Orbits of a maximal torus in a flag variety

Let G be a semisimple Lie group, P its parabolic subgroup, and F=G/P the corresponding flag variety. Let X be the closure of a generic orbit of a maximal torus T acting on F. We give a decomposition of X over Schubert cycles. In the case of the classical Grassmannian of vector subspaces of given dimension, the coefficients of the above decomposition have a simple combinatorial interpretation. Some applications to enumerative geometry will be discussed.