1.1. An Overview
1.1.1. Langlands Functoriality. In the 1960’s, Robert Langlands conjec-
tured that there exists a canonical (not necessarily one-to-one) correspondence
between Galois representations and automorphic representations of reductive al-
gebraic groups, and that this correspondence coincides with the Artin reciprocity
law when the underlying group is GL(1). More precisely, for every reductive group
H over a number field F with Langlands dual
it is believed that each homomor-
phism from the Langlands group LF —a conjectural group generalizing the Galois
and Weil groups—to
parameterizes an L-packet of automorphic representations
of H.
Suppose there are two reductive groups H, G with L-groups
tively, such that there is a homomorphism from LH to LG. Then, each homomor-
phism from the Langlands group to LH induces one from the Langlands group to
LG. Consequently, one should expect a canonical lifting of the automorphic repre-
sentations of H to those of G. A more precise formulation of this principle is as
Conjecture (Langlands Functoriality). Let H, G be reductive groups over a
number field such that there is an L-homomorphism ξ from
Then, for
every automorphic representation π of H, there is a packet {Π} of automorphic
representations of G, parameterized by the same Frobenius-Hecke classes at all but
finitely many places, such that the Frobenius-Hecke classes in
parameterizing π
are mapped by ξ to those parameterizing {Π} in
Over the years, various examples of Langlands functorial lifting have been
proved via the comparison of trace formulas, a technique introduced by Langlands
and his collaborators ([JL], [L], [LL]). We list here a few notable cases which are
of particular interest in relation to our work. They are all examples of (twisted)
endoscopic lifting (see [KoSh]).
The lifting from RE/F GL(1) to GL(2), where [E : F ] = 2, proved by
Jacquet and Langlands [JL].
Cyclic base change lifting for GL(2), proved by Langlands [L], and for
GL(n), by Arthur and Clozel [AC].
The lifting from elliptic algebraic tori to SL(2), by Labesse and Langlands
The lifting from RE/F GL(1) to GL(n), E/F cyclic of order n, by Kazhdan
The lifting from RE/F GL(m) to GL(n), where m · [E : F ] = n, by Wald-
spurger [Wa].
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