Computation of Some Leafwise Cohomology Ring

Abstract

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