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When a finite group acts linearly on a complex vector space, the naturalsemi-direct product of the group and the polynomial ring over the space forms askew group algebra. This algebra plays the role of the coordinate ring of theresulting orbifold and serves as a substitute for the ring of invariantpolynomials from the viewpoint of geometry and physics. Its Hochschildcohomology predicts various Hecke algebras and deformations of the orbifold. Inthis article, we investigate the ring structure of the Hochschild cohomology ofthe skew group algebra. We show that the cup product coincides with a naturalsmash product, transferring the cohomology of a group action into a groupaction on cohomology. We express the algebraic structure of Hochschildcohomology in terms of a partial order on the group (modulo the kernel of theaction). This partial order arises after assigning to each group element thecodimension of its fixed point space. We describe the algebraic structure forCoxeter groups, where this partial order is given by the reflection lengthfunction; a similar combinatorial description holds for an infinite family ofcomplex reflection groups.

Finite groups acting linearly: Hochschild cohomology and the cup product