Classification of $A_{\mathfrak{q}}(\lambda)$ modules by their Dirac cohomology for type $D$, $G_2$ and $\mathfrak{sp}(2n,\mathbb{R})$

Abstract

Let $G$ be a connected real reductive group with maximal compact subgroup $K$ of the same rank as $G$. In the recent paper of Huang, Pand\v{z}i\'{c} and Vogan, it was shown that the admissible $\Theta$--stable parabolic subalgebras $\mathfrak{q}$ of $\mathfrak{g}$ are in one-to-one correspodence with the faces of $W \rho$ intersecting the $\mathfrak{k}$--dominant Weyl chamber and that $A_{\mathfrak{q}}(0)$--modules can be classified by their Dirac cohomology in geometric terms. They described in detail the cases when $\mathfrak{g}_0$ is of type $A$, $B$, $F$ and $C$ except for $\mathfrak{g}_0 = \mathfrak{sp}(2n, \mathbb{R})$. We will describe faces corresponding to $A_{\mathfrak{q}}(0)$--modules for $\mathfrak{g}_0 = \mathfrak{sp}(2n, \mathbb{R})$ and for $\mathfrak{g}_0$ of type $D$ and $G_2$.