Twisted moduli spaces and Duistermaat–Heckman measures

Abstract

Following Boalch–Yamakawa, Li-Bland–Ševera 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⋊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↦hgκ(h−1) for κ∈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. © 2020 Elsevier B.V.