De Rham $2$-cohomology of real flag manifolds

Let ${\mathbb{F}}_{\mathrm{\Theta }}=G\mathrm{/}{P}_{\mathrm{\Theta }}$ be a flag manifold associated to a non-compact real simple Lie group $G$ and the parabolic subgroup $% P_{\Theta }$\mathrm{.}Thisisaclosedsubgroupof$G$determinedbyasubset$% \Theta $ of simple restricted roots of $\mathfrak{g}=Lie(G)$. This paper computes the second de Rham cohomology group of ${\mathbb{F}}_{\mathrm{\Theta }}$. We prove that it zero in general, with some rare exceptions. When it is non-zero, we give a basis of $H^2({\mathbb{F}}_{\mathrm{\Theta }},\mathbb{R})$ 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 ${\mathbb{F}}_{\mathrm{\Theta }}$ with coefficients in a ring $R$.