# Geometric Invariant Theory

## Over the Real and Complex Numbers

• Nolan R. Wallach

Part of the Universitext book series (UTX)

1. Front Matter
Pages i-xiv
2. ### Background Theory

1. Front Matter
Pages 1-1
2. Nolan R. Wallach
Pages 3-29
3. Nolan R. Wallach
Pages 31-47
3. ### Geometric Invariant Theory

1. Front Matter
Pages 49-49
2. Nolan R. Wallach
Pages 51-128
3. Nolan R. Wallach
Pages 129-153
4. Nolan R. Wallach
Pages 155-186
4. Back Matter
Pages 187-190

### Introduction

Geometric Invariant Theory (GIT) is developed in this text within the context of algebraic geometry over the real and complex numbers. This sophisticated topic is elegantly presented with enough background theory included to make the text accessible to advanced graduate students in mathematics and physics with diverse backgrounds in algebraic and differential geometry.  Throughout the book, examples are emphasized. Exercises add to the reader’s understanding of the material; most are enhanced with hints.

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.

### Keywords

Hilbert-Mumford theorem Kostant cone Lie theory and invariant theory algebraic geometry geometric invariant theory geometric invariant theory textbook Lie groups algebraic groups affine theory Borel fixed point theorem Kostant quadratic generation theorem GIT Cartan-Helgason theorem

#### Authors and affiliations

• Nolan R. Wallach
• 1
1. 1.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-319-65907-7
• Copyright Information Nolan R. Wallach 2017
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics
• Print ISBN 978-3-319-65905-3
• Online ISBN 978-3-319-65907-7
• Series Print ISSN 0172-5939
• Series Online ISSN 2191-6675