On a Family of Distributions Obtained from Orbits

of distributions. The terms on the right are parametrized by "cuspidal automorphic data", and are defined in terms of Eisenstein series. They have been evaluated rather explicitly in [3]. The terms on the left are parametrized by semisimple conjugacy classes and are defined in terms of related G(A) orbits. The object of this paper is to evaluate these terms. In previous papers we have already evaluated Jo(f) in two special cases. The easiest case occurs when o corresponds to a regular semisimple conjugacy class {a} in G(F). We showed in Section 8 of [1] that for such an o, J(f) could be expressed as a weighted orbital integral over the conjugacy class of a. (We actually assumed that o was "unramified", which is slightly more general.) The most difficult case is the opposite extreme, in which o corresponds to { 1}. This was the topic of [5]. We were able to express the distribution, which we denoted by Junip, as a finite linear combination of weighted orbital integrals over unipotent conjugacy classes. The general case is a mixture of these two. If o corresponds to an arbitrary semisimple conjugacy class {a), let GO be the connected component of the centralizer of a in G. In this paper we shall reduce the study of Jo(f) to the unipotent case on subgroups of G,. We will then be able to appeal to the results of [5]. Suppose that S is a finite set of valuations of F which contains the Archimedean places. We can embed C°(G(Fs)1) into C°(G(A)1) by multiplying any function e Cc°(G(FS)1) by the characteristic function of a maximal compact subgroup of IvyZs G(FV). Suppose that M is a Levi component of a parabolic subgroup of G which is defined over F. If y is any point in M(Fs), the weighted orbital integral JM(y, f) is the integral of f over the G(Fs)-conjugacy class of y, with respect to a certain noninvariant measure. The measure is easily defined in terms of the