CONVEXITY AND THIMM’S TRICK

In this paper we study topological properties of maps constructed by Thimm's trick with Guillemin and Sternberg's action coordinates on a connected Hamiltonian G -manifold M . Since these maps only generate a Hamiltonian torus action on an open dense subset of M , convexity and fibre-connectedness of such maps does not follow immediately from Atiyah–Guillemin–Sternberg's convexity theorem, even if M is compact. The core contribution of this paper is to provide a simple argument circumventing this difficulty. In the case where the map is constructed from a chain of subalgebras we prove that the image is given by a list of inequalities that can be computed explicitly in many examples. This generalizes the fact that the images of the classical Gelfand–Zeitlin systems on coadjoint orbits are Gelfand–Zeitlin polytopes. Moreover, we prove that if such a map generates a completely integrable torus action on an open dense subset of M , then all its fibres are smooth embedded submanifolds.