  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.
