Abstract. Matching of (twisted) orbital integrals of corresponding spherical functions on
a reductive p-adic group G and its (twisted) endoscopic group if is a prerequisite to lifting
representations from H to G by means of a comparison of trace formulae. Kottwitz-Shelstad
[KS] conjectured the precise form that the matching takes. This matching statement is called
"the fundamental lemma" by Langlands, who proved it for GL(2). It can be reduced to the
case of the unit element of the Hecke algebra, denoted by 1KG and IKH- The case of the unit
elements also implies the transfer of orbital integrals for general locally constant compactly
supported functions; see Waldspurger [W2] (in the non twisted case).
Kazhdan [K] initiated the study of the fundamental lemma in the higher rank case, proving
the matching for G = GL(n), H an elliptic torus, on introducing a decomposition of a com-
pact element as a product of commuting absolutely semi simple and topologically unipotent
elements. In [F7] a twisted analogue of Kazhdan decomposition is introduced, and used to
prove the matching for the symmetric square lifting from SL(2) to PGL(3). The present
paper establishes the matching for the unit elements \KG and \KH on G = GL(4) and its
endoscopic group H = GSp(2), with respect to the twisting by "transpose-inverse" (as in [F7],
with GL(4) instead of GL(S)), in the stable case. A crucial ingredient is again the twisted
Kazhdan decomposition, as is the definition of the norm in [KS], which we use.
Another key idea we borrow from Weissauer [We], who proved the matching between I K on
GSp(2) and IK on its endoscopic group GL{2) x GL(2)/Z, on noticing that any elliptic torus
in GSp(2) lies in an intermediate subgroup GL(2)'(= SO(4)) of GSp(2), and reducing the
computation to one in GL(2)f by means of a double coset decomposition GL{2)r\GSp{2)/K.
Thus by means of the twisted Kazhdan decomposition we reduce our twisted orbital integrals
(in the main case, where the element is topologically unipotent) to the group of fixed points
of the twisting, namely Sp(2). Then we compute the integrals by means of the double coset
decomposition GL(2)'\Sp(2)/K', and apply a similar analysis to the non twisted integrals on
Our work is entirely explicit. We exhibit a set of representatives for the twisted conjugacy
classes in (7, in families of types which we call (I), (II), (III), and (IV). We list those in the same
stable twisted conjugacy class. The listing is done on computing the Galois hypercohomology
groups used in [KS], or simply on using low brow Galois cohomology, but it is important for us
to exhibit explicit representatives, not just to describe the abstract structure of the conjugacy
classes within the stable class. Further we describe the norm map explicitly for each type,
and find representatives for the stable conjugacy classes and the conjugacy classes in it, for
GSp(2). The stable orbital integral is simply the sum over the orbits in the stable orbit.
Thus our computations can be used to compute the unstable orbital integrals. In the case of
GSp{2) we recover the results of Weissauer [We]. In the twisted case, this is done here too
for all unstable twisted endoscopic groups. We compute all unstable orbital integrals of IK on
the group Sp(2), which has more endoscopic groups than G5p(2), and deduce all endoscopic
transfers of orbital integrals.
Key words: (twisted) orbital integrals, (twisted) endoscopic groups, trace formula, symplectic group,
(twisted) stable conjugacy, Galois cohomology, absolutely semi simple, topologically unipotent compact ele-
ments, double coset decomposition.
1991 Mathematics Subject Classification: 11F70, 11F72, 11F85, 11R39, 20G25, 22E35
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