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.
Keywords
K-orbit, convex, normal
element, complex semisimple Lie algebra