Suppose G is a connected
semisimple Lie group. Then the tempered spectrum of G consists of families of
representations induced unitarily from 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 showed
that these wave packets are Schwartz class functions and that each K-finite
Schwartz function is a finite sum of wave packets. Thus he obtained a complete
characterization of K-finite functions in the Schwartz space in terms of their Fourier
transforms.
Now suppose that G has infinite center. Then every K-compact Schwartz function
decomposes naturally as a finite sum of wave packets. A new feature of the infinite
center case is that the wave packets into which it decomposes are not necessarily
Schwartz class functions. This is because of interference 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 conditions are also sufficient. This gives a complete
characterization of K-compact functions in the Schwartz space in terms of their
Fourier transforms.
