Let $Gamma$ be a compact group acting on a smooth, compact manifold $M$, let $P in psi^m(M; E_0, E_1)$ be a $Gamma$-invariant, classical pseudodifferential operator acting between sections of two equivariant vector bundles $E_i to M$, $i = 0,1$, and
let $alpha$ be an irreducible representation of the group $Gamma$. Then $P$ induces a map $pi_alpha(P) : H^s(M; E_0)_alpha to H^{s-m}(M; E_1)_alpha$ between the $alpha$-isotypical components of the corresponding Sobolev spaces of sections. When $Gamma$ is finite, we explicitly characterize the operators $P$ for which the map $pi_alpha(P)$ is Fredholm in terms of the principal symbol of $P$ and the action of $Gamma$ on the vector bundles $E_i$. When $Gamma = {1}$, that is, when there is no group, our result extends the classical characterization of Fredholm (pseudo)differential operators on compact manifolds. The proof is based on a careful study of the symbol $C^*$-algebra and of the topology of its primitive ideal spectrum. We also obtain several results on the structure of the norm closure of the algebra of invariant pseudodifferential operators and their relation to induced representations. Whenever our results also hold for non-discrete groups, we prove them in this greater generality. As an illustration of the generality of our results, we provide some applications to Hodge theory and to index theory of singular quotient spaces.
