THE SCHWARTZ SPACE OF A GENERAL SEMISIMPLE LIE GROUP .5. SCHWARTZ CLASS WAVE-PACKETS

Suppose G is a connected semisimple Lie group. Then the tempered spect rum of G consists of families of representations induced unitarily fro m cuspidal parabolic subgroups. In the case that G has finite center, Harish-Chandra used Eisenstein integrals to construct wave packets of matrix coefficients for each series of tempered representations. He sh owed that these wave packets are Schwartz class functions and that eac h K-finite Schwartz function is a finite sum of wave packets. Thus he obtained a complete characterization of K-finite functions in the Schw artz space in terms of their Fourier transforms. Now suppose that G ha s infinite center. Then every K-compact Schwartz function decomposes n aturally as a finite sum of wave packets. A new feature of the infinit e center case is that the wave packets into which it decomposes are no t necessarily Schwartz class functions. This is because of interferenc e between different series of representations when a principal series representation decomposes as a sum of limits of discrete series. There are matching conditions between the wave packets which are necessary in order that the sum be a Schwartz class function when the individual terms are not. In this paper it is shown that these matching conditio ns are also sufficient. This gives a complete characterization of K-co mpact functions in the Schwartz space in terms of their Fourier transf orms.