{
  "abstract": "Let $F$ be a non-archimedean local field of characteristic zero. Let $M$ be a Levi subgroup of $G = \\mathrm{SL}(n,F)$. Consider the action on the discrete series of $M$ of the group of unitary unramified characters of $M$ (where a character acts by twisting). Given an orbit $\\mathcal{O}$ for that action, let $W_\\mathcal{O}$ be the global stabilizer of $\\mathcal{O}$ in the Weyl group of $M$. We construct families of orbits $\\mathcal{O}$ for which the action of $W_\\mathcal{O}$ has no fixed point. More precisely, let $n \\geq 8$ be an integer that is not prime and not equal to $9$. We attach families of orbits without fixed points to any divisor $m \\geq 2$ of $n$ such that $m \\;| \\; q-1$ where $q$ is the order of the residue field of $F$. The Levi subgroup $M \\subset G$ for the corresponding families has $m$ blocks of size $k\\geq 2$ and $2$ blocks of size $m$. The core of the paper is the construction of superculpidal representations of $\\mathrm{GL}(m,F)$ and $\\mathrm{GL}(k,F)$ satisfying some conditions that we require to construct the families of orbits $\\mathcal{O}$ that we expects. In addition, we study the fixed point problem for the dimensions $n$ that we excluded above, i.e., prime or small dimensions. We prove that there is always a fixed point under the action of $W_\\mathcal{O}$ in those cases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05055",
  "author_total": 1,
  "authors": [
    {
      "name": "Marie Dautheville"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5505a1s5v4e7c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05055",
  "fulltext_url": null,
  "id": "5505a1s5v4e7c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05055",
  "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": "W7211884773",
  "pdf_url": "https://arxiv.org/pdf/2609.05055",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5505a1s5v4e7c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.05055",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05055"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On a basepoint issue in the tempered dual of $p$-adic $SL(n)$",
  "title_text": "On a basepoint issue in the tempered dual of p-adic SL(n)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/5505a1s5v4e7c",
  "venue": "arXiv",
  "versions": []
}