 
Summary: ON A CLASS OF DOUBLE COSETS IN REDUCTIVE ALGEBRAIC
GROUPS
JIANGHUA LU AND MILEN YAKIMOV
Abstract. We study a class of double coset spaces RA \G 1 × G 2 /R C , where G 1 and
G2 are connected reductive algebraic groups, and RA and R C are certain spherical sub
groups of G 1 ×G2 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 Pois
son homogeneous spaces of complex reductive groups with the BelavinDrinfeld Poisson
structures. They also appear in orbit decompositions of the De ConciniProcesi compact
ifications of semisimple groups of adjoint type. We find explicit parametrizations of the
double coset spaces and describe the double cosets as homogeneous spaces of RA × R C .
We further show that all such double cosets give rise to settheoretical solutions to the
quantum YangBaxter equation on unipotent algebraic groups.
1. The setup
1.1. The setup. Let G 1 and G 2 be two connected reductive algebraic groups over an alge
braically closed base field k. For i = 1, 2, we will fix a maximal torus H i in G i and a choice
# +
i of positive roots in the set # i of all roots for G i with respect to H i . For each # # # i ,
we will use U #
i to denote the oneparameter unipotent subgroup of G i determined by #. Let
