Star-shapedness and K-orbits in Complex Semisimple Lie Algebras

Given a complex semisimple Lie algebra g = k + ik (k is a compact real form of g), let π : g → h be the orthogonal projection (with respect to the Killing form) onto the Cartan subalgebra h := t + it, where t is a maximal abelian subalgebra of k. Given x ∈ g, we consider π(Ad(K)x), where K is the analytic subgroup G corresponding to k, and show that it is star-shaped. The result extends a result of Tsing. We also consider the generalized numerical range f(Ad(K)x), where f is a linear functional on g. We establish the star-shapedness of f(Ad(K)x) for simple Lie algebras of type B.