Let G be a real reductive Lie
group and π a tempered invariant eigendistribution on G. Given a natural
ordering on the set of conjugacy classes of Cartan subgroups of G, π 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 π 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.