$B^{\mathrm{\prime }}$-orbits on flag varieties and symmetry breaking

Motivated by branching problems for principal series representations of the Lie group $G=GL(n,\mathbb{R})$, we consider all pairs $(G^{\mathrm{\prime }},P)$ with $G^{\mathrm{\prime }}$ being the Levy factor of a parabolic subgroup of $G$ and $P$ a parabolic subgroup of $G$ for which a Borel subgroup $B^{\mathrm{\prime }}$ of $G^{\mathrm{\prime }}$ has finitely many orbits on $G\mathrm{/}P$. We classify all such pairs $(G^{\mathrm{\prime }},P)$ for which $B^{\mathrm{\prime }}$-orbits on the generalized flag variety $G\mathrm{/}P$ are determined by invariant functions inspired from the Bruhat decomposition. We also describe explicitly the double coset space $B^{\mathrm{\prime }}\mathrm{\backslash }G\mathrm{/}P$ as well as the closed $B^{\mathrm{\prime }}$-orbits on $G\mathrm{/}P$ whenever $B^{\mathrm{\prime }}$-orbits are computed by these invariant functions.