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 convolution operator restricted to $\pi^\bot$ is compact, hence this part decomposes discretely.