{
  "id": "2406.08976",
  "version": "v1",
  "published": "2024-06-13T10:14:19.000Z",
  "updated": "2024-06-13T10:14:19.000Z",
  "title": "Tits groups of affine Weyl groups",
  "authors": [
    "Radhika Ganapathy"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2107.01768",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a connected, reductive group over a non-archimedean local field $F$. Let $\\breve F$ be the completion of the maximal unramified extension of $F$ contained in a separable closure $F_s$. In this article, we construct a Tits group of the affine Weyl group of $G(F)$ when the derived subgroup of $G_{\\breve F}$ does not contain a simple factor of unitary type. If $G$ is a quasi-split ramified odd unitary group, we show that there always exist representatives in $G(F)$ of affine simple reflections that satisfy Coxeter relations (which is weaker than asking for the existence of a Tits group). If $G = U_{2r}, r \\geq 3,$ is a quasi-split ramified even unitary group, we show that there don't even exist representatives in $G(F)$ of the affine simple reflections that satisfy Coxeter relations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-13T10:14:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50",
      "20C08"
    ],
    "keywords": [
      "affine weyl group",
      "tits group",
      "affine simple reflections",
      "satisfy coxeter relations",
      "quasi-split ramified odd unitary group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}