We study Dirac cohomology $H_D^{ \mathfrak{ g},\mathfrak{ h}}(M)$ for modules belonging to category $\mathcal{ O}$ of a finite dimensional complex semisimple Lie algebra. We prove Vogan's conjecture, a nonvanishing result for $H_D^{ \mathfrak{ g},\mathfrak{ h}}(M)$ while we show that in the case of a Hermitian symmetric pair $(\mathfrak{ g},\mathfrak{ k})$ and an irreducible unitary module $M\in\mathcal{ O}$, 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 Pand\v{ z}i\'c and Somberg satisfy nice homological properties for $M\in\mathcal{ O}$.
Categories: math.RT