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: