the examples, is construction of Haar measure and the invariant integral, and the
discussion of convolution product and the Lebesgue spaces.
Part 2, "Representation Theory and Compact Groups", also provides back-
ground, but at a slightly higher level. It contains a discussion of the Mackey
Little-Group method and its application to Heisenberg groups, and a proof of the
Peter-Weyl Theorem. It also contains a discussion of the Cartan highest weight
theory with applications to the Borel-Weil Theorem and to recent results on in-
variant function algebras. Part 2 ends with a discussion of the action of a locally
compact group G on L2(G/T), where Y is a co-compact discrete subgroup.
Part 3, "Introduction to Commutative Spaces", is a fairly complete introduc-
tion, describing the theory up to its resurgence. That resurgence began slowly
in the 1980's and became rapid in the 1990's. After the definitions and a num-
ber of examples, we introduce spherical functions in general and positive definite
ones in particular, including the unitary representation associated to a positive
definite spherical function. The application to harmonic analysis on G/K consists
of a discussion of the spherical transform, Bochner's theorem, the inverse spher-
ical transform, the Plancherel theorem, and uncertainty principles. Part 3 ends
with a treatment of harmonic analysis on locally compact abelian groups from the
viewpoint of commutative spaces.
Part 4, "Structure and Analysis for Commutative Spaces", starts with rie-
mannian symmetric space theory as a sort of role model, and then goes into recent
research on commutative spaces oriented toward similar structural and analytical
results. The structure and classification theory for commutative pairs (G,K), G
reductive, includes the information that (G, K) is commutative if and only if it is
weakly symmetric, and this is equivalent to the condition that (GC,KC) is spher-
ical. Except in special cases the problem of determining the spherical functions,
for these reductive commutative spaces, remains open. The structure and classi-
fication theory for commutative pairs (G, K), where G is the semidirect product
of its nilradical N with the compact group K, is also complete, and in most cases
here the theory of square integrable representations of nilpotent Lie groups leads
to information on the spherical functions. The structure and classification in gen-
eral depends on the results for the reductive and the nilmanifold cases; it consists
of methods for starting with a short list of pairs (G, K) and constructing all the
others. Finally there is a discussion of just which commutative pairs are weakly
At this point I should point out two areas that are not treated here. The
first, already mentioned, is the general theory of weakly symmetric spaces, and the
closely related areas of geodesic orbit spaces and naturally reductive riemannian
homogeneous spaces. That beautiful topic, touched momentarily in Section 13.1C,
has an extensive literature.
The second area not treated here consists of certain extensions of (at least parts
of) the theory of commutative spaces. This includes the extensive but somewhat
technical theory of semisimple symmetric spaces, (the pseudo-riemannian analogs
of riemannian symmetric spaces of noncompact type), the theory of generalized
Gelfand pairs (G,H), and the study of irreducible unitary representations of G
that have an iif-fixed distribution vector. It also includes several approaches to
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