Harmonic analysis on semisimple groups - modern treatment

Question

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 I find it a rather hard read and I feel it's a bit outdated now.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.

Answer 1

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.

Answer 2

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.

Answer 3

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.