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Representations of Algebraic Groups: Second Edition
Jens Carsten Jantzen, Aarhus University, Denmark
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Mathematical Surveys and Monographs
2003; 576 pp; softcover
Volume: 107
ISBN-10: 0-8218-4377-X
ISBN-13: 978-0-8218-4377-2
List Price: US$104
Member Price: US$83.20
Order Code: SURV/107.S
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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

"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

From reviews of the first edition:

"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

Part I. General theory
  • Schemes
  • Group schemes and representations
  • Induction and injective modules
  • Cohomology
  • Quotients and associated sheaves
  • Factor groups
  • Algebras of distributions
  • Representations of finite algebraic groups
  • Representations of Frobenius kernels
  • Reduction mod \(p\)
Part II. Representations of reductive groups
  • Reductive groups
  • Simple \(G\)-modules
  • Irreducible representations of the Frobenius kernels
  • Kempf's vanishing theorem
  • The Borel-Bott-Weil theorem and Weyl's character formula
  • The linkage principle
  • The translation functors
  • Filtrations of Weyl modules
  • Representations of \(G_rT\) and \(G_rB\)
  • Geometric reductivity and other applications of the Steinberg modules
  • Injective \(G_r\)-modules
  • Cohomology of the Frobenius kernels
  • Schubert schemes
  • Line bundles on Schubert schemes
  • Truncated categories and Schur algebras
  • Results over the integers
  • Lusztig's conjecture and some consequences
  • Radical filtrations and Kazhdan-Lusztig polynomials
  • Tilting modules
  • Frobenius splitting
  • Frobenius splitting and good filtrations
  • Representations of quantum groups
  • References
  • List of notations
  • Index
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