Advertisement

Mathematische Annalen

, Volume 337, Issue 1, pp 197–232 | Cite as

Cartan decomposition of the moment map

  • Peter Heinzner
  • Gerald W. Schwarz
Article

Abstract

We investigate a class of actions of real Lie groups on complex spaces. Using moment map techniques we establish the existence of a quotient and a version of Luna’s slice theorem as well as a version of the Hilbert–Mumford criterion. A global slice theorem is proved for proper actions. We give new proofs of results of Mostow on decompositions of groups and homogeneous spaces.

Mathematics Subject Classification (2000)

32M05 57S20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abels H. (1975). Parallelizability of proper actions, global K-slices and maximal compact subgroups. Math. Ann. 212:1–19MathSciNetCrossRefGoogle Scholar
  2. 2.
    Azad, H., Loeb, J.J.: Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces. Indag. Math. N.S. 3–4, 365–375 (1992).Google Scholar
  3. 3.
    Azad H., Loeb J.J. (1993). Plurisubharmonic functions and the Kempf–Ness Theorem. Bull. Lond. Math. Soc. 25:162–168MATHMathSciNetGoogle Scholar
  4. 4.
    Azad H., Loeb J.J. (1999).Some applications of plurisubharmonic functions to orbits of real reductive groups. Indag. Math. N.S. 10: 473–482MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Birkes D. (1971). Orbits of linear algebraic groups. Ann. Math. 93:459–475MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Chevalley C. (1946). Theory of Lie Groups. Princeton University Press, PrincetonMATHGoogle Scholar
  7. 7.
    Guillemin V., Stenzel M. (1991). Grauert tubes and the homogeneous Monge-Ampère equation I. J. Diff. Geom. 34:561–570MATHMathSciNetGoogle Scholar
  8. 8.
    Heinzner P. (1991). Geometric invariant theory on Stein spaces. Math. Ann. 289:631–662MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Heinzner P. (1993). Equivariant holomorphic extensions of real analytic manifolds. Bull. Soc. Math. France 121:445–463MATHMathSciNetGoogle Scholar
  10. 10.
    Heinzner P., Huckleberry A. (1996). Kählerian potentials and convexity properties of the moment map. Invent. Math. 126:65–84MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Heinzner, P., Huckleberry, A.:Complex geometry of Hamiltonian actions (to appear)Google Scholar
  12. 12.
    Heinzner, P., Huckleberry, A., Kutzschebauch, F.: A real analytic version of Abels’ theorem and complexifications of proper Lie group actions. In: Complex Analysis and Geometry (Trento, 1993), Lecture Notes in Pure and Applied Mathematics 173, pp. 229–273 Dekker, New York (1996)Google Scholar
  13. 13.
    Heinzner P., Huckleberry A., Loose F. (1994). Kählerian extensions of the symplectic reduction. J. Reine und Angew. Math. 455:123–140MATHMathSciNetGoogle Scholar
  14. 14.
    Heinzner P., Loose F. (1994). Reduction of complex Hamiltonian G-spaces. Geom. Funct. Anal. 4:288–297MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Heinzner P., Migliorini L., Polito M. (1998). Semistable quotients. Ann. Scuola Norm. Sup. Pisa 26:233–248MATHMathSciNetGoogle Scholar
  16. 16.
    Heinzner, P., Stötzel, H.: Semistable points with respect to real forms. (preprint, 2005)Google Scholar
  17. 17.
    Helgason S. (1978). Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New YorkMATHGoogle Scholar
  18. 18.
    Hochschild G. (1965). The structure of Lie groups. Holden-Day, San FranciscoMATHGoogle Scholar
  19. 19.
    Kempf, G., Ness, L.: The length of vectors in representation spaces. In: Lecture Notes in Math vol. 732, pp. 233–243 Springer, Berlin Heidelberg New York (1978)Google Scholar
  20. 20.
    Kirwan F. (1984). Cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes vol. 31. Princeton University Press, PrincetonMATHGoogle Scholar
  21. 21.
    Lempert L., Szöke R. (1991). Global solutions of the homogeneous complex Monge-Ampère equation and complex structures on the tangent bundle of Riemannian manifolds. Math. Ann. 290:689–712MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Luna D. (1973). Slices étales. Bull. Soc. Math. France, Mémoire 33: 81–105MATHGoogle Scholar
  23. 23.
    Luna D. (1975). Sur certaines opérations différentiables des groupes de Lie. Am. J. Math. 97:172–181MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Mostow G.D. (1955). Some new decomposition theorems for semisimple groups. Memoirs Am. Math. Soc. 14:31–54MATHMathSciNetGoogle Scholar
  25. 25.
    Mostow G.D. (1955). On covariant fiberings of Klein spaces. Am. J. Math. 77:247–278MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Mumford D., Fogarty J., Kirwan F. (1994). Geometric invariant theory, 3rd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete 34. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  27. 27.
    Narasimhan R. (1962). The Levi problem for complex spaces, II. Math. Ann. 146:195–216MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    O’Shea L., Sjamaar R. (2000). Moment maps and Riemannian symmetric pairs. Math. Ann. 317:415–457MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Palais R.S. (1961). On the existence of slices for actions of non-compact Lie groups. Ann. Math. 73:295–323MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Richardson R.W., Slodowy P. (1990). Minimum vectors for real reductive algebraic groups. J. Lond. Math. Soc. 42:409–429MATHMathSciNetGoogle Scholar
  31. 31.
    Schwarz, G.W.: The topology of algebraic quotients. In: Kraft, H. et al. (eds.) Topological methods in algebraic transformation groups. Progress in Mathematics vol. 80, pp. 135–152 Birkhäuser Verlag, Basel-Boston (1989)Google Scholar
  32. 32.
    Sjamaar R. (1995). Holomorphic slices, symplectic reduction and multiplicities of representations. Ann. Math. 141(2):87–129MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Fakultät für MathematikRuhr Universität BochumBochumGermany
  2. 2.Department of MathematicsBrandeis UniversityWalthamUSA

Personalised recommendations