Gelfand-Cetlin abelianizations of symplectic quotients

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.