Kac-Moody groups, analytic regularity conditions and cities

TL;DR

This paper introduces twin cities, a new geometric structure for affine analytic Kac-Moody groups, enabling applications in infinite-dimensional differential geometry and extending the concept of twin buildings.

Contribution

It constructs twin cities for affine analytic Kac-Moody groups, bridging a gap caused by the loss of twin building structures in analytic completions.

Findings

Twin cities consist of two sets of buildings associated with affine analytic Kac-Moody groups.

Points in isoparametric submanifolds correspond to chambers in a city.

The work extends classical finite-dimensional differential geometry results to infinite dimensions.

Abstract

The relationship between minimal algebraic Kac-Moody groups and twin buildings is well known as is the relationship between formal completions in one direction and affine buildings. Nevertheless, as the completion of a Kac-Moody group in one direction destroys the opposite BN-pair, there exists no longer a twin building. For similar reasons, there are so far no buildings at all for analytic completions of affine Kac-Moody groups. In this article we construct a new type of twin buildings, called twin cities, that are associated to affine analytic Kac-Moody groups over the real or complex numbers. Twin cities consist of two sets of buildings. We describe applications of cities in infinite dimensional differential geometry by proving infinite dimensional versions of classical results from finite dimensional differential geometry: For example, we show that points in an isoparametric submanifold in a Hilbert space correspond to all chambers in a city. In a sequel we will describe the theory of twin cities for formal completions.