Unitary representations with Dirac cohomology: A finiteness result for complex Lie groups

Abstract

Abstract Let G be a connected complex simple Lie group, and let G ^ d {\widehat{G}^{\mathrm{d}}} be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that G ^ d {\widehat{G}^{\mathrm{d}}} consists of two parts: finitely many scattered representations, and finitely many strings of representations. Moreover, the strings of G ^ d {\widehat{G}^{\mathrm{d}}} come from L ^ d {\widehat{L}^{\mathrm{d}}} via cohomological induction and they are all in the good range. Here L runs over the Levi factors of proper θ-stable parabolic subgroups of G. It follows that figuring out G ^ d {\widehat{G}^{\mathrm{d}}} requires a finite calculation in total. As an application, we report a complete description of F ^ 4 d {\widehat{F}_{4}^{\mathrm{d}}} .