1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. KNOWN BOUNDS FOR THE CUSPIDAL SPECTRUM 73

The rest of the proof proceeds as in Burger-Sarnak [11]. Write r,r

L i=l, ... droi (oi E G(Q)), d = deg(T); analogous decomposition of T = L:T"I.

Then

Tf(x) = L f(oi (x) (fa function on I'\Gs).

Lemma 3.11 easily implies that for f E Cc(I'\Gs), fnf(x) converges, as a

function of x E r\ Gs, to f^0 ( x) = fr\ 0 s f (g) dg ("constant term of f"); the con-

vergence is uniform for x in a compact subset of I' \Gs (see [11]). Now we consider

6. \Hs C I' \ Gs, where 6. = r n Hs is a congruence subgroup of Hs. We assume
   that dh is a Haar measure on Hs normalized so Jb.\ Hs dn = l. Then we can set

(μ, f) = ( f(h)dh (f E Cc(I'\Gs)).

jb.\Hs

μ is a probability measure on r\ Gs. We can define, dually,

(Tμ, f) = (μ, T* !).

The previous remarks imply that

(3.7) fnμ tends (for the weak convergence of measures) to the (normal-

ized) Haar measure dg.

Now we can prove Theorem 3.10. Using Theorem 3.4 and Lemma 3.5 we see

that we are reduced to the following. Let f E Cc(I'\Gs), and consider the coefficient

c(h) = (f, R(h)f) = ( f(g)f(gh)dg (h E Hs).

Jr\ Gs

We must show that c(h) is a limit of positive linear combinations of coefficients

of automorphic representations of Hs. By (3.7) c(h) is the limit for n---> oo of

(3.8) (Tnμ, f(-)f(-h)) = d~ r L f(c:ix)f(c:ixh)dx'

jb.\ Hs i

where { c:i} - depending of n - is an expression of the iterated operator rn and

d = d(T). The convergence is uniform (in h) on compacta. But now if we set

rn =II I'77j6. , T/i E G(Q)
j
(expression of the double coset rn as a r-6. coset, finite since rn is a finite union
of left r-cosets), we have (for some j) C:i E r T/j 6. and each term in (3.8) can be
written as

(3.9) Aj r f(T/jX)f(T/jXh)dx,

Jb.;\Hs

with 6.j = { o E 6.j77p577j^1 E r} a congruence subgroup of 6., and Aj = dn[l,b.;J >

0. Since the function f(T/jX) belongs to L^2 (6.j\ Hs), (3.9) is a coefficient of this
   representation; the theorem now follows from Theorem 3.4.

3.4. Certain (semi- ) local consequences
Taken together, Arthur's conjectures - chiefly Conjecture 2 of § 2.3 - and the
results of Burger, Li and Sarnak have remarkable, even disquieting, consequences