Stratification of the Helffer-Nourrigat cone
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909957762318336 |
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| author | Cren, Clément |
| author_facet | Cren, Clément |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_11434 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stratification of the Helffer-Nourrigat cone Cren, Clément Analysis of PDEs Functional Analysis Operator Algebras Representation Theory 35H10 (Primary) 22E25, 22E27, 53C17, 47G30 (Secondary) 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. |
| title | Stratification of the Helffer-Nourrigat cone |
| topic | Analysis of PDEs Functional Analysis Operator Algebras Representation Theory 35H10 (Primary) 22E25, 22E27, 53C17, 47G30 (Secondary) |
| url | https://arxiv.org/abs/2512.11434 |