Dirac cohomology and Dirac induction

Abstract

Let G be a connected semisimple Lie group with a maximal compact group K of equal rank. We use the Dirac cohomology of the unitary representations to define Dirac-induction from a representation of K to the discrete series of G. This is closely related to the Dirac induction for the reduced group C * -algebras C * red (G) and a geometric construction of discrete series for semisimple Lie groups. Furthermore, we use Dirac cohomology of the Kostant's cubic Dirac operator to define Dirac-induction for compact Lie groups. This induction for compact Lie groups is simpler than the Bott's induction and is easier for calculation.