Juan Tirao - Böcker

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.