G-Complete Reducibility, Geometric Invariant Theory and Spherical Buildings

The aim of this textbook is to introduce readers at a graduate level to G-complete reducibility and explain some of its many applications across pure mathematics. It is based on the Oberwolfach Seminar of the same name which took place in 2022.

The notion of G-complete reducibility for subgroups of a reductive algebraic group is a natural generalisation of the notion of complete reducibility in representation theory. Since its introduction in the 1990s, complete reducibility has been widely studied, both as an important concept in its own right, with applications to the classification and structure of linear algebraic groups, and also as a useful tool with applications in representation theory, geometric invariant theory, the theory of buildings, and number theory.

Provides a comprehensive review of the field of G-complete reducibility Develops basic theory and advanced applications Provides links of the core theory to a diverse range of pure mathematical applications Gives a graduate audience access to a wide-ranging set of modern tools

Autorentext

The authors Michael Bate, Benjamin Martin and Gerhard Röhrle have a longstanding collaboration and friendship (20 years and counting). Together they have written 20 papers in and around this subject area, with a lasting impact on the field of algebraic groups (including subgroup structure, representation theory, geometric invariant theory, spherical buildings) and applications to other areas such as metric geometry and number theory.

Inhalt

Chapter 1. Introduction.- Chapter 2. Preliminaries.- Chapter 3. Geometric invariant theory.- Chapter 4. G-complete reducibility: first definitions and properties .- Chapter 5. The geometric approach.- Chapter 6. Finiteness, rationality and rigidity results .- Chapter 7. The spherical building of G .- 162 Chapter 8. The optimality formalism.- Chapter 9. Applications to G-complete reducibility .- Chapter 10. Large versus small characteristic .- Chapter 11. G-complete reducibility over an arbitrary field .- Chapter 12.Chapter 12. Variations, applications and future directions .- Chapter 13. Solutions to Exercises.- Bibiliography.- Index.