  The aim of this paper is to put some recent results of Huang-Pandzic
(conjectured by Vogan) and Kostant on Dirac cohomology in a broader
perspective.This is achieved by introducing an induction functor in the
noncommutative equivariant cohomology. In this context, the results of
Huang-Pandzic and Kostant are interpreted as special cases (corresponding to
the manifold being a point) of more general results on noncommutative
equivariant cohomology introduced by Alekseev-Meinrenken.
