SELBERG TRACE FORMULA 7
in [Po]. Our error terms are optimal, at least if t^ - 1/2 1/20, but are
only proved for somewhat special r. Ve would expect such error terms to hold
for all r 6 C^(r\G) (for example), but it would be difficult to extend our
results that far (cf. [Z.2,§0].
Our proof of (0.4) is based on a generalization of the Selberg trace
formula (cf. [Z.2-3]). As reviewed in §1, the standard trace formula
r 2
evaluates traces tr R of convolution operators on L (r\G) in terms of the
Harish-Chandra (or, conjugacy class) transform Ity. The generalization is to
r
consider the composition rR,, where a denotes multiplication by this
r
automorphic form. If the K-weights of a and p are adjusted properly, rR, will
2 r
operate on L (r\h). Tr rR, can then be evaluated in terms of a generalized
Harish-Chandra transform of ip, the (m,s)-parameters of a and the periods a.
J7
Ve then imitate the proofs of (0.2) which avoid the use of the Selberg zeta
function. Sums of periods a naturally replaces sums of lengths, leading to
(0.4).
Some new issues inevitably arise. The most significant involve the
analogues of the Veyl law which plays a key role in the proof of (0.2). This
involves the eigenvalue counting function
(0.6i) Definition Np(T) = #{j: |r.| T} and its continuous spectral
analogue
T
(0.6ii) Mr(T) = - ,1 f Jl^ir)dr.
J-T
Here, A(s) is the determinant of the scattering matrix for T (A10); [Y,
§3.5]). The Veyl law states:
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