Geometric quantization and Zuckerman models of semisimple Lie groups

Abstract

Let G be a real semisimple Lie group. We equip invariant presymplectic forms omega to some fibration X over the flag domain of G. By applying geometric quantization to (X, omega), we obtain a unitary G-representation H whose subrepresentations are infinitesimally equivalent to the Zuckerman modules. The occurence of the Zuckerman modules H are controlled by the image of the moment map of omega. This leads to our notion of Zuckerman model.