Stationary measures for SL₂(ℝ)-actions on homogeneous bundles over flag varieties

Abstract

Let 𝐺 be a real semisimple Lie group with finite centre and without compact factors, Q < G Q<G a parabolic subgroup and 𝑋 a homogeneous space of 𝐺 admitting an equivariant projection on the flag variety G / Q G/Q with fibres given by copies of lattice quotients of a semisimple factor of 𝑄. Given a probability measure 𝜇, Zariski-dense in a copy of H = SL 2 ⁡ ( R ) H=\operatorname{SL}_{2}(\mathbb{R}) in 𝐺, we give a description of 𝜇-stationary probability measures on 𝑋 and prove corresponding equidistribution results. Contrary to the results of Benoist–Quint corresponding to the case G = Q G=Q , the type of stationary measures that 𝜇 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.