Gelfand–Cetlin abelianizations of symplectic quotients

Peter Crooks and Jonathan Weitsman

Vol. 333 (2024), No. 2, 253–271

DOI: 10.2140/pjm.2024.333.253

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 ${\mathfrak{𝔤}}^{\ast }$, the dual of the Lie algebra of $G$. Examples of such integrable systems include the Gelfand–Cetlin systems of Guillemin and Sternberg (1980; 1983) in the case of unitary and special orthogonal groups, and certain integrable systems constructed for all compact connected Lie groups by Hoffman and Lane (2023). Our abelianization result holds for smooth quotients, and more generally for quotients which are stratified symplectic spaces in the sense of Sjamaar and  Lerman (1991).

Keywords

symplectic quotient, Gelfand–Cetlin system, stratified symplectic space

Mathematical Subject Classification

Primary: 53D20

Secondary: 17B80

Milestones

Received: 6 March 2023

Revised: 7 December 2024

Accepted: 8 December 2024

Published: 28 December 2024