Dirac cohomology and translation functors

We study the relationship between the Dirac cohomology of a (g,K)-module X and the Dirac cohomology of a Jantzen–Zuckerman translate of X. More precisely, we show that if X is unitary, and if some submodule X′ of a translate of X has nonzero Dirac cohomology, then X has nonzero Dirac cohomology. We also show that the space of harmonic spinors (i.e., the kernel of the Dirac operator) related to X′ embeds into a certain product of harmonic spinors for X and harmonic spinors for the finite-dimensional module used to define the translation. This generalizes, with a simpler proof, results of Mehdi and Parthasarathy (2008) [MP1], and Mehdi and Parthasarathy (2010) [MP2].