  We study a class of double coset spaces R_A \backslash G_1 \times G_2 /R_C,
where G_1 and G_2 are connected reductive algebraic groups, and R_A and R_C are
certain spherical subgroups of G_1 \times G_2 obtained by ``identifying'' Levi
factors of parabolic subgroups in G_1 and G_2. Such double cosets naturally
appear in the symplectic leaf decompositions of Poisson homogeneous spaces of
complex reductive groups with the Belavin-Drinfeld Poisson structures. They
also appear in orbit decompositions of the De Concini-Procesi compactifications
of semi-simple groups of adjoint type. We find explicit parametrizations of the
double coset spaces and describe the double cosets as homogeneous spaces of R_A
\times R_C. We further show that all such double cosets give rise to
set-theoretical solutions to the quantum Yang-Baxter equation on unipotent
algebraic groups.

Representation Theory;Combinatorics