{
  "id": "2211.10847",
  "version": "v1",
  "published": "2022-11-20T02:24:44.000Z",
  "updated": "2022-11-20T02:24:44.000Z",
  "title": "Parabolic Simple $\\mathscr{L}$-Invariants",
  "authors": [
    "Yiqin He"
  ],
  "comment": "59 pages. arXiv admin note: text overlap with arXiv:1807.10862, arXiv:2109.06696 by other authors",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let $L$ be a finite extension of $\\mathbf{Q}_p$. Let $\\rho_L$ be a potentially semi-stable non-crystalline $p$-adic Galois representation such that the associated $F$-semisimple Weil-Deligne representation is absolutely indecomposable. In this paper, we study Fontaine-Mazur parabolic simple $\\mathscr{L}$-invariants of $\\rho_L$, which was previously only known in the trianguline case. Based on the previous work on Breuil's parabolic simple $\\mathscr{L}$-invariants, we attach to $\\rho_L$ a locally $\\mathbf{Q}_p$-analytic representation $\\Pi(\\rho_L)$ of $\\mathrm{GL}_{n}(L)$, which carries the information of parabolic simple $\\mathscr{L}$-invariants of $\\rho_L$. When $\\rho_L$ comes from a patched automorphic representation of $\\mathbf{G}(\\mathbb{A}_{F^+})$ (for a define unitary group $\\mathbf{G}$ over a totally real field $F^+$ which is compact at infinite places and $\\mathrm{GL}_n$ at $p$-adic places), we prove under mild hypothesis that $\\Pi(\\rho_L)$ is a subrepresentation of the associated Hecke-isotypic subspace of the Banach spaces of (patched) $p$-adic automophic forms on $\\mathbf{G}(\\mathbb{A}_{F^+})$, this is equivalent to say that the Breuil's parabolic simple $\\mathscr{L}$-invariants are equal to Fontaine-Mazur parabolic simple $\\mathscr{L}$-invariants.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-20T02:24:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "invariants",
      "breuils parabolic simple",
      "study fontaine-mazur parabolic simple",
      "semisimple weil-deligne representation",
      "adic galois representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 59,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}