Lowest degree invariant second-order PDEs over rational homogeneous contact manifolds

Abstract

For each simple Lie algebra [Formula: see text] (excluding, for trivial reasons, type [Formula: see text]), we find the lowest possible degree of an invariant second-order PDE over the adjoint variety in [Formula: see text], a homogeneous contact manifold. Here a PDE [Formula: see text] has degree [Formula: see text] if [Formula: see text] is a polynomial of degree [Formula: see text] in the minors of [Formula: see text], with coefficient functions of the contact coordinate [Formula: see text], [Formula: see text], [Formula: see text] (e.g., Monge–Ampère equations have degree 1). For [Formula: see text] of type [Formula: see text] or [Formula: see text], we show that this gives all invariant second-order PDEs. For [Formula: see text] of types [Formula: see text] and [Formula: see text], we provide an explicit formula for the lowest-degree invariant second-order PDEs. For [Formula: see text] of types [Formula: see text] and [Formula: see text], we prove uniqueness of the lowest-degree invariant second-order PDE; we also conjecture that uniqueness holds in type [Formula: see text].