Abelianization and the Duistermaat-Heckman theorem

2023/02/05 by Peter Crooks, Crooks, Peter, Jonathan Weitsman +1 · 1 citation
Mathematics · #Geometry and complex manifolds #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2302.02421

Abstract

Fix a compact connected Lie group G with Lie algebra \mathfrakg, as well as a strong Gelfand-Cetlin datum on \mathfrakg^*. 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 ℝ^b, where b=(1)/(2)(dim G+rank\hspace2ptG). 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.

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