{
  "id": "2108.10545",
  "version": "v1",
  "published": "2021-08-24T06:56:13.000Z",
  "updated": "2021-08-24T06:56:13.000Z",
  "title": "The wave front set correspondence for dual pairs with one member compact",
  "authors": [
    "M. McKee",
    "A. Pasquale",
    "T. Przebinda"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1405.2431",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let W be a real symplectic space and (G,G') an irreducible dual pair in Sp(W), in the sense of Howe, with G compact. Let $\\widetilde{\\mathrm{G}}$ be the preimage of G in the metaplectic group $\\widetilde{\\mathrm{Sp}}(\\mathrm{W})$. Given an irreducible unitary representation $\\Pi$ of $\\widetilde{\\mathrm{G}}$ that occurs in the restriction of the Weil representation to $\\widetilde{\\mathrm{G}}$, let $\\Theta_\\Pi$ denote its character. We prove that, for the embedding $T$ of $\\widetilde{\\mathrm{Sp}}(\\mathrm{W})$ in the space of tempered distributions on W given by the Weil representation, the distribution $T(\\check\\Theta_\\Pi)$ has an asymptotic limit. This limit is an orbital integral over a nilpotent orbit $\\mathcal O_m\\subseteq \\mathrm{W}$. The closure of the image of $\\mathcal O_m$ in $\\mathfrak{g}'$ under the moment map is the wave front set of $\\Pi'$, the representation of $\\widetilde{\\mathrm{G}'}$ dual to $\\Pi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-24T06:56:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E45",
      "22E46",
      "22E30"
    ],
    "keywords": [
      "wave front set correspondence",
      "member compact",
      "weil representation",
      "real symplectic space",
      "orbital integral"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}