{
  "abstract": "Let $\\mathcal O$ be the ring of integers of a non-Archimedean local field with uniformizer $\\varpi$, and let $\\mathcal O_\\ell=\\mathcal O/\\varpi^\\ell\\mathcal O$ for $\\ell\\geq1$. In this article, we study representations of $\\mathrm{GL}_n(\\mathcal O_\\ell)$ parabolically induced from characters of the Borel subgroup $\\mathrm{B}_n(\\mathcal O_\\ell)$. For $n=3$, we establish a criterion for the irreducibility of these induced representations in terms of the relative conductors of the inducing characters. Motivated by this result, we formulate a conjecture characterizing the irreducibility of such representations for arbitrary $n$. We further show that when all inducing characters are equal, the resulting induced representation is not multiplicity-free, implying that the corresponding triple $(\\mathrm{GL}_n(\\mathcal O_\\ell),\\mathrm{B}_n(\\mathcal O_\\ell),χ)$ is not a Gelfand pair. Finally, we study the irreducible decomposition of these representations for $\\mathrm{GL}_3(\\mathcal O_\\ell)$ when two relative conductors attain their maximal values. Using the decomposition theory for $\\mathrm{GL}_2(\\mathcal O_\\ell)$, we describe the irreducible constituents and determine their dimensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15762",
  "author_total": 1,
  "authors": [
    {
      "name": "Prem Dagar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bm6dwp15yg286.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15762",
  "fulltext_url": null,
  "id": "bm6dwp15yg286",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15762",
  "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": "W7221790152",
  "pdf_url": "https://arxiv.org/pdf/2610.15762",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bm6dwp15yg286.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15762",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15762"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Parabolically induced representations of $GL_n(O_l)$",
  "title_text": "Parabolically induced representations of GL_n(O_l)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/bm6dwp15yg286",
  "venue": "arXiv",
  "versions": []
}