The real locus of an involution map on the moduli space of flat connections on a Riemann surface

It is known that every nonorientable surface $\mathrm{\Sigma }$ has an orientable double cover $\tilde{\mathrm{\Sigma }}$. The covering map induces an involution on the moduli space $\tilde{\text{M}}$ of gauge equivalence classes of flat $G$-connections on $\tilde{\mathrm{\Sigma }}$. We identify the relation between the moduli space $\text{M}$ and the fixed point set of the moduli space $\tilde{\text{M}}$. In particular, $\text{M}$ is isomorphic to the fixed point set of $\tilde{\text{M}}$ if and only if the order of the center of $G$ is odd. One important application is that we give a way to construct a minimal Lagrangian submanifold of the moduli space $\tilde{\text{M}}$.