{
  "id": "2402.12106",
  "version": "v3",
  "published": "2024-02-19T12:55:00.000Z",
  "updated": "2024-07-16T13:39:43.000Z",
  "title": "The sign of linear periods",
  "authors": [
    "U. K. Anandavardhanan",
    "Hengfei Lu",
    "Nadir Matringe",
    "Vincent Sécherre",
    "Chang Yang"
  ],
  "comment": "We extended the main result from $p$-adic to all local fields of characteristic zero in Section 7, thanks to the new Appendix D by M. Suzuki and H. Tamori which classifies Archimedean standard modules with a linear model",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a group with subgroup $H$, and let $(\\pi,V)$ be a complex representation of $G$. The natural action of the normalizer $N$ of $H$ in $G$ on the space $\\mathrm{Hom}_H(\\pi,\\mathbb{C})$ of $H$-invariant linear forms on $V$, provides a representation $\\chi_{\\pi}$ of $N$ trivial on $H$, which is a character when $\\mathrm{Hom}_H(\\pi,\\mathbb{C})$ is one dimensional. If moreover $G$ is a reductive group over a local field, and $\\pi$ is smooth irreducible, it is an interesting problem to express $\\chi_{\\pi}$ in terms of the possibly conjectural Langlands parameter $\\phi_\\pi$ of $\\pi$. In this paper we consider the following situation: $G=\\mathrm{GL}_m(D)$ for $D$ a central division algebra of dimension $d^2$ over a local field $F$ of characteristic zero, $H$ is the centralizer of a non central element $\\delta\\in G$ such that $\\delta^2$ is in the center of $G$, and $\\pi$ has generic Jacquet-Langlands transfer to $\\mathrm{GL}_{md}(F)$. In this setting the space $\\mathrm{Hom}_H(\\pi,\\mathbb{C})$ is at most one dimensional. When $\\mathrm{Hom}_H(\\pi,\\mathbb{C})\\simeq \\mathbb{C}$ and $H\\neq N$, we prove that the value of the $\\chi_{\\pi}$ on the non trivial class of $\\frac{N}{H}$ is $(-1)^m\\epsilon(\\phi_\\pi)$ where $\\epsilon(\\phi_\\pi)$ is the root number of $\\phi_{\\pi}$. Along the way we extend many useful multiplicity one results for linear and Shalika models to the case of non split $G$. When $F$ is $p$-adic we also classify standard modules with linear periods and Shalika models, which are new results even when $D=F$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2024-07-16T13:39:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50",
      "11F70"
    ],
    "keywords": [
      "linear periods",
      "shalika models",
      "local field",
      "possibly conjectural langlands parameter",
      "invariant linear forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}