Pseudodifferential operators on filtered manifolds as generalized fixed points

TL;DR

This paper develops a pseudodifferential calculus for filtered manifolds using generalized fixed point algebras, where ellipticity is characterized by the Rockland condition and the algebra's K-theory is computed.

Contribution

It introduces a novel pseudodifferential extension for filtered manifolds that captures their unique order structure and provides a K-theoretic analysis of the principal symbol algebra.

Findings

Constructed a pseudodifferential calculus reflecting filtered manifold structure

Replaced ellipticity with the Rockland condition for Fredholmness

Computed the K-theory of the algebra of principal symbols

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.