Symplectic geometry of Cartan–Hartogs domains

Abstract

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 domains and their dual. Further, we compute their symplectic capacity and show that a Cartan–Hartogs domain admits a symplectic duality if and only if it reduces to be a complex hyperbolic space.