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

TL;DR

This paper classifies all irreducible unitary representations with non-zero Dirac cohomology for the group $E_{7(7)}$, advancing understanding of its representation theory and supporting conjectures in the field.

Contribution

It provides a complete classification of Dirac series for $E_{7(7)}$, improving bounds and exploring properties like Dirac index and multiplicities of spin lowest $K$-types.

Findings

Partial support for Vogan's FPP conjecture.

Identification of Dirac series with multiplicities in spin lowest $K$-types.

Enhanced bounds for $E_{7(7)}$ representations.

Abstract

This paper classifies all the Dirac series (that is, irreducible unitary representations having non-zero Dirac cohomology) of $E_{7(7)}$. Enhancing the Helgason-Johnson bound in 1969 for the group $E_{7(7)}$ is one key ingredient. Our calculation partially supports Vogan's fundamental parallelepiped (FPP) conjecture. As applications, when passing to Dirac index, we continue to find cancellation between the even part and the odd part of Dirac cohomology. Moreover, for the first time, we find Dirac series whose spin lowest $K$-types have multiplicities.