Kac–Moody symmetric spaces

In the present article we introduce and study a class of topological reflectionspaces that we call Kac–Moody symmetric spaces. These are associated with split realKac–Moody groups and generalize Riemannian symmetric spaces of noncompact split type.Based on work by the third-named author, we observe...

Authors: Freyn, Walter
Hartnick, Tobias
Horn, Max
Köhl, Ralf
Document types:Article
Media types:Text
Publication date:2020
Date of publication on miami:12.03.2020
Modification date:13.03.2020
Source:Münster Journal of Mathematics, 13 (2020), S. 1-114
Publisher: Mathematisches Institut (Universität Münster)
Edition statement:[Electronic ed.]
DDC Subject:510: Mathematik
License:InC 1.0
Language:English
Format:PDF document
URN:urn:nbn:de:hbz:6-32109569931
Permalink:http://nbn-resolving.de/urn:nbn:de:hbz:6-32109569931
Other Identifiers:DOI: 10.17879/32109566476
Digital documents:mjm_2020_13_1-114.pdf

In the present article we introduce and study a class of topological reflectionspaces that we call Kac–Moody symmetric spaces. These are associated with split realKac–Moody groups and generalize Riemannian symmetric spaces of noncompact split type.Based on work by the third-named author, we observe that in a non-spherical Kac–Moody symmetric space there exist pairs of points that do notlie on a common geodesic;however, any two points can be connected by a chain of geodesic segments. We moreoverclassify maximal flats in Kac–Moody symmetric spaces and study their intersection patterns,leading to a classification of global and local automorphisms. Some of our methods apply togeneral topological reflection spaces beyond the Kac–Moodysetting.Unlike Riemannian symmetric spaces, non-spherical non-affine irreducible Kac–Moodysymmetric spaces also admit an invariant causal structure.For causal and anti-causal geo-desic rays with respect to this structure we find a notion of asymptoticity, which allows usto define a future and past boundary of such Kac–Moody symmetric space. We show thatthese boundaries carry a natural polyhedral cell structureand are cellularly isomorphic togeometric realizations of the two halves of the twin buildings of the underlying split real Kac–Moody group. We also show that every automorphism of the symmetric space is uniquelydetermined by the induced cellular automorphism of the future and past boundary.The invariant causal structure on a non-spherical non-affineirreducible Kac–Moody sym-metric space gives rise to an invariant pre-order on the underlying space, and thus toa subsemigroup of the Kac–Moody group.We conclude that while in some aspects Kac–Moody symmetric spaces closely resembleRiemannian symmetric spaces, in other aspects they behave similarly tomasures, their non-Archimedean cousins