74 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMSin the representation theory of local semi-simple groups over real or p-adic fields.
We describe them for restriction (i .e ., Thm. 3.10), but similar phenomena appear
for induction.
We consider a pair ( G, H) as in Theorem 3.10 and consider first a finite prime
p and an unramified representation 7rp of Gp which occurs as a factor of an auto-
morphic representation -rr of G(A). By Conjecture 2 7rp is associated to an Arthur
parameter 'l/Jp : W<Qlv x SL(2, q ----> G , 'lf;p being of course unramified. We will call
the map SL(2, q ----> G (which matters only modulo conjugation in G) the SL(2) -
type of 'lf;p or -rr P-
Consider the restriction of -rrp to Hp, of course a continuous integral as in
Theorem 3.3. Let Hp,nr be the set of unramified representations of Hp, an open
subset of Hp. We will only be interested in the part of -rrplHv supported on Hp,nr·
Denote by Gp,Ar the subset of Gp,nr associated to Arthur parameters. This
is clearly a closed subset of Gp,nr· The same notation applies to H. Moreover
it seems likely that any -rr E Gp,Ar belongs to Gp,Aut for a suitable choice of G /Q
yielding Gp at the place p (see the discussion after Conjecture 7). Since by Theorem
3.10 applied to S = {p} each representation Tp in the support of -rrplHv is limit of
automorphic representations, Arthur's conjecture would then have the following,
purely local, consequence.
Conjecture 6. - Assume 7rp E Gp,Ar· If a representation Tp E Hp,nr occurs in
the restriction of-rrp, Tp E Hp,Ar·
A simpler - and less strongly motivated - version is the following. (Refer to
Conjecture 3 in§ 2.3 for terminology). We suppose that the local groups cannot see
the difference between Arthur parameters and the more general holobaric repre-
sentations. We would then have, writing Gp,hol for the set of unramified holobaric
representations:
Conjecture 7. - Assume -rrp E Gp,hol· If Tp (unramified) occurs in the restriction
of -rrp, Tp is holobaric.
Note that the conjectures are purely lo cal. When -rrp can be obtained by unitary
induction from a unitary representation of a subgroup of Gp, they are a priori
accessible by Mackey theory.
We now move to generalize this, replacing p by a finite set S of finite primes.How can we define G s,Ar? Assume 7r = Q9 7r P. Then we must certainly have
pES
7rp E Gp,Ar (p E S). Moreover the SL(2)-type should be t he same for all p. It
seems plausible that there should be no further restriction. For instance if '¢1sL( 2 ,q
is trivial - so the 7rp should be tempered - it is known that the set of @-rrp (-rrp
pES
tempered unramified) that are factors of automorphic -rr are dense in the tempered,
unramified dual of Gs (Cor. to Thm. 3.9).
So we define Gs,Ar by this condition: -rrp E Gp,Ari and constancy of the SL(2)-
type. We now have the analogue of Conjecture 6, to wit:
Conjecture 8. - Assume -rrs E Gs,Ar· If Ts - a representation of Hs -occurs in-rrs, Ts E Hs,Ar·