{
  "id": "1808.08944",
  "version": "v1",
  "published": "2018-08-27T17:46:32.000Z",
  "updated": "2018-08-27T17:46:32.000Z",
  "title": "An induction theorem for groups acting on trees",
  "authors": [
    "Martin H. Weissman"
  ],
  "comment": "6 pages",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "If $G$ is a group acting on a tree $X$, and ${\\mathcal S}$ is a $G$-equivariant sheaf of vector spaces on $X$, then its compactly-supported cohomology is a representation of $G$. Under a finiteness hypothesis, we prove that if $H_c^0(X, {\\mathcal S})$ is an irreducible representation of $G$ and all other cohomology groups vanish, then $H_c^0(X, {\\mathcal S})$ arises by induction from a vertex or edge stabilizing subgroup. If $G$ is a reductive group over a nonarchimedean local field $F$, then Schneider and Stuhler realize every irreducible supercuspidal representation of $G(F)$ in the degree-zero cohomology of a $G(F)$-equivariant sheaf on its reduced Bruhat-Tits building $X$. When the derived subgroup of $G$ has relative rank one, $X$ is a tree. An immediate consequence is that every such irreducible supercuspidal representation arises by induction from a compact-mod-center open subgroup.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-27T17:46:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G25",
      "20E08",
      "22E50"
    ],
    "keywords": [
      "induction theorem",
      "groups acting",
      "equivariant sheaf",
      "irreducible supercuspidal representation arises",
      "compact-mod-center open subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}