SELBERG TRACE FORMULA 5

M *kT

!r(T) = S -£- +

Q(Te3/4T),

1

k=0

sk

This theorem is analysed in detail in [He 1-2], [I] and [V]. It can

surely be generalized to finite volume qoutients of rank one symmetric spaces

(for some results of this kind, see [DeG], [G-V] and [S]). (No generalization

to containing compact regular flats in higher rank compact quotients is known

at present [B-K]).

Our generalization of (0.2) is to consider the above sums

(0.3) Definition

*r('T)

=

s

f *

L T J 7

where a is an automorphic form. Our main result is:

(0.4) Theorem (Theorems 5A,B) Let T c PSL2(R) be cofinite. Then fp(r,T)

M skT 5§T

E 7, Op(r)uv,uv e + 0(e ),for certain constants % and

0p(r)uk,uk .

The constants j* are explicitly computable in terms of the weight m

and Casimir-eigenvalue parameter s of a. The constants Op(r)ut ,Ui are

2

matrix coefficeints of an operator Op((r) on L (r\h) associated to a. More

precisely, Op(r) is the pseudo-differential operator associated to a in the

calculus defined in [Z.6]. It is most easily introduced through its action on

functions on h itself. Therefore, consider the Laplace eigenfunctions

exp(iA z,b) on h, where b is on the boundary of h, X € C, and z,b is the

distance to i of the horocircle determined by (z,b) [He]. When b = oo,