Kac-Moody geometry

TL;DR

This paper explores the infinite-dimensional differential geometry of Kac-Moody groups, including symmetric spaces, isoparametric submanifolds, polar actions, and twin buildings, extending classical Lie group geometries.

Contribution

It introduces the geometric structures associated with affine Kac-Moody groups, generalizing finite-dimensional Lie group geometries to an infinite-dimensional setting.

Findings

Description of Kac-Moody symmetric spaces

Construction of isoparametric submanifolds in Hilbert space

Analysis of polar actions and twin buildings

Abstract

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.