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Symplectic convexity theorems and coadjoint orbits. (English) Zbl 0819.22006
In this far-ranging paper the authors consider Hamiltonian actions \(G \times M \to M\) of a Lie group \(G\) on a symplectic manifold \(M\). Such an action gives rise in a natural way to a moment map \(\Phi\), and the authors propose and develop a unified approach to various convexity theorems in the general framework of such moment maps. They establish first of all a local convexity theorem for the case that \(G = T\) is a torus, and then show how such local theorems and the assumption that \(\Phi\) is proper yield global convexity theorems. This line of argument yields strengthened versions of the Atiyah-Guillemin-Sternberg as well as the Duistermaat convexity theorems on Hamiltonian actions of compact toral groups and provides a generalization of the convexity theorems of Paneitz and Olafsson in the noncompact setting. In the last sections of the paper the authors give applications of their results to the case of coadjoint orbits in the dual of a Lie algebra and characterize those closed coadjoint orbits which contain no line in their convex hull. They also derive some convexity theorems for the case that \(G\) is compact, but not a toral group.

MSC:
22E30 Analysis on real and complex Lie groups
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57S20 Noncompact Lie groups of transformations
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References:
[1] Arnal, D. , and J. Ludwig , La convexité de l’application moment d’un groupe de Lie , J. Funct. Anal. 105 (1992), 256-300 · Zbl 0763.22006
[2] Atiyah, M. , Convexity and commuting hamiltonians , Bull. London Math. Soc. 14 (1982), 1-15 · Zbl 0482.58013
[3] Van Den Ban, E. , A Convexity Theorem for Semisimple Symmetric Spaces , Pacific Journal of Math. 124(1986), 21-55 · Zbl 0599.22014
[4] Bourbaki, N. , ”Topologie Générale”, Chap. 1-10 , Hermann, Paris, 1971 · Zbl 0249.54001
[5] Bourbaki, N. , Groupes et algèbres de Lie, Chap. 9 , Masson, Paris, 1982 · Zbl 0505.22006
[6] Bröcker, T. , and K. Jänich , ” Einführung in die Differentialtopologie ”, Springer Verlag, Berlin, Heidelberg, 1973 · Zbl 0269.57010
[7] Condevaux, M. , P. Dazord and P. Molino , Geometrie du moment, in Sem . Sud-Rhodanien 1988
[8] Tom Dieck, T. , ” Topologie ”, de Gruyter, Berlin, New York, 1991 · Zbl 0731.55001
[9] Duistermaat, J. , Convexity and tightness for restrictions of hamiltonian functions to fixed point sets of antisymplectic involution , Trans AMS.275 (1983), 417-429. · Zbl 0504.58020
[10] Guillemin, V. , and S. Sternberg , Convexity properties of the moment mapping I , Invent. Math. 67 (1982), 491-513 · Zbl 0503.58017
[11] Guillemin, V. , and S. Sternberg , Symplectic techniques in physics , Cambridge Univ. Press, 1984 · Zbl 0576.58012
[12] Helgason, S. , Differential geometry, Lie groups, and symmetric spaces , Acad. Press, London, 1978 · Zbl 0451.53038
[13] Hilgert, J. , Vorlesung über symplektische Geometrie , Erlangen, 1991
[14] Hilgert, J. , K.H. Hofmann , and J.D. Lawson , Lie Groups, Convex Cones, and Semigroups , Oxford University Press, 1989 · Zbl 0701.22001
[15] Hilgert, J. , and K.-H. Neeb , Lie semigroups and their applications , Lecture Notes in Math. 1552, Springer Verlag, 1993 · Zbl 0807.22001
[16] Hilgert, J. , and K.-H. Neeb , Non-linear Convexity Theorems and Poisson Lie groups , in preparation · Zbl 0912.58013
[17] Kirwan, F. , Convexity properties of the moment mapping III , Invent. Math.77 (1984), 547-552 · Zbl 0561.58016
[18] Kostant, B. , On convexity, the Weyl group and the Iwasawa decomposition , Ann. Sci. Ecole Norm. Sup. 6(1973), 413-455 · Zbl 0293.22019
[19] Leichtweiß, K. , Konvexe Mengen , Springer Verlag, Heidelberg, 1980 · Zbl 0442.52001
[20] Libermann, P. , and C. Marle , Symplectic geometry and analytical mechanics , Reidel, Dordrecht, 1987 · Zbl 0643.53002
[21] Loos, O. , ” Symmetric Spaces I : General Theory ”, Benjamin, New York, Amsterdam, 1969 · Zbl 0175.48601
[22] Lu, J. , and T. Ratiu , On the nonlinear convexity theorem of Kostant , Journal of the AMS 4(1991), 349-363 · Zbl 0785.22019
[23] Meyer, K.R. , Hamiltonian systems with a discrete symmetry , J. Diff. Eq. 41(1981), 228-238 · Zbl 0438.70022
[24] Neeb, K.-H. , A convexity theorem for semisimple symmetric spaces , Pac. J. Math., 162 (1994), 305-349. · Zbl 0809.53058
[25] Neeb, K.-H. , Invariant subsemigroups of Lie groups , Mem. of the AMS 499, 1993 · Zbl 0786.22001
[26] Neeb, K.-H. , On closedness and simple connectedness of coadjoint orbits , Manuscripta Math., 82 (1994), 51-56. · Zbl 0815.22004
[27] Neeb, K.-H. , Kähler structures and convexity properties of coadjoint orbits , Forum Math., to appear · Zbl 0823.22017
[28] Neeb, K.-H. , Holomorphic representation theory II , Acta Math., to appear. · Zbl 0842.22004
[29] Neeb, K.-H. , On the convexity of the moment mapping for a unitary highest weight representation , Journal of Funct. Anal., to appear · Zbl 0829.22012
[30] Neeb, K.-H. , Locally polyhedral sets , unpublished note
[31] Olafsson, G. , Causal Symmetric Spaces , Mathematica Gottingensis 15, Preprint, 1990
[32] Paneitz, S. , Determination of invariant convex cones in simple Lie algebras , Arkif för Mat.21(1984), 217-228 · Zbl 0526.22016
[33] Plank, W. , Konvexität in der symplektischen Geometrie , Diplomarbeit, Erlangen, 1992
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