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

For the flag manifoldX=G/B of a complex semi-simple Lie groupG, we make connections between the Kostant harmonic forms onG/B and the geometry of the Bruhat Poisson structure. We show that on each Schubert cell, the corresponding Kostant harmonic form can be described using only data coming from the Bruhat Poisson structure. We do this by using an explicit set of coordinates on the Schubert cell.Research partially supported by an NSF Postdoctorial Fellowship.