Topic
Reductive group
About: Reductive group is a research topic. Over the lifetime, 2645 publications have been published within this topic receiving 65306 citations.
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01 Jan 1991
TL;DR: Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions are discussed in this paper.
Abstract: Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions.
2,919 citations
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01 Jan 1975
TL;DR: A survey of rationality properties of semisimple groups can be found in this paper, where a survey of rational properties of algebraic groups is also presented, as well as a classification of reductive groups representations.
Abstract: Algebraic geometry affine algebraic groups lie algebras homogeneous spaces chracteristic 0 theory semisimple and unipoten elements solvable groups Borel subgroups centralizers of Tori structure of reductive groups representations and classification of semisimple groups survey of rationality properties.
2,070 citations
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01 Jan 1987TL;DR: In this article, the Steinberg modules are used to represent the Frobenius kernels of finite algebraic groups and reduce them to reduction mod $p$ by using a simple reductive group.
Abstract: 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 group 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.
2,009 citations
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01 Jan 1992
TL;DR: Adeles and Ideles as discussed by the authors gave a generalization of the Strong Approximation Theorem for algebraic groups over locally compact fields and showed that the strong and weak approximations in algebraic numbers of groups are equivalent.
Abstract: (Chapter Heading): Algebraic Number Theory. Algebraic Groups. Algebraic Groups over Locally Compact Fields. Arithmetic Groups and Reduction Theory. Adeles. Galois Cohomology. Approximation in Algebraic Groups. Class Numbers andClass Groups of Algebraic Groups. Normal Structure of Groups of Rational Points of Algebraic Groups. Appendix A. Appendix B: Basic Notation. Algebraic Number Theory: Algebraic Number Fields, Valuations, and Completions. Adeles and Ideles Strong and Weak Approximation The Local-Global Principle. Cohomology. Simple Algebras over Local Fields. Simple Algebras over Algebraic Number Fields. Algebraic Groups: Structural Properties of Algebraic Groups. Classification of K-Forms Using Galois Cohomology. The Classical Groups. Some Results from Algebraic Geometry. Algebraic Groups over Locally Compact Fields: Topology and Analytic Structure. The Archimedean Case. The Non-Archimedean Case. Elements of Bruhat-Tits Theory. Results Needed from Measure Theory. Arithmetic Groups and Reduction Theory: Arithmetic Groups. Overview of Reduction Theory: Reduction in GLn(R).Reduction in Arbitrary Groups. Group-Theoretic Properties of Arithmetic Groups. Compactness of GR/GZ. The Finiteness of the Volume of GR/GZ. Concluding Remarks on Reduction Theory. Finite Arithmetic Groups. Adeles: Basic Definitions. Reduction Theory for GA Relative to GK. Criteria for the Compactness and the Finiteness of Volume of GA/GK. Reduction Theory for S-Arithmetic Subgroups. Galois Cohomology: Statement of the Main Results. Cohomology of Algebraic Groups over Finite Fields. Galois Cohomology of Algebraic Tori. Finiteness Theorems for Galios Cohomology. Cohomology of Semisimple Algebraic Groups over Local Fields and Number Fields. Galois Cohomology and Quadratic, Hermitian, and Other Forms. Proof of Theorems 6.4 and 6.6: Classical Groups. Proof of Theorems 6.4 and 6.6: Exceptional Groups. Approximation in Algebraic Groups: Strong and Weak Approximation in Algebraic Varieties. The Kneser-Tits Conjecture. Weak Approximation in Algebraic Groups. The Strong Approximation Theorem. Generalization of the Strong Approximation Theorem. Class Numbers and Class Groups of Algebraic Groups: Class Numbers of Algebraic Groups and Number of Classes in a Genus. Class Numbers and Class Groups of Semisimple Groups of Noncompact Type The Realization Theorem. Class Numbers of Algebraic Groups of Compact Type. Estimating the Class Number for Reductive Groups. The Genus Problem. Normal Subgroup Structure of Groups of Rational Points of Algebraic Groups: Main Conjecture and Results. Groups of Type An. The Classical Groups. Groups Split over a Quadratic Extension. The Congruence Subgroup Problem (A Survey). Appendices: Basic Notation. Bibliography. Index.
1,268 citations