Derived functor modules, dual pairs and $U(\mathfrak{g})^K$-actions

Abstract

Derived functors (or Zuckerman functors) play a very important role in the study of unitary representations of real reductive groups. These functors are usually applied on highest weight modules in the so-called good range and the theory is well-understood. On the other hand, there were several studies on the irreducibility and unitarizability, in which derived functors are applied to singular modules. See Enright et al. (Acta. Math. 1985), for example. In this article, we apply derived functors to certain modules arising from the formalism of local theta lifting, and investigate the irreducible sub-quotients of resulting modules. The key technique is to understand $U(\mathfrak{g})^K$-actions in the setting of a see-saw pair. Our results strongly suggest that derived functor constructions are compatible with local theta lifting.