arXiv subjects

Clément Cren

Publications and source records attributed to Clément Cren.

8 recordsLinked to original sources

arXiv2026

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.

math.AP↗

arXiv2026

Families of Toeplitz operators, family index and deformation quantization

Given a contact fibration, we construct smooth families of formal Szegö projections on the fibers. This allows us to define smooth families of Toeplitz operators. We apply these operators to construct a deformation quantization of prequantizable symplectic fibrations, recovering a result of Kravchenko in an analytic way. This also leads to a comparison between various forms of prequantizations and their geometric implications. We also derive a family index for these families of Toeplitz operators. To this end, we generalize an index formula of Baum and van Erp to families.

math.DG↗

arXiv2026

Fields of Toeplitz algebras form the principal symbol of regular 2-step nilpotent groups

We show that the C*-algebra of a regular 2-step nilpotent lie group can be recovered using continuous fields of Toeplitz algebras and a crossed product. We generalize this result to polycontact manifolds in the sense of van Erp which are endowed with fields of such groups. We also investigate those manifolds with a more rigid structure, namely those modeled on H-type groups. In all those cases, there is a certain pseudodifferential calculus named filtered calculus, we show that the algebra of principal symbols can also be recovered from the field of Toeplitz algebras.

math.OA↗

arXiv2026

Symplectic algebroids, groupoid Toeplitz operators and deformation quantization

We use Toeplitz operators to define a star-product on Poisson manifolds whose Poisson structure is induced by a symplectic Lie algebroid. The Toeplitz operators we consider are defined on groupoids whose algebroid can be endowed with a Heisenberg group structure on the fibers. This generalizes an approach due to Guillemin and Melrose in the symplectic case.

math.SG↗

arXiv2025

Filtered Calculus and crossed products by R-actions

We show an isomorphism between the kernel of the C*-algebra of the tangent groupoid of a filtered manifold and the crossed product of the order 0 pseudodifferential operators in the associated filtered calculus by a natural R-action. This isomorphism is constructed in the same way as in the classical pseudodifferential calculus by Debord and Skandalis. The proof however relies on a structure result for the C*-algebra of graded nilpotent Lie groups which did not appear in the commutative case. A consequence of this structure result is a decomposition of the principal symbol algebra, generalizing the decomposition of Epstein and Melrose in the case of contact manifolds.

math.OA↗

arXiv2025

Transverse parabolic structures and transverse BGG sequences

Manifolds endowed with a parabolic geometry in the sense of Cartan come with natural sequences of differential operators and their analysis provide the so called (curved) BGG sequence of {\v C}ap, Slov{á}k and Sou{\v c}ek. The sequences involved do not form an elliptic complex in the sense of Atiyah but enjoy similar properties. The proper framework to study these operators is the filtered calculus associated to the natural filtration of the tangent bundle induced by the parabolic geometry. Such analysis was carried over by Dave and Haller in a very general setting. In this article we use their methods associated with the transversal index theory for filtered manifolds developped by the author in a previous paper to derive curved BGG sequences for foliated manifolds with transverse parabolic geometry.

math.AP↗

arXiv2025

A transverse index theorem in the calculus of filtered manifolds

We use filtrations of the tangent bundle of a manifold starting with an integrable subbundle to define transverse symbols to the corresponding foliation, define a condition of transversally Rockland and prove that transversally Rockland operators yield a K-homology class. We construct an equivariant KK-class for transversally Rockland transverse symbols and show a Poincare duality type result linking the class of an operator and its symbol.

math.DG↗

arXiv2024

Toeplitz algebras and the Heisenberg group

We show an isomorphism between an algebra which is naturally constructed from the Toeplitz algebra generated by d-shifts, and an ideal of the C * -algebra of the (2d + 1)-dimensional Heisenberg group. This is a particular case of a more general result for graded nilpotent Lie groups involving symbols in the filtered calculus. The proof presented here however only involves basic functional analysis while still showcasing the ideas of the proof in the general setting.

math.DG↗