Hochschild cohomology governs deformations of algebras, and its graded Lie
structure plays a critical role. We explore this structure for a finite
group G acting on an algebra S by automorphisms. We capture the group
together with its action with the natural semi-direct product, S#G,
known as the "skew group algebra" or "smash product algebra". For example,
when G acts linearly on a complex vector space V, it induces an action on
the symmetric algebra S(V), a polynomial ring. The semi-direct product
S(V)#G is a surrogate for the ring of invariant polynomials on V; it
serves as the coordinate ring for the orbifold arising from the action of
G on V. Deformations of this skew group algebra S(V)#G play a prominent
role in representation theory. Such deformations include graded Hecke
algebras (originally defined independently by Drinfeld and by Lusztig),
symplectic reflection algebras (investigated by Etingof and Ginzburg in
the study of orbifolds), and rational Cherednik algebras (introduced to
solve Macdonald's inner product conjectures). We explore the graded Lie
structure (or Gerstenhaber bracket) of the Hochschild cohomology of skew
group algebras with an eye toward deformation theory. For abelian groups
acting linearly, this structure can be described in terms of inner
products of group characters. (Joint work with Sarah Witherspoon.)