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

TL;DR

This paper classifies all irreducible unitary representations with non-zero Dirac cohomology for the group $E_{7(-5)}$, improving bounds and revealing cancellation phenomena, thus advancing understanding of its unitary dual.

Contribution

It provides a complete classification of certain unitary representations of $E_{7(-5)}$ using the Helgason-Johnson bound and explores Dirac cohomology properties.

Findings

Classification of irreducible unitary representations with non-zero Dirac cohomology.

Identification of cancellation between even and odd parts of Dirac cohomology.

Improved Helgason-Johnson bound assuming integral infinitesimal character.

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.