Schubert cells and cohomology of the spaces $G/P$

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 $G/P$.