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 ABS(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 πG/Q on G/Q is presented, in each of the coordinate charts of ABS(G/Q), as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl–Yakimov, making (G/Q,πG/Q,ABS(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 πG/Q. Examples of G/Q include G itself, G/T, G/B, and G/N, where T⊂G is a maximal torus, B⊂G a Borel subgroup, and N the uniradical of B.