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

Sam Evens·Jiang-Hua Lu
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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 $ρ$. 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.

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Sam Evens, Jiang-Hua Lu. 1997-11-26. Poisson harmonic forms, Kostant harmonic forms, and the $S^1$-equivariant cohomology of $K/T$. https://arxiv.org/abs/dg-ga/9711019

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