 
Summary: Vol. 56, No. 2 DUKE MATHEMATICAL JOURNAL © April 1988
THE LOCAL BEHAVIOUR OF WEIGHTED ORBITAL
INTEGRALS
JAMES ARTHUR
Introduction. Let G be a reductive algebraic group over a local field F of
characteristic 0. The invariant orbital integrals
JG(,f) = D(y) I112 f(xyx) dx, y e G(F), fe Cc(G(F)),'G(F)\G(F)
are obtained by integrating f with respect to the invariant measure on the
conjugacy class of y. They are of considerable importance for the harmonic
analysis of G(F). Invariant orbital integrals are also of interest because they
occur on the geometric side of the trace formula, in the case of compact quotient.
For the general trace formula, the analogous terms are weighted orbital integrals
[3]. They are obtained by integrating f over the conjugacy class of y, but with
respect to a measure which is not in general invariant. Weighted orbital integrals
may also play a role in the harmonic analysis of G(F), but this is not presently
understood. Our purpose here is to study the weighted orbital integrals as
functions of y. In particular, we shall show that they retain some of the basic
properties of ordinary orbital integrals.
Recall a few of the main features of the invariant orbital integrals. If F is an
Archimedean field, they satisfy the differential equations
