Dirac Operators in Representation Theory

Abstract

Dirac cohomology is a new tool to study unitary and admissible representations of semisimple Lie groups. Vogan's conjecture on Dirac cohomology reveals an algebraic nature of Dirac operators. In this paper, we explain the joint work with Padzic on a proof of Vogan's conjecture. As applications, we also describe how to simplify the proof of the Atiyah-Schmid theorem on geometric constructions of discrete series and sharpen the Langlands-Hotta-Parthasarathy formula on automorphic forms. We also indicate the relation between Dirac cohomology and Lie algebra cohomologies.