Geometric invariance of the semi-classical calculus on nilpotent graded Lie groups

TL;DR

This paper studies how semi-classical pseudodifferential operators on nilpotent graded Lie groups behave under certain diffeomorphisms, revealing invariance properties linked to Pansu differentiability and geometric structures.

Contribution

It demonstrates that pull-backs by filtration-preserving diffeomorphisms preserve the semi-classical calculus via Pansu differentiability, extending geometric invariance results.

Findings

Pull-back of operators expressed via Pansu differential.

Invariance of semi-classical symbols under specific diffeomorphisms.

Geometric interpretation in filtered manifold setting.

Abstract

In this paper, we consider the semi-classical setting constructed on nilpotent graded Lie groups by means of representation theory. We analyze the effects of the pull-back by diffeomorphisms on pseudodifferential operators. We restrict to diffeomorphisms that preserve the filtration and prove that they are Pansu differentiable. We show that the pull-back of a semi-classical pseudodifferential operator by such a diffeomorphism has a semi-classical symbol that is expressed at leading order in terms of the Pansu differential. We interpret the geometric meaning of this invariance in the setting of filtered manifolds.