  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.
