  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
\omega given 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 \omega on the moduli space.
