The symplectic reduction of pure spherically symmetric (Schwarzschild) classical gravity in D space-time dimensions yields a two-dimensional phase space of observables consisting of the mass M (>0) and a canonically conjugate (Killing) time variable T. Imposing (mass-dependent) periodic boundary conditions in time on the associated quantum-mechanical plane waves which represent the Schwarzschild system in the period just before or during the formation of a black hole yields an energy spectrum of the hole which realizes the old Bekenstein postulate that the quanta of the horizon A(D-2) are multiples of a basic area quantum. In the present paper it is shown that the phase space of such Schwarzschild black holes in D space-time dimensions is symplectomorphic to a symplectic manifold S = {(phi is an element of R mod 2 pi, p proportional to A(D-2) is an element of R+)} With the symplectic form d phi boolean AND dp. As the action of the group SOup arrow(1,2) on that manifold is transitive, effective and Hamiltonian, it can be used for a group theoretical quantization of the system. The area operator (p) over cap for the horizon corresponds to the generator of the compact subgroup SO(2) and becomes quantized accordingly: The positive discrete series of the irreducible unitary representations of the group SOup arrow(1,2) yields an (horizon) area spectrum proportional to(k+n), where k=1,2,..., characterizes the representation and n=0,1,1,..., the number of area quanta. If one employs the unitary representations of the universal covering group of SOup arrow(1,2), the number k can take any fixed positive real value (theta parameter). The unitary representations of the positive discrete series provide concrete Hilbert spaces for quantum Schwarzschild black holes.
