Kirwan, Frances Convexity properties of the moment mapping. III. (English) Zbl 0561.58016 Invent. Math. 77, 547-552 (1984). This paper is concerned with the moment mappings associated to compact Lie group actions on compact symplectic manifolds, which are functions from the manifolds to the duals of the Lie algebras of the groups acting on them. In Part I and II by V. Guillemin and S. Sternberg [ibid. 67, 491-513 (1982; Zbl 0503.58017) and ibid. 77, 533-546 (1984; see the paper reviewed above)] it is proved that the intersection of the image of a moment map with any Weyl chamber is a convex polytope in the cases when either the group is a torus or the manifold is Kähler. Guillemin and Sternberg conjectured that the same is true in general. The purpose of this paper is to prove their conjecture. Cited in 5 ReviewsCited in 102 Documents MSC: 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 53D50 Geometric quantization 53C80 Applications of global differential geometry to the sciences Keywords:compact Lie group actions on compact symplectic manifolds Citations:Zbl 0561.58015; Zbl 0503.58017 PDFBibTeX XMLCite \textit{F. Kirwan}, Invent. Math. 77, 547--552 (1984; Zbl 0561.58016) Full Text: DOI EuDML References: [1] Atiyah, M.F.: Convexity and commuting Hamiltonians. Bull. London Math. Soc.14, 1-15 (1982) · Zbl 0482.58013 · doi:10.1112/blms/14.1.1 [2] Borel, A.: Kählerian coset spaces of semisimple Lie groups. Proc. Nat. Acad. Sci. U.S.A.40, 1147-1151 (1954) · Zbl 0058.16002 · doi:10.1073/pnas.40.12.1147 [3] Guillemin, V., Sternberg, S.: Convexity properties of the moment mapping. Invent. Math.67, 491-513 (1982) · Zbl 0503.58017 · doi:10.1007/BF01398933 [4] Guillemin, V., Sternberg, S.: Convexity properties of the moment mapping, II. Invent. Math.67, 533-546 (1984) · Zbl 0561.58015 · doi:10.1007/BF01388837 [5] Guillemin, V., Sternberg, S.: Symplectic techniques in physics. Cambridge: Cambridge University Press 1984 · Zbl 0576.58012 [6] Arnold, V.: Méthodes mathématiques de la mechanique classique. Éditions de Moscou, 1976 [7] Kirwan, F.C.: The cohomology of quotient spaces in algebraic and symplectic geometry. Thesis, Oxford (1983), to appear in Mathematical Notes, Princeton University Press Yellow Series [8] Ness, L.: A stratification of the null cone via the moment map, with an appendix by D. Mumford, to appear in Am. J. Math. · Zbl 0604.14006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.