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

## TL;DR

This paper proves a finiteness property for the set of irreducible unitary representations with non-zero Dirac cohomology in complex simple Lie groups, classifying them into finitely many parts and providing explicit descriptions.

## Contribution

It establishes a finiteness result for the representations with Dirac cohomology and describes their structure via cohomological induction from Levi subgroups.

## Key findings

- Finite classification of representations with Dirac cohomology
- Representation strings originate from Levi factors via cohomological induction
- Complete description of the set for the group F4

## Abstract

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 $\widehat{F}_4^{\mathrm{d}}$.