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Kähler structures and convexity properties of coadjoint orbits. (English) Zbl 0823.22017
Author’s abstract: Let \(\mathfrak g\) be a finite dimensional real Lie algebra and \({\mathfrak g}^*\) its dual. We show that every coadjoint orbit in \({\mathfrak g}^*\) which is contained in the relative interior of a closed convex invariant set containing no lines carries a uniquely determined Kähler structure which is compatible with the natural symplectic structure. We also characterize those orbits by data associated to a root decomposition with respect to a compactly embedded Cartan algebra, and characterize them among the other Kähler orbits meeting the dual of a compactly embedded Cartan algebra.
Reviewer: F.Rouvière (Nice)

MSC:
22E60 Lie algebras of Lie groups
53C55 Global differential geometry of Hermitian and Kählerian manifolds
17B05 Structure theory for Lie algebras and superalgebras
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