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

## TL;DR

This paper develops geometric and algebraic parameterizations of Dirac cohomology for simple modules in category O, enabling classification and new criteria for module simplicity, with applications to Verma modules.

## Contribution

It introduces novel geometric and algebraic parameterizations of Dirac cohomology for modules in category O, and applies these to classify modules and establish simplicity criteria.

## Key findings

- Dirac cohomology of simple modules is parameterized by Weyl group orbits.
- Any simple module in al Oan be uniquely identified by its Dirac cohomology.
- New simplicity criteria for Verma and parabolic Verma modules are established.

## Abstract

In this paper, we show that the Dirac cohomology $H_{D}(L(\lambda))$ of a simple highest weight module $L(\lambda)$ in $\mathcal{O}^\mathfrak{p}$ can be parameterized 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 up to isomorphism by its Dirac cohomology. We describe four parameterizations of $H_D(L(\lambda))$ when $\lambda$ 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.