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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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