 
Summary: Proceedings of the International Congress of Mathematicians
August 1624, 1983, Warszawa
JAMES ARTHUR
The Trace Formula for Noncompact Quotient
1. In [12] and [13] Selberg introduced a trace formula for a compact,
locally symmetric space of negative curvature. There is a natural algebra
of operators on any such space which commute with the Laplacian. The
Selberg trace formula gives the trace of these operators. Selberg also
pointed out the importance of deriving such a formula when the symmetric
space is assumed only to have finite volume. Then the Laplace operator
will have continuous as well as discrete spectrum; it is the trace of the
restriction of the operator to the discrete spectrum that is sought. Selberg
gave such a formula for the quotient of the upper half plane by SL(2, Z).
(See also [6] and [8].) Selberg also suggested how to extend the formula
to any locally symmetric space of rank 1. Spaces of rank 1 are the easiest
noncompact ones to handle for they can be compactified in a natural
way by adding a finite number of points. I have recently obtained a trace
formula for spaces of higher rank. In this article I shall illustrate the
formula by looking at a typical example.
2. Let X be the space of n by n symmetric positive definite matrices of
