  This paper studies the geometry of Cartan-Hartogs domains from the symplectic
point of view. Inspired by duality between compact and noncompact Hermitian
symmetric spaces, we construct a dual counterpart of Cartan-Hartogs domains and
give explicit expression of global Darboux coordinates for both Cartan-Hartogs
and their dual. Further, we compute their symplectic capacity and show that a
Cartan-Hartogs admits a symplectic duality if and only if it reduces to be a
complex hyperbolic space.
