 
Summary: ContemporaryMathematics
Volume 53, 1986
Dedicated to A. Selberg
THE FOURIER TRANSFORM OF WEIGHTED ORBITAL
INTEGRALS ON SL(2,IR)
3
J s ~ r t h u r ; Rebecca A. Herb, and Paul J. Sally, Jr.
1. INTRODUCTION. A study of the adelic version of the Selberg trace for
mula leads naturally to the analysis of certain tempered distributions on re
ductive groups over local fields [la], [4], [8]. The invariant distributions
which arise in this context appear mainly as ordinary orbital integrals and
their limits. The Fourier analysis of ordinary orbital integrals has been
studied extensively over the past decade, and, in the case of real reductive
groups, this analysis may now be regarded as essentially complete [2], [7b],
[lo1
The situation for padic groups is much less satisfactory, and consid
erable work remains to be done in that case (see [9] for more details).
Along with the ordinary orbital integrals, certain noninvariant dis
tributions called weighted orbital integrals arise as additional terms in the
trace formula. The Fourier analysis of weighted orbital integrals i s more
