Planar Convexity of Isotropy Orbits via Cohomology and Common Zeros

Let g be a noncompact simple real Lie algebra with a Cartan decomposition g=k. 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 and every real linear map L:p², the linear image L(K· C) of the orbit K· C is convex in R², except for the circle orbits when g₂(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.