Configuration Poisson Groupoids of Flags

Abstract

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