Simultaneous linear symplectic reduction and orbit fibrations

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.

Figures (3)

• Figure A.1: The Grassmannians $\operatorname{Gr}(\vec{n})$ are tabulated with vertical coordinate $n_+ - n_-$ and horizontal coordinate $n_0$. The rows describe the partition of the ordinary Grassmannians by the Grassmannians $\operatorname{Gr}(\vec{n})$. Grassmannians of, respectively, symplectic, isotropic, coisotropic subspaces are located on, respectively, left side, lower right, and upper right side of the triangle. Only the isotropic and coisotropic Grassmannians are compact. Taking symplectic complements identifies orbits symmetric about the horizontal line $n_+ = n_-$.
• Figure A.2: The $G$-orbits $\operatorname{Lag}^\mathbb{C}(\vec{n})$ are tabulated with vertical position $n_+ - n_-$ and horizontal position $n_0$. The shaded regions indicate $G$-orbits $\operatorname{Lag}^\mathbb{C}(\vec{n})$ in the closures of $\operatorname{Lag}^\mathbb{C}(0, 3, 1)$, $\operatorname{Lag}^\mathbb{C}(1, 1, 2)$ (and $\operatorname{Lag}^\mathbb{C}(2, 1, 1)$). Only $\operatorname{Lag}^\mathbb{C}(4, 0, 0)$ is compact. Complex conjugation identifies the orbits symmetric about the horizontal line $n_+ = n_-$.
• Figure A.3: Example of Corollary \ref{['cor:homotopytype']}. $Gr(\vec{n}; \mathbb{R}^2)$ and $\operatorname{Lag}^\mathbb{C}(\vec{n};\mathbb{R}^2)$ are homotopy equivalent.

Theorems & Definitions (74)

• Corollary : Corollary \ref{['cor:homotopytype']}
• Theorem : Theorem \ref{['thm:vectorbundle']}, Proposition \ref{['prop:compactfibrations']}
• Theorem : Takeuchi, Wolf, Wolf1
• Definition 2.1: Associated splittings
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Proposition 2.4
• proof