Diagonal systems of differential operators on compact Lie groups
Paulo L. Dattori da Silva, Alexandre Kirilov, Ricardo Paleari da Silva
TL;DR
This work develops a Diophantine-symbolic framework to classify global regularity and solvability for left-invariant operator systems on compact Lie groups, with a primary focus on diagonal and triangular structures. Global solvability is tied to a Diophantine condition on the symbol, while global hypoellipticity requires the Diophantine condition together with a finite set of symbol-vanishing frequencies. The authors provide complete characterizations for diagonal systems, including systems of vector fields on products of tori and spheres (notably on $\mathbb{T}^r\times\mathbb{S}^{3s}$ and $SU(2)$), and extend the theory to triangular systems under a bounded-dimension condition, yielding concrete criteria and explicit examples. The results enhance spectral analysis on compact Lie groups and offer practical criteria for determining when global regularity and solvability hold in these noncommutative settings.
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.