Semisimple Lie groups, and their algebraic analogues over fields
other than the reals, are of fundamental importance in geometry,
analysis, and mathematical physics. Three independent,
self-contained volumes, under the general title Lie Theory, feature
survey work and original results by well-established researchers in
key areas of semisimple Lie theory.
Harmonic Analysis on Symmetric Spacesa "General Plancherel
Theorems presents extensive surveys by E.P. van den Ban, H.
Schlichtkrull, and P. Delorme of the spectacular progress over the
past decade in deriving the Plancherel theorem on reductive
symmetric spaces.
Van den Bana (TM)s introductory chapter explains the basic setup
of a reductive symmetric space along with a careful study of the
structure theory, particularly for the ring of invariant
differential operators for the relevant class of parabolic
subgroups. Advanced topics for the formulation and understanding of
the proof are covered, including Eisenstein integrals, regularity
theorems, Maassa "Selberg relations, and residue calculus for root
systems. Schlichtkrull provides a cogent account of the basic
ingredients in the harmonic analysis on a symmetric space through
the explanation and definition of the Paleya "Wiener theorem.
Approaching the Plancherel theorem through an alternative
viewpoint, the Schwartz space, Delorme bases his discussion and
proof on asymptotic expansions of eigenfunctions and the theory of
intertwining integrals.
Well suited for both graduate students and researchers in
semisimple Lie theory and neighboring fields, possibly even
mathematical cosmology, Harmonic Analysis on Symmetric Spacesa
"General Plancherel Theorems provides abroad, clearly focused
examination of semisimple Lie groups and their integral importance
and applications to research in many branches of mathematics and
physics. Knowledge of basic representation theory of Lie groups as
well as familiarity with semisimple Lie groups, symmetric spaces,
and parabolic subgroups is required.
General
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