On longitudinal differential operators and Nash blowups

In this short note, we describe the Helffer-Nourrigat cone of a singular foliation in terms of the Nash algebroid associated to the foliation. Along the way, we show that the Helffer-Nourrigat cone is a union of symplectic leaves of the canonical Poisson structures on the dual of the holonomy Lie algebroids. We also provide, within this framework, a characterization of longitudinally elliptic differential operators on a singular foliation $\mathcal{F}$, generalizing results previously known in the literature.