{
  "id": "2410.17125",
  "version": "v1",
  "published": "2024-10-22T15:59:37.000Z",
  "updated": "2024-10-22T15:59:37.000Z",
  "title": "Construction of irreducible $\\mathcal{U}(\\mathfrak{g})^{G'}$-modules and discretely decomposable restrictions",
  "authors": [
    "Masatoshi Kitagawa"
  ],
  "comment": "42 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "In this paper, we study the irreducibility of $\\mathcal{U}(\\mathfrak{g})^{G'}$-modules on the spaces of intertwining operators in the branching problem of reductive Lie algebras, and construct a family of finite-dimensional irreducible $\\mathcal{U}(\\mathfrak{g})^{G'}$-modules using the Zuckerman derived functors. We provide criteria for the irreducibility of $\\mathcal{U}(\\mathfrak{g})^{G'}$-modules in the cases of generalized Verma modules, cohomologically induced modules, and discrete series representations. We treat only discrete decomposable restrictions with certain dominance conditions (quasi-abelian and in the good range). To describe the $\\mathcal{U}(\\mathfrak{g})^{G'}$-modules, we give branching laws of cohomologically induced modules using ones of generalized Verma modules when $K'$ acts on $K/L_K$ transitively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-22T15:59:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "discretely decomposable restrictions",
      "generalized verma modules",
      "construction",
      "cohomologically induced modules",
      "discrete series representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}