Barbasch–Pandžić conjecture on Dirac series for complex simple Lie groups

Let $G$) be a connected complex simple Lie group. Write $\widehat{G}^d$ for the representations of $G$ with nonzero Dirac cohomology, and call a representation unipotent as in the paper. A unitary character is a one-dimensional unitary representation.

Barbasch–Pandžić conjecture. If $\pi\in\widehat{G}$ has regular, half-integral infinitesimal character, then

$$\pi\in\widehat{G}^d$$

if and only if $\pi$ is parabolically induced from a unipotent representation with nonzero Dirac cohomology, tensored with a unitary character.

This conjecture characterizes the Dirac series at regular half-integral infinitesimal character in terms of unipotent representations and unitary parabolic induction. The paper proves it for complex connected classical Lie groups and the Spin groups; the general case, including exceptional groups, is not established here.