A Measure on the Unipotent Variety

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 consequence (Corollary 8.3) of our final formula for Junip We shall work indirectly. From [3] we understand the behaviour of Junip under conjugation. If y is any point in G(A)', we have