Integrable systems on the dual of nilpotent Lie subalgebras and $T$-Poisson cluster structures. Part 2

Let $\mathfrak g$ be a semisimple Lie algebra and $\mathfrak g = \mathfrak n \oplus \mathfrak h \oplus \mathfrak n_-$ a triangular decomposition.  Motivated by a construction of Kostant-Lipsman-Wolf, we construct an integrable system on the dual space of $\mathfrak n_-$ equipped with the Kirillov-Kostant Poisson structure.  The Bott-Samelson coordinates on the open Bruhat cell (equipped with the standard Poisson structure) makes it into a symmetric Poisson CGL extension, hence giving rise to a $T$-Poisson seed on it.  We explain a relation between our integrable system and this $T$-Poisson seed.  This is joint work in progress with Yanpeng Li and Jiang-Hua Lu.