{
  "abstract": "Let $\\mathfrak{g}$ be a noncompact simple real Lie algebra with a Cartan decomposition $\\mathfrak{g}=\\mathfrak{k}\\oplus\\mathfrak{p}$. Let $K$ denote the connected Lie group corresponding to $\\mathfrak{k}$. Then $K$ acts on $\\mathfrak{p}$ via the adjoint representation. In this paper, we prove that for every $C\\in\\mathfrak{p}$ and every real linear map $L:\\mathfrak{p}\\to\\mathbb{R}^2$, the linear image $L(K\\cdot C)$ of the orbit $K\\cdot C\\subset\\mathfrak{p}$ is convex in $\\mathbb{R}^2$, except for the circle orbits when $\\mathfrak{g}\\cong\\mathfrak{sl}_2(\\mathbb{R})$ and $C\\ne0$. By the two-piece property of isotropy orbits, any point omitted from the interior of the convex hull of a planar projection induces a nonzero class in the first integral cohomology of the orbit. This class vanishes directly in the non-Hermitian case, whereas in the Hermitian case we show that it vanishes by using the central circle action and a common-zero theorem of Gichev for Laplace eigenfunctions. Our approach treats classical and exceptional noncompact simple real Lie algebras uniformly and extends to real semisimple Lie algebras via Minkowski sums.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02575",
  "author_total": 2,
  "authors": [
    {
      "name": "Yongzhi Luan"
    },
    {
      "name": "Tin-Yau Tam"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/459pqbb6ykxmx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02575",
  "fulltext_url": null,
  "id": "459pqbb6ykxmx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02575",
  "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.FA"
    ],
    [
      "arxiv",
      "math.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02575",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/459pqbb6ykxmx.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02575",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02575"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Planar Convexity of Isotropy Orbits via Cohomology and Common Zeros",
  "title_text": "Planar Convexity of Isotropy Orbits via Cohomology and Common Zeros",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/459pqbb6ykxmx",
  "venue": "arXiv",
  "versions": []
}