6 TRACES OF HECKE OPERATORS
(2-7) " E £ f(g)x(det(g))dg.
The terms (2.1)-(2.4) constitute the geometric side of the trace formula,
and consist of orbital integrals or weighted orbital integrals over the con-
jugacy classes in G(Q). The remaining terms form the spectral side of the
trace formula. We now give a brief elaboration of each geometric term:
(2.1) This is the identity term coming from the conjugacy class {f
n 1
)}.
(2.2) Elliptic terms: 7 G G(Q) is elliptic when it is not conjugate to an
upper triangular matrix (over Q), or equivalently, when the eigenvalues of
7 lie outside Q. For any element 7 G G(Q), [7] is the G(Q)-conjugacy class
of 7, and G7(Q) is the centralizer of 7 in G(Q). The sum is taken over all
elliptic conjugacy classes in G(Q).
(2.3) Hyperbolic terms: An element of G(Q) is hyperbolic if it is con-
jugate to a nonscalar diagonal matrix in G(Q). The sum is taken over all
hyperbolic conjugacy classes in G(Q). Note that G7(A) = M(A) when 7
is diagonal. The weight function v is defined by v(g) = H(g) + H(wg),
where w = (
1 n
J, and H is the height function defined in Section 7.
(2.4) Unipotent term: Here
and f.p.Zp(s) denotes the "finite part" at s = 1 (i.e. the constant term of
s=l
the Laurent expansion about s 1) of the meromorphic zeta-function
ZF(s)= [
F(a)\a\sd*a
JA*
defined by Tate.
The remaining terms (2.5)-(2.7) are the noncuspidal spectral terms.
These terms do not contribute to the traces of Hecke operators on holo-
morphic cusp forms.
(2.5) Continuous terms: This is the contribution of the continuous ker-
nel. We follow [G2] for the notation. The summation is over pairs of Hecke
characters (xi, X2) such that X1X2 = ^ and p(x, s) denotes the induced rep-
resentation space IndB/Au^4^y |§|^), where we write b (
n
, 1 G B(A).
(For the definition of this induced representation, see page 390 of [Knl],
or [GJ] §4A.) Letting
\w
= (X2Xi) M(s) = M(s,x) is the intertwining
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