Analysing the Schmid Operator for SL(2,R) and SU(2,1)

Abstract

The study of holomorphic discrete series on the special linear group SL(2, R) has been very fruitful in many areas in mathematics and physics, so there is a natural question to ask if they can be generalised to other Lie groups. There exists a differential operator called the Schmid operator which can be used to define discrete series representations of semisimple Lie groups. Understanding the Schmid operator is one of the main goals of this thesis. A second goal is to describe the Schmid operator in a tensorial formalism which is more closely related to differential geometry. We show that the Schmid operator before the projection is equivalent to a covariant derivative in the tensorial formalism. The Schmid operator is given explicitly for the Lie groups SL(2, R) and SU(2, 1). In the case of SL(2, R) the conditions for holomorphic and anti-holomorphic discrete series are retrieved. In the case of SU(2, 1) the conditions for holomorphic, antiholomorphic, and quaternionic discrete series are retrieved.