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.

中文翻译：

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扭曲模空间和 Duistermaat-Heckman 测度

继 Boalch-Yamakawa 和 Meinrenken 之后，我们从准汉密尔顿的角度考虑了有边曲面上的某类模空间。对于给定的李群 $G$，这些字符变体参数化了“扭曲”局部系统上的平面 $G$-connections，从某种意义上说，转换函数采用 $G\rtimes\mathrm{Aut}(G)$ 中的值。在回顾了讨论扭曲准哈密尔顿流形的必要工具之后，我们在 $G$ 上构建了一个 Duistermaat-Heckman (DH) 测度，它在扭曲共轭作用 $g\mapsto hg\kappa(h^{-1}) $ 为 $\kappa\in\mathrm{Aut}(G)$，并通过给出其傅立叶系数的定位公式来表征它。然后，我们通过确定扭曲模空间的 DH 度量来说明我们的结果。