1.1. Acknowledgement by Robert Steinberg.

These notes are dedicated to my wife, Maria. They might also have been dedicated to the Yale Mathematics Department typing staff whose uniformly high standards will become apparent to anyone reading the notes. Needless to say, I am greatly indebted to John Faulkner and Robert Wilson for writing up a major part of the notes. Finally, it is a pleasure to acknowledge the great stimulus derived from my class and from many colleagues, too many to mention by name, during my stay at Yale.

1.2. Guide to the Reader.

These notes presuppose the theory of complex semisimple Lie algebras through the classification as may be found, for example, in the books of E B Dynkin, N Jacobson, or J-P Serre, or in the notes of Seminaire Sophus Lie. An appendix dealing with the most frequently needed results about finite reflection groups and root systems has been included. The reader is advised to read this part first. This he can do rather quickly.

1.3. Review by: J E Humphreys.
Mathematical Reviews MR0466335 (57 #6215).

These notes, based on the author's 1967-68 lectures at Yale, have been for a decade the standard reference for Chevalley groups over arbitrary (but especially finite) fields. In addition to the simple (adjoint type) groups treated by C Chevalley, the various central extensions and twisted analogues of these groups are described here. ...

The author presupposes some acquaintance with complex semisimple Lie algebras, but provides in an appendix a nice self-contained treatment of Weyl groups and root systems. Within this framework his treatment of the basic aspects of Chevalley groups is complete; however, in a few later sections he just sketches arguments found elsewhere (including some of his own fundamental contributions). The main topics are: construction of Chevalley groups, via a Chevalley basis of a complex semisimple Lie algebra and the associated Kostant Z-form of its universal enveloping algebra; structure: BN-pair (Tits system), simplicity; relationship with algebraic groups; generators and relations; central extensions and Milnor's $K_{2}$ functor; further study of the Bruhat decomposition over various rings and fields, e.g., closures of Bruhat cells; orders of the finite groups, via a study of Weyl group invariants; automorphisms and isomorphisms; twisted groups; modular and ordinary representations of the finite groups.

In his concise style, the author has compressed an enormous amount of useful material into relatively few pages; this of course makes some demands on the reader. For the neophyte, Carter's book offers a more leisurely and detailed exposition of the adjoint groups, while covering only about half of the material covered by the author. (Carter does, however, include a complete determination of the degrees of basic invariants for Weyl groups, which are a key ingredient in the formulas for the orders of the finite groups.) The format of these lecture notes is inconveniently bulky, and tracking down previous theorems or lemmas (numbered consecutively) can be time-consuming. Even so, they would be a bargain at twice the price.

2.1. From the Introduction.

The following is the substance of a set of lectures given at the Tata Institute of Fundamental Research during November and December of 1972. The notes are divided roughly into two parts. The first part attempts an a priori development of the basic properties of affine algebraic groups with emphasis on those needed in the study of conjugacy classes of elements of reductive groups: the semisimple-unipotent decomposition, conjugacy of Borel subgroups and of maximal tori, completeness of the variety of Borel subgroups, etc. The second part is devoted to the classification and characterisation of various such classes of elements: semisimple, unipotent, regular, subregular, etc. For a more detailed outline the reader may consult the table of contents. All of this is over an algebraically closed field. I had planned to include two talks on rationality questions, but this aim was not realised. Because of time limitations there had to be gaps in the actual development. In the first part the most serious of these is the omission of a large part of the proof of the existence of a quotient of a group by a closed subgroup. Also the principal structural and conjugacy results about connected solvable groups are used without proof, but this is not so serious since the Lie-Kolchin theorem is proved and from there on the proofs, by induction, follow fairly classical lines. In the second part the Bruhat lemma for reductive groups is used without proof (but a fairly complete proof is indicated for the classical groups) as are various properties of root systems and reflection groups (for which a comprehensive treatment may be found in Bourbaki's book). Modulo a few other points left to be checked by the reader I have attempted a coherent development.

It is a pleasure to thank my colleagues at the Tata Institute, young and old, for their hospitality and friendship to my wife and me during our visit and for their stimulating influence on my talks. It is a special pleasure to be able to thank here Shri Vinay V Deodhar who in addition has written up these notes.

2.2. Review by: Andy R Magid.
Bulletin of the American Mathematical Society 81 (2) (1975), 397-400.

Suppose $A$ and $B$ are invertible $n \times n$ complex matrices. How can one tell if $A$ and $B$ are conjugate - that is, if there is an invertible $n \times n$ complex matrix C such that $B = CAC^{-1}$? The solution, of course, is to examine the Jordan canonical form of the two matrices. But now suppose $A$ and $B$ lie in some algebraic subgroup of the group of all invertible $n \times n$ complex matrices - that is, in a subgroup determined by some polynomial conditions on the matrix entries, for example the matrices of determinant one. Now how can one tell if $A$ and $B$ are conjugate in that subgroup?

In more formal language, suppose $G$ is a linear algebraic group over an algebraically closed field $k$, and let $G$ act on itself by conjugation. The quotient set $V$ of $G$ under this action is the set of conjugacy classes of $G$. What one really wants to know is the structure of $V$, or of some subsets of it determined by looking at conjugacy classes of elements of special types. It is to the analysis of this structure that Robert Steinberg's lecture notes Conjugacy classes in algebraic groups are directed.
...

In summary, Steinberg has mapped out an efficient route through algebraic groups from first principles to the deep theorems on conjugacy classes. Readers with some acquaintance with algebraic geometry and algebraic groups, who are prepared for an occasional ipsedixitism or feeling of serendipity, will find this a stimulating and rewarding volume. Novices who choose these notes as their first introduction to algebraic groups may find the going pretty tough at points and perhaps should take a look at Borel's book first.

These notes are taken from lectures the author gave at the Tata Institute of Fundamental Research during two months of 1972, and the volume reflects this lecture note style. There are a few misprints, typing errors, misspellings, omitted references, etc., all minor, and the sort of thing one expects in a volume of this type. Perhaps more important, the volume, like most lecture notes, seems to be more or less a transcription of what the lecturer wrote on the blackboard. One wishes that occasionally the insights and explanations most lecturers give orally as side comments would also make their way into lecture notes. In this volume, for example, we get a brief two-page introduction by the author at the beginning, and only rare hints in the body of the book as to what direction is being taken, why it works, and what the importance of the results are. Of course, the persevering reader will find this out in the end anyway, but some help from the author would have been useful.

2.3. Review by: Sergei I Gelfand.
Mathematical Reviews MR0352279 (50 #4766).

This book, based on a series of lectures given at the Tata Institute, presents a self-contained treatment of some aspects of the theory of affine algebraic groups. The first and second chapters of the book may be considered as preliminary.

Chapter I contains some basic definitions and facts concerning affine algebraic varieties and affine algebraic groups.

The first part of Chapter II is devoted to the study of unipotent and semisimple elements in an affine algebraic group G. Here the author proves the theorem about the Jordan decomposition of elements of $G$ and Kolchin's theorem about the structure of any unipotent subgroup of $Aut(V)$ ($V$ being a finite dimensional vector space). He then discusses some properties of diagonalisable subgroups of affine algebraic groups. The second part of Chapter II is devoted to the properties of quotients of affine algebraic groups by their closed subgroups. Then these properties are applied to the study of Borel subgroups (i.e., maximal connected solvable subgroups). The author proves that all Borel subgroups are conjugate to each other and if $B$ is one of them, then $G/B$ is a complete projective variety. At the end of the chapter the author states the Bruhat lemma about the structure of the set $B∖G/B$ for any Borel subgroup in a connected algebraic group $G$.

Chapter III contains results on the structure of conjugacy classes in semisimple algebraic groups. Some of these results (as well as other results on the theme) can be found in other papers by the author ... . The notion of regular element is introduced and some properties of regular elements are proved. In particular, the author constructs an algebraic cross-section of the collection of all regular conjugacy classes for any simply connected semisimple algebraic group $G$. Then the author proves some results about unipotent classes. Among the most interesting is the construction of a desingularisation of the variety $V$ of unipotent elements in $G$. ... He also gives a classification of the unipotent subregular classes.

3.1. Foreword by Jean-Pierre Serre.

The present volume contains the Collected Papers of Robert Steinberg.

It includes all his published papers on group theory, with commentaries by the author himself.

Among these, one will find those on the "special representations" (now called Steinberg representations), on tensor products of representations, on finite reflection groups, on regular elements of algebraic groups, on Galois cohomology, on universal central extensions, etc.

I have always been an avid reader of Steinberg's papers. It gives me great pleasure to thank him, and the AMS, for making them easily available to a large audience.

3.2. Papers appearing in the volume.

1. A geometric approach to the representations of the full linear group over a Galois field, Trans. Amer. Math. Soc. 71 (1951), 274-282.

2. The representations of $GL(3, q), GL (4, q), PGL (3, q)$, and $PGL (4, q)$, Canad. J. Math. 3 (1951), 225-235.

3. Prime power representations of finite linear groups, Canad. J. Math. 8 (1956), 580-591.

4. Prime power representations of finite linear groups II, Canad. J. Math. 9 (1957), 347-351.

5. On the number of sides of a Petrie polygon, Canad. J. Math. 10 (1958), 220-221.

6. (with A Horn) Eigenvalues of the unitary part of a matrix, Pacific J. Math. 9 (1959), 541-550.

7. Finite reflection groups, Trans. Amer. Math. Soc. 91 (1959), 493-504.

8. Variations on a theme of Chevalley, Pacific J. Math. 9 (1959), 875-891.

9. The simplicity of certain groups, Pacific J. Math. 10 (1960), 1039-1041.

10. Automorphisms of finite linear groups, Canad. J. Math. 12 (1960), 606-615.

11. Invariants of finite reflection groups, Canad. J. Math. 12 (1960), 616-618.

12. Automorphisms of classical Lie algebras, Pacific J. Math. 11 (1961), 1119-1129.

13. A general Clebsch-Gordan theorem, Bull. Amer. Math. Soc. 67 (1961), 406-407.

14. Generators for simple groups, Canad. J. Math. 14 (1962), 277-283.

15. A closure property of sets of vectors, Trans. Amer. Math. Soc. 105 (1962), 118-125.

16. Complete sets of representations of algebras, Proc. Amer. Math. Soc. 13 (1962), 746-747.

17. Generators, relations and coverings of algebraic groups, Colloquium on Algebraic Groups (Brussels, 1962), 113-127.

18. Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33-56.

19. Differential equations invariant under finite reflection groups, Trans. Amer. Math. Soc. 112 (1964), 392-400.

20. Regular elements of semisimple algebraic groups, Publ. Sci. I.H.E.S. 25 (1965), 49-80.

21. On the Galois cohomology of linear algebraic groups, Proc. International Conf. on the Theory of Groups, Canberra (1965) (Gordon and Breach, 1967), 315-319.

22. Classes of elements of semisimple algebraic groups, Proc. International Congress of Mathematicians, Moscow (1966), 277-284.

23. Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. 80 (1968), 1-108.

24. Algebraic groups and finite groups, Illinois J. Math. 13 (1969), 81-86.

25. (with T A Springer) Conjugacy classes, Seminar in algebraic groups and related finite groups, Lecture Notes in Math., Springer-Verlag 131 (1970), 167-266; Addendum to second corrected printing (1986), 322-323.

26. Abstract homomorphisms of simple algebraic groups (after A Borel and J Tits), Séminaire Bourbaki, Lecture Notes in Math., Springer-Verlag 383 (1972), 307-326.

27. Torsion in reductive groups, Advances in Math. 15 (1975), 63-92.

28. On a theorem of Pittie, Topology 14 (1975), 173-177.

29. On the desingularization of the unipotent variety, Invent. Math. 36 (1976), 209-224.

30. On theorems of Lie-Kolchin, Borel and Lang, Contributions to Algebra (Academic Press, 1977), 349-354.

31. Conjugacy in semisimple algebraic groups, J. Algebra 55 (1978), 348-350.

32. Kleinian singularities and unipotent elements, Proc. Sympos. Pure Math. of Amer. Math. Soc. 37 (1980), 265-270.

33. Generators, relations and coverings of algebraic groups II, J. Algebra 71 (1981), 527-543.

34. Finite subgroups of SU2, Dynkin diagrams and affine Coxeter elements, Pacific J. Math. 118 (1985), 587-598.

35. Some consequences of the elementary relations of SL_n, Contemp. Math. 45 (1985), 335-350.

36. Tensor product theorems, Proc. Sympos. Pure Math. of Amer. Math. Soc. 47 (1987), 331-338.

37. On Dickson's theorem on invariants, J. Faculty of Sciences, Univ. of Tokyo 34 (1987), 699-707.

38. An occurrence of the Robinson-Schensted correspondence, J. Algebra 113 (1988), 523-528.

39. (with R Richardson and G Röhrle) Parabolic subgroups with Abelian unipotent radical, Invent. Math. 110 (1992), 649-671.

40. Nagata's example, in Algebraic groups and Lie groups, Australian Math. Soc. Lecture Series 9 (Cambridge University Press, 1997), 375-384.

3.3. Papers and books not Appearing in Volume.

41. Solution to problem 4065, Amer. Math. Monthly 51 (1944), 169-170.

42. (with R M Redheffer) The Laplacian and mean values, Quart. Appl. Math. 9 (1951), 315-317.

43. (with R M Redheffer) Analytic proof of the Lindemann theorem, Pacific J. Math. 2 (1952), 231-242.

44. (with R M Redheffer) Simultaneous trigonometric approximations, J. Math. Phys. 31 (1953), 260-266.

45. An example in functional equations, Math. Magazine 29 (1956),129-130.

46. Note on a theorem of Hadwiger, Pacific J. Math. 6 (1956), 775-777.

47. Note on linear differential equations, Amer. Math. Monthly 64 (1957), 35-36.

48. Generalizations of the theorem of Chasles, Amer. Math. Monthly 64 (1957), 352-353.

49. The isomorphism and isogeny theorems for the reductive algebraic groups.

50. Lectures on Chevalley groups, previously available from Yale Univ. Math. Dept. (1968), 1-277, now being prepared as a book; Russian translation by J N Bernstein and N N Yakovlev, Mir, Moscow, 1975.

51. Conjugacy classes in algebraic groups, based on lectures given at the Tata Institute in Bombay, Lecture Notes in Math., Springer-Verlag 366 (1974), 1-159.

3.4. From the Preface by Robert Steinberg.

Collected in this volume are most of my published papers. All of them are reproduced directly from the journals in which they first appeared with only cosmetic changes, except for (23) and (35) (of the list 3.2 above) which have been retyped with only minor changes and (40) which is still in press. Not included are eight published papers and two sets of lecture notes, one of which has appeared in mimeographed form and the other in book form; all of these are listed on 3.3 above. I have also listed (as (41)) a published solution to a problem in the American Mathematical Monthly, one of many in my early days, since it represents the first appearance of my work in print, and it gives, in my opinion, the "book proof" of a significant, although elementary, result. Also I still remember fondly how this "publication" came into being. I was taking a class in projective geometry at the University of Toronto at the time. It was being taught by Professor H S M Coxeter who seemed to have a never-ending supply of interesting problems. One of these was to show that if finitely many points in the real projective plane have the property that the line through two of them always contains a third then there exists a line containing all of them. I solved it and showed my solution to Professor Coxeter. He was obviously delighted with it and asked my permission to type it up and send it in to the Monthly, where, it turned out, the problem had come from, and I of course agreed. I had not heard of the Monthly until then, but soon after I began sending in solutions on my own.

Also included in this volume is a section, "Comments on the Papers," which comes directly after the papers themselves. It contains minor corrections and clarifications of the papers and indications of how their main ideas and results have evolved and been used since they first appeared. My remarks here are far from complete and there are undoubtedly many relevant works that I should have mentioned and did not. To the authors of all of these I offer my sincere apologies. At the suggestion of several of my colleagues, I have included, in my comments on (4), an extended discussion of what has come to be known as the Steinberg representation.

When Donald Babbitt, the publisher of the AMS (American Mathematical Society) and my former colleague at U.C.L.A., first suggested, in a telephone call, the possibility of doing this book, my first thought was that he had made a mistake and reached the wrong party. But no, he assured me that he had dialled the right number. Needless to say, I am delighted that he had, and I thank him and the AMS for this great honour. I also thank Christine Thivierge and Thomas Costa of the publishing office of the AMS for their work in the production of this book. J-P Serre, the official editor of this book, has also been very helpful, especially in matters mathematical, and I thank him warmly.

Many people have contributed, directly and indirectly, to my mathematical work and my continuing mathematical education. I thank all of them and two of them especially: Richard Brauer, who was my teacher at the University of Toronto, and Claude Chevalley, whose mathematics long ago struck a resonant chord which is with me still.

I also thank my colleagues, past and present, at my home institution U.C.L.A., where most of my work has been done, for providing a stimulating, yet relaxed and unhurried, atmosphere, an atmosphere ideally suited to my temperament.

Most of all I thank my wife Maria who over the years has supported all of my endeavours, mathematical and other, in every possible way.

3.4. Review by: H H Andersen.
Mathematical Reviews MR3437767.

This volume contains almost all work by Steinberg published over the last 45 years. Not included are his well-known "Lecture notes on Chevalley groups"[1967/1968] (now being prepared as a book) and his book Conjugacy classes in algebraic groups [1974] as well as a few minor papers dealing with various topics other than group and representation theory. At the end of the volume one finds some pages with Steinberg's comments on his various papers. In his own words these pages contain "minor corrections and clarifications of the papers and indications of how their main ideas and results have evolved and been used since they appeared".

It is difficult (and maybe not even proper in a short review like this) to single out one or two among the forty papers in this volume. I think all readers of Steinberg's work are struck by the deep insight which the results never fail to reveal and the elegance with which they are proved. My personal favourite is his theorem now known as the Steinberg tensor product theorem (contained in paper 18 in the volume; see also paper 36 where the result is set into perspective). It reduces (for a given prime p) the a priori infinite problem of finding the irreducible representations of a semisimple algebraic group G over an algebraically closed field of characteristic p (and via C W Curtis' theorem [J. Math. Mech. 9 (1960)] also the corresponding representations of the associated family $G(p^{n})$ of finite Chevalley groups) to a finite problem. This reduction is crucial for the famous Lusztig conjecture [G Lusztig, in The Santa Cruz Conference on Finite Groups, 1979] which (for primes greater than the Coxeter number for $G$) gives an algorithm for the characters in the finite set of so-called restricted irreducible representations. By Steinberg's theorem this is enough to get all irreducible characters.

In the modular representation theory of algebraic groups and of the corresponding finite Chevalley groups there is a special representation which nowadays is universally called the Steinberg representation. It has played a big role in many different contexts and in fact several other representation theories (e.g. for $p$-adic groups, quantum groups, etc.) now feature representations of a similar nature also often named Steinberg representations.

Steinberg considered this important representation in paper 4 (from the finite group angle). In his comments at the end of the volume he gives a very nice account of some of the history concerning this as well as some of the related situations in which it figures prominently.

4.1. Foreword by Robert R Snapp.

In 1997 the American Mathematical Society published Robert Steinberg, The Collected Papers, which contains forty key publications selected by the author. One notable omission, however, was his Lectures on Chevalley Groups, which has only appeared in mimeograph form, printed by the Yale Mathematics Department. It was my uncle's constant desire to publish a revised version of this work. As the author's namesake, it is a privilege to assist in the completion of this posthumous edition: Robert Steinberg was a devoted uncle and source of inspiration to all of his nieces and nephews.

The current volume incorporates a new introductory chapter, corrections contributed by astute readers of the original edition, and changes that Robert Steinberg pencilled into his personal copies. Footnotes marked by "(*)" are English translations of footnotes from the 1975 Russian edition of the work (edited by A A Kirillov).

This edition would not be possible without the generous assistance of Christopher Drupieski, who painstakingly transcribed the original mimeographed edition into LATEX, nor without the editorial contributions of my uncle's dear colleague and friend Raja V S Varadarajan. I would also like to acknowledge the editorial staff of the American Mathematical Society, and especially Sergei Gelfand for creating the bibliography and the index, and for translating the editorial footnotes from the Russian edition.

Since its original publication, several books have appeared that may help the reader approach these lectures, including Roger W Carter, Simple Groups of Lie Type, 1989; James E. Humphreys, Introduction to Lie Algebras and Representation Theory, 1972; and V S Varadarajan, Lie Groups, Lie Algebras, and Their Representations, 1984.

4.2. Review by: Robert M Guralnick and Gary M Seitz.
The American Mathematical Monthly 125 (10), 2018), 953-955.

Robert Steinberg spent a sabbatical at Yale University during the 1967-68 academic year and gave a course on Chevalley groups. There were two note-takers in the course (John Faulkner and Robert Wilson), and Steinberg wrote up his lectures based on the notes. These notes were then mimeographed and circulated among group theorists and others for decades. Steinberg kept a record of some minor alterations, but never published the notes officially. Finally, after his death on his 92nd birthday in 2012, a revised version was prepared (supervised by Steinberg's nephew, Robert Snapp, a physicist at the University of Vermont) which included these changes. They also incorporated some small remarks by various mathematicians and some footnotes that had appeared in the Russian edition of the notes. The American Mathematical Society (AMS) has now republished the notes, making them readily available to a new generation of group theorists.

The influence of Steinberg's notes has been remarkable and has impacted almost everyone who does serious work in the area. Steinberg's elegant and comprehensible writing style made his work accessible to a very wide audience. While there are more recent and advanced books in the area (see particularly Jantzen [Representations of Algebraic Groups, 2003]), they mostly assume that one knows all the material in Steinberg's book. There is still no substitute for Steinberg's work. Some aspects of these notes are covered in the very nice book of Carter [Simple Groups of Lie Type, 1972] (with much of his book influenced by Steinberg's work). One should also consult Steinberg's collected works [1997]. The book gives a wonderful exposition of Chevalley's seminal work [Sur certains groupes simple, 1955] as well as fundamental extensions by Steinberg himself [Variations on a theme of Chevalley, 1959].

The theory of finite Chevalley groups, and more generally finite groups of Lie type, is of central importance in group theory, but also in number theory, theoretical physics, and many other areas. Recall that the classification of finite simple groups, completed in 2004, states that a nonabelian finite simple group is either one of 26 sporadic groups, an alternating group, or a finite group of Lie type (i.e., a Chevalley group or a variation due to Steinberg, Suzuki, and Ree). Thus, if one wants to understand finite groups, it is essential to understand finite groups of Lie type. Indeed, many of the properties developed by Steinberg and others (including some developed in the book) are used in the proof of the classification of finite simple groups. Given a possibly unknown simple group, one wants to identify it as a group of Lie type using various properties, such as generators and relations, centralisers of unipotent elements, involutions, etc.