SELBERG TRACE FORMULA 13
geodesic periods J a are best studied through the traces tr aK,.
In the compact case, the trace formulae have the form:
(1.1) Tr dlj = (spec) = (hyp):
(1.2) (spec) = S(0p(Ouj, UjJS^Sj)
(see , § 2) and where
(1.3) (hyp) = S (J a) H rf(a(7)).
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
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)