Extrinsically Immersed Symplectic Symmetric Spaces

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Let (V,) be a symplectic vector space and let ϕ: M V be a symplectic immersion. We show that ϕ(M) ⊂ V is (locally) an extrinsic symplectic symmetric space (e.s.s.s.) in the sense of CGRS if and only if the second fundamental form of ϕ is parallel. Furthermore, we show that any symmetric space which admits an immersion as an e.s.s.s. also admits a full such immersion, i.e., such that ϕ(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.