Dirac series for some real exceptional Lie groups

Abstract

Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the following simple real exceptional Lie groups: EI=E6(6),EIV=E6(−26),FI=F4(4),FII=F4(−20). Along the way, we find an irreducible unitary representation of F4(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.