Skip to main content
SpringerLink
Log in

Complexity of nilpotent orbits and the Kostant-Sekiguchi correspondence

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

Let G be a connected linear semisimple Lie group with Lie algebra , and let be the complexified isotropy representation at the identity coset of the corresponding symmetric space. Suppose that Ω is a nilpotent G-orbit in and is the nilpotent -orbit in associated to Ω by the Kostant-Sekiguchi correspondence. We show that the corank of the Hamiltonian K-space Ω is twice the complexity of the variety .

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Guillemin, V., Sternberg, S.: Multiplicity-free spaces. J. Differ. Geom. 19, 31–56 (1984)

    Google Scholar 

  2. Guillemin, V., Sternberg, S.: Symplectic techniques in physics, Cambridge University Press, 1984

  3. Heckman, G.J.: Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups. Invent. Math. 67, 333–356 (1982)

    Article  Google Scholar 

  4. Huckleberry, A.T., Wurzbacher, T.: Multiplicity-free complex manifolds. Math. Ann. 286, 261–280 (1990)

    Article  Google Scholar 

  5. Karshon, Y.: Hamiltonian actions of Lie groups. PhD thesis, Harvard University, Cambridge, Massachusetts, 1993

  6. King, D.R.: Spherical Nilpotent Orbits and the Kostant-Sekiguchi Correspondence. Trans. Am. Math Soc. 354 (12), 4909–4920 (2002)

    Article  Google Scholar 

  7. Mykytyuk, I.V.: Actions of Borel Subgroups on Homogeneous Spaces of Reductive Complex Lie Groups and Integrability. Compos. Math. 127 (1), 55–67 (2001)

    Article  Google Scholar 

  8. Panyushev, D.I.: Complexity and nilpotent orbits. Manuscr. Math. 83, 223–237 (1994)

    Google Scholar 

  9. Panyushev, D.I.: Complexity and rank of actions in invariant theory. J. Math. Sci. New York 95 (1), 1925–1985 (1999)

    Google Scholar 

  10. Panyushev, D.I.: Complexity and rank of homogeneous spaces. Geom. Dedicata 34, 249–269 (1990)

    Article  Google Scholar 

  11. Richardson, R.W. Jr.: Principal orbit types for algebraic transformation spaces in characteristic zero. Invent. Math. 16, 6–14 (1972)

    Article  Google Scholar 

  12. Schmid, W., Vilonen, K.: The geometry of nilpotent orbits. Asian J. Math. 3 (1), 233–274 (1999)

    Google Scholar 

  13. Vergne, M.: Instantons et correspondance de Kostant-Sekiguchi. C. R. Acad. Sci. Paris Ser. 1 320, 901–906 (1995)

    Google Scholar 

  14. Vergne, M.: Polynomes de Joseph et représentation de Springer, Ann. Sci. Ec. Norm. Super. iv. Ser. 23 (4), 543–562 (1990)

    Google Scholar 

  15. Vinberg, E.B.: Commutative homogeneous spaces and co-isotropic symplectic actions. Russ. Math. Surv. 56 (1), 1–60 (2001)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Northeastern University, 567 Lake Hall, Boston, MA, 02115, USA

    Donald R. King

Authors
  1. Donald R. King
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Donald R. King.

Rights and permissions

Reprints and permissions

About this article

Cite this article

King, D. Complexity of nilpotent orbits and the Kostant-Sekiguchi correspondence. manuscripta math. 118, 121–134 (2005). https://doi.org/10.1007/s00229-005-0584-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-005-0584-z

Mathematics Subject Classification (2000)

Search

Navigation