1549380232-Automorphic_Forms_and_Applications__Sarnak_

58 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS

where B is a Borel subgroup, x is a quasi-character of T(Qp) ~ (Q; )", TC B is a

maximal torus, x is unramified and the induction is unitary induction.

Recall that La nglands's construction of the L-group yields a connected reduc-

tive group G ove r e, and a maximal torus T C G with a canonical identification:

of the lattices of cocharacters/characters. If .A E X*(T), x(.A(p)) E ex. This yields

a morphism X*(T) -+ ex, so an element of T(C); this identifies t he unramified

characters x of T(Qp) with elements of T = T(C).

In fact x is defined by (2.1) mod the Weyl group W = W(G, T) which is

canonically identified with W(G,T). Thus x yield a class t11: E T/W. This will be

called the Hecke matrix of 7r.

Example: G = GL(n), G = GL(n, q : t11: is a diagonal matrix

(

t1

J

(modulo permutation of the entries).

Now if 7r = ©7rv is a continuous representation of G(A), 7rp is unramified for

almost all p so t11:,, : = t11: ,p is defined a · e · (p). If G © Qp is split for almost all primes,

the previous definitions still apply.

Recall that the tempered representations of G(Qp) correspond to parameters

t E Tc/W, where Tc~ U(l)" is the maximal compact subgroup of T.

In .Ac we have the space of cusp forms .Ac,cusp - see Borel and Bernstein's

lectures for definitions. It admits a discrete decomposition:

.Ac ,cusp = Ef)'H11:

7r

into irreducibles under G(A) (of course a - completed - Hilbert sum). So the

simple-minded analogue of the Ramanujan conjecture would be:

"Conjecture 1". - If 7r = ©7rv C .Ac,cusp, the components 7rv are tempered.

If 7r P is unramified, "tempered" is defined in the previous paragraph. For the

other representations -and the Archimedean prime - see [13, 59].

For purely formal reasons this Conjecture does not hold. Indeed let B be

a quaternion algebra over Q (# M 2 (Q)) and G the Qi-group Bx. Then G is

anisotropic and .Ac,cusp = .Ac. The trivial representation of G(A) is in .Ac,cusp

(for w = 1) and is not tempered - see Lecture l. A conjecture which fails in such a

simple case cannot be true - even if we avoid the obvious subspace of L^2 given by

constant functions - for higher groups, and counter-examples were soon found (see

[28]). We retain, however, the

Conjecture 1. - "Conjecture 1" is true for GL(n).

Return to a general group G, assume 7r C .Ac (say, 7r non-trivial or non-

Abelian), 7r unramified at p. Then t he "Conjecture" would be that t11:,p E Tc

(mod W). If G = GL(2) and 7r is associated to a holomorphic form (see lecture 1)

the eigenvalues (t 1 , t 2 ) of t11:,p are equal to p-^1 /^2 x (eigenvalues of frobp in some