Invariant almost Hermitian structures on flag manifolds. (Q1408804)

Let \(\mathfrak g\) be a complex semi-simple Lie algebra of a complex Lie group \(G\), and let \(\mathbb{F}=G/P\) be a maximal flag manifold of \(G\), where \(P\) is a minimal parabolic (Borel) subgroup. For any maximal compact subgroup \(U\) of \(G\) the manifold \(\mathbb{F}\) can be written as \(\mathbb{F}=U/T\), where \(T\subset U\) is a maximal torus. In this paper, the authors study \(U\)-invariant almost Hermitian structures on \(\mathbb{F}\). A pair \((J,\Gamma)\), where \(J\) is an invariant almost complex structure and \(\Gamma\) is an invariant Riemannian metric, is such a structure. An invariant almost complex structure \(J\) is \((1,2)\)-admissible if there is a metric \(\Gamma\) such that \((J,\Gamma)\) is \((1,2)\)-symplectic. The author presents different characterizations of \((1,2)\)-admissible invariant almost complex structures. It is shown that the \((1,2)\)-symplectic structures are naturally related to the affine Weyl groups. A special form for them, involving abelian ideals of a Borel subalgebra, are derived. \((1,2)\)-symplectic structures are also studied to show applications to harmonic maps through a theorem by Gray. Finally, from the \((1,2)\)-symplectic structures a classification of the whole set of invariant structures is provided showing, in particular, that nearly Kähler invariant structures are Kähler.