Rational orbits in some prehomogeneous vector spaces associated to $Sp_{6}$ revisited

TL;DR

This paper classifies rational orbits of a prehomogeneous vector space related to $Sp_{6}$, linking them to composition and Freudenthal algebras over fields with characteristic not 2, providing new arithmetic insights.

Contribution

It establishes a correspondence between $Sp_{6}$-orbits in a specific vector space and maximal flags of composition algebras, offering an arithmetic interpretation of orbit spaces in certain prehomogeneous vector spaces.

Findings

Maximal flags of composition algebras correspond to $k$-rational $Sp_{6}$-orbits.

Orbit spaces encode all reduced Freudenthal algebras of dimensions 6 and 9.

Provides an arithmetic interpretation for semi-stable orbit spaces.

Abstract

Let $k$ be a field with $\text{char}(k)\neq 2$. We prove that all maximal flags of composition algebras over $k$, appear as the $k$-rational $Sp_{6}$-orbits in a Zariski-dense $Sp_{6}$-invariant subset $V^{ss}\subset V=\wedge^{3}V_{6}$, where $V_{6}$ is the standard $6$-dimensional irreducible representation of $Sp_{6}$. This gives an arithmetic interpretation for the orbit spaces of the semi-stable sets in the prehomogeneous vector spaces $(Sp_{6}\times GL_{1}^{2},V)$ and $(GSp_{6}\times GL_{1}^{2},V)$. We also get all reduced Freudenthal algebras of dimensions $6$ and $9$, represented by the same orbit spaces.