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

TL;DR

This paper classifies and describes stationary measures for certain group actions on homogeneous bundles over flag varieties, revealing how their nature depends on the subgroup position, with implications for equidistribution in homogeneous dynamics.

Contribution

It provides a detailed description of $ ext{SL}_2(R)$-stationary measures on homogeneous bundles over flag varieties, extending previous results and analyzing the influence of subgroup positioning.

Findings

Classification of stationary measures depending on subgroup position

Description of equidistribution phenomena for these measures

Extension of Eskin-Mirzakhani and Eskin-Lindenstrauss techniques

Abstract

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.