A uniqueness theorem for tempered invariant eigendistributions

Abstract

Let G be a real reductive Lie group and a tempered invariant eigendistribution on G. Given a natural ordering on the set of conjugacy classes of Cartan subgroups of G, is called extremal if it has a unique maximal element in its support. T. Hirai has proved for a restricted class of real simple Lie groups that if is extremal and satisfies certain regularity conditions, it is uniquely determined by its restriction to the maximal element in its support. The purpose of this paper is to show that Hirai's theorem is true without restriction of the type of G.