The K-orbit of a normal element in a complex semisimple Lie algebra

Tin-Yau Tam and Wai-Shun Cheung

Vol. 238 (2008), No. 2, 387–398

DOI: 10.2140/pjm.2008.238.387

Abstract

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

Mathematical Subject Classification 2000

Primary: 22E10

Secondary: 17B20

Milestones

Received: 15 December 2007

Revised: 29 February 2008

Accepted: 3 March 2008

Published: 1 December 2008