Summary: 2 The Trace Formula
and Hecke Operators
This lecture is intended as a general introduction to the trace formula.
We shall describe a formula that is a natural generalization of the
Selberg trace formula for compact quotient. Selberg also established
trace formulas for noncompact quotients of rank 1, and our formula
can be regarded as an analogue for general rank of these. As an
application, we shall look at the "finite case" ofthe trace formula. We
shall describe a finite closed formula for the traces of Hecke operators
on certain eingenspaces.
A short introduction of this nature will by necessity be rather
superficial. The details of the trace formula are in [l(e)] (and the
references there), while the formula for the traces ofHecke operators is
proved in [l(f)]. There are also other survey articles [1(c)], [l(d)], ,
and [l(g)], where some ofthe topics in this paper are discussed in more
detail and others are treated from a different point of view.
Suppose that G is a locally compact group that is unimodular and that
r is a discrete subgroup of G. There is a right G-invariant measure on