Geometric and algebraic parameterizations for Dirac cohomology of simple modules in $\mathcal{O}^\mathfrak{p}$ and their applications
In this paper, we show that the Dirac cohomology $H_{D}(L(λ))$ of a simple highest weight module $L(λ)$ in $\mathcal{O}^\mathfrak{p}$ can be parameterized by a specific set of weights: a subset $\mathcal{W}_I(λ)$ of the orbit of the Weyl group $W$ acting on $λ+ρ$. As an application, we show that any simple module in $\mathcal{O}^\mathfrak{p}$ is determined up to isomorphism by its Dirac cohomology. We describe four parameterizations of $H_D(L(λ))$ when $λ$ is regular. Two of these parameterizations are geometric in terms of a partial ordering on the dual of the Cartan subalgebra and a generalization of strong linkage, respectively. Using these geometric parameterizations, we derive two algebraic parameterizations in terms of the multiplicities of the composition factors of a Verma module and the embeddings between Verma modules, respectively. As an application, for Verma modules with regular infinitesimal character, we obtain an extended version of the Verma-BGG Theorem. We also investigate Dirac cohomology of Kostant modules. Using Dirac cohomology, we give a new proof of the simplicity criterion for Verma modules and describe a new simplicity criterion for parabolic Verma modules with regular infinitesimal character.