70 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS
We also note an obvious corollary of Theorem 3.4.Lemma 3.6. -Assume H C G is also reductive. Let (n, H) be a unitary repre-
sentation of G and 7rlH its restriction to H. If the trivial representation of G is
weakly contained inn, the trivial representation of H is weakly contained in 7rlH.
We will need two further results. Let us denote by ilc E G the trivial repre-
sentation.
Lemma 3.7. - Assume 1fn E G and 7rn , ilc (for the topology of G). Let Sn CH
be the support ofnnlH. Then il.H belongs to the closure of USn.
n
Proof: The assertion "nn , ilc" means that there exist vectors Vn E H(nn),
with llvnll = 1, such that Cvn ___, 1 on G uniformly~ compacta. The same is true
on H. By Theorem 3.4, ilH is weakly contained in ffi7rnlH, whose support is USn
n
[22, 3.4.2]. So ilH E USn.
Proposition 3.8. - Assume G is simple, simply connected (as an algebraic group)
and non compact. If 7f is an irreducible, non-trivial representation of G, the coeffi-
cients of 7f tend to 0 at infinity.
For the proof, see Howe and Moore [28].
Finally, we note that all the results here extend easily to the case where G =
IT G(Qp) is a finite product of local groups (including possibly p = oo).
pES3.3. The Burger-Li-Sarnak methodAssume G is a semi-simple group over Q. We will in fact assume G simply
connected. Let S be a finite set of primes of Q (possibly containing oo), such that
G(As) is non-compact. We consider the action of G(As) on L^2 (G(Q)\G(A)). Fol-
low~ Burger, Li and Sarnak we denote by Gs,aut the support of this representation
in G(As)·
Let H C G be a semi-simple subgroup (over Q). We can analogously define
Hs,aut·Theorem 3.9. - If 1fH E Hs,auti the support of ind~~~~~ (7rH) is contained in
Gs aut·
Corollary. - Any tempered representation of Gs is automorphic (i.e., belongs to
Gs,aut)·
Indeed take H = {l}.
For S = { oo }, this is due to Burger, Li and Sarnak [10], who give no proof. A
proof has been announced by Shalom [53].Theorem 3.10. - If ?re E Gs,aut 1 the support of?rclH(As ) is contained in Hs,aut·
This is due to Burger and Sarnak [11] for S = { oo}, to Ullmo and the author
in the general case [19a].
We will sketch a proof of both results.
We write Gs, Hs for G(As), H(As).