 
Summary: Can. J. Math., Vol. XXXVIII, No. 1, 1986, pp. 179214
ON A FAMILY OF DISTRIBUTIONS OBTAINED
FROM ORBITS
JAMES ARTHUR
Introduction. Suppose that G is a reductive algebraic group defined over
a number field F. The trace formula is an identity
2 o(f) = Jx(f), f E C°(G(A)1),OE0 XEGr
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
