2. TH E ARTHUR-SELBERG TRAC E FORMULA FO R GL(2) 5
orthogonal Hilbert space direct sum of countably many irreducible repre-
sentations ([GGPS], Chapter 3 §4.6):
* ) =
©*•
(Here 0 denotes an orthogonal Hilbert space direct sum, i.e. the closure
of the usual algebraic direct sum.) See also Chapter 3 of [Bu], or [Knl].
An irreducible representation of G(A) with central character u is cuspidal
if it occurs in this decomposition. Jacquet and Langlands proved that the
multiplicity of each cuspidal representation in the above sum is one (cf. [JL],
[PS], or Chapter 3 of [Bu]).
Let
Cc(G(A),u~1)
be the space of continuous functions / : G(A) C
with compact support modulo Z(A) satisfying f(zg)
ou(z)~1f(g).
Then
we define an operator R(f) on
L2(u)
by
R(f)p(9) = L f(x)(p(gx)dx,
JG(A)
for / G
CC{G(A),UJ~1)
and tp G
L2(u).
In general, R(f) is an infinite
rank operator and not of trace class. However its restriction Ro(f) to the
subspace of cuspidal functions is of trace class, so tri?o(/) is meaningful
([GJ] Corollary 2.4 and [Knl] Corollary 6.3).
The proof of the theorem below can be found in [G2] and [GJ], which
specialize Arthur's formula for a general reductive group ([Arl],[Ar4]) to
the case GL(2). The statement is from [GJ], Theorem 6.33. See also Theo-
rem 7.14 in the survey [Knl]. Haar measures will be fixed in Section 7.
THEOREM
2.1 (Arthur-Selberg Trace Formula for GL2). For
feCdGiA),*-1),
tr/Jo(/) =
(2.1) meas(G(Q)\G(A))/(l)
(2.2) + E £ -
f(9-l79)dg
[7]CS(Q) elliptic ^ W ) \ ^ ( A )
(2.3) -^meas(Q*\A
1
) E £- - f^lQ^Wg
2
1 / 7 G
M ( Q ) MA)\G(A )
(2.4) +f.p;ZF(s)
s=l
1 f°°
(2'5)
+ i ^ E / tr(M(-it)M'(it)p(X,it)(f))dt
XlX2
= u ;
(2-6) - ^ t r ( M ( 0 ) p (
X
, 0 ) ( / ) )
4
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