{
  "id": "math/0407345",
  "version": "v1",
  "published": "2004-07-20T18:53:37.000Z",
  "updated": "2004-07-20T18:53:37.000Z",
  "title": "Distribution of lattice orbits on homogeneous varieties",
  "authors": [
    "Alexander Gorodnik",
    "Barak Weiss"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Given a lattice \\Gamma in a locally compact group G and a closed subgroup H of G, one has a natural action of \\Gamma on the homogeneous space V=H\\G. For an increasing family of finite subsets {\\Gamma_T: T>0}, a dense orbit v\\Gamma, v\\in V, and compactly supported function \\phi on V, we consider the sums S_{\\phi,v}(T)=\\sum_{\\gamma\\in \\Gamma_T} \\phi(v \\gamma). Understanding the asymptotic behavior of S_{\\phi,v}(T) is a delicate problem which has only been considered for certain very special choices of H, G and {\\Gamma_T}. We develop a general abstract approach to the problem, and apply it to the case when G is a Lie group and either H or G is semisimple. When G is a group of matrices equipped with a norm, we have S_{\\phi,v}(T) \\sim \\int_{G_T} \\phi(vg) dg, where G_T={g\\in G:||g||<T} and \\Gamma_T = G_T \\cap \\Gamma. We also show that the asymptotics of S_{\\phi,v}(T) is governed by \\int_V \\phi d\\nu, where \\nu is an explicit limiting density depending on the choice of v and the norm.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-07-20T18:53:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57S30",
      "37A17",
      "22E40",
      "11H55"
    ],
    "keywords": [
      "lattice orbits",
      "homogeneous varieties",
      "distribution",
      "general abstract approach",
      "lie group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......7345G"
    }
  }
}