{
  "id": "1505.07185",
  "version": "v1",
  "published": "2015-05-27T04:35:33.000Z",
  "updated": "2015-05-27T04:35:33.000Z",
  "title": "Deligne--Lusztig constructions for division algebras and the local Langlands correspondence, II",
  "authors": [
    "Charlotte Chan"
  ],
  "comment": "27 pages",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive $p$-adic groups, analogous to Deligne-Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig's program. Precisely, let $X$ be the $p$-adic Deligne-Lusztig ind-scheme associated to a division algebra $D$ of invariant k/n over a non-Archimedean local field $K$. We give a description of $X$ as a mirabolic-analogue of an affine Deligne-Lusztig variety and study its homology groups by establishing a Deligne-Lusztig theory for families of finite unipotent groups that arise as subquotients of $D^\\times$. The homology of $X$ induces a natural correspondence between quasi-characters of the (multiplicative group of the) unramified degree-$n$ extension of $K$ and representations of $D^{\\times}$. For a broad class of characters $\\theta,$ we show that the representation $H_\\bullet(X)[\\theta]$ is irreducible and concentrated in a single degree. Moreover, we show that this correspondence matches the bijection given by local Langlands and Jacquet-Langlands. As a corollary, we obtain a geometric realization of Jacquet-Langlands transfers between representations of division algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-27T04:35:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local langlands correspondence",
      "division algebra",
      "deligne-lusztig constructions",
      "deligne-lusztig theory",
      "non-archimedean local field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150507185C"
    }
  }
}