The Mathieu Property for Compact Connected Lie Groups

Let $G$ be a compact connected Lie group, let $\mathcal R(G)$ denote its algebra of representative functions, and let $\mathcal I_G(f)=\int_G f(g)\,\mathrm d g$ be normalized Haar integration. We prove that $\ker \mathcal I_G$ is a Mathieu--Zhao subspace of $\mathcal R(G)$ if and only if $G$ is a to