  Let A be a symmetrizable hyperbolic generalized Cartan matrix with Kac-Moody
algebra g = g(A) and (adjoint) Kac-Moody group G = G(A)=$\langle\exp(ad(t
e_i)), \exp(ad(t f_i)) \,|\, t\in C\rangle$ 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 $(\mathcal B^+,\mathcal B^-)$ 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 of G and the Kac-Moody algebra g 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 k contains the compact Cartan t = k
$\cap$ h. We show that a real bilinear form $(\cdot,\cdot)$ is Lorentzian with
signatures $(1, \infty)$ on k, and $(1, n -1)$ on t. We define $\{x\in {\rm k}
\,|\, (x, x) \leq 0\}$ 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 $SU(2)/U(1)\cong S^2$. 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.

Group Theory;Representation Theory