{
  "id": "2002.09551",
  "version": "v1",
  "published": "2020-02-21T20:58:21.000Z",
  "updated": "2020-02-21T20:58:21.000Z",
  "title": "On the Stable Transfer for $\\mathrm{Sym}^{n}$ Lifting of $\\mathrm{GL}_{2}$",
  "authors": [
    "Daniel Johnstone",
    "Zhilin Luo"
  ],
  "comment": "Comments are welcome",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "Following the paradigm of \\cite{MR3117742}, we are going to explore the stable transfer factors for $\\mathrm{Sym}^{n}$ lifting from $\\mathrm{GL}_{2}$ to $\\mathrm{GL}_{n+1}$ over any local fields $F$ of characteristic zero with residue characteristic not equal to $2$. When $F=\\mathbb{C}$ we construct an explicit stable transfer factor for any $n$. When $n$ is odd, we provide a reduction formula, reducing the question to the construction of the stable transfer factors when the $L$-morphism is the diagonal embedding from $\\mathrm{GL}_{2}(\\mathbb{C})$ to finitely many copies of $\\mathrm{GL}_{2}(\\mathbb{C})$ under mild assumptions on the residue characteristic of $F$. With the assumptions on the residue characteristic, the reduction formula works uniformly over any local fields of characteristic zero, except that for $p$-adic situation we need to exclude the twisted Steinberg representations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-21T20:58:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "residue characteristic",
      "characteristic zero",
      "local fields",
      "explicit stable transfer factor",
      "reduction formula works"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}