An equivariant noncommutative residue

Abstract

Let $\gp$ be a finite group acting on a compact manifold $M$ and $\maA(M)$ denote the algebra of classical complete symbols on $M$. We determine all traces on the cross-product algebra $\maA(M) \rtimes \Gamma$. These traces appear as residues of certain meromorphic 'zeta' functions and can be considered as equivariant generalization of the non-commutative residue trace. The local formula for these traces depends on more than one component of the complete asymptotic expansion. For instance, the local formula for these traces depends also on derivatives in the normal directions to fixed point manifolds of higher order components. As an application, we obtain a formula for the asymptotic occurrence of an irreducible representation of $\gp$ in the eigenspaces of an invariant positive elliptic operator. We also obtain an new construction for Dixmier trace of an invariant operator.