Planar Convexity of Isotropy Orbits via Cohomology and Common Zeros

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.