Hi, I have a short question concerning the spectral theory of automorphic forms. What is the main property of the unipotent group $N$, which consist of matrices in the form `\begin{pmatrix} 1 & t \\ 0 &1 \end{pmatrix}`

in $GL(2)$, which provides that the cuspidal spectrum decomposes discretely?

The background: Consider $G = GL_2(\mathbb{A})$, $\Gamma =GL_2(\mathbb{Q})$ and $Z$ the centrum of $G$, then we decompose $$Ind_{ \Gamma Z}^G 1 = \pi \oplus \pi^\bot,$$ where $\pi = Ind_{N\Gamma Z}^G$. The projection onto $\pi$ is given in terms of the integral $$ P : \phi(g) \mapsto \int\limits_{N(\mathbb{A})} \phi(ng) d g.$$ Now given a bounded function $f$ on $\Gamma \backslash G / \Gamma$, with suitable decay properties, we can define the operator

$$Tf : \phi \mapsto f * \phi.$$

Why is $(1-P)Tf(1-P)$ a Hilbert Schmidt operator?

My guess is that an answer should include the Iwasawa decomposition and an according decomposition of the integral operator.