{
  "id": "math/0510033",
  "version": "v1",
  "published": "2005-10-03T07:47:12.000Z",
  "updated": "2005-10-03T07:47:12.000Z",
  "title": "Intersection of harmonics and Capelli identities for symmetric pairs",
  "authors": [
    "Soo Teck Lee",
    "Kyo Nishiyama",
    "Akihito Wachi"
  ],
  "comment": "20 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "We consider a see-saw pair consisting of a Hermitian symmetric pair (G_R, K_R) and a compact symmetric pair (M_R, H_R), where (G_R, H_R) and (K_R, M_R) form real reductive dual pairs in a large symplectic group. In this setting, we get Capelli identities which explicitly represent certain K_C-invariant elements in U(Lie(G)_C) in terms of H_C-invariant elements in U(Lie(M)_C). The corresponding H_C-invariant elements are called Capelli elements. We also give a decomposition of the intersection of O_{2n}-harmonics and Sp_{2n}-harmonics as a module of GL_n = O_{2n} \\cap Sp_{2n}, and construct a basis for the GL_n highest weight vectors. This intersection is in the kernel of our Capelli elements.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-10-03T07:47:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B35",
      "22E46",
      "16S32",
      "15A15"
    ],
    "keywords": [
      "capelli identities",
      "intersection",
      "capelli elements",
      "form real reductive dual pairs",
      "highest weight vectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....10033T"
    }
  }
}