LECTURE 1. MOSTLY SL(2) 55(1) Spectral information about differential operators on r\G(IR) - e.g. the
Laplacian on r \G(IR)/ K=, K= C G(IR) being maximal compact - is contained in
t he (analysis of the) representation of G (IR) on A.
(2) the spectral properties of Hecke operators are described by the representa-
tion of G(Qp) - p a prime -on A.
For instance, using (1.15), one can analyze the action of Tp by considering the
representations of GL(2, Qp) [23]. By the way, the reason we considered GL(2) is
that the standard Tp 's are not realized "on" SL(2, A).
If we choose a cha racter w of the center Z(A) of GL(2, A), and define A as
above, the theory of Eisenstein series yields a decomposition as a representation of
G(A):(1.17) A= E§c EB E97r EB ( j Eisenstein)
exactly similar to (1.1). Here:
E runs over Abelian characters of G(A), invaria nt by G(Q), and restricting to
w.
7r runs over t he cuspidal representations of G(A) occurring in A.
J Eisenstein is a continuous integral (see § 3.2) of representations, translating
the integral term of (1.1).
The typical summand (discrete, or continuous in the last term of (1.17)) of A
is a unitary irreducible representation 7r of G(A). Given the topology of this group,
7r decomposes as an infinite tensor product:7!"= © 1l"v
v=p,oo
where 1l"v is a unitary irreducible representation of G(Qv)· For almost all primes
p < oo, 1l"p is unramified, i.e. it has a vector fixed by G(Zp); this vector is then
unique up to scalars.
So far we have implicitly assumed G = GL(2). Assume G is a general reductive
group over Q. We take G split for simplicity of exposit ion , but this is not necessary
(for the general case see [CJ). A decomposit ion simila r to (1.17) still exists - although
it is more complicated. The representations of G(A) h ave the same structure.
(Recall that G b eing split has a natural model over Z as a Chevalley group. This
defines G ( Zp) for all p.)
In § 2.2 we will review t he parametrization of unra mified representations of
G(Qp) in terms of the dual group G. We give, for GL(2), the translation in these
terms of the classical Hecke operators. Assume for instance that r = r 0 (N), f is
an eigenform, pf N, and (ap, {3p) are associated to Tp as in§ 1.3-1.4. Let 7r = 0 v1l"v
be the cuspidal representation associated to f. Then 1l"p is unramified for pf N.
For G = GL(2, Q), G = GL(2, q. As we will see in § 2.2, the Langla nds
parametrization associates to 1l"p a diagonal matrix t-rr,p E GL(2, C).Then this matrix is ( a.P {3p) (modulo permutation of the entries; indeed only
{ a.P, (3p} was defined by Tp.) Thus all spectral properties we considered are encoded
in t-rr,p· There is a similar description at infinity, relating the eigenvalue .A of l:::.. (for
Maass forms) to 7r =.