Abstract: Complex reflection groups are important objects with algebraic, geometric, and combinatorial properties that have been well-studied for several decades. We shall consider a reflection group G acting on the lattice of its hyperplane arrangement in C^n, and elements X in the lattice (which are subspaces of C^n). The goal is to characterize conditions on X that guarantee every polynomial on X that is invariant under the action of the normalizer of X in G can be extended to a G-invariant polynomial on C^n. A recent (2010) result of Douglass and Röhrle finds such a characterization for the action of Coxeter groups (ie, real reflection groups) in terms of the exponents of the associated hyperplane arrangements. In this talk we will explore current work toward a generalization of this result to all complex reflection groups.