I. BASIC CONSTRUCTIONS
Chapter I is concerned with preliminary constructions. The groups are not
assumed to be quasi-split. The groups are taken over a p-adic field F of charac-
teristic zero. The germs are studied near the group identity. We are not always
careful in distinguishing a group G from its elements over F so that expressions
such as g G G should be interpreted as g £ G(F).
1. Background Information.
Igusa has introduced a method of studying asymptotic expansions of integrals
over a local field. The expansion holds in the following context. A variety is
fibred over a punctured neighborhood of a point p on a curve. Let A be a local
parameter at p. The integral is taken over a fibre and consequently depends on
the parameter A. Igusa theory gives, provided a number of technical conditions
are satisfied, an asymptotic expansion of the integral as A tends to 0. The
theory gives explicit formulas for the coefficients of the asymptotic expansion.
The locus of A = 0 in the variety is to be a union of divisors. The coefficients of
the asymptotic expansion are given as principal value integrals over the divisors.
R.P. Langlands [15] applies Igusa theory to the study of /c-orbital integrals
by constructing a variety and a curve such that the integral taken over the fibre
of the curve is equal to a /c-orbital integral. Chapter I is devoted to a study
of the variety he constructs. Some useful coordinates are defined that simplify
computations in the variety.
2. The Igusa Variety.
This section reviews the construction of the variety Yi introduced in [15].
The variety Y\ and its resolution Yr are constructed using a number of auxiliary
varieties. First I will make a list of these varieties for reference, and then give
the definitions.
is the variety of regular stars
S is the variety of stars, the closure of
S' is the subvariety of S such that for each simple root a there is at least one
chamber W(w) for which z(W{w),a) / 0
Si (Boo, B0) is the fibred product Si (Boo, Bo) = S'(Boo, Bo) XT0
Ar.
Si is a first resolution of S
S" is the open subvariety of Si given on each open patch S"(Boo, Bo) by
S^Boo.Bo) xTo A
r
CS
/
(B
o o
,B
0
) xTo
Ar
=Si(Boo,B
0
).
Received by the editor February 20, 1989
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