The space of linear anti-symplectic involutions is a homogenous space

TL;DR

This paper proves that the set of all linear anti-symplectic involutions forms a homogeneous space, specifically $Gl(n, ) ext{Sp}(n)$, motivated by applications to symmetric periodic orbits in celestial mechanics.

Contribution

It establishes a new geometric characterization of linear anti-symplectic involutions as a homogeneous space, linking symplectic geometry with group actions.

Findings

The space of linear anti-symplectic involutions is isomorphic to $Gl(n, )\Sp(n)$.

Provides a geometric framework for studying symmetric periodic orbits.

Connects symplectic involutions with group theory and homogeneous spaces.

Abstract

In this note we prove that the space of linear anti-symplectic involutions is the homogenous space $Gl(n,\R)\Sp(n)$. This result is motivated by the study of symmetric periodic orbits in the restricted 3-body problem.