# How do we study the theory of reductive groups? [closed]

I am interested in the theory of reductive groups which is useful in the theory of automorphic forms. But the trouble boring me so long time is that I don't know the appropriate material for beginners or outsiders who wants to pave in this field and learn more about automorphic forms. So my question is that what kind of introduction materials, books or papers, fits me as an introduction and for further reading?

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## closed as no longer relevant by Harry Gindi, Yemon Choi, Tom Leinster, Qiaochu Yuan, Gjergji ZaimiMar 9 '10 at 23:23

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It would help if you provide more about your background: do you need reference for preliminary topics like Lie groups, representations of Lie algebras? –  Ilya Nikokoshev Dec 3 '09 at 13:58
The question is too vague to admit a useful answer. You haven't told us anything about what you already know or anything specific about what you want to learn, so how could we figure out what fits you? –  Pete L. Clark Dec 3 '09 at 14:42
-1. See what Pete L. Clark said. I'll happily change my vote if the question is edited so that it's possible to give an answer that isn't a stab in the dark. –  Anton Geraschenko Dec 3 '09 at 19:04
-1. What Anton said. Some of us are only doctors, not miracle workers, Jim. –  Yemon Choi Dec 3 '09 at 19:05
voting to close since the question is still very vague and the answers are just becoming a big list of Stuff Which Might Help Or Might Not –  Yemon Choi Mar 9 '10 at 21:44

Sit at a table with the books of Borel, Humphreys, and Springer. Bounce around between them: if a proof in one makes no sense, it may be clearer in the other. For example, Springer's book develops everything needed about root systems from scratch, and has lots of nice exercises relate to that stuff. On the other hand, Borel is better about systematically allowing general ground fields from early on (so one doesn't have to redo the proofs all over again upon discovering that it is a good idea to allow ground fields like $\mathbf{R}$, $\mathbf{Q}$, $\mathbf{F}_ p$, and $\mathbf{F} _p(t)$). Pay attention to the power of inductive arguments with centralizers and normalizers (especially of tori).

Unfortunately, none makes good use of schemes, which clarifies and simplifies many things related to tangent space calculations, quotients, and positive characteristic. (For example, the definition of central isogeny in Borel's book looks a bit funny, and if done via schemes becomes more natural, though ultimately equivalent to what Borel does.) So if some proofs feel unnecessarily complicated, it may be due to lack of adequate technique in algebraic geometry. (Everyone has to choose their own poison!) Waterhouse's book has nothing serious to say about reductive groups, but the theory of finite group schemes that he discusses (including Cartier duality and structure in the infinitesimal case) is very helpful for a deeper understanding isogenies between reductive groups in positive characteristic. The exposes in SGA3 on quotients and Grothendieck topologies (etale, fppf, etc.) are helpful a lot too (some of which is also developed in the book "Neron Models"). Galois cohomology is also useful when working with rational points of quotients.

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Do you know of any books on the subject that do make use of schemes? Maybe something not at an introductory level? –  Harry Gindi Mar 9 '10 at 21:29
Someone (hint hint) should write such a book. –  Ravi Vakil Mar 9 '10 at 22:10
I've asked this as a proper question: mathoverflow.net/questions/17662/… –  Harry Gindi Mar 9 '10 at 22:13
I think it's hilarious that although this question was posted on Dec. 3 and hung around quietly from Dec. 5 onwards minding its own business, now 3+epsilon months later as it got some activity again it suddenly got closed as "no longer relevant". Whatever... –  BCnrd Mar 10 '10 at 7:07
The main reason is that more people now have the ability to vote to close, so when this question got pushed back up to the top, more people jumped on it. By the way, if you register your account, you will also gain this ability, since you already have over 3000 points. –  Harry Gindi Mar 11 '10 at 3:15

You can look at rt.representation-theory or automophic-forms questions on Math Overflow. Here are some that may be relevant:

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Actually, I have little knowledge about representation theory which is needed for learning reductive groups. I just know some basic definitions of group representations... –  Alex Dec 4 '09 at 4:47

I've heard very good things about the book Linear Algebraic Groups by Springer, though I've only worked through the first few chapters so far.

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Yeah, I have this book, but I have not decide to read it yet. I hope we could discuss the questions of Springer's book in future:) –  Alex Dec 4 '09 at 4:38

Depending on what you're asking, you might want to check out my old question on learning representation theory.

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Thanks for your answer! Well, my major is number theory. So what I care about is how to apply reductive groups theory into the theory of automorphic forms. My ultimate goal is to learn automorphic forms and automorphic representations. –  Alex Dec 4 '09 at 4:33
And also I don't know representation theory~ –  Alex Dec 4 '09 at 4:35
Well, I know nothing about automorphic forms, but pretty much everything I know about reductive groups is via their representations (after all, the condition of being linearly reductive is a condition that these representations behave nicely.) –  Charles Siegel Dec 4 '09 at 6:04

I found the following introductory book very useful: Waterhouse, Introduction to Affine Group Schemes.

It is short, clear and fun to read. The book doesn't assume the previous knowledge of algebraic geometry, and depending on your background it can be either advantage or a disadvantage.

However, it's not going deep into theory of reductive groups. More advanced standard textbooks on algebraic groups are Springer, Borel and Humphreys.

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I learned the basics of Lie/algebraic groups and Lie algebras, including reductive groups, from the book "Lie groups and algebraic groups" by Vinberg and Onishchik (I've read the old Russian edition, but now a new expanded Russian edition has been translated into English). See e.g. here and here.

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