  We investigate the quantum aspects of three-dimensional gravity with a
positive cosmological constant. The reduced phase space of the
three-dimensional de Sitter gravity is obtained as the space which consists of
the Kerr-de Sitter space-times and their Virasoro deformations. A quantization
of the phase space is carried out by the geometric quantization of the
coadjoint orbits of the asymptotic Virasoro symmetries. The Virasoro algebras
with real central charges are obtained as the quantum asymptotic symmetries.
The states of globally de Sitter and point particle solutions become the
primary states of the unitary irreducible representations of the Virasoro
algebras. It is shown that those states are perturbatively stable at the
quantum level. The Virasoro deformations of these solutions correspond to the
excited states in the unitary irreducible representations. In view of the
dS/CFT correspondence, we also study the relationship between the Liouville
field theory obtained by a reduction of the SL(2;$\mathbb{C}$) Chern-Simons
theory and the three-dimensional gravity both classically and quantum
mechanically. In the analyses of the both theories, the Kerr-de Sitter
geometries with nonzero angular momenta do not give the unitary representations
of the Virasoro algebras.
