{
  "id": "1201.3776",
  "version": "v1",
  "published": "2012-01-18T13:02:32.000Z",
  "updated": "2012-01-18T13:02:32.000Z",
  "title": "Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials",
  "authors": [
    "Michael Voit"
  ],
  "journal": "Coll. Math. 123, 149-179 (2011)",
  "categories": [
    "math.CA",
    "math.RT"
  ],
  "abstract": "Let $p,q$ positive integers. The groups $U_p(\\b C)$ and $U_p(\\b C)\\times U_q(\\b C) $ act on the Heisenberg group $H_{p,q}:=M_{p,q}(\\b C)\\times \\b R$ canonically as groups of automorphisms where $M_{p,q}(\\b C)$ is the vector space of all complex $p\\times q$-matrices. The associated orbit spaces may be identified with $\\Pi_q\\times \\b R$ and $\\Xi_q\\times \\b R$ respectively with the cone $\\Pi_q$ of positive semidefinite matrices and the Weyl chamber $\\Xi_q={x\\in\\b R^q: x_1\\ge...\\ge x_q\\ge 0}$. In this paper we compute the associated convolutions on $\\Pi_q\\times \\b R$ and $\\Xi_q\\times \\b R$ explicitly depending on $p$. Moreover, we extend these convolutions by analytic continuation to series of convolution structures for arbitrary parameters $p\\ge 2q-1$. This leads for $q\\ge 2$ to continuous series of noncommutative hypergroups on $\\Pi_q\\times \\b R$ and commutative hypergroups on $\\Xi_q\\times \\b R$. In the latter case, we describe the dual space in terms of multivariate Laguerre and Bessel functions on $\\Pi_q$ and $\\Xi_q$. In particular, we give a non-positive product formula for these Laguerre functions on $\\Xi_q$. The paper extends the known case $q=1$ due to Koornwinder, Trimeche, and others as well as the group case with integers $p$ due to Faraut, Benson, Jenkins, Ratcliff, and others. Moreover, it is closely related to product formulas for multivariate Bessel and other hypergeometric functions of R\\\"osler.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-18T13:02:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A62",
      "33C67",
      "33C52",
      "43A90",
      "43A20"
    ],
    "keywords": [
      "product formula",
      "multivariate laguerre polynomials",
      "multidimensional heisenberg convolutions",
      "vector space",
      "multivariate bessel"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.3776V"
    }
  }
}