De Rham 2-Cohomology of Real Flag Manifolds

Abstract

Let FΘ = G/PΘ be a flag manifold associated to a non-compact real simple Lie group G and the parabolic subgroup PΘ. This is a closed subgroup of G determined by a subset Θ of simple restricted roots of g = Lie(G). This paper computes the second de Rham cohomology group of FΘ. We prove that it is zero in general, with some rare exceptions. When it is non-zero, we give a basis of H²(FΘ, 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 FΘ with coefficients in a ring R.