On a Class of Double Cosets in Reductive Algebraic Groups

We study a class of double coset spaces RA\G1 × G2/RC , where G1 and G2 are connected reductive algebraic groups, and RA and RC are certain spherical subgroups of G1×G2 obtained by “identifying” Levi factors of parabolic subgroups in G1 and G2. 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 RA × RC . We further show that all such double cosets give rise to set-theoretical solutions to the quantum Yang–Baxter equation on unipotent algebraic groups.