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

Abstract

Let A be a symmetrizable hyperbolic generalized Cartan matrix with Kac-Moody algebra 𝖌 = 𝖌(A) and (adjoint) Kac-Moody group 𝐺 = 𝐺(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 𝐺 and let (𝓑⁺, 𝓑⁻) be the corresponding twin building with Weyl group W and natural 𝐺-action, which respects the usual W-valued distance and codistance functions. This work connects the twin building (𝓑⁺, 𝓑⁻) of 𝐺 and the Kac-Moody algebra 𝖌 = 𝖌(A) in a new geometrical way. The Cartan-Chevalley involution, ω, of 𝖌 has a fixed point real subalgebra, 𝔨, the 'compact' (unitary) real form of 𝖌, and 𝔨 contains the compact Cartan t = 𝔨 ∩ h. We show that a real bilinear form (⋅,⋅) is Lorentzian with signatures (1,∞) on 𝔨, and (1, n−1) on t. We define {k ∈ 𝔨 | (k,k) ≤0 } to be the lightcone of 𝔨, and similarly for t. Let K be the compact (unitary) real form of 𝐺, that is, the fixed point subgroup of the lifting of ω to 𝐺. We construct a K-equivariant embedding of the twin building of 𝐺 into the lightcone of the compact real form 𝔨 of 𝖌. 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.