{
  "id": "1506.06229",
  "version": "v1",
  "published": "2015-06-20T11:05:45.000Z",
  "updated": "2015-06-20T11:05:45.000Z",
  "title": "Decomposition of modules over invariant differential operators",
  "authors": [
    "Rikard Bögvad",
    "Rolf Källström"
  ],
  "comment": "33 pages",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Let $G$ be a finite subgroup of the linear group of a finite-dimensional complex vector $V$, $B={\\operatorname S}(V)$ be the symmetric algebra, ${\\mathcal D}=\\mathcal D^G_B$ the ring of $G$-invariant differential operators, and ${\\mathcal D}^-$ its subring of negative degree operators. We prove that $M\\mapsto M^{ann}= {\\operatorname Ann}_{\\mathcal D^-}(M)$ defines an isomorphism between the category of ${\\mathcal D}$-submodules of $B$ and a category of modules formed as lowest weight spaces. This is applied to a construction of simple ${\\mathcal D}$-submodules of $B$ when $G$ is a generalized symmetric group, to show that $B^{ann}$ is a so-called Gelfand model, and to prove branching rules. We also get a new construction of Young bases for representations of the symmetric group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-20T11:05:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F10",
      "20C30"
    ],
    "keywords": [
      "invariant differential operators",
      "decomposition",
      "finite-dimensional complex vector",
      "lowest weight spaces",
      "linear group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150606229B"
    }
  }
}