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 $P\in {\psi }^m(M;E_0,E_1)$ be a $G$--invariant, classical pseudodifferential operator acting between sections of two vector bundles $E_i\to M$, $i=0,1$, and let $\alpha$ be an irreducible representation of the group $G$. 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. We prove that the map ${\pi }_{\alpha }(P)$ is Fredholm if, and only if, $P$ is {\em transversally $\alpha$-elliptic}, a condition defined in terms of the principal symbol of $P$ and the action of $G$ on the vector bundles $E_i$.

eight pages, it explains the main differences in the discrete and non-discrete cases