Fredholm conditions for invariant operators: finite abelian groups and boundary value problems

Let Γ be a finite abelian group acting on a smooth, compact manifold M without boundary and let P∈ψm(M;E0,E1) be a Γ-invariant, classical, pseudodifferential operator acting between sections of two Γ-equivariant vector bundles. Let α be an irreducible representation of Γ. We obtain necessary and sufficient conditions for the restriction πα(P):Hs(M;E0)α→Hs−m(M;E1)α of P between the α-isotypical components of Sobolev spaces to be Fredholm.