  In the first part of this paper, we consider, in the context of an arbitrary
hyperplane arrangement, the map between compactly supported cohomology to the
usual cohomology of a local system. A formula (i.e., an explicit algebraic de
Rham representative) for a generalized version of this map is obtained.
  These results are applied in the second part to invariant theory: Schechtman
and Varchenko connect invariant theoretic objects to the cohomology of local
systems on complements of hyperplane arrangements. The first part of this paper
is then used, following and completing arguments of Looijenga, to determine the
image of invariants in cohomology. In suitable cases (e.g., corresponding to
positive integral levels), the space of invariants is shown to acquire a mixed
Hodge structure over a cyclotomic field. We investigate the Hodge filtration on
the space of invariants, and characterize the subspace of conformal blocks in
Hodge theoretic terms.
