Deformations of quasi-Hamiltonian spaces

We introduce a notion of deformations of quasi-Hamiltonian $G$-spaces to Hamiltonian $G$-spaces and provide several examples. In particular, we show that the double $G\times G$ of a Lie group, viewed as a quasi-Hamiltonian $G\times G$-space, deforms smoothly to the cotangent bundle $T^{\ast }G$. Likewise, any conjugacy class of $G$ sufficiently close to the identity deforms to a coadjoint orbit. We further show that the moduli space of flat $G$-connections on a compact oriented surface of genus $g$ with $r+1$ boundary components deforms to $T^{\ast }{G}^{r+g}$.