A symplectic approach to van den Ban's convexity theorem

Abstract

Let $G$ be a complex semisimple Lie group and $\tau$ a complex antilinear involution that commutes with a 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 then leads to van den Ban's convexity result for (complex) semisimple symmetric spaces.