Poisson harmonic forms, Kostant harmonic forms, and the S1-equivariant cohomology of K/T

We characterize the harmonic forms on a nag 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. By using a family of symplectic structures on K/T, we also show that the Kostant harmonic forms are limits of the more familiar Hedge harmonic forms with respect to a Family of Hermitian metrics on K/T. (C) 1999 Academic Press.