Overview

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.

Full Product Details

Author:   Michael Bate ,  Benjamin Martin ,  Gerhard Röhrle
Publisher:   Springer Nature Switzerland AG
Imprint:   Birkhauser
ISBN:  

9783032088659

ISBN 10:   3032088658
Pages:   343
Publication Date:   09 February 2026
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Forthcoming
Availability:   Not yet available  
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Table of Contents

1. Preliminaries: basic theory of (non-connected) reductive groups over algebraically closed fields, root system, R-parabolic and Levi subgroups, characterisation in terms of cocharacters.- 2. G-complete reducibility: first definition and properties.- 3. The Geometric Approach: introduction to some ideas from GIT, closed orbits and the Hilbert-Mumford Theorem, G-cr subgroups correspond to closed orbits.- 4. The Optimality Formalism: results of Kempf-Rousseau-Hesselink, consequences for G-complete reducibility, Levi ascent and descent.- 5. G-cr and the building of G: links between complete reducibility in the building and in the group, the Centre Conjecture.- 6. G-cr over a field: the notion of a cocharacter-closed orbit, links to G-cr, Galois ascent and descent.- 7. Applications and open problems: a survey of modern applications of the theory, and some open problems.- 8. Appendices, including hints and further references for exercises.

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Author Information

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.

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