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

Let $A$ be a symmetrizable hyperbolic generalized Cartan matrix with Kac–Moody algebra $\mathfrak g=\mathfrak g(A)$ and (adjoint) Kac–Moody group $G = G(A)=\langle \exp({\rm ad}(t e_i)), \exp({\rm ad}(t f_i)) \,|\, t\in\mathbb{C} \rangle $ where $e_i$ and $f_i$ are the simple root vectors. Let $\big(B^+, B^-, N\big)$ be the twin $BN$-pair naturally associated to $G$ and let $\big(\mathcal B^+,\mathcal B^-\big)$ 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 $\big(\mathcal B^+,\mathcal B^-\big)$ of $G$ and the Kac–Moody algebra $\mathfrak g=\mathfrak g(A)$ in a new geometrical way. The Cartan–Chevalley involution, $\omega$, of $\mathfrak g$ has fixed point real subalgebra, $\mathfrak k$, the ‘compact’ (unitary) real form of $\mathfrak g$, and $\mathfrak{f}$ contains the compact Cartan $\mathfrak t = \mathfrak k \cap \mathfrak h$. We show that a real bilinear form $(\cdot,\cdot)$ is Lorentzian with signatures $(1, \infty)$ on $\mathfrak k$, and $(1, n -1)$ on $\mathfrak t$. We define $\{k \in \mathfrak{f} \,|\, (k, k) \leq 0\}$ to be the lightcone of $\mathfrak k$, and similarly for $\mathfrak t$. Let $K$ be the compact (unitary) real form of $G$, that is, the fixed point subgroup of the lifting of $\omega$ to $G$. We construct a $K$-equivariant embedding of the twin building of $G$ into the lightcone of the compact real form $\mathfrak k$ of $\mathfrak g$. 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 ${\rm SU}(2)$-orbit of chambers stabilized by ${\rm U}(1)$ which is thus parametrized by a Riemann sphere ${\rm SU}(2)/{\rm U}(1)\cong S^2$. 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.