Poisson Harmonic Forms, Kostant Harmonic Forms, and the S1-Equivariant Cohomology ofK/T
✍ Scribed by Sam Evens; Jiang-Hua Lu
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 326 KB
- Volume
- 142
- Category
- Article
- ISSN
- 0001-8708
No coin nor oath required. For personal study only.
✦ Synopsis
dedicated to professor bertram kostant for his 70th birthday
We characterize the harmonic forms on a flag manifold KÂT 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 KÂT. 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 KÂT, where the S 1 -action is defined by . Using a simple argument in equivariant cohomology, we recover the connection between the Kostant harmonic forms and the Schubert calculus on KÂT that was found by Kostant and Kumar in 1986. By using a family of symplectic structures on KÂT, 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 on KÂT. 1999 Academic Press Contents 1. Introduction. 2. Poisson harmonic forms. 3. The Bruhat Poisson structure and the Kostant harmonic forms. 3.1. The Bruhat Poisson structure ? and the Koszul Brylinski operator . 3.2. The mixed complex (0(KÂT) K , d, ). 3.3. Kostant's operator versus the Koszul Brylinski operator . 3.4. Poisson geometrical proofs of Kostant's theorems. 4. The S 1 -equivariant cohomology of KÂT. 5. Kostant's harmonic forms as limits of Hodge harmonic forms. 5.1. The family of symplectic structures ? * . 5.2. Poisson harmonic forms for ? * . 5.3. The Kostant operator as a limit of adjoint operators of d. 5.4. Another proof of the disjointness of d and . 5.5. The Kostant harmonic forms as limits of Hodge harmonic forms. 6. Appendix: The Schouten bracket.