Dirac operators and lie algebra cohomology

Dirac cohomology is a new tool used to study unitary and admissible representations of semisimple Lie groups. It was introduced by Vogan and further studied by Kostant and ourselves. The aim of this paper is to study the Dirac cohomology of unitary modules for the Kostant cubic Dirac operator and its relation to nilpotent Lie algebra cohomology. We show that the Dirac cohomology coincides with the corresponding nilpotent Lie algebra cohomology in some cases. Along the way we prove some properties of Dirac cohomology that make it more accessible for calculation.