Star-Shapedness and K-Orbits in Complex Semisimple Lie Algebras

Abstract

AbstractGiven a complex semisimple Lie algebra is a compact real form of g), let be the orthogonal projection (with respect to the Killing form) onto the Cartan subalgebra , where t is a maximal abelian subalgebra of . Given x ∈ g, we consider π(Ad(K)x), where K is the analytic subgroup G corresponding to , 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.