On Tam's star-shapedness conjecture for numerical ranges of compact adjoint orbits

We settle Tam's star-shapedness conjecture for numerical ranges associated with compact adjoint orbits of complex reductive Lie algebras, including its original formulation in terms of the inner product $B_θ$. Let $\mathfrak{g}$ be a complex semisimple Lie algebra, and let $K$ be a connected compact Lie group whose Lie algebra is a compact real form of $\mathfrak{g}$. For any $x\in\mathfrak{g}$ and any complex linear functional $f\in\mathfrak{g}^*$, we prove that $\{ f( \operatorname{Ad} (k) x): k \in K \}$ is star-shaped about zero. In the reductive case, the star center is given by the value of $f$ on the central component of $x$, which coincides with the $B_θ$-pairing of the central components in Tam's original formulation. Our proof is uniform for all complex semisimple Lie algebras. Indeed, using the adjoint Casimir operator, we show that all coordinate functions of the orbit map share a common Laplace eigenvalue; we then use a nodal-hypersurface argument to prove star-shapedness for complex-valued functions in a single eigenspace. The geometric argument relies on a two-sided hypersurface-separation property, which holds whenever the first real cohomology group vanishes. In particular, we do not need simple connectivity or real analyticity. More generally, for finite-dimensional representations of compact connected semisimple Lie groups, we establish star-shapedness whenever the nonfixed component of the orbit vector belongs to a single positive Casimir eigenspace.