Abstract
The main reason that harmonic analysis on semisimple groups can be developed in great depth is of course that the irreducible representations and their matrix coefficients can be constructed quite explicitly. In the setting that is of interest to us this means the construction of the principal series representations which in turn leads to integral representations of the associated elementary spherical functions. The fundamental theorem is that all elementary spherical functions are obtained by this method. This theorem sets the stage for the definition and study of the Harish-Chandra spherical transform of functions and distributions on G//K.
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