{
  "id": "2112.13524",
  "version": "v1",
  "published": "2021-12-27T05:45:16.000Z",
  "updated": "2021-12-27T05:45:16.000Z",
  "title": "Whittaker category for the Lie algebra of polynomial vector fields",
  "authors": [
    "Yufang Zhao",
    "Genqiang Liu"
  ],
  "journal": "Journal of Algebra2022",
  "doi": "10.1016/j.jalgebra.2022.04.025",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "For any positive integer $n$, let $A_n=\\mathbb{C}[t_1,\\dots,t_n]$, $W_n=\\text{Der}(A_n)$ and $\\Delta_n=\\text{Span}\\{\\frac{\\partial}{\\partial{t_1}},\\dots,\\frac{\\partial}{\\partial{t_n}}\\}$. Then $(W_n, \\Delta_n)$ is a Whittaker pair. A $W_n$-module $M$ on which $\\Delta_n$ operates locally finite is called a Whittaker module. We show that each block $\\Omega_{\\mathbf{a}}^{\\widetilde{W}}$ of the category of $(A_n,W_n)$-Whittaker modules with finite dimensional Whittaker vector spaces is equivalent to the category of finite dimensional modules over $L_n$, where $L_n$ is the Lie subalgebra of $W_n$ consisting of vector fields vanishing at the origin. As a corollary, we classify all simple non-singular Whittaker $W_n$-modules with finite dimensional Whittaker vector spaces using $\\mathfrak{gl}_n$-modules. We also obtain an analogue of Skryabin's equivalence for the non-singular block $\\Omega_{\\mathbf{a}}^W$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-12-27T05:45:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B15",
      "17B65"
    ],
    "keywords": [
      "polynomial vector fields",
      "finite dimensional whittaker vector spaces",
      "lie algebra",
      "whittaker category"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}