Configuration Poisson groupoids of flags

Abstract

Let G be a connected complex semi-simple Lie group and B its flag variety. For every positive integer n, we introduce a Poisson groupoid over B n , 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 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. Contents 1. Introduction and statements of results 1 2. A series of Poisson groupoids associated to Poisson Lie groups 9 3. The total configuration Poisson groupoids of flags and isomorphic models 13 4. Special configuration Poisson groupoids of flags and isomorphic models 17 5. Configuration symplectic groupoids of flags 22 Appendix A. Mixed product Poisson structures 25 Appendix B. The Poisson isomorphism J n 29 Appendix C. Generalized double Bruhat cells 32 Appendix D. Symplectic leaves of (O w e × T, π n ⊲⊳ 0) 40 References 55