Lectures on Chevalley Groups

This is the first formal publication of the well-known set of notes from lectures given by R. Steinberg at Yale University in 1967 and initially issued in mimeographed form. Despite their limited circulation, these notes have served as the standard reference for an introduction to the theory of Chevalley groups. Before this publication, to get your hands on this masterly exposition, you had to hope that your local library had a copy or a copy of a copy of it.

The lectures assume a basic knowledge of complex semisimple Lie algebras and their classification in terms of Weyl groups and root systems, something that now can be found, for example in Humphreys’ Introduction to Lie Algebras and Representation Theory (Springer, 1972 and reprinted several times) or in Samelson’s Notes on Lie Algebras (Van Nostrand, 1969, reprinted in 1990 by Springer).

With this background, quickly summarized in the first few pages, the book proceeds to give a thorough treatment of Chevalley groups over an arbitrary field, from their construction using the associated Konstant \(\mathbb Z\)-form, to their description in terms of Tits’s BN-pairs and the Bruhat decomposition. It goes on to discuss their presentation in terms of the universal Chevalley group in terms of generators and relations, their simplicity and relationship with algebraic groups.

A chapter is devoted to calculate the orders of the Chevalley groups when the ground field is finite. Isomorphisms and automorphisms of Chevalley groups over perfect fields are also treated, using symmetries of the associated root systems. Some twisted Chevalley groups are studied as fixed-point subgroups of an automorphism of a Chevalley group.

Central extensions of Chevalley groups are given a detailed treatment in one chapter, including their relation to Milnor’s construction of the \(K_2\)-groups via the Steinberg relations.

The last section considers the representation theory of Chevalley groups, both in the ordinary and modular cases. An appendix summarizes basic facts about finite reflection groups, their root systems, and presentation as Coxeter groups.

I must emphasize that, as anyone involved in these subject knows, that the exposition in these lectures includes many original contributions of the author that have now become the standard ones which one finds in the literature. Even for the Appendix, on such a classic subject, the proof that every reflection group is a Coxeter group given here is by the author and has become the standard one in any exposition on reflection groups.

The AMS edition of these notes has been reset in \(\mathrm{\LaTeX}\) and includes some corrections and a few additions by Steinberg himself.

There is another wonderful introduction to this theory, R. W. Carter’s Simple Groups of Lie Type, Wiley (1972, reprinted in 1989), which is twice as long as Steinberg’s book. It gives a more detailed treatment of the so-called adjoint case, analogue of the centerless complex Lie groups case. In addition to this case, Steinberg’s treatment includes the twisted versions to these groups as well as central extensions of them. But this comes at a cost, making the exposition more concise, sometimes just sketching an argument.

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Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.