Bott-Samelson atlases, total positivity, and Poisson structures on some homogeneous spaces

Abstract

Let $G$ be a connected and simply connected complex semisimple Lie group. For a collection of homogeneous $G$-spaces $G/Q$, we construct a finite atlas ${\\mathcal{A}}_{\\rm BS}(G/Q)$ on $G/Q$, called the Bott-Samelson atlas, and we prove that all of its coordinate functions are positive with respect to the Lusztig positive structure on $G/Q$. We also show that the standard Poisson structure $\\pi_{G/Q}$ on $G/Q$ is presented, in each of the coordinate charts of ${\\mathcal{A}}_{\\rm BS}(G/Q)$, as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl-Yakimov, making $(G/Q, \\pi_{G/Q}, {\\mathcal{A}}_{\\rm BS}(G/Q))$ into a Poisson-Ore variety. Examples of $G/Q$ include $G$ itself, $G/T$, $G/B$, and $G/N$, where $T \\subset G$ is a maximal torus, $B \\subset G$ a Borel subgroup, and $N$ the uniradical of $B$.