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 θ-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}}}.

The Genuine Omega-regular Unitary Dual of the Metaplectic Group

We classify all genuine unitary representations of the metaplectic group whose infinitesimal character is real and at least as regular as that of the oscillator representation. In a previous paper we exhibited a certain family of representations satisfying these conditions, obtained by cohomological induction from the tensor product of a one-dimensional representation and an oscillator representation. Our main theorem asserts that this family exhausts the genuine omega-regular unitary dual of the metaplectic group.