  In previous work jointly with Geske, Maxim and Wang, we constructed a mixed
Hodge structure (MHS) on the torsion part of Alexander modules, which
generalizes the MHS on the cohomology of the Milnor fiber for weighted
homogeneous polynomials. The cohomology of a Milnor fiber carries a monodromy
action, whose semisimple part is an isomorphism of MHS. The natural question of
whether this result still holds for Alexander modules was then posed. In this
paper, we give a positive answer to that question, which implies that the
direct sum decomposition of the torsion part of Alexander modules into
generalized eigenspaces is in fact a decomposition of MHS. We also show that
the MHS on the generalized eigenspace of eigenvalue 1 can be constructed
without passing to a suitable finite cover (as is the case for the MHS on the
torsion part of the Alexander modules), and compute it under some purity
assumptions on the base space.
