Conjugacy classes in semisimple algebraic groups. (English) Zbl 0834.20048

Humphreys, James E.

Mathematical Surveys and Monographs. 43. Providence, RI: American Mathematical Society (AMS). xviii, 196 p. (1995).

The author has collected a wealth of material on conjugacy classes in semi-simple algebraic groups. The book starts out as a monograph, explaining the basics in full detail, and it ends as a quick survey. This works rather well. Topics that are covered range from dimensions of centralizers to Springer’s Weyl group representations. Along the way one has learnt about regular elements, Richardson orbits, the dual group of a reductive group, the unipotent variety, the Bala-Carter classification of nilpotent orbits, the use of Lang’s theorem in the classification of conjugacy classes in finite groups of Lie type, and much much more. The exposition tries to be complementary to that in the books of R. W. Carter [Finite groups of Lie type: Conjugacy classes and complex characters (Wiley, New York, 1985; Zbl 0567.20023)] and D. Collingwood, W. M. McGovern [Nilpotent orbits in semisimple Lie algebras (Van Nostrand, New York, 1993; Zbl 0972.17008)]. The book serves as a useful guide to the literature. The author has called my attention to one more reference [W. Borho, Abh. Math. Semin. Univ. Hamb. 51, 1-4 (1981; Zbl 0495.20019)].
According to the AMS, the volume was printed directly from copy provided by the author. This did not prevent them from getting the page numbers out of sync with those used in the index. Fortunately the discrepancy is constant (four).

Reviewer: W.van der Kallen (Utrecht)

MSC:

20G15	Linear algebraic groups over arbitrary fields
20-02	Research exposition (monographs, survey articles) pertaining to group theory
17B20	Simple, semisimple, reductive (super)algebras
22E10	General properties and structure of complex Lie groups
22E46	Semisimple Lie groups and their representations

Keywords:

semisimple conjugacy classes; unipotent conjugacy classes; semi-simple algebraic groups; dimensions of centralizers; Weyl group representations; regular elements; Richardson orbits; unipotent variety; nilpotent orbits; conjugacy classes; finite groups of Lie type