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.
Unitary Representations and Compactifications of Symmetric
Spaces, a self-contained work by A. Borel, L. Ji, and T. Kobayashi,
focuses on two fundamental questions in the theory of semisimple
Lie groups: the geometry of Riemannian symmetric spaces and their
compactifications; and branching laws for unitary representations,
i.e., restricting unitary representations to (typically, but not
exclusively, symmetric) subgroups and decomposing the ensuing
representations into irreducibles.
Ji's introductory chapter motivates the subject of symmetric
spaces and their compactifications with carefully selected
examples. A discussion of Satake and Furstenberg boundaries and a
survey of the geometry of Riemannian symmetric spaces in general
provide a good background for the second chapter, namely, the
Borela "Ji authoritative treatment of various types of
compactifications useful for studying symmetric and locally
symmetric spaces. Borela "Ji further examine constructions of
Oshima, De Concini, Procesi, and Melrose, which demonstrate the
wide applicability of compactification techniques.
Kobayashi examines the important subject of branching laws.
Important concepts from modern representation theory, such as
Harisha "Chandra modules, associated varieties, microlocal
analysis, derived functor modules, and geometric quantization
areintroduced. Concrete examples and relevant exercises engage the
reader.
Knowledge of basic representation theory of Lie groups as well
as familiarity with semisimple Lie groups and symmetric spaces is
required of the reader.
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