Dirac cohomology and $\Theta$-correspondence for complex dual pairs

TL;DR

This paper investigates how Dirac cohomology behaves under Howe's $ heta$-correspondence for complex reductive dual pairs, providing explicit computations for the Dirac cohomology of lifted modules.

Contribution

It characterizes when Dirac cohomology is preserved under $ heta$-lifting and explicitly computes the Dirac cohomology of the lifted modules in complex dual pairs.

Findings

Identifies conditions for nonzero Dirac cohomology preservation under $ heta$-lifting.

Provides explicit formulas for Dirac cohomology of lifted modules.

Enhances understanding of Dirac cohomology in the context of complex dual pairs.

Abstract

We study the behavior of Dirac cohomology under Howe's $\Theta$-correspondence in the case of complex reductive dual pairs. More precisely, if $(G_1,G_2)$ is a complex reductive dual pair with $G_1$ and $G_2$ viewed as real groups, we describe those Harish-Chandra modules $\pi_1$ of $G_1$ with nonzero Dirac cohomology whose $\Theta$-liftings $\Theta(\pi_1)$ still have nonzero Dirac cohomology. In this case, we compute explicitly the Dirac cohomology of $\Theta(\pi_1)$.