Analytic Extension of the Holomorphic Discrete Series

Abstract

Introduction. The embedding R ' naturally divides into two half-planes, each with R as boundary. These domains together with their function theory play an important role in the harmonic analysis of R. As but one example there is the theorem of Paley and Wiener that describes any square integrable function on R as a sum of boundary values of holomorphic functions (in Hardy spaces) on these domains. Much of the theory has natural extensions to R, to tube domains in C based on cones, and even to more general Siegel domains. However, the dominant influence is abelian harmonic analysis. On the other hand, if G is a semisimple Lie group over R it may often be embedded in its natural complexification Gc. Does any of the abelian