{
  "id": "1611.09258",
  "version": "v1",
  "published": "2016-11-28T17:41:46.000Z",
  "updated": "2016-11-28T17:41:46.000Z",
  "title": "Local Langlands correspondence and ramification for Carayol representations",
  "authors": [
    "Colin J. Bushnell",
    "Guy Henniart"
  ],
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with Weil group $\\Cal W_F$. Let $\\sigma$ be an irreducible smooth complex representation of $\\Cal W_F$, realized as the Langlands parameter of an irreducible cuspidal representation $\\pi$ of a general linear group over $F$. In an earlier paper, we showed that the ramification structure of $\\sigma$ is determined by the fine structure of the endo-class $\\varTheta$ of the simple character contained in $\\pi$, in the sense of Bushnell-Kutzko. The connection is made via the {\\it Herbrand function} $\\Psi_\\varTheta$ of $\\varTheta$. In this paper, we concentrate on the fundamental Carayol case in which $\\sigma$ is totally wildly ramified with Swan exponent not divisible by $p$. We show that, for such $\\sigma$, the associated Herbrand function satisfies a certain symmetry condition or functional equation, a property that essentially characterizes this class of representations. We calculate $\\Psi_\\varTheta$ explicitly, in terms of a classical Herbrand function coming from the Bushnell-Kutzko classification of simple characters. We describe exactly the class of functions arising as Herbrand functions $\\Psi_\\varXi$, as $\\varXi$ varies over totally wild endo-classes of Carayol type. In a separate argument, we get a complete description of $\\sigma$ restricted to any ramification subgroup. This provides a different, more Galois-centred, view on $\\Psi_\\varTheta$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-28T17:41:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50",
      "11S37",
      "11S15"
    ],
    "keywords": [
      "local langlands correspondence",
      "carayol representations",
      "ramification",
      "simple character",
      "general linear group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}