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

GSp{2).

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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