Geometric Invariant Theory: Over the Real and Complex Numbers

The exposition is divided into two parts. The first part, ‘Background Theory’, is organized as a reference for the rest of the book. It contains two chapters developing material in complex and real algebraic geometry and algebraic groups that are difficult to find in the literature. Chapter 1 emphasizes the relationship between the Zariski topology and the canonical Hausdorff topology of an algebraic variety over the complex numbers. Chapter 2 develops the interaction between Lie groups and algebraic groups. Part 2, ‘Geometric Invariant Theory’ consists of three chapters (3–5). Chapter 3 centers on the Hilbert–Mumford theorem and contains a complete development of the Kempf–Ness theorem and Vindberg’s theory. Chapter 4 studies the orbit structure of a reductive algebraic group on a projective variety emphasizing Kostant’s theory. The final chapter studies the extension of classical invariant theory to products of classical groups emphasizing recent applications of the theory to physics.