Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Cohomology of symplectic reductions of generic coadjoint orbits


Authors: R. F. Goldin and A.-L. Mare
Journal: Proc. Amer. Math. Soc. 132 (2004), 3069-3074
MSC (2000): Primary 53D20, 14M15
DOI: https://doi.org/10.1090/S0002-9939-04-07443-X
Published electronically: June 2, 2004
MathSciNet review: 2063128
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\mathcal{O}_\lambda$ be a generic coadjoint orbit of a compact semi-simple Lie group $K$. Weight varieties are the symplectic reductions of $\mathcal{O}_\lambda$ by the maximal torus $T$ in $K$. We use a theorem of Tolman and Weitsman to compute the cohomology ring of these varieties. Our formula relies on a Schubert basis of the equivariant cohomology of $\mathcal{O}_\lambda$, and it makes explicit the dependence on $\lambda$ and a parameter in $Lie(T)^*=:\mathfrak{t}^*$.


References [Enhancements On Off] (What's this?)

  • [Bi] S. Billey, Kostant polynomials and the cohomology of $G/B$, Duke Math. J. 96 (1999) 205-224. MR 2000a:14060
  • [Br] M. Brion, Equivariant cohomology and equivariant intersection theory, in Representation Theory and Algebraic Geometry, Kluwer Acad. Publ. (1998) 1-37. MR 99m:14005
  • [Go1] R. F. Goldin, The cohomology ring of weight varieties and polygon spaces, Adv. in Math. 160 (2001) No. 2, 175-204. MR 2002f:53139
  • [Go2] R. F. Goldin, An effective algorithm for the cohomology ring of symplectic reductions, Geom. and Funct. Anal., Vol. 12 (2002), 567-583. MR 2003m:53148
  • [Ki] F. C. Kirwan, Cohomology of Quotients in Complex and Algebraic Geometry, Mathematical Notes 31, Princeton University Press, Princeton, N. J. (1984). MR 86i:58050
  • [Ko] R. R. Kocherlakota, Integral homology of real flag manifolds and loop spaces of symmetric spaces, Adv. in Math. 110 (1995) no. 1, 1-46. MR 96a:57066
  • [KK] B. Kostant and S. Kumar, The nil Hecke ring and cohomology of $G/P$ for a Kac-Moody group $G$, Adv. in Math. 62 (1986) no. 3, 187-237. MR 88b:17025b
  • [TW] S. Tolman and J. Weitsman, The cohomology rings of symplectic quotients Comm. Anal. Geom. 11 (2003), no. 4, 751-773.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53D20, 14M15

Retrieve articles in all journals with MSC (2000): 53D20, 14M15


Additional Information

R. F. Goldin
Affiliation: Mathematical Sciences, George Mason University, MS 3F2, 4400 University Dr., Fairfax, Virginia 22030
Email: rgoldin@gmu.edu

A.-L. Mare
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
Email: amare@math.toronto.edu

DOI: https://doi.org/10.1090/S0002-9939-04-07443-X
Keywords: Coadjoint orbits, symplectic reduction, Schubert classes
Received by editor(s): November 8, 2002
Published electronically: June 2, 2004
Additional Notes: The first author was supported by NSF-DMS grant number 0305128
Communicated by: Rebecca Herb
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society