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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2020
• Main Authors: Carbone, Lisa, Feingold, Alex J., Freyn, Walter
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2020
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/210705
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group. Lisa Carbone, Alex J. Feingold and Walter Freyn. SIGMA 16 (2020), 045, 47 pages

Let A be a symmetrizable hyperbolic generalized Cartan matrix with Kac-Moody algebra = (A) and (adjoint) Kac-Moody group = (A) = ⟨exp(ad(teᵢ)),exp(ad(tfᵢ)) | t ∈ ℂ⟩ where ei and fi are the simple root vectors. Let (B⁺, B⁻, N) be the twin BN-pair naturally associated to and let (⁺, ⁻) be the corresponding twin building with Weyl group W and natural -action, which respects the usual W-valued distance and codistance functions. This work connects the twin building (⁺, ⁻) of and the Kac-Moody algebra = (A) in a new geometrical way. The Cartan-Chevalley involution, ω, of has a fixed point real subalgebra, , the 'compact' (unitary) real form of , and contains the compact Cartan t = ∩ h. We show that a real bilinear form (⋅,⋅) is Lorentzian with signatures (1,∞) on , and (1, n−1) on t. We define {k ∈ | (k,k) ≤0 } to be the lightcone of , and similarly for t. Let K be the compact (unitary) real form of , that is, the fixed point subgroup of the lifting of ω to . We construct a K-equivariant embedding of the twin building of into the lightcone of the compact real form of . Our embedding gives a geometric model of part of the twin building, where each half consists of infinitely many copies of a W-tessellated hyperbolic space glued together along hyperplanes of the faces. Locally, at each such face, we find an SU(2)-orbit of chambers stabilized by U(1), which is thus parametrized by a Riemann sphere SU(2)/U(1) ≅ S². For n = 2, the twin building is a twin tree. In this case, we construct our embedding explicitly, and we describe the action of the real root groups on the fundamental twin apartment. We also construct a spherical twin building at infinity, and construct an embedding of it into the set of rays on the boundary of the lightcone.