{
  "id": "0903.1413",
  "version": "v2",
  "published": "2009-03-08T13:06:53.000Z",
  "updated": "2011-06-17T13:16:51.000Z",
  "title": "Multiplicity one theorems: the Archimedean case",
  "authors": [
    "Binyong Sun",
    "Chen-Bo Zhu"
  ],
  "comment": "To appear in Annals of Mathematics",
  "journal": "Ann. of Math. (2) 175 (2012), no. 1, 23-44",
  "doi": "10.4007/annals.2012.175.1.2",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be one of the classical Lie groups $\\GL_{n+1}(\\R)$, $\\GL_{n+1}(\\C)$, $\\oU(p,q+1)$, $\\oO(p,q+1)$, $\\oO_{n+1}(\\C)$, $\\SO(p,q+1)$, $\\SO_{n+1}(\\C)$, and let $G'$ be respectively the subgroup $\\GL_{n}(\\R)$, $\\GL_{n}(\\C)$, $\\oU(p,q)$, $\\oO(p,q)$, $\\oO_n(\\C)$, $\\SO(p,q)$, $\\SO_n(\\C)$, embedded in $G$ in the standard way. We show that every irreducible Casselman-Wallach representation of $G'$ occurs with multiplicity at most one in every irreducible Casselman-Wallach representation of $G$. Similar results are proved for the Jacobi groups $\\GL_{n}(\\R)\\ltimes \\oH_{2n+1}(\\R)$, $\\GL_{n}(\\C)\\ltimes \\oH_{2n+1}(\\C)$, $\\oU(p,q)\\ltimes \\oH_{2p+2q+1}(\\R)$, $\\Sp_{2n}(\\R)\\ltimes \\oH_{2n+1}(\\R)$, $\\Sp_{2n}(\\C)\\ltimes \\oH_{2n+1}(\\C)$, with their respective subgroups $\\GL_{n}(\\R)$, $\\GL_{n}(\\C)$, $\\oU(p,q)$, $\\Sp_{2n}(\\R)$, $\\Sp_{2n}(\\C)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-06-17T13:16:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E30",
      "22E46"
    ],
    "keywords": [
      "archimedean case",
      "multiplicity",
      "irreducible casselman-wallach representation",
      "jacobi groups",
      "classical lie groups"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Princeton University and the Institute for Advanced Study",
      "journal": "Ann. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.1413S"
    }
  }
}