Abstract
We study the homological properties of the factor space G/P, where G is a complex semisimple Lie group and P a parabolic subgroup of G. To this end we compare two descriptions of the cohomology of such spaces. One of these makes use of the partition of G/P into cells (Schubert cells), while the other consists in identifying the cohomology of G/P with certain polynomials on the Lie algebra of the Cartan subgroup H of G. The results obtained are used to describe the algebraic action of the Weyl group W of G on the cohomology of G/P.