An elementary fact in the theory of finite groups states that the vector space
spanned by irreducible characters of representations of a group coincides with the
vector space spanned by characteristic functions of conjugacy classes. To give
an explicit formula for an irreducible character is to express a character as a lin-
ear combination of the basis vectors formed by characteristic functions. For Lie
groups or p-adic groups, irreducible characters must be interpreted as distribu-
tions. Similarly the characteristic functions must be replaced by distributions,
orbital integrals, supported on the conjugacy classes if one hopes to develop
systematically a theory of characters on Lie and p-adic groups. A successful
character theory of these groups should give relations expressing distribution
characters as linear combinations of orbital integrals, and orbital integrals as
sums of characters. This work is concerned with the study of orbital integrals on
p-adic groups, needed for eventual applications to automorphic representation
theory and the trace formula.
These orbital integrals have a notoriously complicated structure. As the con-
jugacy class is allowed to vary, the orbital integrals possess an asymptotic ex-
pansion called the Shalika germ expansion. In contrast to what the terminology
might suggest, the asymptotic expansion has only finitely many terms and for
p-adic groups actually gives an exact formula for the orbital integral in a suf-
ficiently small neighborhood of the identity element. Moreover, by inductive
arguments the behavior of an orbital integral may be understood once its be-
havior near the identity element of the group is understood. Consequently most
questions we might have about orbital integrals can be answered from Shalika's
expansion. Unfortunately, Shalika's existence proof of an asymptotic expansion
has not resulted in explicit formulas for the germs except in a few elementary
A basic problem of harmonic analysis on reductive p-adic groups is then to
develop expressions for the terms of the Shalika expansion of orbital integrals.
This work uses a geometrical approach, introduced by Langlands and Shelstad, to
calculate the first two terms of the Shalika expansion. These terms are called the
regular and subregular terms of the expansion. The first term of the expansion
is, with suitable normalizations of measures, an invariant integral over the stable
regular unipotent class. The second term of the expansion, as we will see, is a
sum of integrals of the form
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