An extension theorem for embedded Riemannian symmetric spaces of non-compact type and an application to their universal property

TL;DR

This paper establishes an extension theorem for embedded non-compact Riemannian symmetric spaces, revealing their universal properties and canonical group actions, with implications for Kac-Moody symmetric spaces.

Contribution

It extends known results to arbitrary embedded symmetric spaces, demonstrating their universal property and canonical group actions without differential structure assumptions.

Findings

Embedded hyperbolic planes induce canonical subgroup actions.

Riemannian symmetric spaces satisfy a universal property.

Results apply to Kac-Moody symmetric spaces G/K.

Abstract

It is known that a geodesic Y in an abstract reflection space X in the sense of Loos, without any assumption of differential structure, canonically admits an action of a 1-parameter subgroup of the group of transvections of X. In this article, we prove an analog of this result stating that, if X contains an embedded hyperbolic plane H, then this yields a canonical action of a subgroup of the transvection group of X isomorphic to a perfect central extension of PSL(2,R). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact type embedded in X and can be used to prove that a Riemannian symmetric space and, more generally, the Kac-Moody symmetric space G/K for an algebraically simply connected two-spherical Kac-Moody group G satisfies a universal property similar to the universal property that the group G satisfies itself.