CHAPTER 1

Introduction

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

LH,

it is believed that each homomor-

phism from the Langlands group LF —a conjectural group generalizing the Galois

and Weil groups—to

LH

parameterizes an L-packet of automorphic representations

of H.

Suppose there are two reductive groups H, G with L-groups

LH, LG,

respec-

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

follows:

Conjecture (Langlands Functoriality). Let H, G be reductive groups over a

number field such that there is an L-homomorphism ξ from

LH

to

LG.

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

LH

parameterizing π

are mapped by ξ to those parameterizing {Π} in

LG.

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

[LL].

• The lifting from RE/F GL(1) to GL(n), E/F cyclic of order n, by Kazhdan

[K].

• The lifting from RE/F GL(m) to GL(n), where m · [E : F ] = n, by Wald-

spurger [Wa].

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