Stratification of the Helffer-Nourrigat Cone

Given a singular filtration on a manifold, e.g., a sub-Riemannian setting, one can understand the regularity problems through the Androulidakis-Mohsen-Yuncken pseudodifferential 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 sub-Riemannian 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 Pukánszky 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.