Induction functor in noncommutative equivariant cohomology and Dirac cohomology

The aim of this paper is to put some recent results of Huang and Pandz̆ić (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-Pandz̆ić 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 and Meinrenken.