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

Alexandre Baldare; Rémi Côme; Victor Nistor

doi:10.5802/crmath.257

Operator theory

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

Alexandre Baldare ¹; Rémi Côme ²; Victor Nistor ²

¹ Institut fur Analysis, Welfengarten 1, 30167 Hannover, Germany
² Université Lorraine, 57000 Metz, France

Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1135-1143.

Abstract

Let $G$ be a compact Lie group acting smoothly on a smooth, compact manifold $M$ , let $P \in ψ^{m} (M; E_{0}, E_{1})$ be a $G$ –invariant, classical pseudodifferential operator acting between sections of two $G$ -equivariant vector bundles $E_{i} \to M$ , $i = 0, 1$ , and let $α$ be an irreducible representation of the group $G$ . Then $P$ induces a map $π_{α} (P) : H^{s} {(M; E_{0})}_{α} \to H^{s - m} {(M; E_{1})}_{α}$ between the $α$ -isotypical components. We prove that the map $π_{α} (P)$ is Fredholm if, and only if, $P$ is transversally $α$ -elliptic, a condition defined in terms of the principal symbol of $P$ and the action of $G$ on the vector bundles $E_{i}$ .

Received: 2020-12-07
Revised: 2021-03-25
Accepted: 2021-08-10
Published online: 2021-11-03

DOI: 10.5802/crmath.257

Classification: 47A53, 58J40, 57S15, 47L80, 46N20

Author's affiliations:

Alexandre Baldare ¹; Rémi Côme ²; Victor Nistor ²

¹ Institut fur Analysis, Welfengarten 1, 30167 Hannover, Germany
² Université Lorraine, 57000 Metz, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2021__359_9_1135_0,
     author = {Alexandre Baldare and R\'emi C\^ome and Victor Nistor},
     title = {Fredholm conditions for operators invariant with respect to compact {Lie} group actions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1135--1143},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {9},
     year = {2021},
     doi = {10.5802/crmath.257},
     language = {en},
}

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Alexandre Baldare; Rémi Côme; Victor Nistor. Fredholm conditions for operators invariant with respect to compact Lie group actions. Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1135-1143. doi : 10.5802/crmath.257. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.257/

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