2007-11-16

Pavle Pandzic, University of Zagreb and Cornell University

Dirac operators and unitary representations

In the 1970's, R.Parthasarathy introduced a version of the Dirac operator D attached to a real reductive group, and used it to construct the discrete series representations. He also obtained a useful necessary condition, Dirac operator inequality, for unitarizability of an irreducible Harish-Chandra module. In 1997 Vogan studied a purely algebraic version of D and used it to attach an invariant, called Dirac cohomology, to a Harish-Chandra module X. He conjectured that Dirac cohomology, if nonzero, determines the infinitesimal character of X. This conjecture was proved by J.-S.Huang and myself in 2002. Subsequently, various authors obtained analogues of this result in several other settings (non-commutative equivariant cohomology, Lie superalgebras, affine Lie algebras). In this talk I will give an overview of this and some related more recent results (joint with J.-S.Huang and D.Renard). I will finish by describing some open questions we are currently trying to settle.