# Alternative or reprint of Carter's “Finite Groups of Lie Type: Conjugacy Classes and Complex Characters”

I would like to learn about character theory of finite groups of Lie type and some Deligne-Lusztig theory. The classic textbook on the subject seems to be Roger W. Carter's Finite Groups of Lie Type: Conjugacy Classes and Complex Characters (Wiley 1985, reprinted in 1993 in Wiley Classics). Sadly, this book is out of print, and while second-hand copies can be found, they sell at a ridiculously high price (e.g., this site has one for over 1200€). To make things more problematic, the same author wrote another (earlier) book with a confusingly similar title, Simple Groups of Lie Type (which I have), and which tends to come up whenever one searches for the Finite Groups book.

So, two questions:

1. Is there a (finite!) set of textbooks and/or introductory articles whose union covers the same material as Carter's Finite Groups of Lie Type? I have a copy of Digne & Michel's Representations of Finite Groups of Lie Type, but it seems (a) much harder to read, and (b) less complete (which makes sense as it is over three times shorter); are there other texts which might fill the gaps?

2. (This is not a mathematical question, but I hope it is nevertheless allowed here:) Is there any hope of persuading someone (whether the publisher or author, whoever holds the rights) to republish this book? The fact that it is still considered a standard reference, especially if question (1) does not have an answer, and the fact that used copies are so expensive, suggests that there is some demand. But I don't know if the author is still mathematically active, or if Wiley still has the rights.

• I've got an e-copy. Email me and I'll send it to you. My contact details are on my user page. – Nick Gill Nov 30 '16 at 13:25
• By the way, have you seen Bonnafe's book on $SL_2(q)$? It is restricted in scope but I really liked the presentation, and found it a good place to start on this stuff. – Nick Gill Nov 30 '16 at 13:26
• you can download it as a djvu file on Library Genesis --- while there are ethical concerns regarding this library, for cases such as this I would think its use acceptable – Carlo Beenakker Nov 30 '16 at 13:43
• To follow up on @CarloBeenakker's comment, I e-mailed the publisher, and they said that the rights had reverted to Carter and I would have to contact him. I contacted him and never heard back. That certainly doesn't automatically justify downloading it elsewise, but it does say that there's not a very obvious legal alternative, and that the answer to at least part of (2) may be "no". – LSpice Dec 1 '16 at 0:51
• In 2011 I looked into acquiring the rights to Carter's book. My plan was to publish it very expensively, perhaps on Google books, and donate the small profit, if any. I learned that due to Carter's ill health it would not be pracitical to work with him. Contrary to what Loren Spice was told, the publisher told me that they owned the rights. Nevertheless we got bogged down in trying to work out an arrangement. It might be worth trying again, I know many people would like to have access to this beautiful book. – Jeffrey Adams Dec 4 '16 at 19:49

Concerning the books by Carter, both have since been reprinted by Wiley in their "classics" series, though availability and cost right now is a serious question mark. The 1972 book is limited in scope, emphasizing Chevalley's 1955 approach to constructing simple groups (but over arbitrary fields). The 1985 book is considerably more expansive, emphasizing the Deligne-Lusztig characters and related matters about algebraic groups such as conjugacy classes. Two drawbacks: there are some mistakes (still uncorrected) and there is no treatment of the many further refinements by Lusztig and others. As Nick points out, Cedric Bonnafe's more recent Springer book on $SL_2$ over finite fields is well worth seeking out, in spite of its limited scope.

Anyway, the most recent MathSciNet reference for the 1985 book is here. Part of the problem with Carter's 1985 book is that he himself left mathematics some years ago and is now quite elderly. But the book does have a lot of detailed information which is not readily found elsewhere. It's a subject which deserves a fresh textbook treatment. Good luck.

ADDED: Maybe it's worthwhile to point out that Geck and Malle are in the process of writing a book of their own about the characters of the finite groups of Lie type: see their preprint of Chapter 1.

• In my opinion, Carter's Finite Groups of Lie Type is an extremely well written book, combining impressive breadth with clear and easy to follow expositions of difficult material. It is one of those books you want to read and learn from rather than just have sitting on a shelf. The fact that its author is now hard to reach does not take away from the usefulness and quality of the book. – A Stasinski Dec 1 '16 at 7:20

Malle, Gunter; Testerman, Donna. Linear algebraic groups and finite groups of Lie type. Cambridge Studies in Advanced Mathematics, 133. Cambridge University Press, Cambridge, 2011. xiv+309 pp. "Originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups, and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas."