Poisson harmonic forms, Kostant harmonic forms, and the $S^1$-equivariant cohomology of $K/T$

Abstract

We characterize the harmonic forms on a flag manifold $K/T$ defined by Kostant in 1963 in terms of a Poisson structure. Namely, they are ``Poisson harmonic" with respect to the so-called Bruhat Poisson structure on $K/T$. This enables us to give Poisson geometrical proofs of many of the special properties of these harmonic forms. In particular, we construct explicit representatives for the Schubert basis of the $S^1$-equivariant cohomology of $K/T$, where the $S^1$-action is defined by $\rho$. Using a simple argument in equivariant cohomology, we recover the connection between the Kostant harmonic forms and the Schubert calculus on $K/T$ that was found by Kostant and Kumar in 1986. We also show that the Kostant harmonic forms are limits of the more familiar Hodge harmonic forms with respect to a family of Hermitian metrics.