This book contains two contributions on closely related subjects: the theory of linear algebraic groups and invariant theory. The first part is written by T. A. Springer, a well-known expert in the first mentioned field. Hc presents a comprehensive survey, which contains numerous sketched proofs and he discusses the particular features of algebraic groups over special fields (finite, local, and global). The authors of part two-E. B. Vinbcrg and V. L. Popov-arc among the most active researchers in invariant theory. The last 20 years have bccn a period of vigorous development in this field duc to the influence of modern methods from algebraic geometry. The book will bc very useful as a reference and research guide to graduate students and researchers in mathematics and theoretical physics.

Introduction

Historical Comments

Chapter 1. Linear Algebraic Groups over an Algebraically

Closed Field

1. Recollections from Algebraic Geometry

1.1. Affine Varieties

1.2. Morphisms

1.3. Some Topological Properties

1.4. Tangent Spaces

1.5. Properties of Morphisms

1.6. Non-Affine Varieties

2. Linear Algebraic Groups, Basic Definitions and Properties

2.1. The Definition of a Linear Algebraic Group

2.2. Some Basic Facts

2.3. G-Spaces

2.4. The Lie Algebra of an Algebraic Group

2.5. Quotients

3. Structural Properties of Linear Algebraic Groups

3.1. Jordan Decomposition and Related Results

3.2. Diagonalizable Groups and Tori

3.3. One-Dimensional Connected Groups

3.4. Connected Solvable Groups

3.5. Parabolic Subgroups and Borel Subgroups

3.6. Radicals, Semi-simple and Reductive Groups

4. Reductive Groups

4.1. Groups of Rank One

4.2. The Root Datum and the Root System

4.3. Basic Properties of Reductive Groups

4.4. Existence and Uniqueness Theorems for Reductive Groups

4.5. Classification of Quasi-simple Linear Algebraic Groups

4.6. Representation Theory

Chapter 2. Linear Algebraic Groups over Arbitrary Ground Fields

1. Recollections from Algebraic Geometry

1.1. F-Structures on Affine Varieties

1.2. F-Structures on Arbitrary Varieties

1.3. Forms

1.4. Restriction of the Ground Field

2. F-Groups, Basic Properties

2.1. Generalities About F-Groups

2.2. Quotients

2.3. Forms

2.4. Restriction of the Ground Field

3. Tori

3.1. F-Tori

3.2. F-Tori in F-Groups

3.3. Split Tori in F-Groups

4. Solvable Groups

4.1. Solvable Groups

4.2. Sections

4.3. Elementary Unipotent Groups

4.4. Properties of Split Solvable Groups

4.5. Basic Results About Solvable F-Groups

5. Reductive Groups

5.1. Split Reductive Groups

5.2. Parabolic Subgroups

5.3. The Small Root System

5.4. The Groups G(F)

5.5. The Spherical Tits Building of a Reductive F-Group

6. Classification of Reductive F-Groups

6.1. Isomorphism Theorem

6.2. Existence

6.3. Representation Theory of F-Groups

Chapter 3. Special Fields

1. Lie Algebras of Algebraic Groups in Characteristic Zero

1.1. Algebraic Subalgebras

2. Algebraic Groups and Lie Groups

2.1. Locally Compact Fields

2.2. Real Lie Groups

3. Linear Algebraic Groups over Finite Fields

3.1. Lang's Theorem and its Consequences

3.2. Finite Groups of Lie Type

3.3. Representations of Finite Groups of Lie Type

4. Linear Algebraic Groups over Fields with a Valuation

4.1. The Apartment and Affine Dynkin Diagram

4.2. The Affine Building

4.3. Tits System, Decompositions

4.4. Local Fields

5. Global Fields

5.1. Adele Groups

5.2. Reduction Theory

5.3. Finiteness Results

5.4. Galois Cohomology

References