A non-vanishing criterion for Dirac cohomology

Abstract

This paper gives a criterion for the non-vanishing of the Dirac cohomology of $\mathcal{L}_S(Z)$, where $\mathcal{L}_S(\cdot)$ is the cohomological induction functor, while the inducing module $Z$ is irreducible, unitarizable, and in the good range. As an application, we give a formula counting the number of strings in the Dirac series. Using this formula, we classify all the irreducible unitary representations of $E_{6(2)}$ with non-zero Dirac cohomology. Our calculation continues to support Conjecture 5.7' of Salamanca-Riba and Vogan [SV]. Moreover, we find more unitary representations for which cancellation happens between the even part and the odd part of their Dirac cohomology.