Lie Theory [electronic resource] : Lie Algebras and Representations / by Jens Carsten Jantzen, Karl-Hermann Neeb ; edited by Jean-Philippe Anker, Bent Orsted
Imprint
Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2004
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. A wide spectrum of topics is treated, with emphasis on the interplay between representation theory and the geometry of adjoint orbits for Lie algebras over fields of possibly finite characteristic, as well as for infinite-dimensional Lie algebras. Also covered is unitary representation theory and branching laws for reductive subgroups, an active part of modern representation theory. Finally, there is a thorough discussion of compactifications of symmetric spaces, and harmonic analysis through a far-reaching generalization of Harish--Chandra's Plancherel formula for semisimple Lie groups. Ideal for graduate students and researchers, Lie Theory provides a broad, clearly focused examination of semisimple Lie groups and their integral importance to research in many branches of mathematics. Lie Theory: Lie Algebras and Representations contains J. C. Jantzen's "Nilpotent Orbits in Representation Theory," and K.-H. Neeb's "Infinite Dimensional Groups and their Representations." Both are comprehensive treatments of the relevant geometry of orbits in Lie algebras, or their duals, and the correspondence to representations
CONTENT
Preface -- Nilpotent Orbits in Representation Theory -- 1 Nilpotent Orbits for Classical Groups -- 2 Some General Results -- 3 Centralizers in the Classical Cases -- 4 Bala-Carter Theory -- 5 Centralizers -- 6 The Nilpotent Cone I -- 7 The Nilpotent Cone II -- 8 Functions on Orbits and Orbit Closures -- 9 Associated Varieties -- 10 Springerโ{128}{153}s Fibers and Steinbergโ{128}{153}s Triples -- 11 Paving Springerโ{128}{153}s Fibers -- 12 ?-adic and Perverse Stuff -- 13 Springerโ{128}{153}s Representations -- References -- Infinite-Dimensional Groups and Their Representations -- I The Finite-Dimensional Case -- II Split Lie Algebras -- III Unitary Highest Weight Modules -- IV Banach-Lie Groups -- V Holomorphic Representations of Classical Banach-Lie Groups -- VI Geometry of Coadjoint Orbits of Banach-Lie Groups -- VII Coadjoint Orbits and Complex Line Bundles for U2(H) -- Appendix: The Topology of Classical Banach-Lie Groups -- References
