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,
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