A symplectic approach to van den Ban's convexity theorem

Let G be a complex semisimple Lie group and \tau a complex antilinear involution that commutes with the Cartan involution. If H denotes the connected subgroup of \tau-fixed points in G, and K is maximally compact, each H-orbit in G/K can be equipped with a Poisson structure as described by Evens and Lu. We consider symplectic leaves of certain such H-orbits with a natural hamiltonian torus action. A symplectic convexity theorem of Hilgert-Neeb-Plank then leads to van den Ban's convexity theorem for (complex) semisimple symmetric spaces.