{
  "id": "1610.00284",
  "version": "v1",
  "published": "2016-10-02T14:36:08.000Z",
  "updated": "2016-10-02T14:36:08.000Z",
  "title": "Whittaker supports for representations of reductive groups",
  "authors": [
    "Raul Gomez",
    "Dmitry Gourevitch",
    "Siddhartha Sahi"
  ],
  "comment": "23 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $F$ be either $\\mathbb{R}$ or a finite extension of $\\mathbb{Q}_p$, and let $G$ be the group of $F$-points of a reductive group defined over $F$. Also let $\\pi$ be a smooth representation of $G$ (Frechet of moderate growth if $F=\\mathbb{R}$). For each nilpotent orbit $\\mathcal{O}$ we consider a certain Whittaker quotient $\\pi_{\\mathcal{O}}$ of $\\pi$. We define the Whittaker support WS$(\\pi)$ to be the set of maximal $\\mathcal{O}$ among those for which $\\pi_{\\mathcal{O}}\\neq 0$. In this paper we prove that all $\\mathcal{O}\\in\\mathrm{WS}(\\pi)$ are quasi-admissible nilpotent orbits, generalizing some of the results in [Moe96,JLS]. If $F$ is $p$-adic and $\\pi$ is quasi-cuspidal then we show that all $\\mathcal{O}\\in\\mathrm{WS}(\\pi)$ are $F$-distinguished, i.e. do not intersect the Lie algebra of any proper Levi subgroup of $G$ defined over $F$. We also give an adaptation of our argument to automorphic representations, generalizing some results from [GRS03,Shen16,JLS,Cai] and confirming a conjecture from [Ginz06]. Our methods are a synergy of the methods of the above-mentioned papers, and of our preceding paper [GGS].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-02T14:36:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G05",
      "20G20",
      "20G25",
      "20G30",
      "20G35",
      "22E27",
      "22E46",
      "22E50",
      "22E55",
      "17B08"
    ],
    "keywords": [
      "reductive group",
      "whittaker support ws",
      "proper levi subgroup",
      "lie algebra",
      "moderate growth"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}