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........ published in NEWSLETTER # 68

By Professor A. Broer, University of Montreal, Montreal (Canada)

This volume published by Kluwer (NATO ASI SERIES C514) contains the Proceedings of the NATO Advanced Study Institute on "Representation Theories and Algebraic Geometry" held July 28 - August 6, 1997 at the Universite de Montreal. This ASI was simultaneously the 36th session of the Seminaire de Mathematiques Superieures.

In the last two decades enormous progress has been made in the representation theories of many different but related mathematical objects such as semisimple groups, finite groups of Lie type, Kac- Moody groups, Weyl groups and other Coxeter groups, Hecke- algebras, universal enveloping algebras and quantum groups.

Much of this progress resulted from the rich interplay with algebraic geometry. This is not so surprising for semisimple groups; in its representation theory the geometry of smooth varieties like the flag manifold or symmetric spaces comes up very naturally. For the other objects it is more surprising and recent. Singular varieties also play essential roles.

We shall give some examples. The technique of intersection cohomology was originally introduced to study singular varieties. It soon turned out that it could be used to obtain classifications of irreducible representations of many different objects. The subsequent development of intersection cohomology was largely motivated by its successes in representation theory. Typical singular varieties that come up in applications are Schubert varieties, nilpotent varieties, Springer varieties, decomposition varieties and quiver varieties.

Secondly, it was realized that several classical objects and constructions could be shaefified in unexpected ways. For example, many primitive quotients of enveloping algebras and their modules can be sheafified to sheafs of rings of differential operators on homogeneous spaces and their so-called D-modules. This led to a rich non-commutative algebraic geometry. The success of the applications of the theory of D-modules in representation theory was again a large motivation for the subsequent development of this new algebraic-geometric theory.

Thirdly, it was discovered that several classical objects of study can be deformed or quantized together with their representation theories. In recent years universal enveloping algebras were succesfully deformed into so-called quantum groups. Quantum groups have become immensely popular already, and their development was again very much stimulated by the successes in representation theory. Even in this less geometric theory algebraic varieties play important roles, like the quiver varieties.

As can be expected, all these new techniques are strongly interconnected and give deep insight into the classical theories as well. They motivate the study of special algebraic varieties in depth, from any algebraic or topological point of view. The revenues of the ongoing study of Schubert varieties and nilpotent varieties have been enormously high, and many more treasures are expected to be found.

In these proceedings the various connections between representation theories and algebraic geometry come out clearly. Several of the writers are among the main instigators and architects of the big successes in the last decades. Their expositions are well-written and highlight many parts of this vast subject of mathematics. Some expositions focus on the geometric side of the story, others on the representation theoretic side and the remaining on their interrelationships.
Reference books: C488, C514, C517

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