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I am seeking for a good first reference on algebraic groups, or even linear algebraic groups, where the general theory could be understood through example for the classical groups.

Understanding the theoretical definitions of tori, unipotent radical, reductive group, root system, parabolic subgroups, Levi decomposition, etc. is one thing. I always find a huge and sometimes unbridgeable gap between those definitions and formal manipulations and... what could I say for a specific example, say some classical linear algebraic groups. I found the notes of Murnaghan in the Clay Math Proceedings which are really understandable and works out lots of examples. However, in such expository notes of some pages the general theory, motivations and further examples are lacking, and I would like to find something more comprehensive to begin studying (and practicing!) algebraic groups.

So I wonder, what reference could you advise for learning algebraic group, illustrated by lots of computable examples?

Here are some precisions: I work with automorphic forms, and consider mainly adelic points of an algebraic group as well as its local components. I would like to understand how to deal with and build an intuition on what that means. For instance: what are the relations between compacity and existence of parabolic subgroups? why are the tori so central and important? what are the usual isomorphism between quotients and sequences? etc. I believe I can encapsulate it in: understanding what is exactly behind Murnaghan's notes for number theorists?

Desiderius Severus
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    It might be a good idea to include some more about why you want to understand algebraic groups in the question. The tags seem to suggest a certain goal, but this is not apparent from the questions itself, and the best recommendation would probably change based on it. – Tobias Kildetoft Jan 20 '17 at 18:14
  • @TobiasKildetoft Sorry for the lack of precisions, here are some precisions: I work with automorphic forms, and consider mainly adelic points of an algebraic group as well as its local components. I would like to understand how to deal with and build an intuition on what that means. For instance: what are the relations between compacity and existence of parabolic subgroups? why are the tori so central and important? what are the usual isomorphism between quotients and sequences? etc. I believe I can encapsulate it in: understanding what is exactly behing Murnaghan's notes for number theorists? – Desiderius Severus Jan 24 '17 at 09:31
  • Please add some of this to the question itself so people will not miss it. Unfortunately, I don't really know anything about the role of algebraic groups in number theory, so I don't think I would the the right person to recommend literature for that purpose (I mainly know about their representation theory in positive characteristic, where the interesting questions are of a completely different nature). – Tobias Kildetoft Jan 24 '17 at 09:48
  • @TobiasKildetoft I updated my post accordingly – Desiderius Severus Jan 24 '17 at 09:56
  • Great, I hope someone here will know more about this side of things and be able to provide some good references. – Tobias Kildetoft Jan 24 '17 at 10:00

2 Answers2

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These notes, especially the worked examples, helped me a lot when I needed to get up to speed with algebraic groups without having the time to learn the theory in depth and from scratch.

ws898989
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Algebraic groups are mainly used as a language, not an end in itself, when talking about automorphic forms. You should think of Murnaghan's article as introducing the vocabulary. The tori are important for defining the root decomposition as well as characters and cocharacters, which appear in the definition of root datum. Unfortunately I've looked and looked and for automorphic forms, the expositions that exist mainly consist of lecture notes and research articles. I don't know of any comprehensive source of information. There is certainly a need for it, since I think that the biggest stumbling block for number theorists in coming to automorphic forms may simply just be the language of algebraic groups.

Not a grad student
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