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?
Thanks in advance!
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.
You can look at rt.representation-theory or automophic-forms questions on Math Overflow. Here are some that may be relevant:
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.
Depending on what you're asking, you might want to check out my old question on learning representation theory.
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.
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.
