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 $\Sigma$ has an orientable double cover $\tilde{\Sigma}$. The covering map induces an involution on the moduli space $\tilde{\M}$ of gauge equivalence classes of flat $G$-connections on $\tilde{\Sigma}$. We identify the relation between the moduli space $\M$ and the fixed point set of the moduli space $\tilde{\M}$. In particular, $\M$ is isomorphic to the fixed point set of $\tilde{\M}$ 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 $\tilde{\M}$.