
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 semisimple Lie group, Gamma its (reasonable) subgroup, and the "spectrum" is meant in the grouprepresentation 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 Spinmodules, Dirac operators and OsborneWarner formula, Zuckermann modules, SelbergGangolli zeta function, generalization of Hübner's formula, etc. Although the author has strived to make the exposition more or less selfcontained, 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 wellpresented, and the book is well typeset and nice to read.
There is a huge literature on the topics this book treats  Hejhal's twovolume 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 nonspecialists or to graduate students with the appropriate background  and, besides, written by an author who is a wellknown researcher in the field. Both libraries and researchers or advanced graduate students interested in Lie theory can safely be recommended buying the book.