{
  "id": "1210.3507",
  "version": "v1",
  "published": "2012-10-12T13:16:08.000Z",
  "updated": "2012-10-12T13:16:08.000Z",
  "title": "Joseph-like ideals and harmonic analysis for osp(m|2n)",
  "authors": [
    "Kevin Coulembier",
    "Petr Somberg",
    "Vladimir Soucek"
  ],
  "doi": "10.1093/imrn/rnt074",
  "categories": [
    "math.RT",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The Joseph ideal in the universal enveloping algebra U(so(m)) is the annihilator ideal of the so(m)-representation on the harmonic functions on R^{m-2}. The Joseph ideal for sp(2n) is the annihilator ideal of the Segal-Shale-Weil (metaplectic) representation. Both ideals can be constructed in a unified way from a quadratic relation in the tensor algebra of g for g equal to so(m) or sp(2n). In this paper we construct two analogous ideals in the tensor algebra of g and U(g) for g the orthosymplectic Lie superalgebra osp(m|2n)=spo(2n|m) and prove that they have unique characterizations that naturally extend the classical case. Then we show that these two ideals are the annihilator ideals of respectively the osp(m|2n)-representation on the spherical harmonics on R^{m-2|2n} and a generalization of the metaplectic representation to spo(2n|m). This proves that these ideals are reasonable candidates to establish the theory of Joseph-like ideals for Lie superalgebras. We also discuss the relation between the Joseph ideal of osp(m|2n) and the algebra of symmetries of the super conformal Laplace operator, regarded as an intertwining operator between principal series representations for osp(m|2n).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-12T13:16:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B35",
      "16S32",
      "58C50"
    ],
    "keywords": [
      "joseph-like ideals",
      "harmonic analysis",
      "joseph ideal",
      "annihilator ideal",
      "super conformal laplace operator"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.3507C"
    }
  }
}