Dirac cohomology for the BGG category O

We study Dirac cohomology $H_D^{g,h}\left(M\right)$ for modules belonging to category $O$ of a finite dimensional complex semisimple Lie algebra. We start by studying the generalized infinitesimal character decomposition of $M\otimes S$, with $S$ being a spin module of $h^{\perp }$. As a consequence, “Vogan’s conjecture” holds, and we prove a nonvanishing result for $H_D^{g,h}\left(M\right)$ while we show that in the case of a Hermitian symmetric pair $(g,k)$ and an irreducible unitary module $M\in O$, Dirac cohomology coincides with the nilpotent Lie algebra cohomology with coefficients in $M$. In the last part, we show that the higher Dirac cohomology and index introduced by Pandžić and Somberg satisfy nice homological properties for $M\in O$.