**Mathematical Surveys and Monographs**

Volume: 107;
2003;
576 pp;
Softcover

MSC: Primary 20;
Secondary 17; 22

Print ISBN: 978-0-8218-4377-2

Product Code: SURV/107.S

List Price: $104.00

Individual Member Price: $83.20

**Electronic ISBN: 978-1-4704-1334-7
Product Code: SURV/107.S.E**

List Price: $104.00

Individual Member Price: $83.20

# Representations of Algebraic Groups: Second Edition

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*Jens Carsten Jantzen*

Back in print from the AMS, the first part of this book is an introduction to the general theory of representations of algebraic group schemes. Here, Janzten describes important basic notions: induction functors, cohomology, quotients, Frobenius kernels, and reduction mod \(p\), among others. The second part of the book is devoted to the representation theory of reductive algebraic groups and includes topics such as the description of simple modules, vanishing theorems, the Borel-Bott-Weil theorem and Weyl's character formula, and Schubert schemes and line bundles on them.

This is a significantly revised edition of a modern classic. The author has added nearly 150 pages of new material describing later developments and has made major revisions to parts of the old text. It continues to be the ultimate source of information on representations of algebraic groups in finite characteristics.

The book is suitable for graduate students and research mathematicians interested in algebraic groups and their representations.

#### Readership

Graduate students and research mathematicians interested in algebraic groups and their representations.

#### Reviews & Endorsements

This is an authoritative [book] which, in its updated form, will continue to be the research worker's main reference. From a practical point of view, the scheme adopted of adding new material in the final chapters and keeping the structure of the rest of the book largely unchanged is extremely convenient for all those familiar with the first edition. We are extremely lucky to have such a superb text.

-- Bulletin of the London Mathematical Society

Very readable … meant to give its reader an introduction to the representation theory of reductive algebraic groups …

-- Zentralblatt MATH

Those familiar with [Jantzen's previous] works will approach this new book … with eager anticipation. They will not be disappointed, as the high standard of the earlier works is not only maintained but exceeded … very well written and the author has taken great care over accuracy both of mathematical details and in references to the work of others. The discussion is well motivated throughout … This impressive and wide ranging volume will be extremely useful to workers in the theory of algebraic groups … a readable and scholarly book.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Representations of Algebraic Groups: Second Edition

- Contents v6 free
- Introduction vii8 free
- Part I. General Theory 116 free
- 1. Schemes 318
- 2. Group Schemes and Representations 1934
- 3. Induction and Injective Modules 3752
- 4. Cohomology 4964
- 5. Quotients and Associated Sheaves 6580
- 6. Factor Groups 85100
- 7. Algebras of Distributions 95110
- 8. Representations of Finite Algebraic Groups 111126
- 9. Representations of Frobenius Kernels 125140
- 10. Reduction mod p 141156

- Part II. Representations of Reductive Groups 151166
- 1. Reductive Groups 153168
- 2. Simple G–Modules 175190
- 3. Irreducible Representations of the Frobenius Kernels 189204
- 4. Kempf's Vanishing Theorem 201216
- 5. The Borel-Bott-Weil Theorem and Weyl's Character Formula 217232
- 6. The Linkage Principle 231246
- 7. The Translation Functors 251266
- 8. Filtrations of Weyl Modules 267282
- 9. Representations of G[sub(r)]T and G[sub(r)]B 291306
- 10. Geometric Reductivity and Other Applications of the Steinberg Modules 315330
- 11. Injective G[sub(r)]–Modules 325340
- 12. Cohomology of the Frobenius Kernels 343358
- 13. Schubert Schemes 353368
- 14. Line Bundles on Schubert Schemes 365380
- A. Truncated Categories and Schur Algebras 385400
- B. Results over the Integers 411426
- C. Lusztig's Conjecture and Some Consequences 419434
- D. Radical Filtrations and Kazhdan-Lusztig Polynomials 439454
- E. Tilting Modules 457472
- F. Frobenius Splitting 479494
- G. Frobenius Splitting and Good Filtrations 501516
- H. Representations of Quantum Groups 515530

- References 531546
- List of Notations 569584
- Index 573588