{
  "abstract": "Let $\\mathcal{O}$ be a complete discrete valuation ring with a finite residue field. We introduce a simple framework of constructing smooth representations of $\\mathrm{GL}_n(\\mathcal{O})$ modulo the knowledge of nilpotent orbits, called Jordan induction, which enjoys several remarkable properties: It yields only irreducible representations, it yields all the even level irreducible representations, and up to conjugation it yields non-isomorphic representations. In the even level case, this gives an explicit realisation of Hill's analogue of Lusztig's Jordan decomposition, as well as a vast generalisation of Gérardin's construction for $\\mathrm{GL}_n(\\mathcal{O})$. As a simple application, we construct an explicit section to the orbit map. We also propose a conjecture linking Jordan induction and Lusztig induction, aiming at generalising the algebraisation theorem obtained in recent joint works with Stasinski.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35393",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhe Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9z3h1af15tw2k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35393",
  "fulltext_url": null,
  "id": "9z3h1af15tw2k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35393",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214714497",
  "pdf_url": "https://arxiv.org/pdf/2609.35393",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9z3h1af15tw2k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.35393",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35393"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Jordan induction for $\\mathrm{GL}_n(\\mathcal{O})$",
  "title_text": "Jordan induction for GL_n(𝒪)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/9z3h1af15tw2k",
  "venue": "arXiv",
  "versions": []
}