Gelfand-Cetlin abelianizations of symplectic quotients

TL;DR

This paper demonstrates that generic symplectic quotients of Hamiltonian G-spaces can be realized as quotients by a torus, using integrable systems like Gelfand-Cetlin, extending the abelianization concept to stratified symplectic spaces.

Contribution

It establishes a new abelianization result for symplectic quotients via integrable systems, applicable to stratified symplectic spaces and broad classes of Lie groups.

Findings

Symplectic quotients can be realized as torus quotients.

The result applies to stratified symplectic spaces.

Integrable systems like Gelfand-Cetlin are central to the construction.

Abstract

We show that generic symplectic quotients of a Hamiltonian $G$-space $M$ by the action of a compact connected Lie group $G$ are also symplectic quotients of the same manifold $M$ by a compact torus. The torus action in question arises from certain integrable systems on $\mathfrak{g}^*$, the dual of the Lie algebra of $G$. Examples of such integrable systems include the Gelfand-Cetlin systems of Guillemin-Sternberg in the case of unitary and special orthogonal groups, and certain integrable systems constructed for all compact connected Lie groups by Hoffman-Lane. Our abelianization result holds for smooth quotients, and more generally for quotients which are stratified symplectic spaces in the sense of Sjamaar-Lerman.