A vanishing theorem for hyperplane cohomology

Let A be a hyperplane arrangement in an arbitrary finite dimensional vector space V and let G less than or equal to GL(V) be an automorphism group of A. If lambda is a complex representation of G such that (lambda,1)(GH) = 0 for all pointwise isotropy groups G(H) (H epsilon A), then we prove the ''local-global'' result that lambda does not appear in the representation of G on the Orlik-Solomon algebra of A. The result is applied to complex reflection groups and to finite orthogonal groups. It may also be viewed as a combinatorial result concerning the homology of the lattice of intersections of A. A more general version of the main result is also discussed.