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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.

Inhaltsverzeichnis

Chapter 1. Algebraic Geometry

Abstract
This chapter is a compendium of results from algebraic geometry that we will use in the later chapters. The emphasis is on the relationship between the two natural topologies on an algebraic variety over the complex numbers, the Zariski topology and the metric topology (or standard topology) that comes from the embedding of certain open subsets in the Zariski topology (those that are isomorphic with affine varieties) as closed subsets in a finite-dimensional vector space. The interaction of these two topologies will be one of the main themes of the later chapters in this book. Many of the basic theorems in algebraic geometry (and differential geometry) are stated with references, many to [GW], while others are given proofs. There are complete proofs of most of the statements that relate to the interplay between algebraic and differential geometry.
Nolan R. Wallach

Chapter 2. Lie Groups and Algebraic Groups

Abstract
In this chapter we will study the relationship between algebraic and Lie groups over $$\mathbb{C}$$ and $$\mathbb{R}$$. As in the last chapter, most of the results are standard and there will be references to the literature for a substantial portion of them. However, at the end of the chapter there will be less well known material related to the Kempf–Ness theorem and Tanaka duality. These results point to Chapters 3 and 4. There is also a proof that maximal compact subgroups of a symmetric subgroup of GL(n, $$\mathbb{R}$$) are conjugate. This important result is usually proved using Cartan’s fixed point theorem for negatively curved spaces (cf. [He]).
Nolan R. Wallach

Chapter 3. The Affine Theory

Abstract
This chapter is the heart of our development of geometric invariant theory in the affine case. We will begin (as indicated below) with basic properties of algebraic groups and Lie group actions. As indicated in the preface two proofs of the Hilbert–Mumford theorem are given. The first is a relatively simple Lie group oriented proof of the original characterization of the elements of the zero set of non-constant homogeneous invariants (the null cone).
Nolan R. Wallach

Chapter 4. Weight Theory in Geometric Invariant Theory

Abstract
In the previous chapter the emphasis was on studying closed orbits in reductive group actions on affine varieties. We saw that this was essentially the same as studying orbits under regular representations. In most examples that we considered the generic orbits were usually closed. In this chapter, we consider similar questions for projective varieties and in particular for the projective quotient of a regular representation. If that representation is irreducible, then there is exactly one closed orbit, the unique minimal orbit. Kostant has given a set of quadratic generators for the homogeneous ideal of polynomials vanishing on this orbit. Much of the exposition in this chapter explains the analogous results for the case of reducible representations (based on Kostant’s methods and including some results of Brion). To carry out Kostant’s ideas we need to recall the theory of roots and weights.
Nolan R. Wallach

Chapter 5. Classical and Geometric Invariant Theory for Products of Classical Groups

Abstract
In this chapter we will study the invariant theory of products of classical groups acting on the tensor product of their defining representation.
Nolan R. Wallach

Backmatter

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