Abstract
It is known that every nonorientable surface Σ has an orientable double cover Σ~. The covering map induces an involution on the moduli space ℳ~ of gauge equivalence classes of flat G-connections on Σ~. We identify the relation between the moduli space ℳ and the fixed point set of the moduli space ℳ~. In particular, ℳ is isomorphic to the fixed point set of 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 ℳ~.
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