Fredholm conditions for operators invariant with respect to compact Lie group actions

Abstract

Let G be a compact Lie group acting smoothly on a smooth, compact manifold M, let P ∈ ψm(M;E0,E1) be a G–invariant, classical pseudodifferential operator acting between sections of two G-equivariant vector bundles Ei → M, i = 0,1, and let α be an irreducible representation of the group G. Then P induces a map πα(P): Hs(M;E0)α → Hs−m(M;E1)α between the α-isotypical components. We prove that the map πα(P) is Fredholm if, and only if, P is transversally α-elliptic, a condition defined in terms of the principal symbol of P and the action of G on the vector bundles Ei