{
  "abstract": "We compute the quadratic Casimir on the $K$-fixed subspace of the Hilbert tensor square of the unitary minimal representation of the split real group $G_{2(2)}$. In an explicit algebraic basis of invariant tensors, the Casimir acts by a three-term recurrence. Normalizing this basis yields a symmetric Jacobi operator whose self-adjoint realization is unitarily equivalent to multiplication by $-\\frac92-8x^2$ in a Wilson-polynomial model with parameters $(\\frac34,\\frac34,\\frac7{12},\\frac{11}{12})$. We give the normalized spectral density and the unitary transform. The resulting spectrum is purely absolutely continuous, equals $(-\\infty,-\\frac92]$, and has scalar spectral multiplicity one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24656",
  "author_total": 1,
  "authors": [
    {
      "name": "Hongrui Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d66e0zz88s0wh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24656",
  "fulltext_url": null,
  "id": "d66e0zz88s0wh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24656",
  "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.FA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7223061149",
  "pdf_url": "https://arxiv.org/pdf/2610.24656",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d66e0zz88s0wh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24656",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24656"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Casimir on the $K$-fixed tensor square of the minimal representation of $G_{2(2)}$",
  "title_text": "The Casimir on the K-fixed tensor square of the minimal representation of G_2(2)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/d66e0zz88s0wh",
  "venue": "arXiv",
  "versions": []
}