{
  "id": "1505.01149",
  "version": "v1",
  "published": "2015-05-05T13:17:17.000Z",
  "updated": "2015-05-05T13:17:17.000Z",
  "title": "The absolute continuity of convolutions of orbital measures in symmetric spaces",
  "authors": [
    "Sanjiv Kumar Gupta",
    "Kathryn E. Hare"
  ],
  "categories": [
    "math.RT",
    "math.CA"
  ],
  "abstract": "We characterize the absolute continuity of convolution products of orbital measures on the classical, irreducible Riemannian symmetric spaces $G/K$ of Cartan type $III$, where $G$ is a non-compact, connected Lie group and $K$ is a compact, connected subgroup. By the orbital measures, we mean the uniform measures supported on the double cosets, $KzK,$ in $G$. The characterization can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements $z$. A consequence of our work is to show that the convolution product of any rank% $G/K,$ continuous, $K$-bi-invariant measures is absolutely continuous in any of these symmetric spaces, other than those whose restricted root system is type $A_{n}$ or $D_{3}$, when rank$G/K$ $+1$ is needed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-05T13:17:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A80",
      "22E30",
      "43A90",
      "53C35"
    ],
    "keywords": [
      "orbital measures",
      "absolute continuity",
      "convolution product",
      "irreducible riemannian symmetric spaces",
      "connected lie group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150501149G"
    }
  }
}