{
  "id": "1910.02525",
  "version": "v1",
  "published": "2019-10-06T21:09:10.000Z",
  "updated": "2019-10-06T21:09:10.000Z",
  "title": "Local Langlands correspondence for the twisted exterior and symmetric square $ε$-factors of $\\textrm{gl}_n$",
  "authors": [
    "Dongming She"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $F$ be a non-Archimedean local field. Let $\\mathcal{A}_n(F)$ be the set of equivalence classes of irreducible admissible representations of $\\textrm{GL}_n(F)$, and $\\mathcal{G}_n(F)$ be the set of equivalence classes of n-dimensional Frobenius semisimple Weil-Deligne representations of $W'_F$. The local Langlands correspondence(LLC) establishes the reciprocity maps $\\textrm{Rec}_{n,F}: \\mathcal{A}_n(F)\\longrightarrow \\mathcal{G}_n(F)$ , satisfying some nice properties. An important invariant under this correspondence is the L- and $\\epsilon$-factors. This is also expected to be true under parallel compositions with a complex analytic representations of $\\textrm{GL}_n(\\mathbb{C})$. J.W. Cogdell, F. Shahidi, and T.-L. Tsai proved the equality of the symmetric and exterior square L- and $\\epsilon$-factors [7] in 2017. But the twisted symmetric and exterior square L- and $\\epsilon$-factor are new and very different from the untwisted case. In this paper we will define the twisted symmetric square L- and $\\gamma$-factors using $\\textrm{GSpin}_{2n+1}$, and establish the equality of the corresponding L- and $\\epsilon$-factors. We will first reduce the problem to the analytic stability of their $\\gamma$-factors for supercuspidal representations, then prove the supercuspidal stability by establishing general asymptotic expansions of partial Bessel function following the ideas in [7].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-06T21:09:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local langlands correspondence",
      "symmetric square",
      "twisted exterior",
      "n-dimensional frobenius semisimple weil-deligne representations",
      "equivalence classes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}