Combinatorics of Vogan diagrams for almost-Kähler manifolds

Abstract

Let [Formula: see text] be a non-compact classical semisimple Lie group and let [Formula: see text] be the adjoint orbit with respect to a fixed element in [Formula: see text]. These manifolds can be equipped with an almost-Kähler structure and we provide explicit formulae for the existence of special almost-complex structures on [Formula: see text] purely in terms of the combinatorics of the associated Vogan diagram. The formulae are given separately for Lie groups whose Lie algebras are of type [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text] denotes the rank of the Lie algebra.