Poisson harmonic forms, Kostant harmonic forms, and the S-1-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 harmoni c" with respect to the so-called Bruhat Poisson structure on K/T. This enab les us to give Poisson geometrical proofs of many of the special properties of these harmonic forms. In particular, we construct explicit representati ves 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 co homology, 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. B y using a family of symplectic structures on K/T, we also show that the Kos tant harmonic forms are limits of the more familiar Hedge harmonic forms wi th respect to a Family of Hermitian metrics on K/T. (C) 1999 Academic Press .