 
Summary: Proc. Nat. Acad. Sci. USA
Vol. 72, No. 12,pp.47184719, December 1975
Mathematics
A theorem on the Schwartzspace of a reductive Lie group
(associatedparabolic subgroups/Plancherel measure/Fourier transform on a reductive Lie group)
JAMES ARTHUR
Departmentof Mathematics, Yale University, New Haven, Connecticut 06520
Communicatedby C.D.Mostow, September 22,1975
ABSTRACT The purpose of this paper is to define the
Fourier transform of an arbitrary tempered distribution
on a reductive Lie group. To this end we define a topologi
cal vector space, e(G),in terms of the classes of irreducible
unitary representations of G ,which plays the roleof a dual
Schwartz space. Our main theorem then asserts that the
usual £ Fourier transform, when restricted to functions
in the Schwartz space, e(G) defined by HarishChandra,A
provides a toooloeical isomorphism from e(G)onto e(G).
Let G be a reductive Lie group with Liealgebra g. We make
the three assumptions on G stated in (ref. 1, 5 3), and adopt
the conventions and terminology of (ref. 1, 2 and 5 3). In
