1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. KNOWN BOUNDS FOR THE CUSPIDAL SPECTRUM 67

Of course now this is weaker than the results of Kim and Shahidi, or Kim-Sarnak

(Lecture 1).

Analogous estimates obtain at the Archimedean primes. Here, for each Archi-

medean prime v of F, 1fv is associated to a representation WFv ----> GL(n, <C) (Lang-

lands functoriality). If v is real, we restrict it to We. Thus in all cases we have a

(semi-simple) representation We = ccx ----> GL(n, <C) isomorphic to

Z f--7 diag( zPI Zq^1 • • • zPn zqn),

with Pi - qi E Z. The Ramanujan conjecture (nv tempered) says that Pi +qi is

imaginary. The Jacquet-Shalika estimate, which follows from example from Vogan's

classification of the generic representations of GL(n, IR) or GL(n, <C), is

1

IRe(pi +qi) I < 2

Luo, Rudnick and Sarnak show:

1

(3.5) IRe(pi + qi)I :S 2 - c(n).

with the same improvement c(n) as in the finite case. For applications of these

bounds, see Lecture 4, and also § 3.6.

Finally, we note that a bound analogous to (3.2) on the Hecke matrices of

generic cuspidal representations has been obtained by Shahidi for most quasi-split

groups [51].

3.2. The rest of the lectures will mostly be devoted to an exposition of the

approximations of the Arthur conjectures which can actually be proved. We mostly

consider groups other than GL(n) since for this group the known estimates result

from the theorems described in § 3.2 for cusp-forms and the description by Moeglin

and Waldspurger of Eisenstein series [45].

In 1991 Burger, Li and Sarnak introduced an efficient method allowing one to

control the spectrum (the support of Ac) of a group Gin terms of the spectrum -

supposed better known - of a Q-subgroup H CG.

This method relies on simple facts of the representation theory of reductive

groups over local fields. Since this theory has gone out of fashion,^1 we will recall it

in this paragraph. There are no proofs, and the results are not so easy to extract

from the literature: the best reference remains Dixmier [22]. The reader interested

in these methods should read § 8 and § 18 of this book. Other references are Margulis

[44, I.5] and Zimmer [60, Ch. II].

Suppose G is a connected reductive group over F ( = IR or Qp; we write simply

G for G(F). We denote by G the dual of G - i.e., the set of equivalence classes

of irreducible unitary representations of G^2 This has a natural topology: [22,

(^1) This being due, of course, to the fantastic progress of the purely arithmetic theory in the last
years.
(^2) As the reader will notice we use the same notation for the unitary dual G = WJ and for the
(Langlands) dual group. For the huma n reader this should cause no confusion. It is expected to
fool a machine that would read the paper (and try to prove the conjectures).