A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group

Document Type


Publication Date



Group Theory, Representation Theory


The twin building of a Kac–Moody group G encodes the parabolic subgroup structure of G and admits a natural G–action. When G is a complex Kac–Moody group of hyperbolic type, we construct an embedding of the twin building of G into the lightcone of the compact real form of the corresponding Kac–Moody algebra. When G has rank 2, we construct an embedding of the spherical building at infinity into the set of rays on the boundary of the lightcone.


This material is based upon work supported by the National Science Foundation under Grant No. 1002477. The first author was supported in part by NSF grant number DMS–1101282. All authors wish to thank the IHES´ for support during various visits in May 2013, 2014 and 2015. The second author wishes to thank the Max-Planck Institute for Gravitational Physics (Albert Einstein Institute), Potsdam, Germany, for support during visits in June 2013, 2014 and 2015