Simultaneous linear symplectic reduction and orbit fibrations

TL;DR

This paper establishes a correspondence between symplectic group orbits and Grassmannians, enabling the computation of homotopy types of symplectic subspace Grassmannians and connecting various classical results.

Contribution

It introduces a novel correspondence linking symplectic group orbits with Grassmannians, facilitating homotopy type calculations and unifying several classical observations.

Findings

Computed homotopy types of symplectic Grassmannians

Unified classical results by Arnold, Oh-Park, and Lee-Leung

Identified symmetries in symplectic and Jacobi form structures

Abstract

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.