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.

23 pages, added a few conjectures

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