  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 $\tilde{\M}$.
