Lowest degree invariant 2nd order PDEs over rational homogeneous contact manifolds

TL;DR

This paper determines the minimal degree of invariant second-order PDEs over adjoint varieties in rational homogeneous contact manifolds for various Lie algebra types, revealing uniqueness and explicit formulas in many cases.

Contribution

It identifies the lowest degree invariant second-order PDEs for all simple Lie algebra types (except C), providing explicit formulas and proving uniqueness in several cases.

Findings

For types A and G, all invariant second-order PDEs are characterized.

Explicit formulas for lowest-degree invariant PDEs are given for types B and D.

Uniqueness of the lowest-degree invariant PDEs is proved for types E and F, with a conjecture for D.

Abstract

For each simple Lie algebra $\mathfrak{g}$ (excluding, for trivial reasons, type ${\sf C}$) we find the lowest possible degree of an invariant second-order PDE over the adjoint variety in $\mathbb{P}\mathfrak{g}$, a homogeneous contact manifold. Here a PDE $F(x^i,u,u_i,u_{ij})=0$ has degree $\le d$ if $F$ is a polynomial of degree $\le d$ in the minors of $(u_{ij})$, with coefficients functions of the contact coordinates $x^i$, $u$, $u_i$ (e.g., Monge-Amp\`ere equations have degree 1). For $\mathfrak{g}$ of type ${\sf A}$ or ${\sf G}$ we show that this gives all invariant second-order PDEs. For $\mathfrak{g}$ of type ${\sf B}$ and ${\sf D}$ we provide an explicit formula for the lowest-degree invariant second-order PDEs. For $\mathfrak{g}$ of type ${\sf E}$ and ${\sf F}$ we prove uniqueness of the lowest-degree invariant second-order PDE; we also conjecture that uniqueness holds in type ${\sf D}$.