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Unitary representations with Dirac cohomology: a finiteness result for complex Lie groups. (English) Zbl 1442.22013
Summary: Let $$G$$ be a connected complex simple Lie group, and let $$\widehat{G}^{\text{d}}$$ be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that $$\widehat{G}^{\text{d}}$$ consists of two parts: finitely many scattered representations, and finitely many strings of representations. Moreover, the strings of $$\widehat{G}^{\text{d}}$$ come from $$\widehat{L}^{\text{d}}$$ 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 $$\widehat{G}^{\text{d}}$$ requires a finite calculation in total. As an application, we report a complete description of $$\widehat{F}_4^{\text{d}}$$.
##### MSC:
 2.2e+47 Semisimple Lie groups and their representations
##### Keywords:
Dirac cohomology; good range; spin norm; unitary representation
##### Software:
Atlas of Lie Groups
Full Text:
##### References:
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