Planar Convexity of Isotropy Orbits via Cohomology and Common Zeros

Abstract

Let g be a noncompact simple real Lie algebra with a Cartan decomposition g=k⊕p. Let K denote the connected Lie group corresponding to k. Then K acts on p via the adjoint representation. In this paper, we prove that for every C∈p and every real linear map L:p→ℝ², the linear image L(K· C) of the orbit K· C⊂p is convex in ℝ², except for the circle orbits when g≅sl₂(ℝ) and C≠0. 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.

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BibTeX RIS

BibTeX

@misc{luan2026planar,
  title = {{Planar Convexity of Isotropy Orbits via Cohomology and Common Zeros}},
  author = {Yongzhi Luan and Tin-Yau Tam},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02575},
  eprint = {2611.02575},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02575},
}

RIS

TY  - UNPB
TI  - Planar Convexity of Isotropy Orbits via Cohomology and Common Zeros
AU  - Yongzhi Luan
AU  - Tin-Yau Tam
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.02575
UR  - https://arxiv.org/abs/2611.02575
AB  - Let g be a noncompact simple real Lie algebra with a Cartan decomposition g=k⊕p. Let K denote the connected Lie group corresponding to k. Then K acts on p via the adjoint representation. In this paper, we prove that for every C∈p and every real linear map L:p→ℝ^2, the linear image L(K· C) of the orbit K· C⊂p is convex in ℝ^2, except for the circle orbits when g≅sl_2(ℝ) and C≠0. 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.
ER  -

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