Twisted moduli spaces and Duistermaat–Heckman measures

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