{
  "id": "2309.14864",
  "version": "v1",
  "published": "2023-09-26T11:39:26.000Z",
  "updated": "2023-09-26T11:39:26.000Z",
  "title": "Symmetry breaking for $\\operatorname{PGL}(2)$ over non-archimedean local fields",
  "authors": [
    "Corina Ciobotaru",
    "Jan Frahm"
  ],
  "comment": "42 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "For a quadratic extension $\\mathbb{E}/\\mathbb{F}$ of non-archimedean local fields we construct explicit holomorphic families of intertwining operators between principal series representations of $\\operatorname{PGL}(2,\\mathbb{E})$ and $\\operatorname{PGL}(2,\\mathbb{F})$, also referred to as symmetry breaking operators. These families are given in terms of their distribution kernels which can be viewed as distributions on $\\mathbb{E}$ depending holomorphically on the principal series parameters. For all such parameters we determine the support of these distributions, and we study their mapping properties. This leads to a classification of all intertwining operators between principal series representations, not necessarily irreducible. As an application, we show that every Steinberg representation of $\\operatorname{PGL}(2,\\mathbb{E})$ contains a Steinberg representation of $\\operatorname{PGL}(2,\\mathbb{F})$ as a direct summand of Hilbert spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-26T11:39:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E35",
      "22E50"
    ],
    "keywords": [
      "non-archimedean local fields",
      "symmetry breaking",
      "principal series representations",
      "construct explicit holomorphic families",
      "steinberg representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}