{
  "id": "2211.06911",
  "version": "v1",
  "published": "2022-11-13T14:23:14.000Z",
  "updated": "2022-11-13T14:23:14.000Z",
  "title": "Stationary measures for $\\mathrm{SL}_2(\\mathbb{R})$-actions on homogeneous bundles over flag varieties",
  "authors": [
    "Alexander Gorodnik",
    "Jialun Li",
    "Cagri Sert"
  ],
  "categories": [
    "math.DS"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-13T14:23:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stationary measures",
      "flag variety",
      "homogeneous bundles",
      "real semisimple lie group",
      "stationary probability measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}