{
  "id": "2404.07894",
  "version": "v1",
  "published": "2024-04-11T16:27:50.000Z",
  "updated": "2024-04-11T16:27:50.000Z",
  "title": "Typical blocks of the category $\\mathcal O$ and Whittaker modules for Takiff superalgebras",
  "authors": [
    "Chih-Whi Chen",
    "Yongjie Wang"
  ],
  "comment": "33 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "We study the simplicity of Kac induced modules over the $\\ell$-th Takiff superalgebras $\\widetilde{\\mathfrak g}_\\ell:= \\widetilde{\\mathfrak g}\\otimes \\mathbb C[\\theta]/(\\theta^{\\ell+1})$, for $\\ell>0$, associated with the Lie superalgebras $\\widetilde{\\mathfrak g}$ of type I. We formulate a general notion of typical weights and typical Jordan blocks of the category $\\mathcal O$ for $\\widetilde{\\mathfrak g}_\\ell$ associated with Lie superalgebras $\\mathfrak{gl}(m|n)$, $\\mathfrak{osp}(2|2n)$ and $\\mathfrak{pe}(n)$. For Lie superalgebras $\\mathfrak{gl}(m|n)$ and $\\mathfrak{osp}(2|2n)$, we establish an equivalence from an arbitrary typical Jordan block of the category $\\mathcal O$ for $\\widetilde{\\mathfrak g}_\\ell$ to a Jordan block of the category $\\mathcal O$ for the even subalgebra of $\\widetilde{\\mathfrak g}_\\ell$. This provides a solution to the problem of determining the composition multiplicities of the Verma modules over $\\widetilde{\\mathfrak g}_\\ell$ with typical highest weights. We also investigate non-singular Whittaker modules over these Takiff superalgebras. In particular, we obtain a classification of non-singular simple Whittaker modules and a criterion for simplicity of non-singular standard Whittaker modules.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-11T16:27:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B55"
    ],
    "keywords": [
      "typical blocks",
      "lie superalgebras",
      "non-singular standard whittaker modules",
      "non-singular simple whittaker modules",
      "arbitrary typical jordan block"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}