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