Infinitesimal homogeneity for proper symplectic subgroups

Determine which special symplectic geometries, including Ricci-type, Bochner–Kähler, Bochner–bi-Lagrangian, and exceptional special symplectic holonomy geometries, are infinitesimally homogeneous with respect to suitable proper subgroups of the real symplectic group \(\operatorname{Sp}(2n,\mathbb R)\).

Background

The paper proves that an infinitesimally homogeneous structure with full structure group $\operatorname{Sp}(2n,\mathbb R)$ must be flat, torsion-free, and symplectic. This follows because there is no nonzero $\operatorname{Sp}(2n,\mathbb R)$-equivariant curvature tensor with the required constant representation in all symplectic frames.

The authors contrast this rigidity with the theory of special symplectic connections, whose curvature is generally determined by auxiliary tensor fields and therefore is not constant in the full symplectic frame bundle. The unresolved direction is to identify which of these geometries acquire infinitesimal homogeneity after reducing the structure group to an appropriate proper subgroup of $\operatorname{Sp}(2n,\mathbb R)$.