A measure on the Unipotent Variety

Abstract

Suppose that G is a reductive algebraic group defined over Q. There occurs in the trace formula a remarkable distribution on G(A)1 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 = T0 of a certain polynomial . We shall recall the precise definition in Section 1.