{
  "id": "2210.01381",
  "version": "v1",
  "published": "2022-10-04T04:42:49.000Z",
  "updated": "2022-10-04T04:42:49.000Z",
  "title": "On generalization of Breuil--Schraen's $\\mathscr{L}$-invariants to $\\mathrm{GL}_n$",
  "authors": [
    "Zicheng Qian"
  ],
  "comment": "72 pages",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let $p$ be prime number and $K$ be a $p$-adic field. We systematically compute the higher $\\mathrm{Ext}$-groups between locally analytic generalized Steinberg representations (LAGS for short) of $\\mathrm{GL}_n(K)$ via a new combinatorial treatment of some spectral sequences arising from the so-called Tits complex. Such spectral sequences degenerate at the second page and each $\\mathrm{Ext}$-group admits a canonical filtration whose graded pieces are terms in the second page of the corresponding spectral sequence. For each pair of LAGS, we are particularly interested their $\\mathrm{Ext}$-groups in the bottom two non-vanishing degrees. We write down an explicit basis for each graded piece (under the canonical filtration) of such an $\\mathrm{Ext}$-group, and then describe the cup product maps between such $\\mathrm{Ext}$-groups using these bases. As an application, we generalize Breuil's $\\mathscr{L}$-invariants for $\\mathrm{GL}_2(\\mathbb{Q}_p)$ and Schraen's higher $\\mathscr{L}$-invariants for $\\mathrm{GL}_3(\\mathbb{Q}_p)$ to $\\mathrm{GL}_n(K)$. Along the way, we also establish a generalization of Bernstein--Zelevinsky geometric lemma to admissible locally analytic representations constructed by Orlik--Strauch, generalizing a result in Schraen's thesis for $\\mathrm{GL}_3(\\mathbb{Q}_p)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-04T04:42:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "invariants",
      "generalization",
      "locally analytic representations",
      "breuil-schraens",
      "second page"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 72,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}