Invariant Generalized Complex Structures on Flag Manifolds

Abstract

Let $G$ be a complex semi-simple Lie group and form its maximal flag manifold $\mathbb{F}=G/P=U/T$ where $P$ is a minimal parabolic subgroup, $U$ a compact real form and $T=U\cap P$ a maximal torus of $U$. The aim of this paper is to study invariant generalized complex structures on $\mathbb{F}$. We describe the invariant generalized almost complex structures on $\mathbb{F}$ and classify which one is integrable. The problem reduces to the study of invariant $4$-dimensional generalized almost complex structures restricted to each root space, and for integrability we analyse the Nijenhuis operator for a triple of roots such that its sum is zero. We also conducted a study about twisted generalized complex structures. We define a new bracket `twisted' by a closed $3$-form $\Omega$ and also define the Nijenhuis operator twisted by $\Omega$. We classify the $\Omega$-integrable generalized complex structure.