  We study the quantum fermionic billiard defined by the dynamics of a
quantized supersymmetric squashed three-sphere (Bianchi IX cosmological model
within D=4 simple supergravity). The quantization of the homogeneous gravitino
field leads to a 64-dimensional fermionic Hilbert space. We focus on the 15-
and 20-dimensional subspaces (with fermion numbers N_F=2 and N_F=3) where there
exist propagating solutions of the supersymmetry constraints that carry (in the
small-wavelength limit) a chaotic spinorial dynamics generalizing the
Belinskii-Khalatnikov-Lifshitz classical "oscillatory" dynamics. By exactly
solving the supersymmetry constraints near each one of the three dominant
potential walls underlying the latter chaotic billiard dynamics, we compute the
three operators that describe the corresponding three potential-wall
reflections of the spinorial state describing, in supergravity, the quantum
evolution of the universe. It is remarkably found that the latter, purely
dynamically-defined, reflection operators satisfy generalized Coxeter relations
which define a type of spinorial extension of the Weyl group of the rank-3
hyperbolic Kac-Moody algebra AE_3.
