A Non-vanishing Criterion for Dirac Cohomology

This paper gives a criterion for the non-vanishing of the Dirac cohomology of L S ( Z ) , where L S ( ⋅ ) 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 (Ann. Math., 148 (3), 1067–1133 1998 ). Moreover, we find more unitary representations for which cancellation happens between the even part and the odd part of their Dirac cohomology.