A general Simonenko local principle and Fredholm condition for isotypical components (Q2134898)

The article under review is concerned with \(G\)-invariant elliptic pseudodifferential operators which act on sections of vector bundles over a \(G\)-manifold \(M\). It generalizes and expands earlier results by the authors, which characterise the Fredhlmness of operators as such. Explicitely, it is shown in full generality that, when \(G\) is a compact Lie group, \(P \in \Psi^m(M;E_0,E_1)^G\) and \(\alpha\) an irreducible unitary representation of \(G\), then the Fredholmness of \(\pi_{\alpha}(P)\) is equivalent to the transversal \(\alpha\)-ellipticity of \(P\), which in turn is equivalent to the local \(\alpha\)-invertibility of \(P\). This type of result is known as Simonenko's equivariant localization principle.