On the solvability of the transvection group of extrinsic symplectic symmetric spaces

Let MM be a symplectic symmetric space, and let ı:M→Vı:M→V be an extrinsic symplectic symmetric immersion in the sense of Krantz and Schwachhöfer (2010) [7], i.e., (V,Ω)(V,Ω) is a symplectic vector space and ıı is an injective symplectic immersion such that for each point p∈Mp∈M, the geodesic symmetry in pp is compatible with the reflection in the affine normal space at ı(p)ı(p).We show that the existence of such an immersion implies that the transvection group of MM is solvable.

► We consider extrinsic symplectic symmetric immersions. ► In all known examples with the ambient space being a vector space, the transvection group is 3-step nilpotent. ► We show that if the ambient space is a vector space, then the transvection group is solvable.