{
  "id": "2002.01540",
  "version": "v1",
  "published": "2020-02-04T21:17:41.000Z",
  "updated": "2020-02-04T21:17:41.000Z",
  "title": "Four examples of Beilinson-Bernstein localization",
  "authors": [
    "Anna Romanov"
  ],
  "comment": "21 pages; preliminary version, comments welcome",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathfrak{g}$ be a complex semisimple Lie algebra. The Beilinson-Bernstein localization theorem establishes an equivalence of the category of $\\mathfrak{g}$-modules of a fixed infinitesimal character and a category of modules over a twisted sheaf of differential operators on the flag variety of $\\mathfrak{g}$. In this expository paper, we give four detailed examples of this theorem when $\\mathfrak{g}=\\mathfrak{sl}(2,\\mathbb{C})$. Specifically, we describe the $\\mathcal{D}$-modules associated to finite-dimensional irreducible $\\mathfrak{g}$-modules, Verma modules, Whittaker modules, discrete series representations of $SL(2,\\mathbb{R})$, and principal series representations of $SL(2,\\mathbb{R})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-04T21:17:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "14F10"
    ],
    "keywords": [
      "complex semisimple lie algebra",
      "beilinson-bernstein localization theorem establishes",
      "discrete series representations",
      "principal series representations",
      "flag variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}