SELBERG TRACE FORMULA 13

geodesic periods J a are best studied through the traces tr aK,.

1 9

In the compact case, the trace formulae have the form:

(1.1) Tr dlj = (spec) = (hyp):

where

(1.2) (spec) = S(0p(Ouj, UjJS^Sj)

(see [12], § 2) and where

(1.3) (hyp) = S (J a) H rf(a(7)).

{7'hyp

°

Here H is (as in Al6(b)) a non-standard HC transform and (s,m) are the

(fl,V) parameters of a (Al). More precisely, (1.3) is correct for weight 0 or

discrete series a but equals just a kind of imaginary part for higher weight,

continuous series forms (cf. [Z2],Proposition 2.11; [Z3],Theorem 5.2). Note

that the identity class term vanishes, since a is orthogonal to the constant

functions.

Similar trace formulae hold for operators of the form Op(r)h(R).

Actually, it is very useful to consider the slightly more general class of

operators Op(£r)h(R), where ^ is a cut-off function on h x h, supported in a

neighborhood of the diagonal (see [Z3], § 1(f)). (1-2) and (1.3) then become:

(1.2') (spec) = S(0p(.r)uj, uj)J^(rj)h(rj)