Geometry of bi-Lagrangian Grassmannian

TL;DR

This paper studies the structure and automorphism orbits of bi-Lagrangian Grassmannians associated with pencils of 2-forms, focusing on cases with Jordan blocks of the same eigenvalue and invariant Lagrangian subspaces.

Contribution

It reduces the analysis to specific Jordan-Kronecker canonical forms and provides a complete description of automorphism orbits in key cases.

Findings

Calculated the dimension of bi-Lagrangian Grassmannians.

Described their open orbit under automorphisms.

Classified automorphism orbits for specific Jordan block configurations.

Abstract

This paper explores the structure of bi-Lagrangian Grassmanians for pencils of $2$-forms on real or complex vector spaces. We reduce the analysis to the pencils whose Jordan-Kronecker Canonical Form consists of Jordan blocks with the same eigenvalue. We demonstrate that this is equivalent to studying Lagrangian subspaces invariant under a nilpotent self-adjoint operator. We calculate the dimension of bi-Lagrangian Grassmanians and describe their open orbit under the automorphism group. We completely describe the automorphism orbits in the following three cases: for one Jordan block, for sums of equal Jordan blocks and for a sum of two distinct Jordan blocks.