{
  "id": "1210.1351",
  "version": "v1",
  "published": "2012-10-04T09:33:55.000Z",
  "updated": "2012-10-04T09:33:55.000Z",
  "title": "Olshanski spherical functions for infinite dimensional motion groups of fixed rank",
  "authors": [
    "Margit Rösler",
    "Michael Voit"
  ],
  "journal": "J. Lie Theory 23 (2013), 899--920",
  "categories": [
    "math.CA",
    "math.RT"
  ],
  "abstract": "Consider the Gelfand pairs $(G_p,K_p):=(M_{p,q} \\rtimes U_p,U_p)$ associated with motion groups over the fields $\\mathbb F=\\mathbb R,\\mathbb C,\\mathbb H$ with $p\\geq q$ and fixed $q$ as well as the inductive limit $p\\to\\infty$,the Olshanski spherical pair $(G_\\infty,K_\\infty)$. We classify all Olshanski spherical functions of $(G_\\infty,K_\\infty)$ as functions on the cone $\\Pi_q$ of positive semidefinite $q\\times q$-matrices and show that they appear as (locally) uniform limits of spherical functions of $(G_p,K_p)$ as $p\\to\\infty$. The latter are given by Bessel functions on $\\Pi_q$. Moreover, we determine all positive definite Olshanski spherical functions and discuss related positive integral representations for matrix Bessel functions. We also extend the results to the pairs $(M_{p,q} \\rtimes (U_p\\times U_q),(U_p\\times U_q))$ which are related to the Cartan motion groups of non-compact Grassmannians. Here Dunkl-Bessel functions of type B (for finite $p$) and of type A (for $p\\to\\infty$) appear as spherical functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-04T09:33:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A90",
      "33C80",
      "43A85",
      "22E66"
    ],
    "keywords": [
      "infinite dimensional motion groups",
      "fixed rank",
      "bessel functions",
      "cartan motion groups",
      "positive definite olshanski spherical functions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.1351R"
    }
  }
}