We derive the explicit differential form for the action of the generators of the SU(1,1) group on the corresponding s-parametrized symbols. This allows us to obtain evolution equations for the phase-space functions on the upper sheet of the two-sheet hyperboloid and analyze their semiclassical limits. Dynamics of quantum systems with SU(1,1) symmetry governed by compact and non-compact Hamiltonians are discussed in both quantum and semiclassical regimes.
Keywords: SU(1,1) group; Wigner function; phase space.