^2$$-space decomposes into irreducibles with finite multiplicity from the fact that for $$\phi \in C_c^\infty(G)$$, the convolved operator $$\pi(\phi) = \int_G \phi(h)\pi(h)\, dg$$ is compact when restricted to $$L^2_0(\Gamma \backslash G, \chi)$$. But I am not sure how to establish the compactness of this operator when $$\chi$$ is not trivial when restricted to the upper-triangular unipotent part of $$\Gamma$$. The proof I learned in Bump's book Automorphic forms and representations, Theorem 3.2.3, assumes that $$\chi|_{\Gamma \cap N} = 1$$, and I am having trouble salvaging the proof when this is not true.

The key step in the proof is the estimate $$\lVert \pi(\phi) f \rVert_{L^\infty} \ll \lVert f \rVert_{L^2}$$. In Bump's proof (which is reproduced from Gelfand--Graev--Pjateckii-Shapiro which also makes this assumption on $$\chi$$) this is done by writing $$\pi(\phi) f(g) = \int_{(\Gamma \cap N)\backslash G}K(g, h)f(h)\, dh,$$ where $$K(g, h) = \sum_{\gamma \in \Gamma \cap N} \chi(\gamma)\phi(g^{-1}\gamma h)$$. When $$\chi(\gamma) = 1$$ in all those terms, Bump uses the Poisson summation formula to rewrite this as a sum of things that are easily bounded, except the value at $$0$$ of the Fourier transform of $$n \mapsto \chi(n)\phi(g^{-1}nh)$$ as a function on $$N$$ (here $$\chi$$ I treat as the unique extension from $$\Gamma \cap N$$ to $$N$$). The point is then that this term has zero contribution by the assumption that $$f$$ and thus $$\pi(\phi)f$$ is cuspidal: $$\int_{(\Gamma \cap N) \backslash N} \int_{(\Gamma \cap N)\backslash G}K(ng, h)f(h)\, dh\, dn = 0$$ means that the contribution of $$\int_{(\Gamma \cap N) \backslash N} \sum_{\gamma \in \Gamma \cap N} \chi(\gamma)\phi(g^{-1}n^{-1}\gamma h)$$ is zero. This only seems helpful when $$\chi$$ is trivial, otherwise this isn't equal to the value at $$0$$ of the Fourier transform above, which is $$\int_{(\Gamma \cap N) \backslash N} \sum_{\gamma \in \Gamma \cap N} \chi(n^{-1}\gamma)\phi(g^{-1}n^{-1}\gamma h)\, dn$$. So how does one establish the desired bound in general? Can the same technique be made to work?