Higher symmetries of symplectic Dirac operator

TL;DR

This paper constructs the higher symmetry algebra of the symplectic Dirac operator in 2D projective geometry, revealing its structure as a primitive ideal related to the minimal nilpotent orbit of sl(3,R).

Contribution

It introduces a novel construction of higher symmetry operators for the symplectic Dirac operator and characterizes their algebraic structure in the context of projective differential geometry.

Findings

Higher symmetry algebra corresponds to a primitive ideal with minimal nilpotent orbit.

Constructs projectively invariant higher order symmetry operators.

Links symmetry algebra structure to representation theory of sl(3,R).

Abstract

We construct in projective differential geometry of the real dimension $2$ higher symmetry algebra of the symplectic Dirac operator ${D}\kern-0.5em\raise0.22ex\hbox{/}_s$ acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overdetermined operators of arbitrarily high order acting on symmetric tensors. The higher symmetry algebra structure corresponds to a completely prime primitive ideal having as its associated variety the minimal nilpotent orbit of $\mathfrak{sl}(3,{\mathbb{R}})$.