On the Dirac cohomology of complex lie group representations

Abstract

This note studies the problem of classifying all the irreducible unitary representations with nonzero Dirac cohomology for a complex Lie group G. We reduce it to the classification of spherical ones with nonzero Dirac cohomology on the Levi level. Then in the spherical unitary dual, by computing spin norm and utilizing Vogan pencil, we show how to further reduce the classification to fairly few candidate representations.