Geometric Invariance of the Semi-classical Calculus on Nilpotent Graded Lie Groups

Abstract

In this paper, we consider the semi-classical setting constructed on nilpotent graded Lie groups by means of representation theory. Our aim is to 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 uniformly 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. Finally, we interpret the geometric meaning of this invariance in the setting of filtered manifolds.