DOUBLE BRUHAT CELLS AND SYMPLECTIC GROUPOIDS

Abstract

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