A modification of the Hodge star operator on manifolds with boundary
If M is a smooth compact oriented Riemannian manifold of dimension n=4k+2, with or without boundary, and F is a vector bundle on M with an inner product and a flat connection, we construct a modification of the Hodge star operator on the parabolic cohomology H^{2k+1}_{par}(M;F). The operator gives a canonical complex structure on H^{2k+1}_{par}(M;F) compatible with the symplectic form ωgiven by the wedge product of forms in the middle dimension. In case when k=0 that gives a canonical almost complex structure on the non-singular part of the moduli space of flat connections on a Riemann surface with or without boundary and monodromies along boundary components belonging to fixed conjugacy classes. The almost complex structure is compatible with the standard symplectic form ωon the moduli space.