Convexity and Thimm's Trick

TL;DR

This paper establishes convexity and connectedness properties for certain Hamiltonian maps derived via Thimm's trick, extending classical results and providing explicit descriptions in special cases like Gelfand-Zeitlin systems.

Contribution

It offers a new, simple proof of convexity and fibre-connectedness for Thimm's trick maps, generalizing Gelfand-Zeitlin systems and explicitly characterizing their images.

Findings

Convexity and fibre-connectedness are proven for Thimm's trick maps.

Explicit inequalities describe the image in cases from chains of subalgebras.

Fibres are smooth embedded submanifolds when generating a torus action.

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.