I am interested in introductory books/papers/reports about the (unitary) representation theory of $SL(2,\mathbb{R})$, with particular emphasis on the principal series representations. My background: I have a basic knowledge of Lie groups, I'm not a master in this subject, while I work with unitary representations (of discrete groups), therefore I am familiar with this last topic. I would be grateful for any advice. Thanks.
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2$\begingroup$ I don't remember what it covers, but there's a book literally called 'SL(2, R)'. $\endgroup$– arsmathCommented Nov 3, 2016 at 8:14
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2$\begingroup$ @arsmath: The book you mention is by Serge Lang (also cited in the linked question posted here by John Mangual), though the book doesn't seem to be well-regarded by Lie group specialists. In any case, the linked question only concerns the principal series tangentially, and the books recommended by twch are reasonable sources for your purpose. $\endgroup$– Jim HumphreysCommented Nov 3, 2016 at 13:50
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As an introduction I can recommand the following two books:
Taylor, Michael Eugene. Noncommutative harmonic analysis. No. 22. American Mathematical Soc., 1986.
In this book he discusses the unitary irreducible representations for several explicit examples. Among others there is a chapter on $SL(2,\mathbb{R})$ (Chapter 8) and a subsection on the principle series. Taylor does only assume very little prerequisits such as elementary facts on Lie groups and Lie algebras and he classifies all irreducible unitary representations of $SL(2,\mathbb R)$ and provides concrete realizations of them.
Knapp, Anthony W. Representation Theory of Semisimple Groups: An Overview Based on Examples (PMS-36). Princeton university press, 2016.
Chapter II--Seciton 5 contains a treatment of the unitary irreducible representations of $SL(2\mathbb R)$. If I remember correctly Knap only gives a list of the unitary irreducible representations and describes their realization and does not discuss the question that those are all possible unitary irreducible representations. However the advantage of this book is that it goes much deeper into the represetation theory of semisimple Lie groups in the subsequent chapters. So if you want to understand $SL(2,\mathbb R)$ as a starting point to understand the representation theory of general semisimple Lie groups, then this book could be very usefull.
