李群

　　《李群（英文版）》Part I covers standard general properties of representations of compactgroups （including Lie groups and other compact groups， such as finite or p-adie ones）. These include Schur orthogonality， properties of matrix coefficientsand the Peter-Weyl Theorem.Part II covers the fundamentals of Lie gronps， by which I mean those sub-jects that I think are most urgent for the student to learn. These include thefollowing topics for compact groups： the fundamental group， the conjngacyof maximal tori （two proofs）， and the Weyl character formula. For noncom-pact groups， we start with complex analytic groups that are obtained bycomplexification of compact Lie groups， obtaining the lwasawa and Bruhatdecompositions. These arc the reductive complex groups. They are of course aspecial case， bnt a good place to start in the noncompact world. More generalnoncompact Lie groups with a Cartan decomposition are studied in the lastfew chapters of Part II. Chapter 31， on symmetric spaces， alternates exampleswith theory， discussing the embedding of a noncompact symmetric space inits compact dnal， the boundary components and Bergman-Shilov boundaryof a symmetric tube domain， anti Cartans classification. Chapter 32 con-structs the relative root system， explains Satake diagrams and gives examplesillustrating the various phenomena that can occur， and reproves the Iwasawadecomposition， formerly obtained for complex analytic groups， in this moregeneral context. Finally， Chapter 33 surveys the different ways Lie groups canbe embedded in oue another.

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