Unitary representations with Dirac cohomology: finiteness in the real case

TL;DR

This paper proves a finiteness theorem for classifying certain irreducible unitary representations with non-zero Dirac cohomology of real forms of complex algebraic groups, and studies their distribution properties.

Contribution

It establishes a finiteness result for irreducible unitary Harish-Chandra modules with non-vanishing Dirac cohomology and analyzes their distribution along Vogan pencils.

Findings

Finiteness theorem for classification of modules with Dirac cohomology.

Analysis of spin norm distribution along Vogan pencils.

Insights into unitarily small convex hulls for specific real groups.

Abstract

Let $G$ be a complex connected simple algebraic group with a fixed real form $\sigma$. Let $G(\mathbb{R})=G^\sigma$ be the corresponding group of real points. This paper reports a finiteness theorem for the classification of irreducible unitary Harish-Chandra modules of $G(\mathbb{R})$ (up to equivalence) having non-vanishing Dirac cohomology. Moreover, we study the distribution of the spin norm along Vogan pencils for certain $G(\mathbb{R})$, with particular attention paid to the unitarily small convex hull introduced by Salamanca-Riba and Vogan.