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.

arXiv comment28 pages, accepted by Journal of Algebra

References in corpus (4)

• Unitary representations with Dirac cohomology: finiteness in the real case
• Unitary Representations with Dirac cohomology: a finiteness result for complex Lie groups
• Unitary Representations with non-zero Dirac cohomology for complex $E_6$
• Dirac cohomology of unipotent representations of Sp(2n,R) and U(p,q)

Cited by in corpus (10)

• Dirac series for $E_{6(-14)}$
• On the Helgason-Johnson bound
• A non-vanishing criterion for Dirac cohomology
• Dirac series of $E_{7(-5)}$
• Dirac series of $E_{7(-25)}$
• Dirac series of $GL(n,\mathbb{R})$
• Dirac series of $E_{7(7)}$
• Dirac index of some unitary representations of $Sp(2n,\mathbb{R})$ and $S{O}^{\ast }(2n)$
• Dirac series of $E_{8(-24)}$
• Vogan's FPP conjecture for complex Lie groups