 
Summary: Proceedings of Symposia in Pure Mathematics
Volume 48 (1988)
Characters, Harmonic Analysis,
and an L2Lefschetz Formula
JAMES ARTHUR
Suppose that U is a locally compact group. It is a fundamental problem to
classify the irreducible unitary representations of U. A second basic problem
is to decompose the Hilbert space of square integrable functions on U, or on
some homogeneous quotient of U, into irreducible Uinvariant subspaces. The
underlying domain is often attached to a natural Riemannian manifold, and
the required decomposition becomes the spectral decomposition of the Laplace
Beltrami operator. Weyl solved both problems in the case of a compact Lie
group. His method, which was simple and elegant, was based on the theory of
characters.
In this lecture, we shall briefly review Weyl's theory for compact groups. We
shall then discuss two newer areas that could claim Weyl's work as a progenitor:
the harmonic analysis on noncompact groups, and the analytic theory of auto
morphic forms. The three areas together form a progression that is natural in
several senses; in particular, the underlying algebraic structures of each could be
characterized as that of an algebraic group over the field C, R, or Q. However,
