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.