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The representation theory of Lie groups plays a central role in
both clas sical and recent developments in many parts of
mathematics and physics. In August, 1995, the Fifth Workshop on
Representation Theory of Lie Groups and its Applications took place
at the Universidad Nacional de Cordoba in Argentina. Organized by
Joseph Wolf, Nolan Wallach, Roberto Miatello, Juan Tirao, and Jorge
Vargas, the workshop offered expository courses on current
research, and individual lectures on more specialized topics. The
present vol ume reflects the dual character of the workshop. Many
of the articles will be accessible to graduate students and others
entering the field. Here is a rough outline of the mathematical
content. (The editors beg the indulgence of the readers for any
lapses in this preface in the high standards of historical and
mathematical accuracy that were imposed on the authors of the
articles. ) Connections between flag varieties and representation
theory for real re ductive groups have been studied for almost
fifty years, from the work of Gelfand and Naimark on principal
series representations to that of Beilinson and Bernstein on
localization. The article of Wolf provides a detailed introduc tion
to the analytic side of these developments. He describes the
construction of standard tempered representations in terms of
square-integrable partially harmonic forms (on certain real group
orbits on a flag variety), and outlines the ingredients in the
Plancherel formula. Finally, he describes recent work on the
complex geometry of real group orbits on partial flag varieties."
Representation theory, and more generally Lie theory, has played a
very important role in many of the recent developments of
mathematics and in the interaction of mathematics with physics. In
August-September 1989, a workshop (Third Workshop on Representation
Theory of Lie Groups and its Applications) was held in the environs
of C6rdoba, Argentina to present expositions of important recent
developments in the field that would be accessible to graduate
students and researchers in related fields. This volume contains
articles that are edited versions of the lectures (and short
courses) given at the workshop. Within representation theory, one
of the main open problems is to determine the unitary dual of a
real reductive group. Although this prob lem is as yet unsolved,
the recent work of Barbasch, Vogan, Arthur as well as others has
shed new light on the structure of the problem. The article of D.
Vogan presents an exposition of some aspects of this prob lem,
emphasizing an extension of the orbit method of Kostant, Kirillov.
Several examples are given that explain why the orbit method should
be extended and how this extension should be implemented."
The representation theory of Lie groups plays a central role in
both clas sical and recent developments in many parts of
mathematics and physics. In August, 1995, the Fifth Workshop on
Representation Theory of Lie Groups and its Applications took place
at the Universidad Nacional de Cordoba in Argentina. Organized by
Joseph Wolf, Nolan Wallach, Roberto Miatello, Juan Tirao, and Jorge
Vargas, the workshop offered expository courses on current
research, and individual lectures on more specialized topics. The
present vol ume reflects the dual character of the workshop. Many
of the articles will be accessible to graduate students and others
entering the field. Here is a rough outline of the mathematical
content. (The editors beg the indulgence of the readers for any
lapses in this preface in the high standards of historical and
mathematical accuracy that were imposed on the authors of the
articles. ) Connections between flag varieties and representation
theory for real re ductive groups have been studied for almost
fifty years, from the work of Gelfand and Naimark on principal
series representations to that of Beilinson and Bernstein on
localization. The article of Wolf provides a detailed introduc tion
to the analytic side of these developments. He describes the
construction of standard tempered representations in terms of
square-integrable partially harmonic forms (on certain real group
orbits on a flag variety), and outlines the ingredients in the
Plancherel formula. Finally, he describes recent work on the
complex geometry of real group orbits on partial flag varieties."
Representation theory, and more generally Lie theory, has played a
very important role in many of the recent developments of
mathematics and in the interaction of mathematics with physics. In
August-September 1989, a workshop (Third Workshop on Representation
Theory of Lie Groups and its Applications) was held in the environs
of C6rdoba, Argentina to present expositions of important recent
developments in the field that would be accessible to graduate
students and researchers in related fields. This volume contains
articles that are edited versions of the lectures (and short
courses) given at the workshop. Within representation theory, one
of the main open problems is to determine the unitary dual of a
real reductive group. Although this probĀ lem is as yet unsolved,
the recent work of Barbasch, Vogan, Arthur as well as others has
shed new light on the structure of the problem. The article of D.
Vogan presents an exposition of some aspects of this probĀ lem,
emphasizing an extension of the orbit method of Kostant, Kirillov.
Several examples are given that explain why the orbit method should
be extended and how this extension should be implemented.
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