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

TL;DR

This paper connects Kostant's harmonic forms and the Bruhat-Poisson structure on flag manifolds, providing explicit coordinate descriptions that relate cohomology, symplectic geometry, and Poisson structures.

Contribution

It explicitly describes Kostant's harmonic forms using moment maps and volume forms within the Bruhat-Poisson framework on flag manifolds.

Findings

Explicit coordinate expressions for harmonic forms on Schubert cells

Relation between Kostant's theorem and Bruhat-Poisson geometry

Explicit formulas linking cohomology and symplectic structures

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.