Twisted Moduli Spaces and Duistermaat-Heckman Measures

Abstract: Following Boalch-Yamakawa and Meinrenken, we consider a certain class of moduli spaces on bordered surfaces from a quasi-Hamiltonian perspective. For a given Lie group $G$, these character varieties parametrize flat $G$-connections on "twisted" local systems, in the sense that the transition functions take values in $G\rtimes\mathrm{Aut}(G)$. After reviewing the necessary tools to discuss twisted quasi-Hamiltonian manifolds, we construct a Duistermaat-Heckman (DH) measure on $G$ that is invariant under the twisted conjugation action $g\mapsto hg\kappa(h^{-1})$ for $\kappa\in\mathrm{Aut}(G)$, and characterize it by giving a localization formula for its Fourier coefficients. We then illustrate our results by determining the DH measures of our twisted moduli spaces.