{
  "id": "2405.12919",
  "version": "v1",
  "published": "2024-05-21T16:41:23.000Z",
  "updated": "2024-05-21T16:41:23.000Z",
  "title": "The $L_p$-dual space of a semisimple Lie group",
  "authors": [
    "Bachir Bekka"
  ],
  "comment": "18 pages",
  "categories": [
    "math.RT",
    "math.DS",
    "math.OA"
  ],
  "abstract": "Let $G$ be a semisimple Lie group. We describe the irreducible representations of $G$ by linear isometries on $L_p$-spaces for $p\\in (1,+\\infty)$ with $p\\neq 2.$ More precisely, we show that, for every such representation $\\pi,$ there exists a parabolic subgroup $Q$ of $G$ such that $\\pi$ is equivalent to the natural representation of $G$ on $L_p(G/Q)$ twisted by a unitary character of $Q.$ When $G$ is of real rank one, we give a complete classification of the possible irreducible representations of $G$ on an $L_p$-space for $p\\neq 2,$ up to equivalence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-21T16:41:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E46",
      "47L10",
      "37A40"
    ],
    "keywords": [
      "semisimple lie group",
      "dual space",
      "irreducible representations",
      "linear isometries",
      "real rank"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}