Duistermaat-Heckman measures for Hamiltonian groupoid actions

TL;DR

This work extends the Duistermaat–Heckman framework to Hamiltonian actions of source-proper, source-connected, regular symplectic groupoids with a proper, fibre-connected moment map. It introduces an affine measure $\mu_{aff}$ on the leaf space $B=M/\mathcal{G}$ and proves that the Duistermaat–Heckman measure $\mu_{DH}$ factors as $\mu_{DH}=\mathrm{vol}\cdot\mathrm{vol}_{\mathrm{red}}\cdot\mu_{aff}$, with $\mathrm{vol}$ and $\mathrm{vol}_{\mathrm{red}}$ polynomial on $B$; Morita invariance is analyzed with $\mu_{aff}$ and $\mathrm{vol}_{\mathrm{red}}$ invariant, while $\mathrm{vol}$ may vary. The paper develops the required background on symplectic groupoids, integral affine structures, and measures on leaf spaces, and provides a pointwise proof via a local model, connecting to classical DH results in the appropriate limits. Overall, the results generalize DH polynomiality to a broad Poisson-geometric setting and unify various Hamiltonian-type constructions under a common leaf-space framework.

Abstract

Consider a source proper, source connected regular symplectic groupoid acting locally freely and effectively in a Hamiltonian way, and assume that the moment map is proper and has connected fibres. In this case there is an associated Duistermaat-Heckman measure on the quotient orbifold. We show that this measure is polynomial with respect to the natural affine measure.