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

TL;DR

This paper classifies certain modules of real reductive groups of types D, G2, and sp(2n,R) using Dirac cohomology, extending previous geometric classifications to these specific Lie algebra types.

Contribution

It extends the classification of $A_{\mathfrak{q}}(0)$--modules via Dirac cohomology to the cases of type $D$, $G_2$, and $\mathfrak{sp}(2n, \mathbb{R})$, which were not previously covered.

Findings

Classified faces of $W \rho$ for $\mathfrak{sp}(2n, \mathbb{R})$

Described faces corresponding to modules for types $D$ and $G_2$

Extended geometric classification to new Lie algebra types

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 $G_2$.