1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 6. CONVERSE THEOREMS AND FUNCTORIALITY 163

field k [68]. Not knowing what this should look like, one still expects to have

local-global compatibility. If one begins with an irreducible automorphic repre-

sentation 7r = <;?/nv of H(.A) then, assuming the Local Langlands Conjecture for

each local group H(kv), one can attach ton the collection {¢v} oflocal parameters

</>v = <l>'lrv : Wt ___, LH given by the local components 'Trv. This system of local pa-

rameters can often be used as a substitute for a global parameter. This collection

of local data is sufficient to define the global £-function and c-factor attached ton

and a representation r: LH ___, GLn(C) by

L(s, n, r) =II L(s, nv, r) =II L(s, r o ¢v)

v v

and

c(s, n, r) =II c(s, 'Trv, r, 'l/Jv) =II c(s, r o </>v, 'l/Jv)

v v

where 'lj; = ®'l/Jv is an additive character of .A trivial on k.

6.1.2. Local Functoriality

We are interested in Functoriality from H to G = GLn. In general, Functoriality

is associated to an £-homomorphism, which in our context is simply a complex

analytic homomorphism u: LH ___,LG= GLn(C).

Local Functoriality is very natural if one assumes the Local Langlands Conjec-

ture for H( kv). In this case, if 7r v is an irreducible admissible representation of H( kv)

then here is associated a parameter or admissible homomorphism <l>v : w~ ___, LH. If

we compose this with the £-homomorphism u we obtain a parameter fo~ GLn(kv),

namely <r>v = u o <f>v : Wt ___, LG = GLn(C). Since the local Langlands cor-

respondence for GLn is bijective, this parameter determines a unique irreducible

admissible representation IIv of GLn(kv):

7r v t------7

W' kv

<P v f-------7 IIv.

The representation IIv is called the local Langlands lift or trans! er of n v associated

to the £-homomorphism u. Note that in terms of local £-functions, we have the

equalities

L(s, 'Trv, u) = L(s, u o </>v) = L(s, <Pv) = L(s, Ilv)

and

c(s, nv, u, 'l/Jv) = c(s, u o </>v, 'l/Jv) = c(s, <Pv, 'l/Jv) = c(s, IIv, 'l/Jv)

as well as equalities for the twisted versions with representations 7r~ of GLm(kv)·

6.1.3. Global Functoriality

We retain our £-homomorphism u : LH ___, LG = GLn(q. If we begin with a
cuspidal representation n = ®'nv of H(.A) then the global Principle of Functoriality