 
Summary: Some tempered distributions on semisimple
groups of real rank one
Introduction
The Selberg trace formula leads naturally to the study of certain
tempered distributions on reductive groups defined over local fields. An
important problem is to calculate the Fourier transforms of these distribu
tions. We shall consider this question for the case that the local field is R
and the group G is semisimple and has real rank one. In this context the
notion of the Fourier transform of a tempered distribution has been defined
in [l(a)].
A distribution T is said to be invariant if
T(f "1 = T(f)
for every fe C:(G) and y e G, where
f qx) = f (yxyl) , x e G .
The invariant distributions which appear in the trace formula have recently
been examined by Sally and Warner. However, the trace formula also
contains some interesting noninvariant distributions. In this paper we shall
calculate the Fourier transforms of the restriction of these distributions to
£[(G)the space of cusp forms on G.
For the case that G = PSL(2, R) these noninvariant distributions have
