{
  "id": "1805.09713",
  "version": "v1",
  "published": "2018-05-24T15:00:42.000Z",
  "updated": "2018-05-24T15:00:42.000Z",
  "title": "Geometry of coadjoint orbits and multiplicity-one branching laws for symmetric pairs",
  "authors": [
    "Toshiyuki Kobayashi",
    "Salma Nasrin"
  ],
  "comment": "Kirillov volume",
  "categories": [
    "math.RT",
    "math.SG"
  ],
  "abstract": "Consider the restriction of an irreducible unitary representation $\\pi$ of a Lie group $G$ to its subgroup $H$. Kirillov's revolutionary idea on the orbit method suggests that the multiplicity of an irreducible $H$-module $\\nu$ occurring in the restriction $\\pi|_H$ could be read from the coadjoint action of $H$ on $O^G \\cap pr^{-1}(O^H)$ provided $\\pi$ and $\\nu$ are \"geometric quantizations\" of a $G$-coadjoint orbit $O^G$ and an $H$-coadjoint orbit $O^H$,respectively, where $pr: \\sqrt{-1} g^{\\ast} \\to \\sqrt{-1} h^{\\ast}$ is the projection dual to the inclusion $h \\subset g$ of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits $O^G$ of a semisimple Lie group $G$ corresponding to highest weight modules of scalar type. We prove that the Corwin--Greenleaf number $\\sharp(O^G \\cap pr^{-1}(O^H))/H$ is either zero or one for any $H$-coadjoint orbit $O^H$, whenever $(G,H)$ is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits $O^H$ with nonzero Corwin-Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as \"classical limits\" of the multiplicity-free branching laws of holomorphic discrete series representations (T.Kobayashi [Progr.Math.2007]).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-24T15:00:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "coadjoint orbit",
      "multiplicity-one branching laws",
      "symmetric pair",
      "holomorphic discrete series representations",
      "highlight specific elliptic orbits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}