Unitary Representations with Dirac cohomology: a finiteness result for complex Lie groups

Let $G$ be a connected complex simple Lie group, and let ${\widehat{G}}^{\mathrm{d}}$ be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that ${\widehat{G}}^{\mathrm{d}}$ consists of two parts: finitely many scattered representations, and finitely many strings of representations. Moreover, the strings of ${\widehat{G}}^{\mathrm{d}}$ come from ${\widehat{L}}^{\mathrm{d}}$ via cohomological induction and they are all in the good range. Here $L$ runs over the Levi factors of proper $\theta$-stable parabolic subgroups of $G$. It follows that figuring out ${\widehat{G}}^{\mathrm{d}}$ requires a finite calculation in total. As an application, we report a complete description of ${\widehat{F}}_4^{\mathrm{d}}$.