DOUBLE BRUHAT CELLS AND SYMPLECTIC GROUPOIDS

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