4 TRACES OF HECKE OPERATORS
where Kp = GL2(Zp) is the standard maximal compact subgroup of G(QP),
and KQQ S0(2). See Proposition 6.3. Setting K ripoo^p
w e a^so
have
G(A) = B(A)K.
Identifying Q with its diagonal image in A, we view G(Q) as a subgroup of
G(A).
The center of G is denoted by Z and consists of the scalar matrices. Let
G = Z\G.
More generally for any subset S C G we let S denote the image of S under
the map G G.
A Hecke character is a continuous multiplicative homomorphism from
A* to C* which is trivial on Q*. Let UJ : Q*\A* C* be a unitary Hecke
character (i.e. |o;(:r)| 1 for all x £ A*). Because Z(A) = A*, we can view
a; as a character of Z(A). We adopt the following convention throughout
this text:
** All Hecke characters are assumed to be unitary **
Define
L
2
H =
L2(G(Q)\G(A),u;)
=
(i) j) is measurable
(ii) j)(zg) = u(z)4(g) for all z G Z(A)
([[[) lG(Q)\G(A)\^9)\2dgoo
where dg is any right G(A)-invariant measure on G(Q)\G(A). As usual two
such functions are equivalent if they agree up to a set of measure zero. An
important subspace of
L2{UJ)
is the set of cuspidal functions, defined as
follows:
LQ(CJ)
= (f) e
L2{u)
I / t(ng)dn = 0 for a.e. g G G(A) ,
[ JN(Q)\N(A) J
where as before N is the subgroup of unipotent elements in G. The integral
of (f(ng) over N(Q)\N(A) is called the constant term of / . This is directly
related to the constant term of a classical modular form at a cusp (compare
(3.21) and (12.15)).
Let R denote the right regular representation of G(A) on
L2(u)\
R(x)(/(g) = t{gx).
This is an infinite-dimensional representation of G(A). Because G(Q)\G(A)
is noncompact, R does not decompose into a direct sum of irreducible rep-
resentations. ^ However, it is easy to check that
LQ(U)
is stable under this
action, and we set
RQ
=
R\L2(UJ)-
This representation
RQ
does split into an
: G(Q)\G(A) - C
^See [GGPS] Chapter 1 §2.3 for a proof of complete reductibility in the compact quotient
case.
http://dx.doi.org/10.1090/surv/133/02
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