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ON A CLASS OF DOUBLE COSETS IN REDUCTIVE ALGEBRAIC JIANG-HUA LU AND MILEN YAKIMOV
 

Summary: ON A CLASS OF DOUBLE COSETS IN REDUCTIVE ALGEBRAIC
GROUPS
JIANG-HUA LU AND MILEN YAKIMOV
Abstract. 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 sub-
groups 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 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 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 Yang­Baxter equation on unipotent algebraic groups.
1. The setup
1.1. The setup. Let G1 and G2 be two connected reductive algebraic groups over an alge-
braically closed base field k. For i = 1, 2, we will fix a maximal torus Hi in Gi and a choice
+
i of positive roots in the set i of all roots for Gi with respect to Hi. For each i,
we will use U
i to denote the one-parameter unipotent subgroup of Gi determined by . Let

  

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

 

Collections: Mathematics