These results give beautifully compact and complete answers in the nilpotent case to most of the questions usually asked about unitary representations of a given group, namely, the classification of its irreducible representations, the decomposition of a representation restricted to a subgroup or induced from a subgroup, the structure of Kronecker products, the Plancherel formula, and the nature of the infinitesimal and distribution characters of the irreducible representations. ...

The book under review, written by a leading figure in the field, constitutes the first treatise devoted to the theory of group representations that presents not only the foundations or some special aspects of the theory but also enough material to give a fair idea of what this domain of contemporary mathematics actually is.

Some 20 years ago in the book "Elements of the Theory of Representations" I wrote that representation theory enters the fourth stage of its evolution, when the three main directions of its development will presumably be the following ones:

- representations of algebraic groups over local fields and adele rings;

- representations of infinite-dimensional Lie groups and Lie algebras;

- representations of Lie supergroups and Lie superalgebras.

These directions, together with the application of the orbit method, formed the main topic of discussion at our seminar.

Of course, real life corrected this prediction. Two new areas of mathematical physical physics, namely the theory of completely integrable systems and string theory, have had a great influence on mathematics in general, and on representation theory in particular. One must also mention here the growing influence of the idea of supersymmetry. Originally this idea appeared in theoretical physics as a means to "put on an equal footing" fermions and bosons, but later it became a general mathematical principle.

This text, based on courses and seminars at Moscow University, consists of three main parts: expository text, problems and hints for solution. The exposition is concise but very clear.

The main goal of the book is to show the meaning of the notion(s) of number in contemporary mathematics, to inform about problems arising in connection with various approaches to the notion and about methods of solving such problems. In each case, the very beginning of the corresponding theory is outlined and suitable literature is indicated for further details. Among other subjects, the basic notions of p-adic and non-standard analysis are presented, and the concepts of a quaternion and a Cayley number are explained. The author also acquaints the reader with the notion of a von Neumann algebra and the idea of "supermathematics", the calculus of anti-commuting variables. The book is addressed to students and scientists who are interested in applications of mathematics.

This is the first systematic and reasonably self-contained exposition of the orbit method for representations of Lie groups to appear in print. It is written by a leading pioneer and major worker in the subject, as an outgrowth of an earlier paper and lectures on the subject, and is aimed at both experts and students. It surveys a large part of the subject, omitting many supplementary details not necessary to follow the main thread. The aim of the orbit method is to unite harmonic analysis and symplectic geometry to describe the representations of a group in geometric terms. This aim has been most successfully realized for nilpotent and solvable Lie groups; most of this book is devoted to their representation theory.

The book offers a nicely written, systematic and readable description of the orbit method for various classes of Lie groups. ... [It] should be on the shelves of mathematicians and theoretical physicists using representation theory in their work.

The name of Kirillov is known to everyone who studies representation theory or uses it in one's research. He is a classical author of the subject, the creator and developer of its basic notions and methods. Kirillov's orbit method, the Kirillov-Kostant bracket, Kirillov's character formula, the Gelfand-Kirillov conjecture, the Gelfand-Kirillov dimension, Kirillov's model, are terms firmly established in the language of mathematics. Kirillov's orbit method is one of the most original and fruitful discoveries in representation theory in its hundred-plus year history. This method soon became widely popular and stimulated a flow of results that has not dried up to these days. Another fundamental direction was called to life by the seminal joint papers of Gelfand and Kirillov on the skew fields of fractions of universal enveloping algebras. Wonderful and as always highly original results were obtained by Kirillov in the theory of infinite-dimensional Lie groups and algebras, and their representations. This work is relatively less known and should yet be carefully read and studied. Kirillov's seminar at Moscow State University gathered for 30 years. It is with deep nostalgia that all its participants remember it. Kirillov has numerous students that were formed not only by the problems he put forward, but even more by his powerful personality. ... Alexander Kirillov has the gift of creating a specific atmosphere that stimulates research and has exquisite mathematical taste. All his students and friends warmly remember his out-of-the-city seminar sessions with their traditional soccer and especially volleyball, and charming charades.