Stratification of the Helffer-Nourrigat cone

TL;DR

This paper introduces a method to desingularize the Helffer-Nourrigat cone, a key phase space in subriemannian geometry, by stratifying its structure to facilitate analysis of elliptic regularity problems.

Contribution

It extends the stratification of the unitary spectrum of nilpotent groups to the Helffer-Nourrigat cone, enabling better understanding of its topology and associated C*-algebras.

Findings

The Helffer-Nourrigat cone can be stratified into locally compact Hausdorff strata.

The C*-algebra of principal symbols is solvable with explicit subquotients.

The approach aids in analyzing elliptic regularity in subriemannian geometry.

Abstract

Given a singular filtration on a manifold, e.g. a subriemannian setting, one can understand the elliptic regularity problems through a special kind of calculus. The principal symbol in this calculus involves the unitary representations of a family of graded nilpotent groups. Not all the irreducible representations of these groups have to be taken into account however, the ones that should be considered form the Helffer-Nourrigat cone. This space thus plays the role of a phase space in subriemannian geometry. Its topology is however very singular, preventing any kind of geometry on it. We propose a way to desingularize it. The unitary spectrum of a nilpotent group can be stratified into strata that are locally compact Hausdorff, following Puckansky and Pedersen. We show how this stratification extends to the whole Helffer-Nourrigat cone. As a byproduct, we show that the C*-algebra of principal symbols and the one of pseudodifferential operators of order 0 are solvable with explicit subquotients.