LECTURE 2. THE SPECTRAL DECOMPOSITION OF L^2 (G(IQ)\ G(A)) 61"purity + Lefschetz" behaviour onrepresentations of G(A) which have no relation
with cohomology - like Maass forms.2.4. Arthur's conjectures: ramified representations
As above we assume G x QP split. The careful reader will have noticed that we
did not really need the Weil group in order to state the Conjectures of § 3 : since
we consider only unramified representations only the image of frobp matters. We
have stated the conjecture in § 3, however, so it extends naturally to the general
case.
According to Langlands one should parametrize tempered representations of
G(Qp) by homomorphisms cp : WP ---+ G with bounded image (equivalently, whose
image has compact closure). Then discrete series representations should correspond
to "irreducible" parameters cp, i.e., cp does not factor: WP ---+ M where Mis a Levi
subgroup of G. (Think of G = GL(n): this means that cp is irreducible as a
representation of WP in this case).
However this is not quite correct because it does not account for the "special"
representations of GL(n) [8a, 13b]. Thus we assume that cp : WP x SL(2, q ---+ G
(direct product). The assumption is that cplWP verifies the previous condition. Let
us write w; for the product WP x SL(2, q. As usual, cp is algebraic on SL(2, q.
Finally, an Arthur parameter, will be a homomorphism(2.6) 'I/; : w; x SL(2, q ___, a.
Associated to 'I/;, it is conjectured that there is a family II('I/;) of representations
of G(Qp) satisfying certain conditions inspired by the stabilization of the trace
formula [4b, Conjecture 6.1]. Then any representation of G(Qp) occuring in Ac
should belong to II('I/;) for some parameter 'lj;.
Assume that the w;-component of 'I/; is unramified (in particular, trivial on
SL(2, q c w;). Then the unramified representation associated to 'I/; by Conjec-
ture 2 should belong to II('I/;). I will chance the following addition to Conjecture 2,
which is not stated by Arthur:
Conjecture 2A. -The unramified representation 7r such that t1'
'l/;(frobp, j (frobp)) is the unique unramified element of IT..p.
This is true when 'I/; is tempered or for the unitary group in 3 variables [49],
taking Kp equal to the hyperspecial subgroup. If true Arthur's local conjectures
would imply Conjecture 2 (this is the "+t"' in the formulation of this conjecture.)
At this point, Arthur's conjectures are far from being proved (!) so this must
be considered as a guide to the spectral decomposition. Note that II('I/;) is not
even defined in general. There are, however, exceptions:
- If G/Q is SL(2) or U(2, 1) -the quasi-split unitary group of rank 3 over
Q associated to a quadratic extension of Q - the local packets II( 'I/;) can be defined,
with the conditions imposed by Arthur. Possibly the same is known of some experts
when G = Sp(4,Q). - Consider the real prime v = oo of Q so Qv = R In this case an Arthur
parameter is simply a morphism