  Hochschild cohomology governs deformations of algebras, and its graded Lie
structure plays a vital role. We study this structure for the Hochschild
cohomology of the skew group algebra formed by a finite group acting on an
algebra by automorphisms. We examine the Gerstenhaber bracket with a view
toward deformations and developing bracket formulas. We then focus on the
linear group actions and polynomial algebras that arise in orbifold theory and
representation theory; deformations in this context include graded Hecke
algebras and symplectic reflection algebras. We give some general results
describing when brackets are zero for polynomial skew group algebras, which
allow us in particular to find noncommutative Poisson structures. For abelian
groups, we express the bracket using inner products of group characters.
Lastly, we interpret results for graded Hecke algebras.
