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Book review

Author(s) Williams, F.L.
Title Lectures on the spectrum of L^2(GammaG)
Publisher Longman
Year of publication 1991
Reviewed by Frantisek Vcelar

A small explanation is perhaps in order - the G in the title denotes a semi-simple Lie group, Gamma its (reasonable) subgroup, and the "spectrum" is meant in the group-representation sense, i.e. one is interested in the multiplicities with which the irreducible representations come into the right regular representation of G on L^2(Gamma\G). The book discusses deep and beautiful cohomological, arithmetical, and geometric interpretations attached to these multiplicities. The main highlights are probably the Selberg trace formula, the Fourier transform on symmetric spaces, and the Eisenstein series, but there is much more: basics of Clifford algebras and Spin-modules, Dirac operators and Osborne-Warner formula, Zuckermann modules, Selberg-Gangolli zeta function, generalization of Hübner's formula, etc. Although the author has strived to make the exposition more or less self-contained, some familiarity with the subject is necessary for reading the book - in particular, some background in Lie theory (root systems, Weyl groups, Cartan subgroups) is needed.

The book is an expanded version of the author's lecture notes, which explains why proofs are frequently omitted or replaced by a reference, and probably also accounts for occasional clumsities in presentation - for instance, equations are occasionally labelled in the text (without being displayed on a separate line), which both makes them difficult to find when referred to, and is sometimes confusing (e.g., apart from =def there is also =b. and it took the reviewer some time to realize that this is just an equation labelled as "b."). The symbol <formula>, used very often but never explained, stands (hopefully) for "such that". But, apart from these trifles, the material is well-presented, and the book is well typeset and nice to read.

There is a huge literature on the topics this book treats - Hejhal's two-volume monograph on the Selberg trace formula, the books of Gangolli and Varadarajan, Warner, Terras, Wallach, or Sugiura (on harmonic analysis on Lie groups), Lang's SL2(R) (dealing with the simplest noncommutative case), Kubota's volume on Eisenstein series, etc. What makes the present book different is that it presents a range of topics, including both the basics and some of the latest results, in a form accessible to non-specialists or to graduate students with the appropriate background - and, besides, written by an author who is a well-known researcher in the field. Both libraries and researchers or advanced graduate students interested in Lie theory can safely be recommended buying the book.