Double Bruhat cells and symplectic groupoids

TL;DR

This paper demonstrates that double Bruhat cells in complex semisimple Lie groups, equipped with a standard Poisson structure, naturally form Poisson groupoids over Bruhat cells, revealing new symplectic and Poisson action structures.

Contribution

It establishes the Poisson groupoid structure of double Bruhat cells and describes their symplectic leaves as symplectic groupoids, with compatible Poisson actions.

Findings

Double Bruhat cells form Poisson groupoids over Bruhat cells.

Symplectic leaves are symplectic groupoids.

There are commuting Poisson actions on double Bruhat cells.

Abstract

Let $G$ be a connected complex semisimple Lie group, equipped with a standard multiplicative Poisson structure $\pi_{{\rm st}}$ 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 cell $G^{v,v} = BvB \cap B_-vB_-$ in $G$, together with the Poisson structure $\pi_{{\rm st}}$, is naturally a Poisson groupoid over the Bruhat cell $BvB/B$ in the flag variety $G/B$. Correspondingly, every symplectic leaf of $\pi_{{\rm st}}$ in $G^{v,v}$ is a symplectic groupoid over $BvB/B$. For $u, v \in W$, we show that the double Bruhat cell $(G^{u,v}, \pi_{{\rm st}})$ has a naturally defined left Poisson action by the Poisson groupoid $(G^{u, u},\pi_{{\rm st}})$ and a right Poisson action by the Poisson groupoid $(G^{v,v}, \pi_{{\rm st}})$, and the two actions commute. Restricting to symplectic leaves of $\pi_{{\rm st}}$, 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.