PAR T I. Preparations.
A. Statement of Theorem.
Let R denote the ring of integers in a local non archimedean field F. Let G be the F-
group G i x G
m
, where G i = GL(4) and G
m
= GL(1). Put fg\ for the transpose of g\ G
Gi- Define w = ( ^ ) , J = ( _
0
^ ) , 0(9l) =
Sg^J'1,
and 0(gue) = (0(9l),e\\9i\\) for
g = (gi,e) G G; \\gi\\ denotes the determinant of g\. Put H = GSp(2) = GSp(J) for the
group {pi G Gi]0(gi) = eg\ for some e = e(g{) G GL(1)} of symplectic similitudes. We write
G = G(F ) and # = H(F ) for the groups of F-points, and K = G(R) and KH = U(R) for
the standard maximal compact subgroups. Similarly we have Gi,Ki,
We choose Haar measures dg, dh,... on G, H,..., and denote by IK 1KG the quotient
by the volume \K\ of K of the characteristic function of K = KG in G, by \KH the analogous
object for KH, l^i for K\ in Gi, etc. Then IK lies in the space C£°(G) of locally constant
compactly supported functions on G. We often omit the subscript of K, when it is clear
from the context. Identify C%°(G) with C™(G6) by f(g) = f(gO), put lnt(g){tO) = gtOg'1 =
gt9(g~l)9, and introduce the orbital integral
*f(tO) = ^(tO;dG/dZGm) = f f((lnt(g))(W))dg/dZGm
JG/ZG(t6)
of / G C%°(G) at t9,t e G (it is also called the 9-orbital integral of / at £). Here
Z
G
( ^ ) = {(/GG;Int((7)(W)=^ }
is the O-centralizer of £ in G, or the centralizer of tO in G.
The elements t,t' of G are called stably O-conjugate if £'0 = lnt(g)(tO) for some g G G ( =
G(F), F = algebraic closure of F) . There are finitely many 0-conjugacy classes (lnt(#)(£0), g G
G) in a stable 0-conjugacy class, and we define the stable orbital integral £y's(£0) of / at
tO to be the sum ^2^f{t'0) over a set of representatives t' for the 0-conjugacy classes within
the stable 0-conjugacy class of t (in G). Note that Z G ( £ 0 ) and Z G ( ^ 0 ) are isomorphic when
t, t' are stably 0-conjugate, this isomorphism is used to relate the measures on these groups.
Similarly we have the stable orbital integral ^f,st(h]dH/dZH(h)) of / G C%°(H) at h G H.
The purpose of this paper is to prove the following.
Theorem. For any strongly 0-regular t G G we have
*i?(t0; dG/dTe) = **£ (Nt; dH/dTe o (1 + 0) o AT 1 ).
An element t of G is called 0-semi-simple if £0 is semi-simple in the group G x (0) (0 is an
automorphism of G of order two). Such an element is called 9-regularii Z G ( £ 0 ) ° , the connected
component of the identity in Z G ( £ 0 ), is a torus. Further it is called strongly 6-regular if Za(tO)
is abelian. In this case Z G ( Z G ( £ 0 ) ° ) is a maximal torus T in G which is stable under lnt(£0),
and ZG{tO) = Tlnt^ (see Kottwitz-Shelstad [KS, 3.3]). According to [KS, Lemma 3.2.A(a)],
we may assume that the strongly 0-regular t lies in a 0-stable F-torus T. Thus t G T = 0(T).
6
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