{
  "abstract": "A well-known result of Hesselink gives a formula for the $q$-character of the nilpotent cone of a semisimple Lie algebra in terms of a $q$-analog of the Kostant partition function. For a reductive Lie algebra $\\mathfrak{g}$, we define a class of \\emph{Hesselink-type representations}, for which we prove an analog of Hesselink's formula under certain additional freeness assumptions. Moreover, we prove that the formula still holds for certain examples of Hesselink-type representations without the freeness assumptions. Using these methods, we obtain $q$-character formulas for the nilpotent cone of a representation of a cyclic quiver with equal dimensions, a representation of a cyclic quiver with two vertices, and a representation of a product of copies of $\\mathfrak{sl}_2$ we call an \\emph{extended quiver representation of trivial type}. In the process, we give a general framework for proving formulas of this type for representations that are not necessarily Hesselink-type. We also provide some counterexamples to natural questions regarding Hesselink-type representations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.03314",
  "author_total": 2,
  "authors": [
    {
      "name": "Vasily Krylov"
    },
    {
      "name": "Frank Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e249mb2xrvyrv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.03314",
  "fulltext_url": null,
  "id": "e249mb2xrvyrv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.03314",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7172520892",
  "pdf_url": "https://arxiv.org/pdf/2608.03314",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e249mb2xrvyrv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2608.03314",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.03314"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Hesselink-type formula for the nilpotent cone of Lie algebra representations",
  "title_text": "A Hesselink-type formula for the nilpotent cone of Lie algebra representations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/e249mb2xrvyrv",
  "venue": "arXiv",
  "versions": []
}