1549380232-Automorphic_Forms_and_Applications__Sarnak_

76 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS

Since the eigenvalues ti of ta,p verify

p-l/2+c(a)::::; ltil::::; pl/2+c(a)

(Theorem 3.2), we see that there is a unique partition 'ljJ of n - namely, n =

b + b + · · · + b (a factors) - such that the eigenvalues T 1 , ... , Tn of tn,p verify, for

some permutation of { 1, ... , n}:

(3.13) P-T/(a) ::::; ITi Ti~JI ::::; pT/(a)

(We have set 17(a) = ~ -.s(a) E]O, ~[.)

Thus 'ljJ is well-defined. Finally, according to Langlands, the Hecke matrix of

any representation p occuring in L^2 (GL(n, Ql)\ GL(n, A)) is a direct sum of matrices

trr;,P associated to the discrete spectrum. This implies that a variant of (3.13)

applies to all representations in Gp,Aut· Namely, if t"v is the Hecke matrix of such a

representation, there exists a partition 'ljJ (non-homogeneous in general) such that,

for the eigenvalues (ti) of t"v and after a suitable permutation:

P-71(n) <- Iti · r -1i,1/J 1 <pT/<nl. -

(Note that 17(n) = Sup 17(a)).

i::;a::=;n

Note that for this to define 'ljJ uniquely (for a representation of Gp,aut, i.e.,

limit of "true" automorphic representations) we need the uniform bound (3.13)

of Theorem 3.2 when O" varies.

We can apply this, with obvious corrections, to representations of SL(n). Now

Theorem 3.10 has the following unconditional consequence, using the arguments

that motivated Conjecture 9 :

Theorem 3.12. - Assume H = SL(m) C G = SL(n) is an arbitrary embedding,

and 'lrp is an unramified representation in Gp,aut· Then the SL(2)-type of an un-

ramified representation Tp E Hp,aut occurring in 'lrp is uniquely determined.

We note that for GL(n) the representations in Gp,Ar are explicitly determined.

Namely, let 'ljJ : n = n 1 + · · · + n r be a partition and let wi be an unramified

character of «Ji;, seen as a representation of GL(ni , Qlp) via the determinant. The

representations of type 'ljJ in Gp,Ar are all representations of the form:

(3 .14) ind~L(n) n 1,···nr (w1 ® · · · ® Wr)

(unitary induction; see Tadic [55]). Thus Conjecture 9, in this case, should be
accessible using Mackey theory.
Finally, we sketch a proof of Conjecture 9 for an embedding SL(m) c SL(n),
when Qlp is replaced by a (local) function field of characteristic p.
Denote such a field by kv, and assume it is the completion at a prime v of
a global function field k. Moeglin and Waldspurger have obtained the spectral
decomposition of L^2 (GL(n, k) \ GL(n, Ak)): the description is identical to that given
for Ql. Moreover, in this case, the Ramanujan conjecture is true, thanks to the work
of Lafforgue [40]. Write G = GL(n). If n E G(kv)Ari 7r is of the form (3.14) and
is limit of local components of automorphic representations. A straightforward
extension of Theorem 3.10 to SL(n) over k, and the fact that the Ramanujan
conjecture is true for SL(m), imply tha t Conjecture 6 is true in this case. Finally,
the (p, q)-argument implies Conjecture 9.