{
  "id": "1212.0002",
  "version": "v1",
  "published": "2012-11-30T12:58:05.000Z",
  "updated": "2012-11-30T12:58:05.000Z",
  "title": "On the product formula on non-compact Grassmannians",
  "authors": [
    "Piotr Graczyk",
    "Patrice Sawyer"
  ],
  "categories": [
    "math.RT",
    "math.PR"
  ],
  "abstract": "We study the absolute continuity of the convolution $\\delta_{e^X}^\\natural \\star \\delta_{e^Y}^\\natural$ of two orbital measures on the symmetric space $SO_0(p,q)/SO(p)\\timesSO(q)$, $q>p$. We prove sharp conditions on $X$, $Y\\in\\a$ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions. We show that the sharp criterion developed for $\\SO_0(p,q)/\\SO(p)\\times\\SO(q)$ will also serve for the spaces $SU(p,q)/S(U(p)\\timesU(q))$ and $Sp(p,q)/Sp(p)\\timesSp(q)$, $q>p$. We also apply our results to the study of absolute continuity of convolution powers of an orbital measure $\\delta_{e^X}^\\natural$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-30T12:58:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "product formula",
      "non-compact grassmannians",
      "orbital measure",
      "absolute continuity",
      "convolution powers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.0002G"
    }
  }
}