  The symplectic reduction of pure spherically symmetric (Schwarzschild)
classical gravity in D space-time dimensions yields a 2-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 a Schwarzschild black hole in D space-time dimensions is
symplectomorphic to a symplectic manifold
  S={(phi in R mod 2 pi, p = A_{D-2} >0)} with the symplectic form d phi wedge
d p. As the action of the group SO_+(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 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 SO_+(1,2) yields
an (horizon) area spectrum proportional k+n, where k =1,2,... characterizes the
representation and n = 0,1,2,... the number of area quanta. If one employs the
unitary representations of the universal covering group of SO_+(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.
