Global analytic hypoellipticity and solvability of certain operators subject to group actions

Abstract

On T × G T \times G , where T T is a compact real-analytic manifold and G G is a compact Lie group, we consider differential operators P P which are invariant by left translations on G G and are elliptic in T T . Under a mild technical condition, we prove that global hypoellipticity of P P implies its global analytic-hypoellipticity (actually Gevrey of any order s ≥ 1 s \geq 1 ). We also study the connection between the latter property and the notion of global analytic (resp. Gevrey) solvability, but in a much more general setup.