Abstract: In this thesis we study a representation theory for semisimple Lie groups which have infinite center and provide a geometric description of the irreducible representations of any such group, G. Our perspective is motivated by the localization theory of A. Beilinson and J. Bernstein; they show that an irreducible representation of G can equivalently be regarded as an algebraic D-module, where D is a twisted sheaf of differential operators on a certain smooth projective variety. A description of irreducible representations of G is thus achieved by constructing "standard" D-modules which have finite length and whose composition factors exhaust the irreducible D-modules.
These standard modules are also endowed with the action of a reductive algebraic group K, and if the center of G were finite, then they would commonly be known as standard Harish-Chandra sheaves. However, in this thesis the usual compatibility condition on the D-module action and K-action for Harish-Chandra sheaves is weakened slightly to accommodate the possibility that the center of G is infinite.
Our principal result gives a simple criterion for irreducibility of a given standard module. As an application, the program is carried out for the case in which G is the universal covering group of SU(n,1). The special case of the universal covering group of SL(2,R) is also discussed in more detail in the expository paper The universal cover of SL(2,R) and its representations by this author.