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: $\mathrm{EI}=E_{6(6)},\mathrm{EIV}=E_{6(-26)},\mathrm{FI}=F_{4(4)},\mathrm{FII}=F_{4(-20)}$. 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.

28 pages, accepted by Journal of Algebra