Abelianization and the Duistermaat-Heckman theorem

TL;DR

This paper extends the Duistermaat-Heckman theorem to non-abelian settings by relating measures on Lie algebra duals to symplectic quotient volumes, using abelianization techniques and Gelfand-Cetlin data.

Contribution

It introduces a non-abelian version of the Duistermaat-Heckman theorem by connecting measures on dual Lie algebras to symplectic quotient volumes.

Findings

Established a measure on bb related to the Lie group action.

Expressed the Radon-Nikodym derivative in terms of symplectic quotient volumes.

Proved a non-abelian generalization of the Duistermaat-Heckman theorem.

Abstract

Fix a compact connected Lie group $G$ with Lie algebra $\mathfrak{g}$, as well as a strong Gelfand-Cetlin datum on $\mathfrak{g}^*$. Let us also fix a connected symplectic manifold $M$ endowed with an effective Hamiltonian $G$-action and proper moment map. We associate to such information a measure on $\mathbb{R}^{\mathrm{b}}$, where $\mathrm{b}=\frac{1}{2}(\dim G+\mathrm{rank}\hspace{2pt}G)$. We also express the Radon-Nikodym derivative of this measure in terms of the volumes of the symplectic quotients of $M$ by $G$, and thereby prove a non-abelian version of the Duistermaat-Heckman theorem.