{
  "id": "1408.3370",
  "version": "v1",
  "published": "2014-08-14T18:10:18.000Z",
  "updated": "2014-08-14T18:10:18.000Z",
  "title": "On special representations of $p$-adic reductive groups",
  "authors": [
    "Elmar Grosse-Klönne"
  ],
  "journal": "Duke Mathematical Journal 163, No. 12, 2179 -- 2216 (2014)",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "Let $F$ be a non-Archimedean locally compact field, let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\\subset G$ and a ring $L$ we consider the $G$-representation on the $L$-module$$(*)\\quad\\quad\\quad\\quad C^{\\infty}(G/Q,L)/\\sum_{Q'\\supsetneq Q}C^{\\infty}(G/Q',L).$$Let $I\\subset G$ denote an Iwahori subgroup. We define a certain free finite rank $L$-module ${\\mathfrak M}$ (depending on $Q$; if $Q$ is a Borel subgroup then $(*)$ is the Steinberg representation and ${\\mathfrak M}$ is of rank one) and construct an $I$-equivariant embedding of $(*)$ into $C^{\\infty}(I,{\\mathfrak M})$. This allows the computation of the $I$-invariants in $(*)$. We then prove that if $L$ is a field with characteristic equal to the residue characteristic of $F$ and if $G$ is a classical group, then the $G$-representation $(*)$ is irreducible. This is the analog of a theorem of Casselman (which says the same for $L={\\mathbb C}$); it had been conjectured by Vign\\'eras. Herzig (for $G={\\rm GL}_n(F)$) and Abe (for general $G$) have given classification theorems for irreducible admissible modulo $p$ representations of $G$ in terms of supersingular representations. Some of their arguments rely on the present work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-14T18:10:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "adic reductive groups",
      "special representations",
      "non-archimedean locally compact field",
      "free finite rank",
      "residue characteristic"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.3370G"
    }
  }
}