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

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

1997-11-26

arxiv.org/abs/dg-ga/9711019v1

Mathematics

Differential Geometry

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Poisson harmonic forms, Kostant harmonic forms, and the $S^1$-equivariant cohomology of $K/T$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Poisson harmonic forms, Kostant harmonic forms, and the $S^1$-equivariant cohomology of $K/T$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Poisson harmonic forms, Kostant harmonic forms, and the $S^1$-equivariant cohomology of $K/T$ will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-503891