Homology of complete symbols and non-commutative geometry

We identify the periodic cyclic homology of the algebra of complete symbols on a differential groupoid $\GR$ in terms of the cohomology of $S^*(\GR)$, the cosphere bundle of $A(\GR)$, where $A(\GR)$ is the Lie algebroid of $\GR$. We also relate the Hochschild homology of this algebra with the homogeneous Poisson homology of the space $A^*(\GR) \smallsetminus 0 \cong S^*(\GR) \times (0,\infty)$, the dual of $A(\GR)$ with the zero section removed. We use then these results to compute the Hochschild and cyclic homologies of the algebras of complete symbols associated with manifolds with corners, when the corresponding Lie algebroid is rationally isomorphic to the tangent bundle.