This book starts with the basic theory of topological groups, harmonic
analysis, and unitary representations. It then concentrates on geometric
structure, harmonic analysis, and unitary representation theory in
commutative spaces. Those spaces form a simultaneous generalization of
compact groups, locally compact abelian groups, and riemannian symmetric
spaces. Their geometry and function theory is an increasingly active topic
in mathematical research, and this book brings the reader up to the
frontiers of that research area with the recent classifications of weakly
symmetric spaces and of Gelfand pairs.

Part 1, “General Theory of Topological Groups”, is an
introduction with many examples, including all of the standard
semisimple linear Lie groups and the Heisenberg groups. It presents
the construction of Haar measure, the invariant integral, the
convolution product, and the Lebesgue spaces.

Part 2, “Representation Theory and Compact Groups”,
provides background at a slightly higher level. Besides the basics, it
contains the Mackey Little-Group method and its application to
Heisenberg groups, the Peter–Weyl Theorem, Cartan's highest
weight theory, the Borel–Weil Theorem, and invariant function
algebras.

Part 3, “Introduction to Commutative Spaces”, describes
that area up to its recent resurgence. Spherical functions and
associated unitary representations are developed and applied to
harmonic analysis on $G/K$ and to uncertainty principles.

Part 4, “Structure and Analysis for Commutative Spaces”,
summarizes riemannian symmetric space theory as a rôle model,
and with that orientation delves into recent research on commutative
spaces. The results are explicit for spaces $G/K$ of nilpotent or
reductive type, and the recent structure and classification theory
depends on those cases.

Parts 1 and 2 are accessible to first-year graduate students. Part
3 takes a bit of analytic sophistication but generally is accessible
to graduate students. Part 4 is 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.

Readership

Graduate students and research mathematicians interested in lie
groups and their representations.