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A convexity theorem for Poisson actions of compact Lie groups. (English) Zbl 0877.58025

When a connected compact symplectic manifold \(X\) admits a Hamiltonian action of a connection compact Lie group \(K\), the image of the moment map from \(X\) to the dual of the Lie algebra of \(K\) meets any positive Weyl chamber in a convex polytope [V. Guillemin and S. Sternberg, Invent. Math. 67, 491–513 (1982; Zbl 0503.58017) and ibid. 77, 533–546 (1984; Zbl 0561.58015), and F. Kirwan, Invent. Math. 77, 547–552 (1984; Zbl 0561.58016)].
The authors generalise this result to Poisson actions of connected compact semisimple groups, as follows. Let \(X\) be a compact connected symplectic manifold, with a Poisson action of a compact connected semisimple Lie group \(K\) and a moment map \(J\) in the sense of J.-H. Lu [Math. Sci. Res. Inst. Publ. 20, 209–226 (1991; Zbl 0735.58004)] from \(X\) to the dual group \(K^d\). If \(A_+\) is the image under the exponential map of a suitable positive Weyl chamber, then \(\log (J(X) \cap A_+)\) is a convex polytope. The method of proof is to reduce to the action of a maximal torus \(T\) and show that the action of \(T\) is in fact Hamiltonian.
Reviewer: F.Kirwan (Oxford)

MSC:

37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
53D20 Momentum maps; symplectic reduction
58B25 Group structures and generalizations on infinite-dimensional manifolds
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References:

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