Convexity and Thimm's Trick

Abstract

In this paper we prove a convexity and fibre-connectedness theorem for proper maps constructed by Thimm's trick on a connected Hamiltonian $G$-space $M$ that generate a Hamiltonian torus action on an open dense submanifold. Since these maps only generate a Hamiltonian torus action on an open dense submanifold of $M$, convexity and fibre-connectedness do 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. This generalizes the famous example of Gelfand-Zeitlin systems on coadjoint orbits introduced by Guillemin and Sternberg. Moreover, we prove that if such a map generates a completely integrable torus action on an open dense submanifold of $M$, then all its fibres are smooth embedded submanifolds.

This final version of the paper is to appear. The introduction has changed to better explain the relevance of this work. A brief section comparing the contents of this paper to symplectic contraction recently introduced by HMM has been reinstated. The details of several examples are expanded. A gap in an earlier proof of Proposition 4 has been corrected, thanks to the keen eye of the referees

References in corpus (4)

• Simplices in Newton-Okounkov bodies and the Gromov width of coadjoint orbits
• Automorphisms of multiplicity free Hamiltonian manifolds
• Multiplicity of compact group representations and applications to Kronecker coefficients
• Contraction of Hamiltonian $K$-spaces

Cited by in corpus (1)

• A completely integrable system on $G_2$ coadjoint orbits