{
  "id": "0910.5664",
  "version": "v1",
  "published": "2009-10-29T15:46:54.000Z",
  "updated": "2009-10-29T15:46:54.000Z",
  "title": "Algebras of invariant differential operators on a class of multiplicity free spaces",
  "authors": [
    "Hubert Rubenthaler"
  ],
  "journal": "Comptes Rendus de l Acad\\'emie des Sciences - Series I - Mathematics (2009) Rubenthaler",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Let G be a connected reductive algebraic group and let G'=[G,G] be its derived subgroup. Let (G,V) be a multiplicity free representation with a one dimensional quotient (see definition below). We prove that the algebra D(V)^{G'} of G'-invariant differential operators with polynomial coefficients on V, is a quotient of a so-called Smith algebra over its center. Over C this class of algebras was introduced by S.P. Smith as a class of algebras similar to the enveloping algebra U(sl(2)) of sl(2). Our result generalizes the case of the Weil representation, where the associative algebra generated by Q(x) and Q(?) (Q being a non degenerate quadratic form on V) is a quotient of U(sl(2)) Other structure results are obtained when (G,V) is a regular prehomogeneous vector space of commutative parabolic type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-10-29T15:46:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "invariant differential operators",
      "multiplicity free spaces",
      "non degenerate quadratic form",
      "multiplicity free representation",
      "regular prehomogeneous vector space"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0910.5664R"
    }
  }
}