Fredholm conditions for invariant operators: finite abelian groups and boundary value problems

TL;DR

This paper characterizes when invariant pseudodifferential operators are Fredholm on specific isotypical components, especially focusing on finite abelian groups, by linking Fredholmness to a notion called alpha-ellipticity.

Contribution

It establishes a precise criterion for Fredholmness of invariant operators on isotypical components in the finite abelian group case, connecting it to alpha-ellipticity based on the principal symbol.

Findings

Fredholmness characterized by alpha-ellipticity for finite abelian groups.

Ellipticity implies alpha-ellipticity, but not vice versa, unless the action is free.

Structural analysis of invariant pseudodifferential operator algebras and their restrictions.

Abstract

We answer the question of when an invariant pseudodifferential operator is Fredholm on a fixed, given isotypical component. More precisely, let $\Gamma$ be a compact group acting on a smooth, compact, manifold $M$ without boundary and let $P \in \psi^m(M; E_0, E_1)$ be a $\Gamma$-invariant, classical, pseudodifferential operator acting between sections of two $\Gamma$-equivariant vector bundles $E_0$ and $E_1$. Let $\alpha$ be an irreducible representation of the group $\Gamma$. Then $P$ induces by restriction a map $\pi_\alpha(P) : H^s(M; E_0)_\alpha \to H^{s-m}(M; E_1)_\alpha$ between the $\alpha$-isotypical components of the corresponding Sobolev spaces of sections. We study in this paper conditions on the map $\pi_\alpha(P)$ to be Fredholm. It turns out that the discrete and non-discrete cases are quite different. Additionally, the discrete abelian case, which provides some of the most interesting applications, presents some special features and is much easier than the general case. We thus concentrate in this paper on the case when $\Gamma$ is finite abelian. We prove then that the restriction $\pi_\alpha(P)$ is Fredholm if, and only if, $P$ is "$\alpha$-elliptic", a condition defined in terms of the principal symbol of $P$. If $P$ is elliptic, then $P$ is also $\alpha$-elliptic, but the converse is not true in general. However, if $\Gamma$ acts freely on a dense open subset of $M$, then $P$ is $\alpha$-elliptic for the given fixed $\alpha$ if, and only if, it is elliptic. The proofs are based on the study of the structure of the algebra $\psi^{m}(M; E)^\Gamma$ of classical, $\Gamma$-invariant pseudodifferential operators acting on sections of the vector bundle $E \to M$ and of the structure of its restrictions to the isotypical components of $\Gamma$. These structures are described in terms of the isotropy groups of the action of the group $\Gamma$ on $E \to M$.