DOUBLE BRUHAT CELLS AND SYMPLECTIC GROUPOIDS

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 υ in the Weyl group W of G, the double Bruhat cell Gυ,υ = BυB Ω B_υB_ in G, together with the Poisson structure πst, is naturally a Poisson groupoid over the Bruhat cell BυB/B in the flag variety G/B. Correspondingly, every symplectic leaf of πst in Gυ,υ is a symplectic groupoid over BυB/B. For u, υ ϵ W, we show that the double Bruhat cell (Gu,υ, πst) has a naturally defined left Poisson action by the Poisson groupoid (Gu,υ, πst) and a right Poisson action by the Poisson groupoid (Gu,υ, πst), and the two actions commute. Restricting to symplectic leaves of πst, one obtains commuting left and right Poisson actions on symplectic leaves in Gu,υ by symplectic leaves in Gu,u and Gυ,υ as symplectic groupoids.