  Let $G$ be the group $SL(2,\mathbb{R})$, $P\subset G$ be the parabolic
subgroup of upper triangular matrices and $\Gamma\subset G$ be a cocompact
lattice. A right action of $P$ on $\Gamma\backslash G$ defines an orbit
foliation $\mathcal{F}_P$. We compute the leafwise cohomology ring
$H^*(\mathcal{F}_P)$ by exploiting non-abelian harmonic analysis on $G$.
