Stationary measures for $\mathrm{SL}_2(\mathbb{R})$-actions on homogeneous bundles over flag varieties

Let $G$ be a real semisimple Lie group with finite centre and without compact factors, $Q<G$ a parabolic subgroup and $X$ a homogeneous space of $G$ admitting an equivariant projection on the flag variety $G/Q$ with fibres given by copies of lattice quotients of a semisimple factor of $Q$. Given a probability measure $\mu$, Zariski-dense in a copy of $H=\mathrm{SL}_2(\mathbb{R})$ in $G$, we give a description of $\mu$-stationary probability measures on $X$ and prove corresponding equidistribution results. Contrary to the results of Benoist-Quint corresponding to the case $G=Q$, the type of stationary measures that $\mu$ admits depends strongly on the position of $H$ relative to $Q$. We describe possible cases and treat all but one of them, among others using ideas from the works of Eskin-Mirzakhani and Eskin-Lindenstrauss.