{
  "id": "1710.07271",
  "version": "v1",
  "published": "2017-10-19T17:59:43.000Z",
  "updated": "2017-10-19T17:59:43.000Z",
  "title": "A minimal representation of the orthosymplectic Lie superalgebra",
  "authors": [
    "Sigiswald Barbier",
    "Jan Frahm"
  ],
  "comment": "45 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "We construct a minimal representation of the orthosymplectic Lie supergroup $OSp(p,q|2n)$, generalising the Schr\\\"{o}dinger model of the minimal representation of $O(p,q)$ to the super case. The underlying Lie algebra representation is realized on functions on the minimal orbit inside the Jordan superalgebra associated with $\\mathfrak{osp}(p,q|2n)$, so that our construction is in line with the orbit philosophy. Its annihilator is given by a Joseph-like ideal for $\\mathfrak{osp}(p,q|2n)$, and therefore the representation is a natural generalization of a minimal representations to the context of Lie superalgebras. We also construct a non-degenerate sesquilinear form for which the representation is skew-symmetric and which is the analogue of an $L^2$-inner product in the supercase, and calculate its Gelfand--Kirillov dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-19T17:59:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B60",
      "22E46",
      "58C50"
    ],
    "keywords": [
      "minimal representation",
      "orthosymplectic lie superalgebra",
      "lie algebra representation",
      "orthosymplectic lie supergroup",
      "minimal orbit inside"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}