{
  "id": "1407.2885",
  "version": "v3",
  "published": "2014-07-10T18:08:49.000Z",
  "updated": "2015-07-28T15:29:07.000Z",
  "title": "Explicit induction principle and symplectic-orthogonal theta lifting",
  "authors": [
    "Xiang Fan"
  ],
  "comment": "34 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "First, an explicit version of induction principle is formulated to compute the local theta correspondence for $(O(p,q),Sp(2n,\\mathbb{R}))$ with $p+q$ even: when $p+q\\leqslant2n$, the Langlands parameters of the theta $(n+k)$-lift of a representation of $O(p,q)$ is read off from the parameters of its theta $n$-lift, if the $n$-lift is nonzero; similarly when $p+q\\geqslant2n+2$, a nonzero theta $(p,q)$-lift of a representation of $Sp(2n,\\mathbb{R})$ determines its theta $(p+k,q+k)$-lift explicitly. Secondly, after reducing computations by our explicit induction principle, a complete and explicit description of the local theta correspondence is obtained for all the dual pairs $(O(p,q),Sp(2n,\\mathbb{R}))$ with $p+q=4$. Our strategy is to determine the theta lifts under consideration by their infinitesimal characters and lowest $K$-types.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-14T15:14:39.000Z",
      "title": "Explicit Induction Principle for the Dual Pairs $(O(p,q),Sp(2n,\\mathbb{R}))$ with $p+q$ even",
      "abstract": "Let $\\pi\\in\\mathcal{R}(O(p,q))$ and $\\pi'\\in\\mathcal{R}(Sp(2n,\\mathbb{R}))$ correspond in the local theta correspondence, with $p+q$ even. If $2n\\geqslant p+q$, then the Langlands parameters of $\\pi'=\\theta_{n}(\\pi)$ satisfy a condition with respect to $p-q$, so that we can get $\\theta_{n+k}(\\pi)$ from $\\theta_{n}(\\pi)$ explicitly in terms of Langlands parameters by the Induction Principle. We call this \"Explicit Induction Principle\" on $n$. Similar \"Explicit Induction Principle\" on $(p,q)$ holds in the opposite direction if $p+q\\geqslant2n+2$, which gives $\\theta_{p+k,q+k}(\\pi')$ from $\\theta_{p,q}(\\pi')$.",
      "comment": "16 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-07-28T15:29:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F27",
      "22E46"
    ],
    "keywords": [
      "explicit induction principle",
      "dual pairs",
      "langlands parameters",
      "local theta correspondence",
      "opposite direction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.2885F"
    }
  }
}