Configuration Poisson Groupoids of Flags

Abstract Let $G$ be a connected complex semi-simple Lie group and ${\mathcal {B}}$ its flag variety. For every positive integer $n$, we introduce Poisson groupoid over ${{\mathcal {B}}}^n$, called the $n$th total configuration of flags $G$, which contains family sub-groupoids whose spaces are generalized double Bruhat cells bases Schubert in {B}}^n$. Certain symplectic leaves these then shown to groupoids cells. We also give explicit descriptions three series varieties associated $G$.