An equivariant noncommutative residue

Abstract

Let $\Gamma$ be a finite group acting on a compact manifold $M$ and let $\mathcal{A}(M)$ denote the algebra of classical complete symbols on $M$. We determine all traces on the cross-product algebra $\mathcal{A}(M)⋊\Gamma$ as residues of certain meromorphic zeta functions. Further we compute the cyclic homology for $\mathcal{A}(M)⋊\Gamma$ in terms of the de Rham cohomology of the fixed point manifolds $S^{\ast }{M}^g$. In the process certain new results on the homologies of general cross-product algebras are obtained.