{
  "id": "1406.5548",
  "version": "v1",
  "published": "2014-06-20T22:07:11.000Z",
  "updated": "2014-06-20T22:07:11.000Z",
  "title": "Branching laws on the metaplectic cover of ${\\rm GL}_{2}$",
  "authors": [
    "Shiv Prakash Patel"
  ],
  "comment": "Ph. D. Thesis",
  "categories": [
    "math.RT"
  ],
  "abstract": "Representation theory of $p$-adic groups naturally comes in the study of automorphic forms and one way to understand representations of a group is by restricting to its nice subgroups. D. Prasad studied the restriction for pairs $({\\rm GL}_{2}(E), {\\rm GL}_{2}(F))$ and $({\\rm GL}_{2}(E), D_{F}^{\\times})$ where $E/F$ is a quadratic equation and $D_{F}$ is the unique quaternion division algebra, and $D_{F}^{\\times} \\hookrightarrow {\\rm GL}_{2}(E)$. Prasad proved a multiplicity one result and a `dichotomy' relating the restriction for the pairs $({\\rm GL}_{2}(E), {\\rm GL}_{2}(F))$ and $({\\rm GL}_{2}(E), D_{F}^{\\times})$ involving the Jacquet-Langlands correspondence. We study a restriction problem involving covering groups. In an analogy to the case of Prasad, we consider pairs $(\\widetilde{{\\rm GL}_{2}(E)}, {\\rm GL}_{2}(F))$ and $(\\widetilde{{\\rm GL}_{2}(E)}, D_{F}^{\\times})$ where $\\widetilde{{\\rm GL}_{2}(E)}$ is the $\\mathbb{C}^{\\times}$-metaplectic covering of ${\\rm GL}_{2}(E)$. We do not have multiplicity one in this case but there is an analogue of dichotomy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-20T22:07:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E35",
      "22E50",
      "11F70",
      "11S37"
    ],
    "keywords": [
      "metaplectic cover",
      "branching laws",
      "unique quaternion division algebra",
      "automorphic forms",
      "multiplicity"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.5548P"
    }
  }
}