Helffer–Nourrigat conjecture for bracket-generating Lie groupoids

Establish the Helffer–Nourrigat conjecture for a Lie groupoid G with sections X_1,\dots,X_r of its Lie algebroid satisfying a Hörmander-type Lie bracket-generating condition, in particular by proving that differential operators of the form \(\sum_{i=1}^r X_i^2\) have an invertible symbol in the associated pseudodifferential calculus.

Background

The paper considers a Lie groupoid $G\rightrightarrows \units G$, its Lie algebroid, and a finite family of sections satisfying a Hörmander-type bracket-generating condition. It constructs the geometric, weighted deformation needed for a corresponding pseudodifferential calculus, extending constructions known for filtered manifolds and pair groupoids.

The stated objective is to use this calculus to address the Helffer–Nourrigat conjecture, specifically the invertibility of the symbol of sums of squares such as $\sum_i X_i^2$. The paper develops the deformation geometry but does not establish the associated analytic symbol result, leaving this conjectural objective unresolved in the presented work.