The K-orbit of a Normal Element in a Complex Semisimple Lie Algebra

Given a complex semisimple Lie algebra g = k+ ik, we consider the converse question of Kostant’s convexity theorem for a normal x ∈ g. Let π : g → h be the orthogonal projection under the Killing form onto the Cartan subalgebra h := t+it where t is a maximal abelian subalgebra of k. If π(Ad(K)x) is convex, then there is k ∈ K such that each simple component of Ad(k)x can be rotated into the corresponding component of t. The result also extends a theorem of Au-Yeung and Tsing on the generalized numerical range.