Abstract: One of the main notion introduced in the study of finite dimensional compact Lie groups is the so-called maximal torus. In 1997, Bao and Ratiu discovered an infinite dimensional subgroup in the group of the volume-preserving diffeomorphisms of the 2-dimensional annulus that can potentially play the role of a maximal torus. They showed this subgroup is a path-connected submanifold which is flat and totally geodesic with respect to the hydrodynamic metric. Moreover it is a maximal abelian subgroup (with a finite Weyl group). This suggested that part of finite dimensional Lie group theory may be extended to the volume-preserving diffeomorphisms of the annulus. Indeed, in a later work, Bloch, Flaschka and Ratiu showed that after an appropriate completion of the spaces considered, a version of Schur-Horn-Kostant convexity theorem holds. El-Hadrami extended these results to the case of the unit sphere and CP^{2}, found a candidate for the maximal torus in the symplectomorphism group of symplectic toric manifolds, and then conjectured that some results in previous works can be extended to those groups. However, a gap in El-Hadrami’s arguments was later discovered.

In two talks we discuss some possible extensions and corrections to El-Hadrami´s work. We also mention the Schur-Horn-Kostant convexity theorem for the symplectomorphism groups of toric manifolds.