62 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS
WIR being still the Weil group, with bounded restriction to WJR. In this case the
construction of II(Vi) is given by Adams, Barbasch and Vogan [l]. However their
description is very complicated; of course it does not imply that the lo cal compo-
nents 7fJR of automorphic representations belong to the set of representations they
describe.
There is a simpler set of representations of G(IR), namely, those which have
non-trivial cohomology (see the end of§ 3). In this case^4 the Arthur packets a re
described in [4b, § 5], using the work of Adams and Johnson.
2.5. Global parameters
Assume now that 7f = @1fv is an irreducible representation of G(A.) that occurs
in Ac. What can we say about 7f or the local components 7f v?
We will not describe this in any detail, as we will not need much of the global
conjectures. Arthur - following Langlands - postulates that there should exist
a group L<Q whose unitary, irreducible representations r of degree n parametrize
cuspidal representations (unitary) of GL(n, A). In, particular L<Q should be endowed
with maps^5 WQv -t L<Q ( v = p, oo) making the following diagram commutative:
r ~1f=®1fv
1 1
rlwQv^1 1----* 1fvwhere r is an (irreducible unitary) representation of L<Q. Note that t he lower hori-
zontal map is well-defined, by the work of Langlands [42d] and Speh if v = oo, and
of Harris and Taylor if v < oo: this is the local Langlands conjecture [25b], [27b].
If now 7f = ®1fv occurs in Ac, Arthur conjectures that there is a map
(2.8) Vi: L<Q x SL(2,C)-) G
such that each component 1fv b elongs to the "Arthur packet" 7r(1/iv), where Viv =
Vilw~ x SL(2, q. Of course this is undefined - except for GL(n) and small groups.
Vague as this general conjecture is, it has immediate spectral consequences: in
fact the restriction of Viv to (the second factor) SL(2, q should not depend on v.
Thus we are led to the very precise conjecture:
Conjecture 4. - Assume 7f = ®1fv occurs in Ac.
(1) if for a finite prime p 1fp is unramified and tempered, all components
of 7f are tempered.
(2) In general the weights of 1fp (in the sense of Conjecture 3) should be
independent of p , when 1fp is unramified.(^4) and under the assumption that G(IR) has a discrete series.
(^5) This is slightly incorrect. Arthur takes W,QP = W<Qp x SU(2), and LIQ compact. Complexify L<Q
to make our statement corect, or use Arthur's W,QP. We prefer our W,QP, with the SL(2, IC) factor,
because the SL(2) factor in W' is associated in the context of Shimura varieties (end of § 2.3) to
bad reduction and monodromy, so to nilpotent elements, which are hard to find in SU(2)! Of
course the two formulations are equivalent.