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Let |$G$| be a connected complex semi-simple Lie group and |${\mathcal {B}}$| its flag variety. For every positive integer |$n$|⁠ , we introduce a Poisson groupoid over |${{\mathcal {B}}}^n$|⁠ , called the |$n$| th 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 |${\mathcal {B}}^n$|⁠. 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 |$G$|⁠. [ABSTRACT FROM AUTHOR]

Let G be a connected and simply connected complex semisimple Lie group. For a collection of homogeneous G-spaces G/Q, we construct a finite atlas A BS (G / Q) on G/Q, called the Bott–Samelson atlas, and we prove that all of its coordinate functions are positive with respect to the Lusztig positive structure on G/Q. We also show that the standard Poisson structure π G / Q on G/Q is presented, in each of the coordinate charts of A BS (G / Q) , as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl–Yakimov, making (G / Q , π G / Q , A BS (G / Q)) into a Poisson–Ore variety. In addition, all coordinate functions in the Bott–Samelson atlas are shown to have complete Hamiltonian flows with respect to the Poisson structure π G / Q . Examples of G/Q include G itself, G/T, G/B, and G/N, where T ⊂ G is a maximal torus, B ⊂ G a Borel subgroup, and N the uniradical of B. [ABSTRACT FROM AUTHOR]