De Rham $2$-cohomology of real flag manifolds

Abstract

Let ${\mathbb{F}}_{\Theta }=G/P_{\Theta }$ be a flag manifold associated to a non-compact real simple Lie group $G$ and the parabolic subgroup . This is a closed subgroup of $G$ determined by a subset of simple restricted roots of $\mathfrak{g}=Lie(G)$. This paper computes the second de Rham cohomology group of ${\mathbb{F}}_{\Theta }$. We prove that it zero in general, with some rare exceptions. When it is non-zero, we give a basis of $H^2({\mathbb{F}}_{\Theta },\mathbb{R})$ through the Weil construction of closed 2-forms as characteristic classes of principal fiber bundles. The starting point is the computation of the second homology group of ${\mathbb{F}}_{\Theta }$ with coefficients in a ring $R$.