{
  "abstract": "Let $\\overline{\\mathbf{Y}}(w)\\to\\overline{\\mathbf{X}}(w)$ be the ramified Galois cover of compactified Deligne--Lusztig varieties. Our objects of study are the Lefschetz numbers for the action of the finite covering group $T^{wF}$ on the coherent cohomology of the structure sheaf of $\\overline{\\mathbf{Y}}(w)$. The Lefschetz numbers for all elements of $T^{wF}$ determine the virtual $T^{wF}$-representation given by the alternating sum of these cohomology groups. In the two cases of Deligne--Lusztig surfaces of type $A_2$ and ${}^{2}A_2$, we give explicit formulas for the Lefschetz numbers for a generator of the stabilizer of an irreducible component. It follows that the equivariant Euler characteristic of the compactification $\\overline{\\mathbf{Y}}(w)$ is not virtually free as a virtual module over the covering group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40335",
  "author_total": 1,
  "authors": [
    {
      "name": "Adithya Sathyanarayana"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b1tkzcmat7jj6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40335",
  "fulltext_url": null,
  "id": "b1tkzcmat7jj6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40335",
  "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.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40335",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b1tkzcmat7jj6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.40335",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40335"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Galois module structure of Deligne--Lusztig varieties for $\\mathrm{GL}_3$",
  "title_text": "Galois module structure of Deligne--Lusztig varieties for GL_3",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/b1tkzcmat7jj6",
  "venue": "arXiv",
  "versions": []
}