Diagonal systems of differential operators on compact Lie groups

TL;DR

This paper characterizes the conditions under which systems of left-invariant differential operators on compact Lie groups are globally hypoelliptic and solvable, focusing on diagonal systems and extending to triangular systems with applications to tori and spheres.

Contribution

It provides necessary and sufficient conditions for global hypoellipticity and solvability of diagonal systems, including Diophantine conditions and finiteness criteria, extending to triangular systems on compact Lie groups.

Findings

Global solvability characterized by Diophantine conditions on symbols.

Global hypoellipticity requires finiteness of certain symbol-dependent sets.

Complete characterization for systems on products of tori and spheres.

Abstract

We investigate the global hypoellipticity and global solvability of systems of left-invariant differential operators on compact Lie groups. Focusing on diagonal systems, we establish necessary and sufficient conditions for these global properties. Specifically, we show that global solvability is characterized by a Diophantine condition on the symbol of the system, while global hypoellipticity further requires that a set depending on the symbol to be finite. As an application, we provide a complete characterization of these properties for systems of vector fields defined on products of tori and spheres. Additionally, we present illustrative examples, including systems involving higher-order differential operators. Finally, we extend our analysis to triangular systems on compact Lie groups, introducing an additional condition related to the boundedness of the dimensions of the representations in these Lie groups.