{
  "id": "1802.02637",
  "version": "v1",
  "published": "2018-02-07T21:15:06.000Z",
  "updated": "2018-02-07T21:15:06.000Z",
  "title": "Doubling constructions: local and global theory, with an application to global functoriality for non-generic cuspidal representations",
  "authors": [
    "Yuanqing Cai",
    "Solomon Friedberg",
    "Eyal Kaplan"
  ],
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "A fundamental difficulty in the study of automorphic representations, representations of $p$-adic groups and the Langlands program is to handle the non-generic case. In this work we develop a complete local and global theory of tensor product $L$-functions of $G\\times GL_k$, where $G$ is a symplectic group, split special orthogonal group or the split general spin group, that includes both generic and non-generic representations of $G$. Our theory is based on a recent collaboration with David Ginzburg, where we presented a new integral representation that applies to all cuspidal automorphic representations. Here we develop the local theory over any field (of characteristic $0$), define the local $\\gamma$-factors and provide a complete description of their properties. We then define $L$- and $\\epsilon$-factors, and obtain the properties of the completed $L$-function. By combining our results with the Converse Theorem, we obtain a full proof of the global functorial lifting for automorphic representations of $G$ to the natural general linear group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-07T21:15:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F70",
      "11F55",
      "11F66",
      "22E50",
      "22E55"
    ],
    "keywords": [
      "non-generic cuspidal representations",
      "global theory",
      "global functoriality",
      "doubling constructions",
      "automorphic representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}