LECTURE 2
Arthur's conjectures Lecture 2. The spectral decomposition of L^2 (G(Q)\G(A)):
Arthur's conjectures
2.1. In this lecture I want to give a conjectural answer to the following kind of
problem. Assume G is a reductive group over Q. We consider the space of functionsAo= L^2 (G(Q)\G(A),w).
Here w is a unitary character of Z(A)/Z(Q) where Z is the center of G. The
group G(A) acts by right translations on Ao, the space of functions such that
f(zg) = w(z)f(g), defining a unitary (Hilbert space) representation. We can ask:
(1) What representations of G(A) occur in A 0?
( 2) In particular, if S is a finite set of primes (per haps containing oo) of Q,
which representations of G(Qs) occur? Here Qs = IT «Jlv; we will also sometimes
vES
write As.
Note that "occurs" has to be defined with some care, because Ao is not in
general a discrete sum of representations (it is if G(Q)Z(A)\ G(A) is compact, i.e.,
if the derived group of G is anisotropic). We'll be careless about this in t his lecture,
returning to the precise definitions in Lecture 3.
2.2. Parametrization of unramified representations, and the Ra-
manujan conjecture.
For simplicity assume G split over Q. Then G is a "group over 'lL", i.e., there is
a smooth model G of G over 'lL.^1 In particular for p a finite prime, KP = G('lLp) C
G(Qp) is defined.
(In general G will be "unramified over Qp" for all primes but a finite number.
Similar definitions apply, see Borel [8b, § 9-10]).
Assume 7r is an irreducible representation of G(Qp) which has a Kp-fixed vector.
(We assume 7r realized in a Hilbert space H , but not necessarily unitary). Then
(see [12]) 7r is the unique subquotient having a KP-fixed vector in
(2.1)
(^1) See Springer's lecture, Reductive groups, in [CJ, vol. I.
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