We give a classification of SBOs between spherical principal series representations of $G$ and $G’$, essentially generalizing the results on $(O(1,n+1),O(1,n))$ of Kobayashi-Speh (2015).

Moreover we show how to decompose unitary representations occurring in (not necessarily) spherical principal series representations of $G$ in terms of unitary $G’$ representations, by making use of the knowledge gathered in the classification of the SBOs and the structure of the open $P’$ orbit in $G/P$ as a homogenous $G’$-space, where $P’$ is a minimal parabolic in $G’$ and $P$ is a minimal parabolic in $G$. This includes the construction of discrete spectra in the restriction of complementary series representations and unitarizable composition factors.

The aim of this lecture is to survey an arithmeticity criterium for discrete subgroups which are not assumed to be lattices. This criterium, obtained with Miquel, generalizes works of Selberg and Hee Oh and solves a conjecture of Margulis. It says: a discrete irreducible Zariski-dense subgroup of $G$ that intersects cocompactly at least one horospherical subgroup of $G$ is an arithmetic group.