Bott-Samelson varieties and Poisson Ore extensions

Abstract

Let $G$ be a connected complex semi-simple Lie group, and let $Z_{{\bf u}}$ be an $n$-dimensional Bott-Samelson variety of $G$, where ${\bf u}$ is any sequence of simple reflections in the Weyl group of $G$. We study the Poisson structure $\pi_n$ on $Z_{\bf u}$ defined by a standard multiplicative Poisson structure $\pi_{\rm st}$ on $G$. We explicitly express $\pi_n$ on each of the $2^n$ affine coordinate charts, one for every subexpression of ${\bf u}$, in terms of the root strings and the structure constants of the Lie algebra of $G$. We show that the restriction of $\pi_n$ to each affine coordinate chart gives rise to a Poisson structure on the polynomial algebra ${\mathbb{C}}[z_1, \ldots, z_n]$ which is an {\it iterated Poisson Ore extension} of $\mathbb{C}$ compatible with a rational action by a maximal torus of $G$. For canonically chosen $\pi_{\rm st}$, we show that the induced Poisson structure on ${\mathbb{C}}[z_1, \ldots, z_n]$ for every affine coordinate chart is in fact defined over ${\mathbb Z}$, thus giving rise to an iterated Poisson Ore extension of any field ${\bf k}$ of arbitrary characteristic. The special case of $\pi_n$ on the affine chart corresponding to the full subexpression of ${\bf u}$ yields an explicit formula for the standard Poisson structures on {\it generalized Bruhat cells} in Bott-Samelson coordinates. The paper establishes the foundation on generalized Bruhat cells and sets up the stage for their applications, some of which are discussed in the Introduction of the paper.