For my finals, I am digging through the book by Varadarajan [An introduction to harmonic analysis on semisimple Lie groups](http://books.google.com/books?id=seycdRenNfoC&lpg=PP1&dq=varadarajan%2520harmonic%2520analysis%2520on%2520semisimple%2520lie%2520groups&pg=PP1#v=onepage&q&f=false%20%22Varadarajan%20-%20An%20introduction%20to%20harmonic%20analysis%20on%20semisimple%20Lie%20groups%22). I find it a rather hard read and I feel it's a bit outdated now.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.



Harmonic analysis on semisimple groups - modern treatment
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. 


