{
  "id": "1906.00934",
  "version": "v1",
  "published": "2019-06-03T17:19:46.000Z",
  "updated": "2019-06-03T17:19:46.000Z",
  "title": "Deligne-Lusztig duality on the stack of local systems",
  "authors": [
    "Dario Beraldo"
  ],
  "comment": "preliminary version",
  "categories": [
    "math.RT",
    "math.AG"
  ],
  "abstract": "In the setting of the geometric Langlands conjecture, we argue that the phenomenon of divergence at infinity on Bun_G (that is, the difference between $!$-extensions and $*$-extensions) is controlled, Langlands-dually, by the locus of semisimple $\\check{G}$-local systems. To see this, we first rephrase the question in terms of Deligne-Lusztig duality and then study the Deligne-Lusztig functor DL_G^\\spec acting on the spectral Langlands DG category IndCoh_N(LS_G). We prove that DL_G^\\spec is the projection IndCoh_N(LS_G) \\to QCoh(LS_G), followed by the action of a coherent D-module St_G which we call the {Steinberg} D-module. We argue that St_G might be regarded as the dualizing sheaf of the locus of semisimple $G$-local systems. We also show that DL_G^\\spec, while far from being conservative, is fully faithful on the subcategory of compact objects.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-03T17:19:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local systems",
      "deligne-lusztig duality",
      "spectral langlands dg category",
      "geometric langlands conjecture",
      "extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}