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Rating: 
Summary: Very detailed with lots of motivating examples
Review: The theory of representations of semisimple Lie groups is very complete from a mathematical perspective and is of enormous importance in high energy physics. This book gives a comprehensive overview of this theory, and deals with both the noncompact and compact cases. My interest was with the noncompact case and in topics such as the Langland's classification, and so I read only chapters 5 - 10. Therefore my review will be confined to these chapters. Throughout the book, G denotes the group in question and K denotes the elements of G fixed under the Cartan involution. The author endeavors, and this is reflected in the title of the book, to employ many examples to illustrate the main results. This makes the book considerably more easy to follow than others that are written in the "Bourbaki" style. The Iwasawa and Bruhat decompositions and the Weyl group construction are shown to hold for non-compact groups in chapter 5. The Borel-Weil theorem is proven for compact connected Lie groups using the results of the chapter. The Harish-Chandra decomposition fo linear connected reductive groups is proven in chapter 6. The author shows clearly the role of holomorphic representations in obtaining this result and the construction of holomorphic discrete series. The principal series representations of SL(2, R) and SL(2, C) are use to motivate the notion of an 'induced representation" in chapter 7. The theory of induced representations involves the Bruhat theory and its use of distribution theory, and relates via the 'intertwining operators', irreducible representations of two subgroups. The author discusses the notion of an admissible representation in chapter 8, which are representations on a Hilbert space by unitary operators and each element in K has finite multiplicity when the representation is restricted to K. Equivalence of admissible representations are discussed via the concept of an "infinitesimal equivalance", which is the usual notion if the representation is unitary and irreducible. The Langlands classification of irreducible admissible representations is discussed in detail. The Langlands program shows to what extent irreducible admissible representations of a group are determined by the parabolic subgroups. The construction of discrete series, used throughout the proof of the Langlands classification, is then done in detail in the next chapter. Ths concept of an admissible infinitesimally unitary representation plays particular importance here. Here the representation operators act like skew-Hermitian operators with respect to an inner product on the space of K-finite vectors. If one reads this chapter from a physics perspective, the representations constructed using discrete series are somewhat 'exotic' and will probably not enter into applications, in spite of the fact that physical considerations do dictate sometimes the use of noncompact groups. Chapter 10 addresses the question as to the completeness of irreducible admissible representations using discrete series. If there not enough discrete series representations this will show up in the Fourier analysis of square integrable functions on the group. In the compact case, Fourier analysis proceeded via the characters of irreducible representations. The author shows how to do this in the noncompact case via 'global characters' of representations, which are well-behaved generalizations of the compact case. The well-behavedness of global characters comes from their being of trace class, with the result of the trace being a distribution. The author gives explicit formulas for the case of SL(2, R), and shows hows differential equations can be used to limit the possibilities for how characters behave. In fact, the author shows to what extent characters are functions, proving that the restriction of any irreducible global character of G to the 'regular set' is a real analytic function.
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