  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.
