Abstract

Given a complex semisimple Lie algebra g = k + ik, we consider the converse question of Kostant’s convexity theorem for a normal x in 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 in 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

Mathematical Subject Classification

Primary: 22E10

Secondary: 17B20

Authors