Kac-Moody geometry

The geometry of symmetric spaces, polar actions, isoparametric submanifolds and spherical buildings is governed by spherical Weyl groups and simple Lie groups. A natural generalization of semisimple Lie groups are affine Kac-Moody groups as they mirror their structure theory and have good explicitely known representations as groups of operators. In this article we describe the infinite dimensional differential geometry associated to Kac-Moody groups: Kac-Moody symmetric spaces, isoparametric submanifolds in Hilbert space, polar actions on Hilbert spaces and universal geometric twin buildings.