Simultaneous linear symplectic reduction and orbit fibrations

We develop a correspondence between the orbits of the group of linear symplectomorphisms of a real finite dimensional symplectic vector space in the complex Lagrangian Grassmannian and the Grassmannians of linear subspaces of the real symplectic vector space. Under this correspondence, orbit fibration maps whose fibers are holomorphic arc components correspond to fibrations from simultaneous linear symplectic reduction. We use this to compute the homotopy types of Grassmannians of linear subspaces of the symplectic vector space in the general case, recovering the observations of Arnold in the Lagrangian case, Oh-Park in the coisotropic case, and Lee-Leung in the symplectic case. Binary octahedral symmetries, symplectic twistor Grassmannians, and symmetries of Jacobi forms appear within this structure.