Dirac series for some real exceptional Lie groups

TL;DR

This paper classifies all irreducible unitary representations with non-zero Dirac cohomology for certain real exceptional Lie groups, revealing new phenomena and disproving a previous conjecture about Dirac cohomology cancellation.

Contribution

It provides a complete classification of Dirac cohomology for specific real exceptional Lie groups and uncovers a counterexample to a 2015 conjecture.

Findings

Classified all irreducible unitary representations with non-zero Dirac cohomology for EI, EIV, FI, FII groups.

Discovered a representation of F4(4) with zero Dirac index but non-zero Dirac cohomology.

Provided evidence against the conjecture that Dirac cohomology's even and odd parts do not cancel.

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