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

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.