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(t epsilon(i))), exp(ad(tf(i))) t is an element of C) where e(i) and f(i) 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, omega, of g has fixed point real subalgebra, k, the 'compact' (unitary) real form of g, and t contains the compact Cartan k = k boolean AND h. We show that a real bilinear form (., .) is Lorentzian with signatures (1, infinity) on k, and (1, n - 1) on f. We define {k is an element of 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 omega 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)congruent to 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.