De Rham 2-Cohomology of Real Flag Manifolds

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