  In this paper we study the Hochschild cohomology ring of convolution algebras
associated to orbifolds, as well as their deformation quantizations. In the
first case the ring structure is given in terms of a wedge product on twisted
polyvectorfields on the inertia orbifold. After deformation quantization, the
ring structure defines a product on the cohomology of the inertia orbifold. We
study the relation between this product and an $S^1$-equivariant version of the
Chen--Ruan product. In particular, we give a de Rham model for this equivariant
orbifold cohomology.
