Lectures on the orbit method / A. A. Kirillov

Bibliogr. p.395-402. Index

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. In more detail, there are six chapters. The first discusses the geometry of coadjoint orbits for arbitrary Lie groups. The second studies the representation theory and geometry of the Heisenberg group in detail, as an introduction to the generalization of this theory to arbitrary nilpotent Lie groups in the next chapter. Chapter 4 examines the solvable case; here the results are more complicated and not quite as clean as in the nilpotent case, but the essence of the theory goes through. There is no general extension of the orbit method beyond these two settings, but there are many partial and highly suggestive results, together with tantalizing open problems. Chapter 5 summarizes the picture of orbits and representations for compact groups while Chapter 6 makes some remarks about semisimple and non-semisimple Lie groups, together with some non-Lie groups. It also gives a number of very interesting open problems. There are five very useful appendices, on topology and category theory, smooth manifolds, Lie groups and homogeneous manifolds, functional analysis, and representation theory. The focus throughout is on a very useful and interesting “user’s guide” for the orbit method, which the author had formulated in an earlier lecture and reproduces in the introduction. There is also a fairly large list of references and a good index. (MathSciNet)