{
  "id": "1506.03107",
  "version": "v1",
  "published": "2015-06-09T21:36:00.000Z",
  "updated": "2015-06-09T21:36:00.000Z",
  "title": "The Capelli problem for $\\mathfrak{gl}(m|n)$ and the spectrum of invariant differential operators",
  "authors": [
    "Siddhartha Sahi",
    "Hadi Salmasian"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "The \"Capelli problem\" for the symmetric pairs $(\\mathfrak{gl}\\times \\mathfrak{gl},\\mathfrak{gl})$ $(\\mathfrak{gl},\\mathfrak{o})$, and $(\\mathfrak{gl},\\mathfrak{sp})$ is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter. In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs $(\\mathfrak{g},\\mathfrak{g}):=(\\mathfrak{gl}(m|2n),\\mathfrak{osp}(m|2n))$ and $(\\mathfrak{gl}(m|n)\\times\\mathfrak{gl}(m|n),\\mathfrak{gl}(m|n))$, acting on $W:=S^2(\\mathbb C^{m|2n})$ and $\\mathbb C^{m|n}\\otimes(\\mathbb C^{m|n})^*$. We also give an affirmative answer to the abstract Capelli problem for these cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-09T21:36:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "05E05"
    ],
    "keywords": [
      "invariant differential operators",
      "abstract capelli problem",
      "shifted jack polynomials",
      "special values",
      "lie superalgebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150603107S"
    }
  }
}