Introduction.
Langlands' principle of functoriality [B] conjectures that there is a parametrization of the
set Repjp(Gf) of admissible [BZ] or automorphic [BJ] representations of a reductive group G
over a local or global field F , by admissible homomorphisms p : WF G x WF Here WF is
a form of the Weil group [T] of F , and G is the connected (complex) Langlands dual group
[B] of G, on which WF acts via the absolute Galois group of F. If H is another reductive
group over F and there is an admissible map H x WF G x WF, then composing with
PH '• WF —+ H x WF we get p : WF G x VT/?, and by the functoriality conjecture we would
expect a "lifting" map RepF(H) —RepF(G)-
The trace formula has been used to establish the lifting in a few cases. For a test func-
tion / = g/„ G C£°(G(A)), the convolution operator r(f) maps cj) in L2(G(F)\G(A)) to
the function whose value at h G G(A) is fG,A\ f(g)f(hg)dg. It is an integral operator with
kernel Kf(x,y) which has geometric expansion XXeG(F) f(x~liy)i a n d spectral expansion
^27rYl(f)r(f)(t)(x)(l)(y)-
Here TT ranges over the set of the irreducible direct summands of
L2
as a module under the action of G(A) by multiplication on the right, and 0 ranges over an
orthonormal basis of smooth vectors. Integrating over x = y G G(F)\G(A) we obtain the trace
formula J2n ^T7r(f) = J1G/~ ^ / ( T ) - Here Gj ~ denotes the set of conjugacy classes in G(F),
and 3/(7) = JG(A)/Z( ) f(x7x~1)dx is an orbital integral of / . In this outline we ignore all
questions of convergence, which make the development of the trace formula such a formidable
task.
To develop a theory of liftings of representations from the group H to G, one develops a
trace formula for a test function / # on H(A), of the form J27VH tr7r#(///) = J2H/~ ^ * / H ( 7 ^ ) -
One then tries to compare the geometric sides of the two trace formulae. For this one needs:
(1) A notion of a norm map N : {G/ ~ } {H/ ~ } , sending a stable conjugacy class 7 in
G(F) to JH in H(F), locally and globally. This has been defined by Kottwitz-Shelstad [KS]
in our context.
(2) A statement of transfer of orbital integrals, asserting that given a test function / G
C
C
° ° ( ( J ( F ) ), where F is a local field, there exists a test function / # , and given / # there is an
/ , with "matching orbital integrals", namely $/(7) = $fH(N^).
The global test function / is a product of local functions which are almost all the unit
element I K of the Hecke algebra of spherical (bi-invariant by a standard maximal compact
subgroup K of the local group G(F) (K is hyperspecial, [Ti, 3.9.1]) functions on G(F). Hence
one must have also the statement that:
(3) $1^(7) = &iK (N^) for all (regular) 7. This statement is called the fundamental
lemma. It is a necessary initial point for the comparison to exist.
Further, the admissible map H x WF G x WF defines a lifting map for unramified
representations from H(F) to G(F), and via the Satake transform a dual map from the Hecke
algebra of G (locally) to the Hecke algebra of i7, and one needs:
(4) An extended fundamental lemma, relating the orbital integrals of the corresponding
spherical functions.
Received by the editor February 5, 1997, and in revised form August 21, 1997.
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