  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.
