1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. FUNCTIONAL EQUATIONS AND MULTIPLICATIVITY 317

Proposition 3.5 [Shl]. Given 1 < i:::; m , there exists a split group G i over F , a

maximal F-parabolic subgroup P i = MiNi and a cuspidal automorphic represen-

tation n' of Mi = Mi(AF), unramified for every v rf_ S, such that if the adjoint

m'

action of L Mi on Lni decomposes as r' = EB rj, then

j=l

(3.10) Ls(s, n, ri ) = Ls(s, n', r~).

Moreover m' < m.

It was observed by Arthur [A], that each Mi can be taken equal to M and

n' = n. More precisely:

Proposition 3.6 [A]. Given i, 1 < i:::; m , there exists a split connected reductive

F-group G i with M as a Levi subgroup and m' < m for which r~ = ri. Each

G i can be taken to be an endoscopic group for G. (Its £-group is the connected

component of the centralizer of a semisimple element in LG.)

Next, we need the following variant of a result of Henniart and Vigneras. In

it, we assume that the defining additive character 'I/; for x is local component of a

global one.

Proposition 3. 7 [Shl]. Let u be an irreducible x-generic supercuspidal represen-
tation of G = G(F), where F is a non-archimedean local field and G is defined over
F. Let B =TU be the Borel subgroup of G defining X · Then there exists a number
field K with a ring integers 0, a split group H over K, a non-degenerate character
x = ®vXv ofUH(K)\ UH(AK), and a globally x-generic cusp form 7r = ®v1rv on
H = H(AK) such that:

(1) Kv 0 = F for some place Vo of K ,

(2) Xv 0 = X,

(3) as a group over F, H = G,

(4) 7rv 0 = u , and finally

(5) for every other finite place v of K , v I-vo, nv is of class one with respect

to a special maximal compact subgroup Qv of H(Kv)· Here UH is the

unipotent radical of a Borel subgroup of H for which UH as a group over

F equals U.

Proposition 3.8 [Shl,Sh4]. Assume either Fv is archimedean or nv has a vector
fixed by an Iwahori subgroup. Let 'Pv: w;,,v ~ L Mv be the homorphism of the
Deligne-Weil group parametrizing 7r v. For each i, let L( s, r i · 'Pv) and c: ( s, r i · 'Pv, 1/Jv)
be the Artin £-function and root number attached tori· 'Pv· Then

rr

m. _ - L(l - is,ri · 'Pv)

(3.11) CxJs,nv)= c:(is,ri·'Pv,1/Jv) L(" - ) ·

i=l is,ri·'Pv

Remark. If G is quasisplit, but not split, then a product of Langlands >.-functions
(Hilbert symbols) will also appear on the right hand side of (3.11) ([KSh], [Sh4]).
Applying these propositions inductively and using the Crude Functional
Equations (3.8) then implies:

Theorem 3.9 [Shl]. Assume G is a split reductive algebraic group over a local field
F of characteristic zero. Let P =MN, P ::) B , be a maximal parabolic subgroup
as before. Let x be a generic character defined by the splitting and 'I/; F E P. Given