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.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/3946