1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 4. APPLICATIONS: CONTROL OF THE SPECTRUM 83

A few remarks are in order. Recall t hat K azhda n 's property T for Gv is the

assertion "The trivial representation is isolated in Gv". It is always true if the rank

of Gv is> 1 (or if Gv is compact). See Margulis [44, ch. 3 J and Zimmer [60]. T hus

the theorem has a "global" content only when rk(Gv) = l. If vis a finite prime,

and rk(Gv) = 1, property Tis always false. For real groups see [44, p. 131]. At any

rate, one sees that property (T) fails only for groups of classical type (i.e., Gv x Qv

is of classical type A, B, C, D).

So assume G is as in the theorem, of classical type and Gv h as rank l. Assume

H C G is a Q-group (semi-simple, simply connected). If the theorem is false we

can find a sequence 7f~ E Gv,aut, 7f~ '/!. ilc v , such that

7f~---+ ilc v (trivial representation).

By }emma 3.7, ilHv belongs to the closure of UnSupp(7r~JHv). If ilHv is isolated

in Hv,aut, Theorem 3. 10 implies that the trivial representation of Hv is discretely

contained in 7f~IHv for n >> 0. This is however impossible, as follows easily from

Prop. 3.8, if Hv is non-compact.

Thus Theorem 4.4 is redu ced to
(a) a combinatorial exercise - find a subgroup Hin G t hat is a minimal semi-simple
Q-subgroup, with moreover rk(Hv) > 0 (so rk(Hv) = 1).

(b) Prove the theorem for such minimal groups H.
If we assume, as we may, that G -so also H -is a nisotropic, it turns out that the
minimal subgroups Hare of two types (with the condition that rk Hv = rk Gv = 1):
(A) SL(l, D) where D is a quaternion algebra over Q. Then Theorem 4.4 is
known thanks to the J acquet-Langlands correspondence and to the known results
for GL(2, Q).

(B) A unitary group associated to a "complex conjugation" (correctly speaking:
an involution of the second kind) on D , where Dis a division algebra of prime degree
on a quadratic extension of Q.

(C) A group of type (A) or (B) over a finite extension k of Q.
Now Theorem 4.4 is reduced to case (B) (possibly over an extension k). It is
a small miracle that this case can be treated directly, t hanks to Langlands's base
change, developed in this case by Kottwitz, Labesse and the author (see [15c,§ 2]).

We now explain the meaning of the t heorem for the Archimedian prime. So
we consider G(IR), which we can assume of rank l. Let K= C G(IR) be a maximal
compact subgroup. The set of spherical representations (those having a vector fixed

by K=, cf. the unramified representations in § 2.2) is open in G(JR). --
On the other hand, X = G (IR)/ K = is a Riemannian variety and so h as a
natural Laplacian we, which is invariant by G(!R). (We choose the Laplacia n which

has positive eigenvalues, so of the form - I: ( 8 ~,) 2+ lower terms). It is easy to

see that Thm. 4.4 (for v = oo) is equivalent to:

Corollary 4.5. - There exists a constant Cc > 0 - depending on G/Q - with
the following property. Assume r C G /Q is an arbitrary congruence subgroup and