The cup product in orbifold Hochschild cohomology

We study the multiplicative structure of orbifold Hochschild cohomology in an attempt to generalize the results of Kontsevich and Calaque-Van den Bergh relating the Hochschild and polyvector field cohomology rings of a smooth variety. We introduce the concept of linearized derived scheme, and we argue that when $X$ is a smooth algebraic variety and $G$ is a finite abelian group acting on $X$, the derived fixed locus $\widetilde{X^G}$ admits an HKR linearization. This allows us to define a product on the cohomology of polyvector fields of the orbifold $[X/G]$. We analyze the obstructions to associativity of this product and show that they vanish in certain special cases. We conjecture that in these cases the resulting polyvector field cohomology ring is isomorphic to the Hochschild cohomology of $[X/G]$. Inspired by mirror symmetry we introduce a bigrading on the Hochschild homology of Calabi-Yau orbifolds. We propose a conjectural product which respects this bigrading and simplifies the previously introduced product.