Introduction

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

L2(G/K)

is multiplicity-free.

Another is that the (convolution) algebra

L1(K\G/K)

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

L2

{G/K) into

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

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