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On the nonlinear convexity theorem of Kostant. (English) Zbl 0785.22019
Let \(G\) be a non-compact semisimple Lie group and \(G=KAN\) an Iwasawa decomposition. Let \(L: G\to{\mathfrak a}={\mathbf L}(A)\) be the corresponding Iwasawa projection and \(a\in A\). Then according to a theorem of Kostant the set \(L(aK)\) is precisely the convex hull of the Weyl group orbit of \(L(a)\). The goal of the paper under review is to present a proof of this fact using the symplectic convexity theorems for moment maps due to Guillemin-Sternberg and Atiyah.
In the case that \(G\) happens to be a complex Lie group it is known that \(K\) and \(AN\) are Poisson Lie groups. Moreover one knows that the symplectic leaves in \(AN\) are precisely the orbits of the right \(K\)-action on \(AN\) induced by the decomposition \(KAN\). The authors show that in this case a suitable maximal torus of \(K\) acts in Hamiltonian fashion and that the moment map is, up to a constant, just the Iwasawa projection.
In order to prove the theorem also for real groups one wants to use a theorem of Duistermaats which deals with images under the moment map of sets of fixed points under anti-symplectic involutions. To this end one complexifies the real group, uses the decomposition and symplectic structure explained above, and takes the complex conjugation as an involution into account. This approach works as long as one can guarantee that the Iwasawa \(K\) and \(AN\) of the complexified group are invariant under the complex conjugation. Unfortunately this is not always the case (the minimal parabolic of the real group has to be solvable).

MSC:
22E46 Semisimple Lie groups and their representations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
32M05 Complex Lie groups, group actions on complex spaces
22E10 General properties and structure of complex Lie groups
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