 
Summary: Can. J. Math., Vol. XXXVII, No. 6, 1985, pp. 12371274
A MEASURE ON THE UNIPOTENT VARIETY
JAMES ARTHUR
Introduction. Suppose that G is a reductive algebraic group defined over
Q. There occurs in the trace formula a remarkable distribution on G(A)l
which is supported on the unipotent set. It is defined quite concretely in
terms of a certain integral over G(Q)\G(A)1. Despite its explicit
description, however, this distribution is not easily expressed locally, in
terms of integrals on the groups G(Qv). For many applications of the trace
formula, it will be essential to do this. In the present paper we shall solve
the problem up to some undetermined constants.
The distribution, which we shall denote by Junip, was defined in [1] and
[3] as one of a family {JO, of distributions. It is the value at T = TO of a
certain polynomial Jnip. We shall recall the precise definition in Section
1. Let us just say here that for f Cc((G(A)1), J Tip(f) is given as an
integral over G(Q)\G(A)' which converges only for T in a certain
chamber which depends on the support of f. This is a source of some
difficulty. For example, since Junip(f) is defined by continuation in T
outside the domain of absolute convergence of the integral Tnip(f), it is
not possible, a priori, to identify Junip with a measure. This will be a
