Filtered Calculus and crossed products by R-actions
Clément Cren
TL;DR
This work extends the Debord–Skandalis tangent-groupoid framework to filtered manifolds by develops a symbol calculus based on graded nilpotent groups, the osculating group bundle, and Pedersen stratification. It constructs an $\,\mathbb{R}$-action on the symbol algebra and demonstrates an isomorphism $\Psi^*_H(M)\rtimes \mathbb{R} \cong C^*_0(\mathbb{T}^+_H M)$, linking order-0 filtered pseudodifferential operators to the crossed product of principal symbols. Key technical advances include a generalized stratification for families of graded groups, a global Taylor-based lifting of symbols, and a Schwartz-algebra framework on the tangent groupoid. The results yield KK-equivalences and Morita equivalences between filtered and classical calculi, showing that the filtered approach does not produce new K-theoretic invariants beyond those from the classical calculus, and outlining the index-theoretic implications via Connes–Thom theory.
Abstract
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.