Equivariant homology for pseudo-differential operators

We compute the cyclic homology for the cross-product al- gebra $A(M)⋊\mathrm{\Gamma }$ of the algebra of complete symbols on a compact man- ifold $M$ with action of a finite group $\mathrm{\Gamma }$. A spectral sequence argument shows that these groups can be identified using deRham cohomology of the fixed point manifolds $S^{\ast }{M}^g$ . In the process we obtain new re- sults about the homologies of general cross-product algebras and provide explicit identification of the homologies for $C^{\mathrm{\infty }}(M)⋊\mathrm{\Gamma }$.