A note on the equivariant Chern character in Noncommutative Geometry

Given a smooth action of a Lie group on a manifold, we give two constructions of the Chern character of an equivariant vector bundle in the cyclic cohomology of the crossed product algebra. The first construction associates a cycle to the vector bundle whose structure maps are closely related to Getzler’s model for equivariant cohomology. The second construction uses a direct map between this model and the (periodic) cyclic cohomology localized at the unit element. Finally, it is shown that the two constructions are equivalent when the action is proper.