Double Bruhat cells and symplectic groupoids

Abstract

Let G be a connected complex semisimple Lie group, equipped with a standard multiplicative Poisson structure πst determined by a pair of opposite Borel subgroups (B, B−). We prove that for each v in the Weyl group W of G, the double Bruhat celltogether with the Poisson structure πst, is naturally a Poisson groupoid over the Bruhat cell BvB/B in the flag variety G/B. Correspondingly, every symplectic leaf of πst in G v,v is a symplectic groupoid over BvB/B. For u, v ∈ W , we show that the double Bruhat cell (G u,v , πst) has a naturally defined left Poisson action by the Poisson groupoid (G u,u , πst) and a right Poisson action by the Poisson groupoid (G v,v , πst), and the two actions commute. Restricting to symplectic leaves of πst, one obtains commuting left and right Poisson actions on symplectic leaves in G u,v by symplectic leaves in G u,u and in G v,v as symplectic groupoids.