Zierau will investigate the realization of singular unitary representation associated to elliptic coadjoint orbits of a semisimple Lie group, i. e. representations on Dolbeault cohomology. He will unitarize these representations using square integrable harmonic forms and the invariant indefinite metric on the orbit. The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics.