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

Abstract

Let $G$ be a connected complex simple Lie group, and let ${\hat{G}}^{\mathrm{d}}$ be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that ${\hat{G}}^{\mathrm{d}}$ consists of two parts: finitely many scattered representations, and finitely many strings of representations. Moreover, the strings of ${\hat{G}}^{\mathrm{d}}$ come from ${\hat{L}}^{\mathrm{d}}$ 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 ${\hat{G}}^{\mathrm{d}}$ requires a finite calculation in total. As an application, we report a complete description of ${\hat{F}}_4^{\mathrm{d}}$.

arXiv comment27 pages

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Cited by in corpus (8)

• Dirac series for some real exceptional Lie groups
• Unitary representations with Dirac cohomology: finiteness in the real case
• Dirac series for $E_{6(-14)}$
• Scattered representations of $SL(n,\mathbb{C})$
• Unitary Representations with non-zero Dirac cohomology for complex $E_6$
• Scattered representations of complex classical Lie groups
• Dirac series for complex $E_7$
• Dirac series of $GL(n,\mathbb{R})$