I am interested in any sources that can be helpful for learning the representation theory of real reductive groups. I am currently reading Wallach book, but I feel that I don't understand the subject properly and want to find an alternative.
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1$\begingroup$ This is a natural question but not at all research-level. Aside from that, there is no single answer that would work for everyone, so community-wiki is appropriate. But did you try math.stackexchange.com/questions? $\endgroup$– Jim HumphreysCommented Apr 12, 2014 at 15:51
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1$\begingroup$ I have seen a lot of textbook recommendations on this site (e.g. mathoverflow.net/questions/2446/… or mathoverflow.net/questions/13/learning-about-lie-groups) so I thought it would be appropriate. Copied the question to math.stackexhange. $\endgroup$– user68061Commented Apr 12, 2014 at 18:19
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First note that there is the book of Vogan (Representation Theory if real reductive groups) which discusses the case of $SL_2(\mathbb{R})$ on a very basic level. I think this is a good start. In my opinion the remainder of the book is not very accessible.
However for the whole theory I would recommend that you first look at the notes from the conference "Computational Theory of Real Reductive Groups"
http://www.math.utah.edu/realgroups/schedule.html
Especially the notes from Adams and Vogan give a good introduction to the subject, don’t require much background and both are filled with examples.
Finally I should add Knapp's Book: Lie Groups Beyond an Introduction. At least the structure of real reductive groups (max. compact subgroups and so on) is discussed there.
In which direction to you want to go? Do you "really" want to do representation theory or are you planning to go to geometric representation theory?
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$\begingroup$ Thank you very much for your answer. I am not quite sure what about the direction of my studies mainly because I don't know much about geometric representation theory. $\endgroup$ Commented Apr 12, 2014 at 12:22
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$\begingroup$ Anyway, if you are interested in learning geometric representation theory the you might find this interesting: mathoverflow.net/questions/160387/… $\endgroup$ Commented Apr 12, 2014 at 12:40
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$\begingroup$ 10 years later but I am curious about the final comment. Is your meaning to say that the representation theory of real reductive groups considered disjoint from geometric representation theory? I see the same names in the literature. $\endgroup$– Song YeCommented Oct 3 at 0:43