{
  "id": "2309.02540",
  "version": "v1",
  "published": "2023-09-05T19:28:46.000Z",
  "updated": "2023-09-05T19:28:46.000Z",
  "title": "Toeplitz operators on the Siegel domain and the Heisenberg group",
  "authors": [
    "Julio A. Barrera-Reyes",
    "Raul Quiroga-Barranco"
  ],
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "For the usual action of the Heisenberg group $\\mathbb{H}_n$ on the Siegel domain $D_{n+1}$ ($n \\geq 1$) denote by $\\mathcal{T}^{(\\lambda)}(L^\\infty(D_{n+1})^{\\mathbb{H}_n})$ the $C^*$-algebra acting on the weighted Bergman space $\\mathcal{A}^2_\\lambda(D_{n+1})$ ($\\lambda > -1$) generated by Toeplitz operators whose symbols belong to $L^\\infty(D_{n+1})^{\\mathbb{H}_n}$ (essentially bounded and $\\mathbb{H}_n$-invariant). We prove that $\\mathcal{T}^{(\\lambda)}(L^\\infty(D_{n+1})^{\\mathbb{H}_n})$ is commutative and isomorphic to $\\mathrm{VSO}(\\mathbb{R}_+)$ (very slowly oscillating functions on $\\mathbb{R}_+$), for every $\\lambda > -1$ and $n \\geq 1$. We achieve this by Lie theory and symplectic geometry methods. We also compute the moment map of the $\\mathbb{H}_n$-action and some of its subgroups. A decomposition of the Bergman spaces $\\mathcal{A}^2_\\lambda(D_{n+1})$ as direct integrals of Fock spaces is also obtained and used to explain our main results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-05T19:28:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B35",
      "22E25",
      "53D20"
    ],
    "keywords": [
      "siegel domain",
      "heisenberg group",
      "toeplitz operators",
      "symplectic geometry methods",
      "weighted bergman space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}