{
  "id": "1008.3292",
  "version": "v2",
  "published": "2010-08-19T13:11:09.000Z",
  "updated": "2011-08-06T20:08:30.000Z",
  "title": "On the endomorphism algebra of generalised Gelfand-Graev representations",
  "authors": [
    "Matthew C. Clarke"
  ],
  "comment": "18 pages. To appear in Transactions of the American Mathematical Society",
  "categories": [
    "math.RT",
    "math.AG",
    "math.GR"
  ],
  "abstract": "Let $G$ be a connected reductive algebraic group defined over the finite field $\\F_q$, where $q$ is a power of a good prime for $G$, and let $F$ denote the corresponding Frobenius endomorphism, so that $G^F$ is a finite reductive group. Let $u \\in G^F$ be a unipotent element and let $\\Gamma_u$ be the associated generalised Gelfand-Graev representation of $G^F$. Under the assumption that $G$ has a connected centre, we show that the dimension of the endomorphism algebra of $\\Gamma_u$ is a polynomial in $q$, with degree given by $\\dim C_G(u)$. When the centre of $G$ is disconnected, it is impossible, in general, to parametrise the (isomorphism classes of) generalised Gelfand-Graev representations independently of $q$, unless one adopts a convention of considering separately various congruence classes of $q$. Subject to such a convention we extend our result.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-08-06T20:08:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "endomorphism algebra",
      "connected reductive algebraic group",
      "corresponding frobenius endomorphism",
      "finite reductive group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.3292C"
    }
  }
}