1549380232-Automorphic_Forms_and_Applications__Sarnak_

80 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS

class, the class of regular unipotent elements. This corresponds to the trivial rep-
resentation.^1
If G = G1(n), the Hecke matrix of the trivial representation is

p-2-n-3

associated to the irreducible representation of degree n of 81(2). According to

Arthur, any non-trivial, automorphic representation will have a Hecke matrix whose

n-2

eigenvalues (or rather their absolute value) are bounded by p-2-. Analogous com-

putations apply to all groups.

If, for example, G = Sp(g) -so G = 80(2g+ 1, q -the "largest" Hecke matrix

(barring the trivial representation) will have eigenvalues

pg-1

( 4.1)

1

pl-g

1

1

as opposed to

(4.2)

1

for the trivial representation.
In turn ( 4.1) yields a natural bound on the eigenvalues of Ta in the correspond-
ing unramified representation. We can try to prove this bound, easier than the
general conjectures of Arthur.
In [19a] an approach was started which depended on the Burger-8arnak method.
Assume H CG is a Q-subgroup, and 1fp is an automorphic representation of G(Qp)·
We want to prove that 1fp is not too "big" - i.e., that the eigenvalues of its Hecke
matrix are not too large. If a representation Tp of H(Qp) occurs in 1fp, it is au-
tomorphic. If we have a good control of the automorphic representations Tp 1-C
of Hp, we can then get upper bounds on the coefficients of 1fp and therefore on
its Hecke matrix. The precise assertions are given in [11, see formula ( 4.6)] for JR.
and in [19a, Thm. 7.1] for Qp. In both references the subgroup His chosen to be
isomorphic to 81(2)9.

(^1) Compare with the parameter associated to the Steinberg representation (before Conjecture 5) ,
and note that we don't consider the same SL(2)!