The orbit method played and will play a very important role in description of basic facts concerning representations of Lie groups. Originally, they were developed for nilpotent Lie groups. The book offers a nicely written, systematic and readable description of the orbit method for various classes of Lie groups. The book starts with the description of the coadjoint action and its orbit structure, including the moment map and polarizations. The study of the case of nilpotent Lie groups starts with the important special case of the Heisenberg group. The orbit method is then fully described for the classical case of nilpotent Lie groups. The next two chapters describe the use of the orbit methods for solvable and compact Lie groups. The last chapter contains a short indication how the method works in other cases, including semisimple Lie groups, Lie groups of general type, infinite-dimensional groups, or in the case of groups over other fields. There are also general intuitive comments on the orbit method and suggestions for further research. To understand the book, the reader needs quite a lot of preliminary knowledge. For the convenience of the reader, the book contains five long appendices on topology, category theory, cohomology; real, complex, symplectic and Poisson manifolds; homogeneous spaces of Lie groups, basics of functional analysis, infinite dimensional representation and induced representations. These appendices form the second part of the book (almost 200 pages). The book is an excellent addition to the existing literature and should be on the shelves of mathematicians and theoretical physicists using representation theory in their work.

Reviewer:

vs