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 $${{\mathcal {A}}}_{{\scriptscriptstyle BS}}(G/Q)$$ 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 $$\pi _{{\scriptscriptstyle G}/{\scriptscriptstyle Q}}$$ on G/Q is presented, in each of the coordinate charts of $${{\mathcal {A}}}_{{\scriptscriptstyle BS}}(G/Q)$$ , as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl–Yakimov, making $$(G/Q, \pi _{{\scriptscriptstyle G}/{\scriptscriptstyle Q}}, {{\mathcal {A}}}_{{\scriptscriptstyle BS}}(G/Q))$$ into a Poisson–Ore variety. In addition, all coordinate functions in the Bott–Samelson atlas are shown to have complete Hamiltonian flows with respect to the Poisson structure $$\pi _{{\scriptscriptstyle G}/{\scriptscriptstyle Q}}$$ . 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.