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

## TL;DR

This paper constructs a positive coordinate atlas called the Bott-Samelson atlas on certain homogeneous spaces of complex semisimple Lie groups, demonstrating their compatibility with Poisson structures and total positivity.

## Contribution

It introduces the Bott-Samelson atlas for homogeneous G-spaces, showing all coordinate functions are positive and compatible with the standard Poisson structure, forming a Poisson-Ore variety.

## Key findings

- All coordinate functions are positive with respect to Lusztig positivity.
- The Poisson structure is presented as a symmetric Poisson CGL extension.
- The framework applies to spaces like G, G/T, G/B, and G/N.

## 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 $B$.