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

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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.

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Notes

  1. This should not be confused with sheaf cohomology since we are not assuming local exactness of \(\mathrm {d}'\).

  2. Notice that when G is commutative these are actually zero: since the basic vector fields (1.2) are then pairwise commutative their dual forms \(\tau _1, \ldots , \tau _n, \zeta _1, \ldots , \zeta _m\) are automatically closed, hence (1.6) vanishes in that case.

  3. We are actually summing over a set of representations of G containing exactly one representative of each class in \({\widehat{G}}\).

  4. See Remark 2.1.

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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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Correspondence to Gabriel Araújo.

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This work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Grant 140838/2012-0) and the São Paulo Research Foundation (FAPESP, Grant 2018/12273-5).

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Araújo, G. Global regularity and solvability of left-invariant differential systems on compact Lie groups. Ann Glob Anal Geom 56, 631–665 (2019). https://doi.org/10.1007/s10455-019-09682-9

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