{
  "id": "2107.09348",
  "version": "v1",
  "published": "2021-07-20T09:07:18.000Z",
  "updated": "2021-07-20T09:07:18.000Z",
  "title": "Symmetry breaking operators for dual pairs with one member compact",
  "authors": [
    "M. McKee",
    "A. Pasquale",
    "T. Przebinda"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1405.2431",
  "categories": [
    "math.RT"
  ],
  "abstract": "We consider a dual pair (G, G'), in the sense of Howe, with G compact acting on $L^2(\\mathbb{R}^n)$, for an appropriate $n$, via the Weil representation $\\omega$. Let $\\tilde{\\mathrm{G}}$ be the preimage of G in the metaplectic group. Given a genuine irreducible unitary representation $\\Pi$ of $\\tilde{\\mathrm{G}}$, let $\\Pi'$ be the corresponding irreducible unitary representation of $\\tilde{\\mathrm{G}'}$ in the Howe duality. The orthogonal projection onto $L^2(\\mathbb{R}^n)_\\Pi$, the $\\Pi$-isotypic component, is the essentially unique symmetry breaking operator in $\\mathrm{Hom}_{\\tilde{\\mathrm{G}}\\tilde{\\mathrm{G}'}}(\\mathcal{H}_\\omega^{\\infty}, \\mathcal{H}_\\Pi^{\\infty}\\otimes \\mathcal{H}_{\\Pi'}^{\\infty})$. We study this operator by computing its Weyl symbol. Our results allow us to compute the wavefront set of $\\Pi'$ by elementary means.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-20T09:07:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E45",
      "22E46",
      "22E30"
    ],
    "keywords": [
      "dual pair",
      "member compact",
      "genuine irreducible unitary representation",
      "essentially unique symmetry breaking operator",
      "orthogonal projection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}