Dirac cohomology of cohomologically induced modules for reductive Lie groups

We extend the setting and a proof of the Vogan's conjecture on Dirac cohomology to a possibly disconnected real reductive Lie group $G$ in the Harish-Chandra class. We show that the Dirac cohomology of cohomologically induced module ${\mathcal L}_S(Z)$ is completely determined by the Dirac cohomology of the inducing module $Z$. More precisely, we prove that if $Z$ is weakly good then the Dirac cohomology $H_D({\mathcal L}_S(Z))$ is equal to ${\mathcal L}^{\widetilde{K}}_S(H_D(Z)\otimes\Bbb{C}_{-\rho(\frak{p}\cap\frak{u})})$. An immediate application is a classification of tempered irreducible unitary representations with nonzero Dirac cohomology.