SIGMA 22 (2026), 091, 24 pages      arXiv:2512.11434      https://doi.org/10.3842/SIGMA.2026.091

Stratification of the Helffer-Nourrigat Cone

Clément Cren
Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstraße 3-5, D-37073 Göttingen, Germany

Received April 02, 2026, in final form September 09, 2026; Published online September 22, 2026

Abstract
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.

Key words: singular filtrations on manifolds; Helffer-Nourrigat cone; pseudodifferential operators; principal symbol.

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