Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the following simple real exceptional Lie groups: ${ \rm EI}=E_{ 6(6)}, { \rm EIV}=E_{ 6(-26)}, { \rm FI}=F_{ 4(4)}, { \rm FII}=F_{ 4(-20)}$. Along the way, we find an irreducible unitary representation of $F_{ 4(4)}$ whose Dirac index vanishes, while its Dirac cohomology is non-zero. This disproves a conjecture raised in 2015 asserting that there should be no cancellation between the even part and the odd part of the Dirac cohomology.
Categories: math.RT