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For my finals, I am digging through the book by Varadarajan An introduction to harmonic analysis on semisimple Lie groups. I find it a rather hard read and I feel it's a bit outdated now. Any recommendation of a more modern (and/or) introductory treatment to the topics covered by this book would be greatly appreciated.

I am more interested in the representation theory, although I find the connection to harmonic analysis intriguing. Having learned finite-dimensional representation theory I wanted to move on to the infinite-dimensional one. From harmonic analysis I know only the classical Pontryagin duality, which I thought is enough to get me started, but I've found Varadarajan's approach based on examples difficult to follow.

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Can you clarify the words "more modern"? The current edition of Varadarajan's book is just over 10 years old after all. Also for reference requests it will help if you give a bit of your background so the recommendations can be more appropriate. –  Willie Wong Oct 7 '10 at 10:57
    
My copy is about 20 years old. I didn't realize that there's more recent edition. –  Vít Tuček Oct 7 '10 at 12:16
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One small piece of advice. The general (locally compact, not necessarily abelian) case is quite complex. Start with the compact case and understand the Peter-Weyl Theorem and Tannaka theorem first. Chevalley's book "Theory of Lie Groups" is old but very well-written and still an excellent place to get started. –  Dick Palais Oct 7 '10 at 15:02
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You might want to read the answers to this question: mathoverflow.net/questions/37021 to get an overview of the representation theory of and harmonic analysis on locally compact groups. –  Emerton Oct 7 '10 at 18:05
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Perhaps, I misunderstood the sentiment expressed in the body of the question, but I don't see how a more general approach to harmonic analysis on semisimple Lie groups can be more accessible to anyone, let alone beginners, than an approach based on examples. The area is notoriously difficult and full of technical subtleties. For more positive outlook, see my post and comment to AnthonyT's. –  Victor Protsak Oct 8 '10 at 2:37

3 Answers 3

up vote 6 down vote accepted

Speaking as a nonexpert, I'd emphasize that the subject as a whole is deep and difficult. Even leaving aside the recent developments for $p$-adic groups, the representation theory of semisimple Lie groups has been studied for generations in the spirit of harmonic analysis. So there is a lot of literature and a fair number of books (not all still in print). Having heard many of Harish-Chandra's lectures years ago, I know that the subject requires enormous dedication and plenty of background knowledge including classical special cases. Some books are certainly more accessible for self-study than others, but a lot depends on what you already know and what you think you want to learn.

Access to MathSciNet is helpful for tracking books and other literature, as well as some insightful reviews. Without attempting my own assessment, here are the most likely books to be aware of besides the corrected paperback reprint of Varadarajan's 1989 Cambridge book (I have the original but not the corrected printing, so don't know how many changes were made):

MR2426516 (2009f:22009), Faraut, Jacques (F-PARIS6-IMJ), Analysis on Lie groups. An introduction. Cambridge Studies in Advanced Mathematics, 110. Cambridge University Press, Cambridge, 2008.

MR1151617 (93f:22009), Howe, Roger (1-YALE); Tan, Eng-Chye (SGP-SING), Nonabelian harmonic analysis. Applications of SL(2,R). Universitext. Springer-Verlag, New York, 1992.

MR0498996 (58 #16978), Wallach, Nolan R., Harmonic analysis on homogeneous spaces. Pure and Applied Mathematics, No. 19. Marcel Dekker, Inc., New York, 1973.

This old book by Wallach as well as another by him on Lie group representations are presumably out of print. In any case, textbooks at an introductory level which emphasize both Lie group representations and harmonic analysis (often in the direction of symmetric spaces) are relatively few and far between. That probably reflects the practical fact that graduate courses aren't often attempted and are inevitably rather advanced. On the other hand, there are some modern graduate-level texts on compact Lie groups and related harmonic analysis as well as books on Lie groups and their representations with less coverage of harmonic analysis and symmetric spaces.

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FWIW I've had the Faraut book recommended to me as a good book to read "If you're a functional analyst". Not that I've had time to follow up on this yet... –  Matthew Daws Oct 7 '10 at 19:11

From my very limited google preview I take it the question is about an introduction to Harish Chandra's work described by emerton in a previous post he references. If you don't like the example approach (I don't either) take a peek from time to time at Varadarajan's (masterly) exposition "Harmonic Analysis on Reductive Groups" (Springer LN #576, 1977). The only examples there are those needed to start induction arguments or to do descent to centralizers. But he does lay out the plan very clearly along the way. For an exposition using a more modern approach to the discrete series, try Wallach's books "Real Reductive Groups." I have only seen Vol I (1988) but already that has some simplifications and appears more appealing to at least some students. For a truly modern approach, that is an interesting question.

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Wallach's book is a great reference and very systematic; however, precisely these qualities make it much less suitable as an introduction for a beginner. Howe and Tan's book, on the other hand, only treats the case of $SL(2,\mathbb{R}),$ but gives you a lot more flavor while being unencumbered with technical subtleties. –  Victor Protsak Oct 8 '10 at 1:51
    
I thought we'd established that the questioner was not overwhelmed by technical subtleties and just wanted a glimpse at the big picture. If you really want to to stick to little examples you are going to be in big trouble if you don't include SU(2,1) as well. –  AnthonyT Oct 8 '10 at 2:13
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Varadarajan's older book is very faithful to Harish-Chandra's original approach. I think that everyone should decide for himself (or herself) what is "worth the effort". I do not completely understand OP's background and motivation, but assuming he wants to get the flavor of non-abelian harmonic analysis, Howe and Tan seems appropriate; if the goal is to see Harish-Chandra's theory then Howe's survey I cited, Varadarajan, Harish-Chandra's papers, and Wallach, in the order of increasing level of generality and detail, are suitable sources. –  Victor Protsak Oct 8 '10 at 9:18
    
I should've phrased my question differently. I'm sorry. I'm not looking for more general or abstract approach per se; what I had in my was more of a top-down approach. Varadarajan builds from the bottom and I find myself unsure a lot of times, for example I often wonder whether G in a current context means U(n) or a general group. Also it is riddled with misprints and small errors (like using Fourier transform on general compact semisimple group without defining it first). Your comment was the most helpful for me. –  Vít Tuček Oct 21 '10 at 16:57

The first thing one should keep in mind is that harmonic analysis on semisimple Lie groups is very different from the "abstract harmonic analysis" a la Loomis or Hewitt and Ross dealing with locally compact abelian groups. The semisimple case was developed in the large part by Harish-Chandra and his papers (reprinted in his 4 volume Collected papers), while considerably older than Varadarajan's book, are still a good source of results and inspiration for many of us.

For an introduction to the subject, I warmly recommend Howe and Tan's book cited by Jim. It treats from the representation theory point of view the simplest nontrivial case, nonabelian harmonic analysis related to $SL(2,\mathbb{R}).$ The book uses elementary tools, and yet it deals with a wide range of topics. Elements of Harish-Chandra's theory for general reductive Lie groups may be found in Howe's survey article A century of Lie theory.

The theory of special functions and harmonic analysis on classical symmetric spaces and reductive symmetric spaces based on representation theoretic approach with different flavor is exposed in the books by Vilenkin, Helgasson, and Heckman and Schlichtkrull, which provide good complementary accounts of this theory.

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