This is a very basic and most definitively a naive question but coming from a student it is probably OK.

I am trying to learn representation theory of (linear) algebraic groups and looking for a an easy resource. The books and notes which I have come across start with extensive knowledge of algebraic geometry (which is essential for a comprehensive treatment) but coming from a weak background of mathematics by the time you absorb all that you are exhausted.

I am wondering if there is another way to get into representation theory of algebraic groups without worrying too much about algebraic geometry part. We have tried to go through the famous books (Humphreys, Springer, Borel), although excellent but too much for a beginner only interested in representation theory part to start with.

Any help will be highly appreciated.

  • 7
    $\begingroup$ "Representation theory" is a fairly large topic, but for algebraic groups in characteristic 0 it's usually much easier to start with the mostly equivalent classical (finite dimensional) representation theory of semisimple Lie algebras. For this there are a number of self-contained books including the text by Fulton and Harris. More ambitious books by Goodman-Wallach, Onishnik-Vinberg combine this with some study of the groups. The problems in prime characteristic go deeper and are less understood, so it's unclear where you draw a line. $\endgroup$ Commented Sep 28, 2016 at 13:23
  • 1
    $\begingroup$ P.S. The transliteration is actually "Onishchik". $\endgroup$ Commented Sep 28, 2016 at 13:38
  • 4
    $\begingroup$ You're going to have to learn the general theory eventually, but a good first overview that avoids technicalities is Carter-Segal-MacDonald's "Lectures on Lie Groups and Lie Algebras". It contains three sets of lecture notes; the third is on linear algebraic groups and the first is on the representation theory of Lie algebras. Given your interests, I suppose that you could skip the chapter on the representation theory of compact Lie groups. $\endgroup$ Commented Sep 28, 2016 at 13:59

2 Answers 2


I would suggest Procesi's Lie Groups, as a text that introduces algebraic groups with minimal prerequisites. Chapter 7 "Algebraic Groups" is

a quick introduction to algebraic groups. In this chapter I make fair use of notions from algebraic geometry, and I try to at least clarify the statements used, referring to standard books for the proofs. In fact it is impossible, without a rather long detour, to actually develop in detail the facts used. I hope that the interested reader who does not have a background in algebraic geometry can still follow the reasoning developed here.

This is followed by Chapter 8 "Representation Theory",

a first look into the representation theory of various groups with extra structure, such as algebraic or compact groups. We will use the necessary techniques from elementary algebraic geometry or functional analysis, referring to standard textbooks. One of the main points is a very tight relationship between a special class of algebraic groups, the reductive groups, and compact Lie groups.


I know close to nothing about algebraic groups, but the little I know comes from this article.

Basically, it assumes that you know what an algebraic variety is, and develops from scratch both the elementary properties of these groups and their actions on varieties (which I think is what motivates their introduction in the first place).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.