TITLE: Text for studying group representations in the context of (abstract) harmonic analysis
QUESTION [11 upvotes]: I would like to study elements of representation theory as I often encounter it when reading texts on harmonic analysis. I was therefore curious if someone could recommend a book for this.
When looking into the books that have "representation theory" as their title, it seems there is a variety of books written by people from different fields. For example, it seems there is almost no overlap in the content of Naimark's "Theory of Group representations", and the books by Serre and Fulton & Harris on representation theory. 
I was therefore wondering which "type of representation theory" is used in harmonic analysis, and how it relates to other "types of representation theory" such as the one studied in abstract algebra. 
Any advice or explanation is highly appreciated.

REPLY [5 votes]: I'd like to make a couple of recommendations at a slightly lower or comparable level than Varadarajan's excellent treatise that I wished I had read before that book.
First, Folland's 'A course in abstract harmonic analysis' gives a clear road map from functional analysis to the representation theory of topological groups.  It treats compact groups carefully with many worked examples, it emphasizes the peculiarities of non-unimodular groups (and even non type I groups, which I found very illuminating), it has a complete treatment o systems of imprimitivity (finally!) and in general touches upon all the subtle analytical issues sometimes glossed over in representation theory textbooks.  This is why I recommend it before Varadarajan.
Secondly, the book by Howe and Tan Non-abelian harmonic analysis has a completely different feel: it is driven by the representation theory of a single group, but because of that it has the luxury of going pretty deep into the analytical aspects of its representation theory.  It is much more readable (but definitely not as complete) than Lang's $\textrm{SL}(2,\mathbb{R})$ and chapter 4 is a spectacular demonstration of the power of representation theory in analysis! (although the one I found most useful was chapter 5 on decay of matrix coefficients).
I recommend these two books, the first for the solid analytical foundations it gives you for further study, and the second for a remarkable example with all sorts of applications that gives an honest, if rank $1$, glimpse into the general theory of representations of semisimple groups.