{
  "abstract": "Let $\\mathrm{D}$ be a quaternion division algebra over a non-Archimedean local field $\\mathrm{F}$ of characteristic zero, and let $\\mathrm{G}_n=\\mathrm{GL}_n(\\mathrm{D})$. Let $\\mathrm{H}_{1,n-1}=\\{\\mathrm{diag}(g_1,g_2)\\in\\mathrm{G}_1,\\ g_2\\in\\mathrm{G}_{n-1}\\}$ and, for $s \\in \\mathbb{R}$, define $χ_s(\\mathrm{diag}(g_1,g_2))=ν(g_1)^{2s}ν(g_2)^{-2s}$. We classify, for $n=3,4$, the irreducible smooth representations of $\\mathrm{G}_n$ admitting a generalized linear period with respect to $(\\mathrm{H}_{1,n-1},χ_s)$. Motivated by these results, we conjecture a complete classification for all $n>2$. Assuming this conjecture, we characterize such representations in terms of Langlands parameters: an irreducible smooth representation $π$ of $\\mathrm{G}_n$ admits such a period if and only if $\\mathfrak{L}(π)$ contains a Weil-Deligne subrepresentation isomorphic to $\\mathfrak{L}(ν^{-2s})$, the $(2n-4)$-dimensional parameter of $ν^{-2s}$ on $\\mathrm{G}_{n-2}$, and the four-dimensional quotient is the Langlands parameter of either the trivial representation of $\\mathrm{G}_2$, with $s=\\pm\\frac{n-2}{2}$, or an irreducible infinite-dimensional $\\mathrm{H}_{1,1}$-distinguished representation of $\\mathrm{G}_2$. We also verify the Lapid-Prasad conjecture in this setting: the $L$-packet of an irreducible representation of $\\mathrm{G}_n$ admitting a linear period with respect to $\\mathrm{H}_{1,n-1}$ is invariant under $ρ\\mapsto\\widetildeρ^θ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34307",
  "author_total": 2,
  "authors": [
    {
      "name": "Prem Dagar"
    },
    {
      "name": "Hariom Sharma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/94g1k76aytmk2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34307",
  "fulltext_url": null,
  "id": "94g1k76aytmk2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34307",
  "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": "W7214802956",
  "pdf_url": "https://arxiv.org/pdf/2609.34307",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/94g1k76aytmk2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34307",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34307"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Representations of $\\mathrm{GL}_n(\\mathrm{D})$ admitting a generalized linear period",
  "title_text": "On Representations of GL_n(D) admitting a generalized linear period",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/94g1k76aytmk2",
  "venue": "arXiv",
  "versions": []
}