  The short-time heat kernel expansion of elliptic operators provides a link
between local and global features of classical geometries. For many geometric
structures related to (non-)involutive distributions, the natural differential
operators tend to be Rockland, hence hypoelliptic. In this paper we establish a
universal heat kernel expansion for formally selfadjoint non-negative Rockland
differential operators on general closed filtered manifolds. The main
ingredient is the analysis of parametrices in a recently constructed calculus
adapted to these geometric structures. The heat expansion implies that the new
calculus, a more general version of the Heisenberg calculus, also has a
non-commutative residue. Many of the well known implications of the heat
expansion such as, the structure of the complex powers, the heat trace
asymptotics, the continuation of the zeta function, as well as Weyl's law for
the eigenvalue asymptotics, can be adapted to this calculus. Other consequences
include a McKean-Singer type formula for the index of Rockland differential
operators. We illustrate some of these results by providing a more explicit
description of Weyl's law for Rumin-Seshadri operators associated with curved
BGG sequences over 5-manifolds equipped with a rank two distribution of Cartan
type.
