1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 2. THE SPECTRAL DECOMPOSITION OF L^2 (G(Q)\G(A)) 59

subspace of H^1 (Xo(N), Qe)).^2 Thus t 1 , t 2 are p-^1 /^2 x (Weil numbers of weight 1),

so unitary (lti l = 1), or, as we will say, of weight 0.

For GL(n), if trr,p = (t1,... , tn) with lti l = 1 we say that 1fp is isobaric

of weight 0. Even if 1fp is ramified, Langlands has defined a notion of isobaric

representation - see [42c], and perhaps [48a, 15a].

2.3. Arthur's conjectures: unramified case

We still assume G split, or at least G x QP spli t when we work over QP. (For

t he general case, see Arthur [4b]). Assume n = ®nv occurs in Ac (the timorous

reader may assume that 7f occurs in Ac,dis, the discrete part of A c regarded as a

representation of G(A) or G(Qs)). Assume p unramified for n, so trr,p E T/W.

From t he discussion at the end of§ 2, it is natural to see trr,p as a "Frobenius

element". So let WP = WQ,, be the Weil group [56], with its canonical exact

sequence:

(2.2)

I p being the inertia.

Recall that t he quotient Z in (2.2) is naturally isomorphic with t he subgroup

of Gal(Fp/1Fp) generated by t he (arithmetic) Frobenius map x f---+ xP. We denote

by frobp the inverse of t he arithmetic Frobenius in Z = Wp/ Ip. (This is not

important).

Given trr,p we obtain a representation

(2.3) <p = <p(n,p): Wp ~TC G,

unramified (i.e., <p(Ip) = 1) and sending frobp to trr,p· It is well-defined modulo
conjugation in G.
In [4b] Arthur has made very general conjectures about the spectrum of G(A) in
Ac. We will describe first what this says in the unramified case. Arthur considers
"parameters", i.e., morphisms

(2.4) 'I/; : Wp x SL(2, q ~ G.

'I/; is supposed continuous; moreover we assume that 'l/JlsL( 2 ,q is a n algebr a ic

representation. We will further assume

(2.5) 'l/Jlw,, is unramified, isobaric of weight 0.

By "isobaric of weight O" (see end of§ 3) we mean that 'l/;(frobp) belongs to a

maximal compact subgroup of G.

We now consider the map j : Wp ~ SL(2, C) which is unramified and s uch that

(

IPl

112

frobp f---+ IPl-1;2 ).

Conjecture 2 (Arthur + c:). - If 1fp is a local, unramified component of a repre-
sentation 7f occurring in Ac, trr,p = 'l/;(frobp,j(frobp)) for a parameter 'I/; verifying
(2.5).
This very simple conjecture is extremely deep and would have important conse-
quences, even limited as here to unramified components. For instance, note that it
implies the Ramanujan conjecture for Maass forms: if f is a Maass form -say, for

(^2) frobp is the "geometric" Frobenius , denoted by <pp in Lecture l.