Bott-Samelson varieties and Poisson Ore extensions

Let $G$ be a connected complex semi-simple Lie group, and let $Z_{\mathbf{u}}$ be an $n$-dimensional Bott-Samelson variety of $G$, where $\mathbf{u}$ is any sequence of simple reflections in the Weyl group of $G$. We study the Poisson structure ${\pi }_n$ on $Z_{\mathbf{u}}$ defined by a standard multiplicative Poisson structure ${\pi }_{\mathrm{s}\mathrm{t}}$ on $G$. We explicitly express ${\pi }_n$ on each of the $2^n$ affine coordinate charts, one for every subexpression of $\mathbf{u}$, 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}$}[z_1, \ldots, z_n]$whichisaniteratedPoissonOreextensionof$$\mathbb{C}$$compatiblewitharationalactionbyamaximaltorusof$G$. For canonically chosen ${\pi }_{\mathrm{s}\mathrm{t}}$, we show that the induced Poisson structure on $\mathbb{C}[z_1,\dots ,z_n]$ for every affine coordinate chart is in fact defined over $\mathbb{Z}$, thus giving rise to an iterated Poisson Ore extension of any field $\mathbf{k}$ 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.