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 BorelBottWeil 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 BorelBottWeil 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 KazhdanLusztig polynomials
 Tilting modules
 Frobenius splitting
 Frobenius splitting and good filtrations
 Representations of quantum groups
 References
 List of notations
 Index
