Coordinates on Schubert cells, Kostant's harmonic forms, and the Bruhat-Poisson structure on $G/B$

Abstract

We relate Kostant's theorem on the cohomology of a flag manifold $G/B$ with the geometry of the Bruhat-Poisson structure. We express Kostant's harmonic forms in terms of the moment maps (for the torus action) and the Liouville volume forms for the symplectic structures on the Schubert cells induced by the Bruhat-Poisson structure. We do this by writing everything down explicitly in some coordinates on each cell.