Complete reducibility in Kac-Moody groups

If G is a reductive group with Borel subgroup B then the coset space G/B is the chamber set of a spherical building whose Weyl distance is governed by the Bruhat order on the Weyl group. For Kac-Moody groups, the two conjugacy classes of standard Borel subgroups give rise to a twin building, with a codistance governed by the reverse Bruhat order. However, infinite root systems admit many more conjugacy classes of sets of positive roots, and these are not captured by the twin building. This thesis defines a new object, called a citadel, designed to organise these non-standard conjugacy classes. A citadel consists of a distinguished central building, called the keep, together with further buildings associated to other conjugacy classes of positive systems. The additional buildings are related to the keep through distance functions defined by Dyer's twisted Bruhat orders. It is proved that a citadel supports notions of projections between residues, leading to a notion of convexity. The thesis proves that a Kac-Moody group possesses two canonical citadels that are shown to form a twin citadel. The set of chambers of the twin citadel can be identified with the disjoint union of coset spaces G/Q_\Psi, where the Q_\Psi are quasi-Borel subgroups generalising standard Borel subgroups. The construction requires the development of quasi-parabolic subgroups, generalising the parabolic subgroups, and an extension of the Bruhat-type decomposition of Billig and Dyer. The thesis concludes by defining a notion of complete reducibility for convex subsets of twin citadels. This leads to a new notion of G-complete reducibility for subgroups of Kac-Moody groups, generalising existing formulations that have been derived using the twin building.