Lie Groups

Reductive Lie groups and their representations form a very broad field. The aim of the book is to select essential topics for a year course for (good) graduate students and it offers several possibilities. The simplest one is to concentrate on compact groups and their representations. The book also includes many other beautiful topics, which can be combined with the main line. A short introductory part quickly explains basic properties of representations of compact groups (Schur orthogonalilty, the Peter-Weyl theorem). The main topics for a course are contained in the second part (Lie groups and the corresponding Lie algebras, universal enveloping algebra, Weyl integration formula, root systems, the Weyl group, Weyl character formula). The second part also contains a discussion of reductive complex groups and real simple Lie groups and symmetric space (including Satake diagrams, Iwasawa decomposition and a discussion of embeddings of Lie groups). The last part adds an array of other very interesting topics. The major theme is the Frobenius-Schur duality and its various applications. It includes random matrix theory, branching formulae, the Cauchy identity, Gel’fand pairs, Hecke algebras, cusp forms and cohomology of flag varieties. The book is nicely written and efficiently organized. There are many books covering basic facts of finite dimensional representations of simple Lie groups and algebras. The presented book brings a beautiful selection of a number of further important additional topics, which are worth including in a course. It is a very important addition to existing literature on the subject.

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