THE IRREDUCIBILITY OF STANDARD MODULES

FOR WEAKENED HARISH-CHANDRA SHEAVES

Robert Shalla

University of Utah, 1995

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).

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