A Non-vanishing Criterion for Dirac Cohomology

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