Dirac series of $E_{7(-5)}$

Abstract

Using the sharpened Helgason-Johnson bound, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology of $E_{7(-5)}$. As an application, we find that the cancellation between the even part and the odd part of the Dirac cohomology continues to happen for certain unitary representations of $E_{7(-5)}$. Assuming the infinitesimal character being integral, we further improve the Helgason-Johnson bound for $E_{7(-5)}$. This should help people to understand (part of) the unitary dual of this group.