Dirac cohomology, branching laws and Wallach modules

Abstract

The idea of using Dirac cohomology to study branching laws was initiated by Huang, et al. (2013) [11]. One of their results says that the Dirac cohomology of π completely determines $\pi {|}_K$, where π is any irreducible unitarizable highest weight $(g,K)$ 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.