Higher symmetries of symplectic Dirac operator

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 $\mathfrak{sl}(3,\mathbb{R})$.

Symplectic Dirac operator, Higher symmetry algebra, Projective differential geometry, Minimal nilpotent orbit, $\mathfrak{sl}(3,\mR)$