{
  "id": "2405.10382",
  "version": "v1",
  "published": "2024-05-16T18:22:08.000Z",
  "updated": "2024-05-16T18:22:08.000Z",
  "title": "Cartan subalgebras for restrictions of $\\mathfrak{g}$-modules",
  "authors": [
    "Masatoshi Kitagawa"
  ],
  "comment": "31 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "In this paper, we deal with the $\\mathcal{U}(\\mathfrak{g})$-action on a $\\mathfrak{g}$-module on which a larger algebra $\\mathcal{A}$ acts irreducibly. Under a mild condition, we will show that the support of the $\\mathcal{Z}(\\mathfrak{g})$-action is a union of affine subspaces in the dual of a Cartan subalgebra modulo the Weyl group action. As a consequence, we propose a definition of a Cartan subalgebra for such a $\\mathfrak{g}$-module. The support of the $\\mathcal{Z}(\\mathfrak{g})$-module is an algebraic counterpart of the support of the measure in the irreducible decomposition of a unitary representation. This consideration is motivated by the theory of the discrete decomposability initiated by T. Kobayashi. Defining a Cartan subalgebra for a $\\mathfrak{g}$-module is motivated by the study of I. Losev on Poisson $G$-varieties. These are related each other through the associated variety and the nilpotent orbit associated to a $\\mathfrak{g}$-module.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-16T18:22:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "restrictions",
      "cartan subalgebra modulo",
      "weyl group action",
      "nilpotent orbit",
      "affine subspaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}