1549380232-Automorphic_Forms_and_Applications__Sarnak_

72 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS

• an adelic Hecke operator on the left-hand side. This is simply the action,
  by the integrated right representation of G^5 = G(.A.^5 ), of the characteristic function
  r.p'Y of K 'Y K (we normalize measures so vol(K) = 1).


• a "classical" Hecke operator T'Y, on the right-hand side : this is the
  action of the double coset r 'Yr; write r 'Yr = L rbi ( Oi E G(Q))' a finite sum,

and define T'Y(f(x)) = L f(oi x), x E r\G(As).

i

Using strong approximation one checks that these operators coincide. The

degree of T'Y is #(f\f'Yr) = fc s r.p'Y(g)dg.

We will choose a prime p f:: S such that G is split over QP and K = Kp KP with

Kp = G(Zp)· We can choose"(, by strong approximation, such that 'YE G(Qp)KP.

Then r.p'Y is essentially an element of the (unramified) Hecke algebra of (Gp, Kp)·

J_

We write L^2 (f \Gs) = L6(f\Gs) EB C , C denoting the constant functions. If T

is a sum of such operators (with integral > 0 coefficients) the degree is defined by

linearity. Finally, T = (degT)-^1 T.

Lemma 3.11. - There exists a Hecke operator T'Y (or a positive integral sum T of

such) such that

for any f E L6(r\Gs).

Proof: If the absolute rank of (the factors of) G is > 1, this is proved in [19a,

Thm. 5.10]. Otherwise we are essentially reduced to the case of a form of SL(2, k)

• k a number field - seen as a Q-group. We have
  2 J_2 J_2
  L (r\Gs) = C EB Lcusp(r\Gs) EB LE;s(r\Gs).
  For a prime p f:: S such that G(Qp) contains a factor SL(2, Qp), take <.pp

char(Kp (P P- 1 ) Kp), Kp = SL(2, Zp)· Write

J_

L~usp(f\Gs) =EB 7rKv ,

a sum over irreducible (unramified) representations of SL(2, Qp)· Then each 7r is
unitary and non-trivial by strong approximation. A simple calculation of Satake
transforms shows that <.pp operates on 7rKv via the scalar >. = p(z + ~) + p - 1 with
p-^1 < lzl < p ; for z = p (associated to the trivial representation) we get the degree
p^2 + p of 'Pp·

We conclude that I.Al < p^2 + p. Now the operator T acts by 1 on C, has

eigenvalues IA'I < 1 in L~usp(r\Gs), and its L^2 operator norm is bounded by 1

[19a, Lemma 5.5]. Lebesgue's dominated convergence theorem then implies that

(T)n f --+ 0 for f E L~uw The same argument applies to L~; 8 , where the estimates

are even better.^5

(^5) This improves on [19a] because we have not used any approximation to the Ramanujan conjecture
for SL(2). This is needed for our §7. This "weak" argument based on dominated convergence was
introduced in [19c].