{
  "id": "1910.14457",
  "version": "v1",
  "published": "2019-10-30T14:48:30.000Z",
  "updated": "2019-10-30T14:48:30.000Z",
  "title": "A criterion for discrete branching laws for Klein four symmetric pairs and its application to $E_{6(-14)}$",
  "authors": [
    "Haian He"
  ],
  "comment": "11 pages, 1 figure. arXiv admin note: text overlap with arXiv:1908.04723",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a noncompact connected simple Lie group, and $(G,G^\\Gamma)$ a Klein four symmetric pair. In this paper, the author shows a necessary condition for the discrete decomposability of unitarizable simple $(\\mathfrak{g},K)$-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for $(G,G^\\Gamma)$, there does not exist any unitarizable simple $(\\mathfrak{g},K)$-module that is discretely decomposable as a $(\\mathfrak{g}^\\Gamma,K^\\Gamma)$-module. As an application, for $G=\\mathrm{E}_{6(-14)}$, the author obtains a complete classification of Klein four symmetric pairs $(G,G^\\Gamma)$ with $G^\\Gamma$ noncompact, such that there exists at least one nontrivial unitarizable simple $(\\mathfrak{g},K)$-module that is discretely decomposable as a $(\\mathfrak{g}^\\Gamma,K^\\Gamma)$-module and is also discretely decomposable as a $(\\mathfrak{g}^\\sigma,K^\\sigma)$-module for some nonidentity element $\\sigma\\in\\Gamma$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-30T14:48:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E46"
    ],
    "keywords": [
      "symmetric pair",
      "discrete branching laws",
      "unitarizable simple",
      "application",
      "noncompact connected simple lie group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}