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

## TL;DR

This paper investigates the global regularity, solvability, and cohomology properties of left-invariant differential systems on compact Lie groups, extending classical methods to systems and various function spaces.

## Contribution

It provides abstract characterizations of these properties for systems, generalizing previous results and offering new insights into the global analysis on compact Lie groups.

## Key findings

- Characterization of regularity and solvability for systems
- Extension of Greenfield and Wallach's methods to systems
- Global versions of involutive structure results

## 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 (1973) to systems, we obtain abstract characterizations for these properties and use them to derive some generalizations of results due to Greenfield (1972), Greenfield and Wallach (1972), as well as global versions of a result of Caetano and Cordaro (2011) for involutive structures.