Let g be a finite-dimensional Lie algebra. The dual g^* of g has a well-known (linear) Poisson structure and a symplectic foliation into coadjoint orbits. In the case of g=gl(n,C), Kostant and Wallach have constructed a completely integrable system on each regular coadjoint orbit. This is a geometric analogue of the classical Gelfand-Tsetlin bases for irreducible representations.

In the context of the (infinite dimensional) direct limit algebra gl(infinity), the situation is somewhat more delicate. Nonetheless, there is still a beautiful Lie-Poisson structure, symplectic foliation, and a polarisation given by Gelfand-Tsetlin systems. I will discuss some of this geometry, based on joint work with Mark Colarusso.