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### Description

Dirac operators were used in the context of Representation Theory by Parthasarathy in 1972, as invariant first order differential operators acting on sections of homogeneous vector bundles over symmetric spaces $G/K$ in order to obtain realizations of the discrete series representations of $G$.

In a series of lectures in 1997, Vogan introduced an algebraic analogue of Parthasarthy's Dirac operator. By using this operator, he defined the so-called Dirac cohomology of $(\mathfrak{g},K)$-modules $X$ and conjectured a relation between the Dirac cohomology of $X$ and its infinitesimal character, proved by Huang and Pandžić in 2001. Since then, Dirac cohomology has been computed for various families of modules, including highest weight modules, $A_{\mathfrak q}(\lambda)$ modules, generalized Enright-Varadarajan modules, unipotent representations, etc.

In this talk, we will present some results concerning Dirac operators for modules belonging to the standard BGG category $\mathcal{O}$ of a complex semisimple Lie algebra $\mathfrak{g}$. This category consists of the finitely generated, locally $\mathfrak{n}$-finite weight modules of $\mathfrak{g}$ and seems to be the "correct" module category to study questions raised by Verma concerning composition series and embeddings of Verma modules, and Jantzen concerning his so-called translation functors.