{
  "id": "2007.14105",
  "version": "v1",
  "published": "2020-07-28T10:22:06.000Z",
  "updated": "2020-07-28T10:22:06.000Z",
  "title": "Homogeneous Hermitian Holomorphic Vector Bundles And Operators In The Cowen-Douglas Class Over The Poly-disc",
  "authors": [
    "Prahllad Deb",
    "Somnath Hazra"
  ],
  "comment": "30 pages, 1 figure",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this article, we obtain two sets of results. The first set of complete results are exclusively for the case of the bi-disc while the second set of results describe in part, which of these carry over to the general case of the poly-disc: * A classification of irreducible hermitian holomorphic vector bundles over $\\mathbb{D}^2$, homogeneous with respect to $\\mbox{M\\\"ob}\\times \\mbox{M\\\"ob}$, is obtained assuming that the associated representations are \\textit{multiplicity-free}. Among these the ones that give rise to an operator in the Cowen-Douglas class of $\\mathbb{D}^2$ of rank $1,2$ or $3$ is determined. * Any hermitian holomorphic vector bundle of rank $2$ over $\\mathbb{D}^n$, homogeneous with respect to the $n$-fold product of the group $\\mbox{M\\\"ob}$ is shown to be a tensor product of $n-1$ hermitian holomorphic line bundles, each of which is homogeneous with respect to $\\mbox{M\\\"ob}$ and a hermitian holomorphic vector bundle of rank $2$, homogeneous with respect to $\\mbox{M\\\"ob}$. * The classification of irreducible homogeneous hermitian holomorphic vector buldles over $\\mathbb{D}^2$ of rank $3$ (as well as the corresponding Cowen-Douglas class of operators) is extended to the case of $\\mathbb{D}^n$, $n>2$. * It is shown that there is no irreducible $n$ - tuple of operators in the Cowen-Douglas class $\\mathrm B_2(\\mathbb{D}^n)$ that is homogeneous with respect $\\mbox{Aut}(\\mathbb{D}^n)$, $n >1$. Also, pairs of operators in $\\mathrm B_3(\\mathbb{D}^2)$ homogeneous with respect to $\\mbox{Aut}(\\mathbb{D}^2)$ are produced, while it is shown that no $n$ - tuple of operators in $\\mathrm B_3(\\mathbb{D}^n)$ is homogeneous with respect to $\\mbox{Aut}(\\mathbb{D}^n)$, $n > 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-28T10:22:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B32",
      "47B13",
      "17B10",
      "20C25",
      "53C07"
    ],
    "keywords": [
      "homogeneous hermitian holomorphic vector bundles",
      "cowen-douglas class",
      "hermitian holomorphic vector buldles",
      "hermitian holomorphic line bundles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}