Symplectic convexity theorems and coadjoint orbits

Joachim Hilgert; Karl-Hermann Neeb; Werner Plank

Compositio Mathematica (1994)

  • Volume: 94, Issue: 2, page 129-180
  • ISSN: 0010-437X

How to cite


Hilgert, Joachim, Neeb, Karl-Hermann, and Plank, Werner. "Symplectic convexity theorems and coadjoint orbits." Compositio Mathematica 94.2 (1994): 129-180. <>.

author = {Hilgert, Joachim, Neeb, Karl-Hermann, Plank, Werner},
journal = {Compositio Mathematica},
keywords = {Hamiltonian actions; Lie groups; symplectic manifolds; moment maps; convexity theorems; coadjoint orbits; Lie algebras},
language = {eng},
number = {2},
pages = {129-180},
publisher = {Kluwer Academic Publishers},
title = {Symplectic convexity theorems and coadjoint orbits},
url = {},
volume = {94},
year = {1994},

AU - Hilgert, Joachim
AU - Neeb, Karl-Hermann
AU - Plank, Werner
TI - Symplectic convexity theorems and coadjoint orbits
JO - Compositio Mathematica
PY - 1994
PB - Kluwer Academic Publishers
VL - 94
IS - 2
SP - 129
EP - 180
LA - eng
KW - Hamiltonian actions; Lie groups; symplectic manifolds; moment maps; convexity theorems; coadjoint orbits; Lie algebras
UR -
ER -


  1. [AL92] Arnal, D., and J. Ludwig, La convexité de l'application moment d'un groupe de Lie, J. Funct. Anal.105 (1992), 256-300 Zbl0763.22006MR1160080
  2. [At82] Atiyah, M., Convexity and commuting hamiltonians, Bull. London Math. Soc.14 (1982), 1-15 Zbl0482.58013MR642416
  3. [vdB86] van den Ban, E., A Convexity Theorem for Semisimple Symmetric Spaces, Pacific Journal of Math. 124(1986), 21-55 Zbl0599.22014MR850665
  4. [Bou71 ] Bourbaki, N., "Topologie Générale", Chap. 1-10, Hermann, Paris, 1971 Zbl0249.54001MR358652
  5. [Bou82] Bourbaki, N., Groupes et algèbres de Lie, Chap. 9, Masson, Paris, 1982 Zbl0505.22006
  6. [BJ73] Bröcker, T., and K. Jänich, "Einführung in die Differentialtopologie ", Springer Verlag, Berlin, Heidelberg, 1973 Zbl0269.57010MR358848
  7. [CDM88] Condevaux, M., P. Dazord and P. Molino, Geometrie du moment, in Sem. Sud-Rhodanien1988 MR1040871
  8. [tD91] tom Dieck, T., "Topologie", de Gruyter, Berlin, New York, 1991 Zbl0731.55001MR1150244
  9. [Dui83] Duistermaat, J., Convexity and tightness for restrictions of hamiltonian functions to fixed point sets of antisymplectic involution, Trans AMS.275 (1983), 417-429. Zbl0504.58020MR678361
  10. [GS82] Guillemin, V., and S. Sternberg, Convexity properties of the moment mapping I, Invent. Math.67 (1982), 491-513 Zbl0503.58017MR664117
  11. [GS84] Guillemin, V., and S. Sternberg, Symplectic techniques in physics, Cambridge Univ. Press, 1984 Zbl0576.58012MR770935
  12. [Hel78] Helgason, S., Differential geometry, Lie groups, and symmetric spaces, Acad. Press, London, 1978 Zbl0451.53038MR514561
  13. [Hil91 ] Hilgert, J., Vorlesung über symplektische Geometrie, Erlangen, 1991 
  14. [HHL89] Hilgert, J., K.H. Hofmann, and J.D. Lawson, Lie Groups, Convex Cones, and Semigroups, Oxford University Press, 1989 Zbl0701.22001MR1032761
  15. [HiNe93a] Hilgert, J., and K.-H. Neeb, Lie semigroups and their applications, Lecture Notes in Math.1552, Springer Verlag, 1993 Zbl0807.22001MR1317811
  16. [HiNe93b] Hilgert, J., and K.-H. Neeb, Non-linear Convexity Theorems and Poisson Lie groups, in preparation 
  17. [Ki84] Kirwan, F., Convexity properties of the moment mapping III, Invent. Math.77 (1984), 547-552 Zbl0561.58016MR759257
  18. [Ko73] Kostant, B., On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ecole Norm. Sup.6(1973), 413-455 Zbl0293.22019MR364552
  19. [Le80] Leichtweiß, K., Konvexe Mengen, Springer Verlag, Heidelberg, 1980 Zbl0442.52001MR586235
  20. [LM87] Libermann, P., and C. Marle, Symplectic geometry and analytical mechanics, Reidel, Dordrecht, 1987 Zbl0643.53002
  21. [Lo69] Loos, O., "Symmetric Spaces I : General Theory", Benjamin, New York, Amsterdam, 1969 Zbl0175.48601MR239005
  22. [LR91 ] Lu, J., and T. Ratiu, On the nonlinear convexity theorem of Kostant, Journal of the AMS4(1991), 349-363 Zbl0785.22019MR1086967
  23. [Me81] Meyer, K.R., Hamiltonian systems with a discrete symmetry, J. Diff. Eq.41(1981), 228-238 Zbl0438.70022MR630991
  24. [Ne92] Neeb, K.-H., A convexity theorem for semisimple symmetric spaces, Pac. J. Math., 162 (1994), 305-349. Zbl0809.53058MR1251904
  25. [Ne93a] Neeb, K.-H., Invariant subsemigroups of Lie groups, Mem. of the AMS499, 1993 Zbl0786.22001MR1152952
  26. [Ne93b] Neeb, K.-H., On closedness and simple connectedness of coadjoint orbits, Manuscripta Math., 82 (1994), 51-56. Zbl0815.22004MR1254139
  27. [Ne93c] Neeb, K.-H., Kähler structures and convexity properties of coadjoint orbits, Forum Math., to appear Zbl0823.22017MR1325561
  28. [Ne93d] Neeb, K.-H., Holomorphic representation theory II, Acta Math., to appear. Zbl0842.22004MR1294671
  29. [Ne93e] Neeb, K.-H., On the convexity of the moment mapping for a unitary highest weight representation, Journal of Funct. Anal., to appear Zbl0829.22012
  30. [Ne93f] Neeb, K.-H., Locally polyhedral sets, unpublished note 
  31. [Ola90] Olafsson, G., Causal Symmetric Spaces, Mathematica Gottingensis15, Preprint, 1990 
  32. [Pa84] Paneitz, S., Determination of invariant convex cones in simple Lie algebras, Arkif för Mat.21(1984), 217-228 Zbl0526.22016MR727345
  33. [Pl92] Plank, W., Konvexität in der symplektischen Geometrie, Diplomarbeit, Erlangen, 1992 

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