Extrinsically Immersed Symplectic Symmetric Spaces

Let $(V,\text{Om})$ be a symplectic vector space and let $\varphi :M\text{ra}V$ be a symplectic immersion. We show that $\varphi (M)\subset V$ is (locally) an extrinsic symplectic symmetric space (e.s.s.s.) in the sense of \cite{CGRS} if and only if the second fundamental form of $\varphi$ is parallel. Furthermore, we show that any symmetric space which admits an immersion as an e.s.s.s. also admits a {\em full} such immersion, i.e., such that $\varphi (M)$ is not contained in a proper affine subspace of $V$, and this immersion is unique up to affine equivalence. Moreover, we show that any extrinsic symplectic immersion of $M$ factors through to the full one by a symplectic reduction of the ambient space. In particular, this shows that the full immersion is characterized by having an ambient space $V$ of minimal dimension.