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 $\pi_\alpha(P) : H^s(M; E_0)_\alpha \to H^{sm}(M; E_1)_\alpha$ 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 $E_i$.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 DOI:
 10.48550/arXiv.2012.03944
 arXiv:
 arXiv:2012.03944
 Bibcode:
 2020arXiv201203944B
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry;
 Mathematics  Representation Theory;
 Mathematics  Spectral Theory
 EPrint:
 eight pages, it explains the main differences in the discrete and nondiscrete cases