The first part (Sections 1-6) of this paper is a survey of some of the recent
developments in the theory of Dirac cohomology, especially the relationship of
Dirac cohomology with (g,K)-cohomology and nilpotent Lie algebra cohomology;
the second part (Sections 7-12) is devoted to understanding the unitary
elliptic representations and endoscopic transfer by using the techniques in
Dirac cohomology. A few problems and conjectures are proposed for further
investigations.Comment: This paper will appear in `Representations of Reductive Groups, in
Honor of 60th Birthday of David Vogan', edited by M. Nervins and P. Trapa,
published by Springe
