Representations of algebraic groups.
2nd ed.

*(English)*Zbl 1034.20041
Mathematical Surveys and Monographs 107. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3527-0/hbk). xiii, 576 p. (2003).

This is the second edition of a work that first appeared in 1987. The first printing was reviewed extensively by V. L. Popov, see Zbl 0654.20039. Let me just say that Jantzen’s book has been an indispensable reference for anyone involved with representations of algebraic groups, in particular of reductive groups in positive characteristic.

The second edition differs substantially from the first. Again it is a book one must have. About one third is new. The old part has been sprinkled with new comments, but has been left intact as much as is feasible. New chapters at the end, identified with capital letters, take care of several developments in the intervening years.

The following topics are now also covered in the familiar clear manner. Chapter A: Truncated categories and Schur algebras. Here the truncations are those of Donkin, associated to a finite ‘saturated’ set of dominant weights. Chapter B: Results over the integers. This globalizes several cohomological results. Chapter C: Lusztig’s Conjecture and some consequences. Chapter D: Radical filtrations and Kazhdan-Lusztig polynomials. These two chapters belong together and discuss various aspects of this central conjecture concerning characters of certain simple modules. Chapter E: Tilting modules. Chapter F: Frobenius splitting. Chapter G: Frobenius splitting and good filtrations. This chapter gives the proof by Mathieu of the theorem about tensor products of modules with good filtration. Logically chapters F and G should be read early. Chapter H: Representations of quantum groups. This chapter is in a different style. It is just a survey.

It is a pity that the author has still not adopted certain widely used terminology. For instance, the usual form of ‘sum formula’ is ‘Jantzen sum formula’.

The second edition differs substantially from the first. Again it is a book one must have. About one third is new. The old part has been sprinkled with new comments, but has been left intact as much as is feasible. New chapters at the end, identified with capital letters, take care of several developments in the intervening years.

The following topics are now also covered in the familiar clear manner. Chapter A: Truncated categories and Schur algebras. Here the truncations are those of Donkin, associated to a finite ‘saturated’ set of dominant weights. Chapter B: Results over the integers. This globalizes several cohomological results. Chapter C: Lusztig’s Conjecture and some consequences. Chapter D: Radical filtrations and Kazhdan-Lusztig polynomials. These two chapters belong together and discuss various aspects of this central conjecture concerning characters of certain simple modules. Chapter E: Tilting modules. Chapter F: Frobenius splitting. Chapter G: Frobenius splitting and good filtrations. This chapter gives the proof by Mathieu of the theorem about tensor products of modules with good filtration. Logically chapters F and G should be read early. Chapter H: Representations of quantum groups. This chapter is in a different style. It is just a survey.

It is a pity that the author has still not adopted certain widely used terminology. For instance, the usual form of ‘sum formula’ is ‘Jantzen sum formula’.

Reviewer: Wilberd van der Kallen (Utrecht)

##### MSC:

20G05 | Representation theory for linear algebraic groups |

20G10 | Cohomology theory for linear algebraic groups |

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |

14L15 | Group schemes |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

14L17 | Affine algebraic groups, hyperalgebra constructions |

20G15 | Linear algebraic groups over arbitrary fields |