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Annals of Global Analysis and Geometry

, Volume 56, Issue 4, pp 631–665 | Cite as

Global regularity and solvability of left-invariant differential systems on compact Lie groups

  • Gabriel AraújoEmail author
Article

Abstract

We are interested in global properties of systems of left-invariant differential operators on compact Lie groups: regularity properties, properties on the closedness of the range and finite dimensionality of their cohomology spaces, when acting on various function spaces, e.g., smooth, analytic and Gevrey. Extending the methods of Greenfield and Wallach (Trans Am Math Soc 183:153–164, 1973) to systems, we obtain abstract characterizations for these properties and use them to derive some generalizations of results due to Greenfield (Proc Am Math Soc 31:115–118, 1972), Greenfield and Wallach (Proc Am Math Soc 31:112–114, 1972), as well as global versions of a result of Caetano and Cordaro (Trans Am Math Soc 363(1):185–201, 2011) for involutive structures.

Keywords

Left-invariant operators Solvability Regularity Differential complexes Locally integrable structures 

Mathematics Subject Classification

35R03 35A01 58J10 

Notes

Acknowledgements

I wish to thank Paulo D. Cordaro and Andrew Raich for discussing parts of this work and their very useful inputs and also especially Max R. Jahnke and Luis F. Ragognette for their active participation in the earlier stages of this work, including helping to set up the original questions and proposing the framework that led to it, as well as many helpful suggestions throughout its development.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Departamento de Matemática, Instituto de Ciências Matemáticas e de ComputaçãoUniversidade de São Paulo (USP)São CarlosBrazil

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