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

We study the behavior of Dirac cohomology under Howe’s $\mathrm{\Theta }$-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 $\mathrm{\Theta }$-liftings $\mathrm{\Theta }({\pi }_1)$ still have nonzero Dirac cohomology. In this case, we compute explicitly the Dirac cohomology of $\mathrm{\Theta }({\pi }_1)$.