{
  "id": "math/0512474",
  "version": "v1",
  "published": "2005-12-20T17:05:14.000Z",
  "updated": "2005-12-20T17:05:14.000Z",
  "title": "Bessel convolutions on matrix cones",
  "authors": [
    "Margit Rösler"
  ],
  "comment": "33 pages",
  "journal": "Compos. Math. 143 (2007), no.3, 749-779",
  "categories": [
    "math.CA",
    "math.RT"
  ],
  "abstract": "In this paper we introduce probability-preserving convolution algebras on cones of positive semidefinite matrices over one of the division algebras $\\b F = \\b R, \\b C$ or $\\b H$ which interpolate the convolution algebras of radial bounded Borel measures on a matrix space $M_{p,q}(\\b F)$ with $p\\geq q$. Radiality in this context means invariance under the action of the unitary group $U_p(\\b F)$ from the left. We obtain a continuous series of commutative hypergroups whose characters are given by Bessel functions of matrix argument. Our results generalize well-known structures in the rank one case, namely the Bessel-Kingman hypergroups on the positive real line, to a higher rank setting. In a second part of the paper, we study structures depending only on the matrix spectra. Under the mapping $r\\mapsto \\text{spec}(r)$, the convolutions on the underlying matrix cone induce a continuous series of hypergroup convolutions on a Weyl chamber of type $B_q$. The characters are now Dunkl-type Bessel functions. These convolution algebras on the Weyl chamber naturally extend the harmonic analysis for Cartan motion groups associated with the Grassmann manifolds $U(p,q)/(U_p\\times U_q)$ over $\\b F$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-12-20T17:05:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C67",
      "33C80",
      "43A62",
      "43A85"
    ],
    "keywords": [
      "bessel convolutions",
      "convolution algebras",
      "weyl chamber",
      "context means invariance",
      "cartan motion groups"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....12474R"
    }
  }
}