Combinatorics of Vogan diagrams for almost-Kähler manifolds

Abstract

Let $G$ be a non-compact classical semisimple Lie group and let $G/V$ be the adjoint orbit with respect to a fixed element in $G$. These manifolds can be equipped with an almost-Kähler structure and we provide explicit formulae for the existence of special almost-complex structures on $G/V$ 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 $A_{\ell }$, $B_{\ell }$, $C_{\ell }$, $D_{\ell }$, where $\ell$ denotes the rank of the Lie algebra.

28 pages