 
Summary: STABILIZATION OF A FAMILY OF DIFFERENTIAL EQUATIONS
James Arthur*
Abstract. Differential equations have a central place in the invariant
harmonic analysis of HarishChandra on real groups. Related differential
equations also play a role in the noninvariant harmonic analysis that
arises from the study of automorphic forms. We shall establish some
interesting identities among these latter equations. The identities we
obtain are likely to be useful for the comparison of automorphic forms
on different groups.
1. The differential equations. Suppose that G is a real, reductive algebraic group,
and that T is a maximal torus in G which is defined over R. We write Treg(R) for the
open dense subset of elements in T(R) that are strongly regular, in the sense that their
centralizer in G equals T. HarishChandra reduced many fundamental questions on the
harmonic analysis of G(R) to the study of a family of functions on Treg(R). We are referring
to the invariant orbital integrals
IG(, f) = JG(, f) = fG() , Treg(R), f C G(R) ,
which can be defined for any function f in HarishChandra's Schwartz space C G(R) by
(1.1) IG(, f) = D()
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