Dirac cohomology, branching laws and Wallach modules

The idea of using Dirac cohomology to study branching laws was initiated by Huang, Pandzi\'c and Zhu in 2013 [HPZ]. One of their results says that the Dirac cohomology of $\pi$ completely determines $\pi|_{K}$, where $\pi$ is any irreducible unitarizable highest weight $(\mathfrak{g}, K)$ module. This paper aims to develop this idea for the exceptional Lie groups $E_{6(-14)}$ and $E_{7(-25)}$: we recover the $K$-spectrum of the Wallach modules from their Dirac cohomology.