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

TL;DR

This paper characterizes when $G$-invariant pseudodifferential operators on compact manifolds are Fredholm by introducing a transversally $ ext{alpha}$-elliptic condition related to the group action and principal symbol.

Contribution

It establishes a Fredholm criterion for $G$-invariant operators using a new transversally $ ext{alpha}$-elliptic condition based on the principal symbol and group action.

Findings

Fredholmness characterized by transversally $ ext{alpha}$-ellipticity.

Provides necessary and sufficient conditions for $G$-invariant operators.

Connects representation theory with pseudodifferential operator analysis.

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 $E_i$.