1549380232-Automorphic_Forms_and_Applications__Sarnak_

164 J.W. COGDELL, £-FUNCTIONS FOR GLn

states that to 7r and u should be associated an automorphic representation IT of

GLn(A) for which L(s, 7r, u) = L(s, IT) among other things.

If we assume the Local Langlands Conjecture for each local group H(kv) then

this global F\mctoriality is easy to formulate. We first take 7r and decompose it into

its local components 7r = &/7rv· For each lo cal representation 7rv we apply our local

lifting diagram:

LH __ u ___. Le

1rv >------>

W' k v

Then piecing the local representations ITv together we obtain an irreducible admis-

sible representation IT = @'ITv of GLn(A). Langlands' Principle of Functoriality

then says that IT should be automorphic. Then IT would be the global Langlands

lift or transfer of 7r associated to the L-homomorphism u. Note that in this case

we would have the equality of L-and €-factors

L(s, 7r, u) =II L(s, 7rv, u) =II L(s, ITv) = L(s, IT)

v v

and

c(s, 7r, u) =II c(s, 7rv, u, 'l/Jv) =II c(s, ITv, 'l/Jv) = c(s, IT)

v v

as well as equalities for the twisted versions with representations 7r^1 of GLm(A)

L(s, 7r x 7r^1 , u 0 L) = L(s, IT x 7r^1 )

for cuspidal automorphic representations of GLm(A), where l: GLm(<C) -+ GLm(<C)

is the identity map viewed as an L-homomorphism, and the related equalities of

€-factors.

In general, as noted above, there will be a finite set S of finite places of k for

which we do not know the local Langlands conjecture for H(kv)· So for any finite

set of finite places S we will call an automorphic representation IT of GLn(A) a

global Langlands lift of 7r if for every v tJ_ S we have that ITv is the lo cal Langlands

lift of 7rv. In particular this will imply an equality of partial L-functions

L^8 (s , 7r,u) = L^8 (s,IT)

as well as the related equalities of €-factors and twisted versions.

6.2. Functoriality and the Converse Theorem

It should now be clear how one can apply the Converse Theorem to establish lift-
ings or transfers from split connected reductive groups H to an appropriate GLN
associated to an L-homomorphism u: LH-+ GLN(<C). There are essentially three
steps.

1. Construction of a candidate lift. We begin with a cuspidal automorphic
   representation 7r = @^1 7rv of H(A). Assume that for each place v we can construct