Abstract
We answer the question of when an invariant pseudodifferential operator is
Fredholm on a fixed, given isotypical component. More precisely, let $\Gamma$
be a compact group acting on a smooth, compact, manifold $M$ without boundary
and let $P \in \psi^m(M; E_0, E_1)$ be a $\Gamma$-invariant, classical,
pseudodifferential operator acting between sections of two $\Gamma$-equivariant
vector bundles $E_0$ and $E_1$. Let $\alpha$ be an irreducible representation
of the group $\Gamma$. Then $P$ induces by restriction 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. We study in this
paper conditions on the map $\pi_\alpha(P)$ to be Fredholm. It turns out that
the discrete and non-discrete cases are quite different. Additionally, the
discrete abelian case, which provides some of the most interesting
applications, presents some special features and is much easier than the
general case. We thus concentrate in this paper on the case when $\Gamma$ is
finite abelian. We prove then that the restriction $\pi_\alpha(P)$ is Fredholm
if, and only if, $P$ is "$\alpha$-elliptic", a condition defined in terms of
the principal symbol of $P$. If $P$ is elliptic, then $P$ is also
$\alpha$-elliptic, but the converse is not true in general. However, if
$\Gamma$ acts freely on a dense open subset of $M$, then $P$ is
$\alpha$-elliptic for the given fixed $\alpha$ if, and only if, it is elliptic.
The proofs are based on the study of the structure of the algebra $\psi^{m}(M;
E)^\Gamma$ of classical, $\Gamma$-invariant pseudodifferential operators acting
on sections of the vector bundle $E \to M$ and of the structure of its
restrictions to the isotypical components of $\Gamma$. These structures are
described in terms of the isotropy groups of the action of the group $\Gamma$
on $E \to M$.
