Combinatorics of Vogan diagrams for almost-K\"ahler manifolds

TL;DR

This paper provides explicit combinatorial criteria based on Vogan diagrams to determine the existence of special almost-complex structures on adjoint orbits of non-compact classical semisimple Lie groups, enriching the understanding of their geometric structures.

Contribution

It introduces explicit formulae linking Vogan diagram combinatorics to the existence of special almost-complex structures on almost-K"ahler manifolds associated with classical Lie groups.

Findings

Explicit criteria for type A, B, C, D Lie algebras

Formulas depend solely on Vogan diagram combinatorics

Applicable to adjoint orbits with almost-K"ahler structures

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-K\"ahler 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.