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 Belavin--Drinfeld Poisson
structures. They also appear in orbit decompositions of the De Concini--Procesi 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 Yang--Baxter 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

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics