Filtered deformations of Lie groupoids

@misc{breton2026filtered,
  title = {{Filtered deformations of Lie groupoids}},
  author = {Paul Le Breton},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29331},
  eprint = {2609.29331},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29331},
}

TY  - UNPB
TI  - Filtered deformations of Lie groupoids
AU  - Paul Le Breton
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29331
UR  - https://arxiv.org/abs/2609.29331
AB  - Let G rightrightarrows G^(0) be a Lie groupoid, A G → G^(0) its Lie algebroid and X_1, …, X_r a family of sections of A G satisfying a Lie bracket generating condition of Hörmander type. We aim to build a pseudodifferential calculus allowing to study a Helffer-Nourrigat's conjecture on the groupoid G; in particular, we want differential operators of the form ∑_i = 1^r X_i^2 to have an invertible symbol. In this article we achieve the geometrical part of this construction by defining a "weighted" version of the deformation to the normal cone DNC(G, G^(0)) rightrightarrows G^(0) × ℝ_+. Heuristically, we deform G around G^(0) with a "zoom" parameter t ∈ ℝ_+ by stretching G by t in the directions of the sections X_i, t^2 along [X_i, X_j], t^3 along [X_i, [X_j, X_k]] etc. In the case where G = M × M we recover a construction of Mohsen, and when the structure is equiregular we recover a construction of van Erp-Yuncken.
ER  -