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Summary: Proc. Indian Acad. Sci. (Math. Sci.), Vol. 97, Nos 1-3, December 1987, pp.3-19.
© Printed in India.
The characters of supercuspidal representations as
weighted orbital integrals
JAMES ARTHUR*
Department of Mathematics, University of Toronto, Toronto, Canada M5S lAI
Abstract. Weighted orbital integrals are the terms which occur on the geometric side of the
trace formula. We shall investigate these distributions on a p-adic group. We shall evaluate
the weighted orbital integral of a supercuspidal matrix coefficient as a multiple of the
corresponding character.
Keywords. Supercuspidal representation; weighted orbital integrals.
1. Introduction
Let G be a reductive algebraic group over a non-Archimedean local field F of
characteristic 0. Suppose that r is a (smooth) supercuspidal representation of G(F) on
a complex vector space V. Let f(x) be a finite sum of matrix coefficients
(nr(x)-'v), xeG(F), veV, leV*.
Then f is a locally constant function on G(F) which is compactly supported modulo
the split component AG of the centre of G. If ,, is the character of n, set
t.(f)= f(X)©(x)O,(x)dx.
AG(F)\G(F)
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