{
  "id": "2102.11247",
  "version": "v1",
  "published": "2021-02-22T18:32:39.000Z",
  "updated": "2021-02-22T18:32:39.000Z",
  "title": "A generalized semi-infinite Hecke equivalence and the local geometric Langlands correspondence",
  "authors": [
    "Alexey Sevostyanov"
  ],
  "comment": "11 pages. arXiv admin note: text overlap with arXiv:math/0004139",
  "categories": [
    "math.RT"
  ],
  "abstract": "We introduce a class of equivalences, which we call generalized semi-infinite Hecke equivalences, between certain categories of representations of graded associative algebras which appear in the setting of semi-infinite cohomology for associative algebras and categories of representations of related algebras of Hecke type which we call semi-infinite Hecke algebras. As an application we obtain an equivalence between a category of representations of a non-twisted affine Lie algebra $\\widehat{\\mathfrak g}$ of level $-2h^\\vee-k$, where $h^\\vee$ is the dual Coxeter number of the underlying semisimple Lie algebra $\\mathfrak g$ and $k\\in \\mathbb{C}$, and the category of finitely generated representations of the W-algebra associated to $\\widehat{\\mathfrak g}$ of level $k$. When $k=-h^\\vee$ this yields an equivalence between a category of representations of $\\widehat{\\mathfrak g}$ of central charge $-h^\\vee$ and the category ${\\rm Coh}({\\rm Op}_{^LG}(D^\\times))$ of coherent sheaves on the space ${\\rm Op}_{^LG}(D^\\times)$ of $^LG$-opers on the punctured disc $D^\\times$, where $^LG$ is the Langlands dual group to the algebraic group of adjoint type with Lie algebra $\\mathfrak g$. This can be regarded as a version of the local geometric Langlands correspondence. The above mentioned equivalences generalize to the case of affine Lie algebras the Skryabin equivalence between the categories of generalized Gelfand-Graev representations of $\\mathfrak g$ and the categories of representations of the corresponding finitely generated W-algebras, and Kostant's results on the classification of Whittaker modules over $\\mathfrak g$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-22T18:32:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B67",
      "81R10",
      "16G99",
      "22E57"
    ],
    "keywords": [
      "local geometric langlands correspondence",
      "generalized semi-infinite hecke equivalence",
      "representations",
      "affine lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}