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In this paper, we show that the Dirac cohomology $H_{D}(L(\lambda))$ of asimple highest weight module $L(\lambda)$ in $\mathcal{O}^\mathfrak{p}$ can beparameterized by a specific set of weights: a subset $\mathcal{W}_I(\lambda)$of the orbit of the Weyl group $W$ acting on $\lambda+\rho$. As an application,we show that any simple module in $\mathcal{O}^\mathfrak{p}$ is determined upto isomorphism by its Dirac cohomology. We describe four parameterizations of$H_D(L(\lambda))$ when $\lambda$ is regular. Two of these parameterizations aregeometric in terms of a partial ordering on the dual of the Cartan subalgebraand a generalization of strong linkage, respectively. Using these geometricparameterizations, we derive two algebraic parameterizations in terms of themultiplicities of the composition factors of a Verma module and the embeddingsbetween Verma modules, respectively. As an application, for Verma modules withregular infinitesimal character, we obtain an extended version of the Verma-BGGTheorem. We also investigate Dirac cohomology of Kostant modules. Using Diraccohomology, we give a new proof of the simplicity criterion for Verma modulesand describe a new simplicity criterion for parabolic Verma modules withregular infinitesimal character.

Geometric and algebraic parameterizations for Dirac cohomology of simple modules in $\mathcal{O}^\mathfrak{p}$ and their applications