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

Abstract

We characterize the harmonic forms on a flag manifoldK/Tdefined by Kostant in 1963 in terms of a Poisson structure. Namely, they are "Poisson harmonic" with respect to the so-called Bruhat Poisson structure onK/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 theS1-equivariant cohomology ofK/T, where theS1-action is defined byρ. Using a simple argument in equivariant cohomology, we recover the connection between the Kostant harmonic forms and the Schubert calculus onK/Tthat was found by Kostant and Kumar in 1986. By using a family of symplectic structures onK/T, 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 onK/T. © 1999 Academic Press.