We consider, in the context of an arbitrary hyperplane arrangement, the map between compactly supported cohomology to the usual cohomology of a rank one 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 to invariants of tensor product of representations: Schechtman and Varchenko connect invariant theoretic objects to the cohomology of local systems on complements of hyperplane arrangements. 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.

This a joint work with P. Belkale and P. Brosnan.