  A degeneration of compact Kaehler manifolds gives rise to a monodromy action
on Betti moduli space H^1(X, G) = Hom(\pi_1(X),G)/G over smooth fibres with a
complex algebraic structure group G being either abelian or reductive. Assume
that the singularities of the central fibre is of normal crossing. When G = C,
the invariant cohomology classes arise from the global classes. This is no
longer true in general. In this paper, we produce large families of locally
invariant classes that do not arise from global ones for reductive G. These
examples exist even when G is abelian, as long as G contains multiple torsion
points. Finally, for general G, we make a new conjecture on local invariant
classes and produce some suggestive examples.
