Approximating Fourier transformation of orbital integrals

In this paper, we are concerned with orbital integrals on a class C of real reductive Lie groups with non-compact Iwasawa K-component. The class C contains all connected semisimple Lie groups with infinite center. We establish that any given orbital integral over general orbits with compactly supported continuous functions for a group G in C is convergent. Moreover, it is essentially the limit of corresponding orbital integrals for its quotient groups in Harish-Chandra’s class. Thus the study of orbital integrals for groups in class C reduces to those of Harish-Chandra’s class. The abstract theory for this limiting technique is developed in the general context of locally compact groups and linear functionals arising from orbital integrals.We point out that the abstract theory can be modified easily to include weighted orbital integrals as well. As an application of this limiting technique, we deduce the explicit Plancherel formula for any group in class C.  2004 Elsevier Inc. All rights reserved