Commutative space theory is a common generalization of the theories of com-
pact topological groups, locally compact abelian groups, riemannian symmetric
spaces and multiply transitive transformation groups. This is an elegant meeting
ground for group theory, harmonic analysis and differential geometry, and it even
has some points of contact with number theory and mathematical physics. It is
fascinating to see the interplay between these areas, as illustrated by an abundance
of interesting examples.
There are two distinct approaches to the theory of commutative spaces: ana-
lytic and geometric. The geometric approach, which is the theory of weakly sym-
metric spaces, is quite beautiful, but slightly less general and is still in a state of
rapid development. The analytic approach, which is harmonic analysis of commu-
tative spaces, has reached a certain plateau, so it is an appropriate moment for a
monograph with that emphasis. That is what I tried to do here.
Commutative pairs (G, K) (or commutative spaces G/K) can be characterized
in several ways. One is that the action of G on
Another is that the (convolution) algebra
of if-bi-invariant functions
on G is commutative. A third, applicable to the case where G is a Lie group, is that
the algebra D(G, K) of G-invariant differential operators on G/K is commutative.
The common ground and basic tool is the notion of spherical function. In the Lie
group case the spherical functions are the (normalized) joint eigenfunctions of the
commutative algebra D(G, K). The result is a spherical transform, which reduces to
the ordinary Fourier transform when G = Rn and K is trivial, an inversion formula
for that transform, and a resulting decomposition of the G-module
irreducible representation spaces for G. In many cases this can be made quite
explicit. But in many others that has not yet been done.
This monograph is divided into four parts. The first two are introductory and
should be accessible to most first year graduate students. The third takes a bit
of analytic sophistication but, again, should be reasonably accessible. The fourth
describes recent results and in intended for mathematicians beginning their research
careers as well as mathematicians interested in seeing just how far one can go with
this unified view of algebra, geometry and analysis.
Part 1, "General Theory of Topological Groups", is meant as an introduction
to the subject. It contains a large number of examples, most of which are used in
the sequel. These examples include all the standard semisimple linear Lie groups,
the Heisenberg groups, and the adele groups. The high point of Part 1, beside