SO(2,1)-invariant quantization of the Liouville theory

The recently proposed SO(2,1)-invariant quantization of the Liouville theory is elaborated. We develop a renormalized perturbation expansion which preserves this symmetry to all orders, but spontaneously breaks Poincar\'e invariance. Some Green's functions and scattering amplitudes are calculated in low perturbative order, and it is established that the S matrix is trivial in the tree approximation. Whether this is also true of the complete S matrix remains an open question.