 
Summary: Contemporary Mathematics
Volume 177, 1994
The Trace Paley Wiener Theorem for Schwartz Functions
James Arthur
Suppose that G is a connected reductive algebraic group over a local field
F of characteristic 0. If f is a function in the Schwartz space C(G(F)), and
ir E Htemp (G(F)) is an irreducible tempered representation of G(F), the operator
fG(r) = hj f (x)(x)dxG(F)
is of trace class. We can therefore map f to the function
fG(7r) = tr(Tr(f))
on 11temp(G(F)) . The object of this note is to characterize the image of the map.
Results of this nature are well known. The case of the Hecke algebra on G(F),
which is in fact more difficult, was established in [3] and [5]. A variant of the
problem for the smooth functions of compact support on a real group was solved
in [4]. For the Schwartz space, one has a choice of several possible approaches. We
shall use the characterization of the operator valued Fourier transform
f (f), f EC(G(F)),
which was solved separately for real and padic groups [2], [9, Part B]. (See also [6,
Lemma 5.2].)
Irreducible tempered representations occur as constituents of induced repre
