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Pseudodifferential operators on filtered manifolds as generalized fixed points

Eske Ewert

TL;DR

This work develops a pseudodifferential calculus on filtered manifolds by embedding the tangent groupoid into a generalized fixed point algebra framework. The principal symbol is realized as a continuous field of fixed-point algebras $\mathrm{Fix}^{\mathbb{R}_{>0}}(\mathbb J_0,\mathcal{R}_0)$, encoding Rockland-type invertibility on osculating group representations and yielding a Fredholm criterion that matches classical ellipticity in the step-1 case. The approach yields Morita equivalences to the ordinary principal symbol algebra $C_0(S^*M)$ and connects to van Erp–Yuncken’s calculus via a fixed-point interpretation, while establishing $K$-theory computability and an index theory framework through Connes–Thom deformation and Kasparov theory. These results extend to vector-bundle-valued operators and provide a robust bridge between noncommutative geometry on the tangent groupoid and traditional index theory on $T^*M$, with applications to filtered manifolds and their hypoelliptic operators.

Abstract

On filtered manifolds one can define a different notion of order for the differential operators. In this paper, we use generalized fixed point algebras to construct a pseudodifferential extension that reflects this behaviour. In the corresponding calculus, the principal symbol of an operator is a family of operators acting on certain nilpotent Lie groups. The role of ellipticity as a Fredholm condition is replaced by the Rockland condition on these groups. Our approach allows to understand this in terms of the representation of the corresponding algebra of principal symbols. Moreover, we compute the $K$-theory of this algebra.

Pseudodifferential operators on filtered manifolds as generalized fixed points

TL;DR

This work develops a pseudodifferential calculus on filtered manifolds by embedding the tangent groupoid into a generalized fixed point algebra framework. The principal symbol is realized as a continuous field of fixed-point algebras , encoding Rockland-type invertibility on osculating group representations and yielding a Fredholm criterion that matches classical ellipticity in the step-1 case. The approach yields Morita equivalences to the ordinary principal symbol algebra and connects to van Erp–Yuncken’s calculus via a fixed-point interpretation, while establishing -theory computability and an index theory framework through Connes–Thom deformation and Kasparov theory. These results extend to vector-bundle-valued operators and provide a robust bridge between noncommutative geometry on the tangent groupoid and traditional index theory on , with applications to filtered manifolds and their hypoelliptic operators.

Abstract

On filtered manifolds one can define a different notion of order for the differential operators. In this paper, we use generalized fixed point algebras to construct a pseudodifferential extension that reflects this behaviour. In the corresponding calculus, the principal symbol of an operator is a family of operators acting on certain nilpotent Lie groups. The role of ellipticity as a Fredholm condition is replaced by the Rockland condition on these groups. Our approach allows to understand this in terms of the representation of the corresponding algebra of principal symbols. Moreover, we compute the -theory of this algebra.

Paper Structure

This paper contains 28 sections, 47 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Filtered manifolds and their groupoids
  3. The osculating groupoid
  4. The tangent groupoid
  5. The zoom action
  6. Functoriality
  7. The groupoid $\mathrm C^*$-algebra of the tangent groupoid
  8. Haar system
  9. The groupoid $\mathrm C^*$-algebra
  10. Continuous field structure
  11. A Schwartz type algebra
  12. Definition of the Schwartz type algebra
  13. Algebra structure
  14. Generalized fixed point algebras for filtered manifolds
  15. Generalized fixed point algebras
  16. ...and 13 more sections

Key Result

Proposition 2.12

The assignment $(M,H)\mapsto T_HM$ and $f\mapsto Tf$ defines a functor from the category of filtered manifolds with filtered manifold maps to the category of Lie groupoids with Lie groupoid homomorphisms.

Theorems & Definitions (131)

  • Definition 2.1: tanaka
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 121 more