Vekua-Type Operators on Compact Lie Groups: Hypoellipticity, Solvability, and Self-Duality

Abstract

We study global hypoellipticity and global solvability for Vekua-type operators associated with diagonal left-invariant operators on compact Lie groups. The conjugation term produces, on the Fourier side, a family of coupled systems whose determinants govern both regularity and solvability. Under a natural non-self-duality assumption, we obtain complete Diophantine-type characterizations of global hypoellipticity and global solvability, the latter on the natural space of admissible data. We then show how the theory must be modified in the self-dual setting by carrying out a complete analysis of the model case \(\mathbb S^3\simeq SU(2)\), where conjugation acts inside each representation block. The resulting criteria exhibit two Fourier-side mechanisms for Vekua-type operators on compact Lie groups: coupling between distinct conjugate representations and coupling inside self-dual representation blocks. We also present examples on product groups illustrating how these mechanisms may coexist and how global solvability may hold even when global hypoellipticity fails.